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[Emacs-diffs] Changes to calc.texi
|
From: |
Glenn Morris |
|
Subject: |
[Emacs-diffs] Changes to calc.texi |
|
Date: |
Thu, 06 Sep 2007 04:34:12 +0000 |
CVSROOT: /sources/emacs
Module name: emacs
Changes by: Glenn Morris <gm> 07/09/06 04:34:11
Index: calc.texi
===================================================================
RCS file: calc.texi
diff -N calc.texi
--- calc.texi 1 Sep 2007 04:11:58 -0000 1.112
+++ /dev/null 1 Jan 1970 00:00:00 -0000
@@ -1,36190 +0,0 @@
-\input texinfo @c -*-texinfo-*-
address@hidden %**start of header (This is for running Texinfo on a region.)
address@hidden smallbook
address@hidden ../info/calc
address@hidden [title]
address@hidden GNU Emacs Calc 2.1 Manual
address@hidden odd
address@hidden %**end of header (This is for running Texinfo on a region.)
-
address@hidden The following macros are used for conditional output for single
lines.
address@hidden @texline foo
address@hidden `foo' will appear only in TeX output
address@hidden @infoline foo
address@hidden `foo' will appear only in non-TeX output
-
address@hidden @expr{expr} will typeset an expression;
address@hidden $x$ in TeX, @samp{x} otherwise.
-
address@hidden
address@hidden texline
address@hidden macro
address@hidden infoline=comment
address@hidden expr=math
address@hidden tfn=code
address@hidden mathit=expr
address@hidden cpi{}
address@hidden@pi{}}
address@hidden macro
address@hidden cpiover{den}
address@hidden@pi/\den\}
address@hidden macro
address@hidden iftex
-
address@hidden
address@hidden texline=comment
address@hidden infoline{stuff}
-\stuff\
address@hidden macro
address@hidden expr=samp
address@hidden tfn=t
address@hidden mathit=i
address@hidden cpi{}
address@hidden
address@hidden macro
address@hidden cpiover{den}
address@hidden/\den\}
address@hidden macro
address@hidden ifnottex
-
-
address@hidden
-% Suggested by Karl Berry <karl@@freefriends.org>
-\gdef\!{\mskip-\thinmuskip}
address@hidden tex
-
address@hidden Fix some other things specifically for this manual.
address@hidden
address@hidden
address@hidden@:=`@: @c Make Calc fractions come out right in math mode
address@hidden
-\gdef\coloneq{\mathrel{\mathord:\mathord=}}
-
-\gdef\beforedisplay{\vskip-10pt}
-\gdef\afterdisplay{\vskip-5pt}
-\gdef\beforedisplayh{\vskip-25pt}
-\gdef\afterdisplayh{\vskip-10pt}
address@hidden tex
address@hidden@kyvpos @kyvpos=0pt
address@hidden@kyhpos @kyhpos=0pt
address@hidden@calcclubpenalty @calcclubpenalty=1000
address@hidden
address@hidden@calcpageno
address@hidden@calcoldeverypar @address@hidden
address@hidden@address@hidden@calcoldeverypar}
address@hidden@address@hidden@address@hidden@fi
address@hidden@address@hidden@address@hidden@fi
address@hidden@\=0 address@hidden
address@hidden
address@hidden @address@hidden@active
address@hidden ignore
address@hidden iftex
-
address@hidden
-This file documents Calc, the GNU Emacs calculator.
-
-Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
-2005, 2006, 2007 Free Software Foundation, Inc.
-
address@hidden
-Permission is granted to copy, distribute and/or modify this document
-under the terms of the GNU Free Documentation License, Version 1.2 or
-any later version published by the Free Software Foundation; with the
-Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
-Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
-Texts as in (a) below. A copy of the license is included in the section
-entitled ``GNU Free Documentation License.''
-
-(a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify
-this GNU Manual, like GNU software. Copies published by the Free
-Software Foundation raise funds for GNU development.''
address@hidden quotation
address@hidden copying
-
address@hidden Emacs
address@hidden
-* Calc: (calc). Advanced desk calculator and mathematical tool.
address@hidden direntry
-
address@hidden
address@hidden 6
address@hidden @titlefont{Calc Manual}
address@hidden 4
address@hidden GNU Emacs Calc Version 2.1
address@hidden [volume]
address@hidden 5
address@hidden Dave Gillespie
address@hidden daveg@@synaptics.com
address@hidden
-
address@hidden 0pt plus 1filll
-Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
- 2005, 2006, 2007 Free Software Foundation, Inc.
address@hidden
address@hidden titlepage
-
-
address@hidden
-
address@hidden [end]
-
address@hidden
-
address@hidden [begin]
address@hidden
address@hidden Top, Getting Started, (dir), (dir)
address@hidden The GNU Emacs Calculator
-
address@hidden
address@hidden is an advanced desk calculator and mathematical tool
-written by Dave Gillespie that runs as part of the GNU Emacs environment.
-
-This manual, also written (mostly) by Dave Gillespie, is divided into
-three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
-``Calc Reference.'' The Tutorial introduces all the major aspects of
-Calculator use in an easy, hands-on way. The remainder of the manual is
-a complete reference to the features of the Calculator.
address@hidden ifnottex
-
address@hidden
-For help in the Emacs Info system (which you are using to read this
-file), type @kbd{?}. (You can also type @kbd{h} to run through a
-longer Info tutorial.)
address@hidden ifinfo
-
address@hidden
-* Getting Started:: General description and overview.
address@hidden
-* Interactive Tutorial::
address@hidden ifinfo
-* Tutorial:: A step-by-step introduction for beginners.
-
-* Introduction:: Introduction to the Calc reference manual.
-* Data Types:: Types of objects manipulated by Calc.
-* Stack and Trail:: Manipulating the stack and trail buffers.
-* Mode Settings:: Adjusting display format and other modes.
-* Arithmetic:: Basic arithmetic functions.
-* Scientific Functions:: Transcendentals and other scientific functions.
-* Matrix Functions:: Operations on vectors and matrices.
-* Algebra:: Manipulating expressions algebraically.
-* Units:: Operations on numbers with units.
-* Store and Recall:: Storing and recalling variables.
-* Graphics:: Commands for making graphs of data.
-* Kill and Yank:: Moving data into and out of Calc.
-* Keypad Mode:: Operating Calc from a keypad.
-* Embedded Mode:: Working with formulas embedded in a file.
-* Programming:: Calc as a programmable calculator.
-
-* Copying:: How you can copy and share Calc.
-* GNU Free Documentation License:: The license for this documentation.
-* Customizing Calc:: Customizing Calc.
-* Reporting Bugs:: How to report bugs and make suggestions.
-
-* Summary:: Summary of Calc commands and functions.
-
-* Key Index:: The standard Calc key sequences.
-* Command Index:: The interactive Calc commands.
-* Function Index:: Functions (in algebraic formulas).
-* Concept Index:: General concepts.
-* Variable Index:: Variables used by Calc (both user and internal).
-* Lisp Function Index:: Internal Lisp math functions.
address@hidden menu
-
address@hidden
address@hidden Getting Started, Interactive Tutorial, Top, Top
address@hidden ifinfo
address@hidden
address@hidden Getting Started, Tutorial, Top, Top
address@hidden ifnotinfo
address@hidden Getting Started
address@hidden
-This chapter provides a general overview of Calc, the GNU Emacs
-Calculator: What it is, how to start it and how to exit from it,
-and what are the various ways that it can be used.
-
address@hidden
-* What is Calc::
-* About This Manual::
-* Notations Used in This Manual::
-* Demonstration of Calc::
-* Using Calc::
-* History and Acknowledgements::
address@hidden menu
-
address@hidden What is Calc, About This Manual, Getting Started, Getting Started
address@hidden What is Calc?
-
address@hidden
address@hidden is an advanced calculator and mathematical tool that runs as
-part of the GNU Emacs environment. Very roughly based on the HP-28/48
-series of calculators, its many features include:
-
address@hidden @bullet
address@hidden
-Choice of algebraic or RPN (stack-based) entry of calculations.
-
address@hidden
-Arbitrary precision integers and floating-point numbers.
-
address@hidden
-Arithmetic on rational numbers, complex numbers (rectangular and polar),
-error forms with standard deviations, open and closed intervals, vectors
-and matrices, dates and times, infinities, sets, quantities with units,
-and algebraic formulas.
-
address@hidden
-Mathematical operations such as logarithms and trigonometric functions.
-
address@hidden
-Programmer's features (bitwise operations, non-decimal numbers).
-
address@hidden
-Financial functions such as future value and internal rate of return.
-
address@hidden
-Number theoretical features such as prime factorization and arithmetic
-modulo @var{m} for any @var{m}.
-
address@hidden
-Algebraic manipulation features, including symbolic calculus.
-
address@hidden
-Moving data to and from regular editing buffers.
-
address@hidden
-Embedded mode for manipulating Calc formulas and data directly
-inside any editing buffer.
-
address@hidden
-Graphics using GNUPLOT, a versatile (and free) plotting program.
-
address@hidden
-Easy programming using keyboard macros, algebraic formulas,
-algebraic rewrite rules, or extended Emacs Lisp.
address@hidden itemize
-
-Calc tries to include a little something for everyone; as a result it is
-large and might be intimidating to the first-time user. If you plan to
-use Calc only as a traditional desk calculator, all you really need to
-read is the ``Getting Started'' chapter of this manual and possibly the
-first few sections of the tutorial. As you become more comfortable with
-the program you can learn its additional features. Calc does not
-have the scope and depth of a fully-functional symbolic math package,
-but Calc has the advantages of convenience, portability, and freedom.
-
address@hidden About This Manual, Notations Used in This Manual, What is Calc,
Getting Started
address@hidden About This Manual
-
address@hidden
-This document serves as a complete description of the GNU Emacs
-Calculator. It works both as an introduction for novices, and as
-a reference for experienced users. While it helps to have some
-experience with GNU Emacs in order to get the most out of Calc,
-this manual ought to be readable even if you don't know or use Emacs
-regularly.
-
-The manual is divided into three major parts:@: the ``Getting
-Started'' chapter you are reading now, the Calc tutorial (chapter 2),
-and the Calc reference manual (the remaining chapters and appendices).
address@hidden [when-split]
address@hidden This manual has been printed in two volumes, the @dfn{Tutorial}
and the
address@hidden @dfn{Reference}. Both volumes include a copy of the ``Getting
Started''
address@hidden chapter.
-
-If you are in a hurry to use Calc, there is a brief ``demonstration''
-below which illustrates the major features of Calc in just a couple of
-pages. If you don't have time to go through the full tutorial, this
-will show you everything you need to know to begin.
address@hidden of Calc}.
-
-The tutorial chapter walks you through the various parts of Calc
-with lots of hands-on examples and explanations. If you are new
-to Calc and you have some time, try going through at least the
-beginning of the tutorial. The tutorial includes about 70 exercises
-with answers. These exercises give you some guided practice with
-Calc, as well as pointing out some interesting and unusual ways
-to use its features.
-
-The reference section discusses Calc in complete depth. You can read
-the reference from start to finish if you want to learn every aspect
-of Calc. Or, you can look in the table of contents or the Concept
-Index to find the parts of the manual that discuss the things you
-need to know.
-
address@hidden Marginal notes
-Every Calc keyboard command is listed in the Calc Summary, and also
-in the Key Index. Algebraic functions, @kbd{M-x} commands, and
-variables also have their own indices.
address@hidden Each
address@hidden In the printed manual, each
-paragraph that is referenced in the Key or Function Index is marked
-in the margin with its index entry.
-
address@hidden [fix-ref Help Commands]
-You can access this manual on-line at any time within Calc by
-pressing the @kbd{h i} key sequence. Outside of the Calc window,
-you can press @kbd{C-x * i} to read the manual on-line. Also, you
-can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{C-x * t},
-or to the Summary by pressing @kbd{h s} or @kbd{C-x * s}. Within Calc,
-you can also go to the part of the manual describing any Calc key,
-function, or variable using @address@hidden k}}, @kbd{h f}, or @kbd{h v},
-respectively. @xref{Help Commands}.
-
address@hidden
-The Calc manual can be printed, but because the manual is so large, you
-should only make a printed copy if you really need it. To print the
-manual, you will need the @TeX{} typesetting program (this is a free
-program by Donald Knuth at Stanford University) as well as the
address@hidden program and @file{texinfo.tex} file, both of which can
-be obtained from the FSF as part of the @code{texinfo} package.
-To print the Calc manual in one huge tome, you will need the
-source code to this manual, @file{calc.texi}, available as part of the
-Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
-Alternatively, change to the @file{man} subdirectory of the Emacs
-source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
-get some ``overfull box'' warnings while @TeX{} runs.)
-The result will be a device-independent output file called
address@hidden, which you must print in whatever way is right
-for your system. On many systems, the command is
-
address@hidden
-lpr -d calc.dvi
address@hidden example
-
address@hidden
-or
-
address@hidden
-dvips calc.dvi
address@hidden example
address@hidden ifnottex
address@hidden Printed copies of this manual are also available from the Free
Software
address@hidden Foundation.
-
address@hidden Notations Used in This Manual, Demonstration of Calc, About This
Manual, Getting Started
address@hidden Notations Used in This Manual
-
address@hidden
-This section describes the various notations that are used
-throughout the Calc manual.
-
-In keystroke sequences, uppercase letters mean you must hold down
-the shift key while typing the letter. Keys pressed with Control
-held down are shown as @kbd{C-x}. Keys pressed with Meta held down
-are shown as @kbd{M-x}. Other notations are @key{RET} for the
-Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
address@hidden for the Delete key, and @key{LFD} for the Line-Feed key.
-The @key{DEL} key is called Backspace on some keyboards, it is
-whatever key you would use to correct a simple typing error when
-regularly using Emacs.
-
-(If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
-the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
-If you don't have a Meta key, look for Alt or Extend Char. You can
-also press @key{ESC} or @kbd{C-[} first to get the same effect, so
-that @kbd{M-x}, @address@hidden x}, and @kbd{C-[ x} are all equivalent.)
-
-Sometimes the @key{RET} key is not shown when it is ``obvious''
-that you must press @key{RET} to proceed. For example, the @key{RET}
-is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
-
-Commands are generally shown like this: @kbd{p} (@code{calc-precision})
-or @kbd{C-x * k} (@code{calc-keypad}). This means that the command is
-normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
-but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
-
-Commands that correspond to functions in algebraic notation
-are written: @kbd{C} (@code{calc-cos}) address@hidden This means
-the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
-the corresponding function in an algebraic-style formula would
-be @samp{cos(@var{x})}.
-
-A few commands don't have key equivalents: @code{calc-sincos}
address@hidden
-
address@hidden Demonstration of Calc, Using Calc, Notations Used in This
Manual, Getting Started
address@hidden A Demonstration of Calc
-
address@hidden
address@hidden Demonstration of Calc
-This section will show some typical small problems being solved with
-Calc. The focus is more on demonstration than explanation, but
-everything you see here will be covered more thoroughly in the
-Tutorial.
-
-To begin, start Emacs if necessary (usually the command @code{emacs}
-does this), and type @kbd{C-x * c} to start the
-Calculator. (You can also use @kbd{M-x calc} if this doesn't work.
address@hidden Calc}, for various ways of starting the Calculator.)
-
-Be sure to type all the sample input exactly, especially noting the
-difference between lower-case and upper-case letters. Remember,
address@hidden, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
-Delete, and Space keys.
-
address@hidden calculation.} In RPN, you type the input number(s) first,
-then the command to operate on the numbers.
-
address@hidden
-Type @kbd{2 @key{RET} 3 + Q} to compute
address@hidden @math{\sqrt{2+3} = 2.2360679775}.
address@hidden the square root of 2+3, which is 2.2360679775.
-
address@hidden
-Type @kbd{P 2 ^} to compute
address@hidden @math{\pi^2 = 9.86960440109}.
address@hidden the value of `pi' squared, 9.86960440109.
-
address@hidden
-Type @key{TAB} to exchange the order of these two results.
-
address@hidden
-Type @kbd{- I H S} to subtract these results and compute the Inverse
-Hyperbolic sine of the difference, 2.72996136574.
-
address@hidden
-Type @key{DEL} to erase this result.
-
address@hidden calculation.} You can also enter calculations using
-conventional ``algebraic'' notation. To enter an algebraic formula,
-use the apostrophe key.
-
address@hidden
-Type @kbd{' sqrt(2+3) @key{RET}} to compute
address@hidden @math{\sqrt{2+3}}.
address@hidden the square root of 2+3.
-
address@hidden
-Type @kbd{' pi^2 @key{RET}} to enter
address@hidden @math{\pi^2}.
address@hidden `pi' squared.
-To evaluate this symbolic formula as a number, type @kbd{=}.
-
address@hidden
-Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
-result from the most-recent and compute the Inverse Hyperbolic sine.
-
address@hidden mode.} If you are using the X window system, press
address@hidden@kbd{C-x * k}} to get Keypad mode. (If you don't use X, skip to
-the next section.)
-
address@hidden
-Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
-``buttons'' using your left mouse button.
-
address@hidden
-Click on @key{PI}, @key{2}, and @tfn{y^x}.
-
address@hidden
-Click on @key{INV}, then @key{ENTER} to swap the two results.
-
address@hidden
-Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
-
address@hidden
-Click on @key{<-} to erase the result, then click @key{OFF} to turn
-the Keypad Calculator off.
-
address@hidden data.} Type @kbd{C-x * x} if necessary to exit Calc.
-Now select the following numbers as an Emacs region: ``Mark'' the
-front of the list by typing @address@hidden or @kbd{C-@@} there,
-then move to the other end of the list. (Either get this list from
-the on-line copy of this manual, accessed by @address@hidden * i}}, or just
-type these numbers into a scratch file.) Now type @kbd{C-x * g} to
-``grab'' these numbers into Calc.
-
address@hidden
address@hidden
-1.23 1.97
-1.6 2
-1.19 1.08
address@hidden group
address@hidden example
-
address@hidden
-The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
-Type @address@hidden R +}} to compute the sum of these numbers.
-
address@hidden
-Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
-the product of the numbers.
-
address@hidden
-You can also grab data as a rectangular matrix. Place the cursor on
-the upper-leftmost @samp{1} and set the mark, then move to just after
-the lower-right @samp{8} and press @kbd{C-x * r}.
-
address@hidden
-Type @kbd{v t} to transpose this
address@hidden @math{3\times2}
address@hidden 3x2
-matrix into a
address@hidden @math{2\times3}
address@hidden 2x3
-matrix. Type @address@hidden u}} to unpack the rows into two separate
-vectors. Now type @address@hidden R + @key{TAB} V R +}} to compute the sums
-of the two original columns. (There is also a special
-grab-and-sum-columns command, @kbd{C-x * :}.)
-
address@hidden conversion.} Units are entered algebraically.
-Type @address@hidden' 43 mi/hr @key{RET}}} to enter the quantity 43
miles-per-hour.
-Type @address@hidden c km/hr @key{RET}}}. Type @address@hidden c m/s
@key{RET}}}.
-
address@hidden arithmetic.} Type @kbd{t N} to get the current date and
-time. Type @kbd{90 +} to find the date 90 days from now. Type
address@hidden' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see
how
-many weeks have passed since then.
-
address@hidden Algebraic entries can also include formulas
-or equations involving variables. Type @address@hidden' [x + y} = a, x y = 1]
@key{RET}}
-to enter a pair of equations involving three variables.
-(Note the leading apostrophe in this example; also, note that the space
-between @samp{x y} is required.) Type @address@hidden S x,y @key{RET}}} to
solve
-these equations for the variables @expr{x} and @expr{y}.
-
address@hidden
-Type @kbd{d B} to view the solutions in more readable notation.
-Type @address@hidden C}} to view them in C language notation, @kbd{d T}
-to view them in the notation for the @TeX{} typesetting system,
-and @kbd{d L} to view them in the notation for the address@hidden typesetting
-system. Type @kbd{d N} to return to normal notation.
-
address@hidden
-Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these
formulas.
-(That's a letter @kbd{l}, not a numeral @kbd{1}.)
-
address@hidden
address@hidden functions.} You can read about any command in the on-line
-manual. Type @kbd{C-x * c} to return to Calc after each of these
-commands: @kbd{h k t N} to read about the @kbd{t N} command,
address@hidden f sqrt @key{RET}} to read about the @code{sqrt} function, and
address@hidden s} to read the Calc summary.
address@hidden ifnotinfo
address@hidden
address@hidden functions.} You can read about any command in the on-line
-manual. Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
-return here after each of these commands: @address@hidden k t N}} to read
-about the @address@hidden N}} command, @kbd{h f sqrt @key{RET}} to read about
the
address@hidden function, and @kbd{h s} to read the Calc summary.
address@hidden ifinfo
-
-Press @key{DEL} repeatedly to remove any leftover results from the stack.
-To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.
-
address@hidden Using Calc, History and Acknowledgements, Demonstration of Calc,
Getting Started
address@hidden Using Calc
-
address@hidden
-Calc has several user interfaces that are specialized for
-different kinds of tasks. As well as Calc's standard interface,
-there are Quick mode, Keypad mode, and Embedded mode.
-
address@hidden
-* Starting Calc::
-* The Standard Interface::
-* Quick Mode Overview::
-* Keypad Mode Overview::
-* Standalone Operation::
-* Embedded Mode Overview::
-* Other C-x * Commands::
address@hidden menu
-
address@hidden Starting Calc, The Standard Interface, Using Calc, Using Calc
address@hidden Starting Calc
-
address@hidden
-On most systems, you can type @kbd{C-x *} to start the Calculator.
-The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch},
-which can be rebound if convenient (@pxref{Customizing Calc}).
-
-When you press @kbd{C-x *}, Emacs waits for you to press a second key to
-complete the command. In this case, you will follow @kbd{C-x *} with a
-letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
-which Calc interface you want to use.
-
-To get Calc's standard interface, type @kbd{C-x * c}. To get
-Keypad mode, type @kbd{C-x * k}. Type @kbd{C-x * ?} to get a brief
-list of the available options, and type a second @kbd{?} to get
-a complete list.
-
-To ease typing, @kbd{C-x * *} also works to start Calc. It starts the
-same interface (either @kbd{C-x * c} or @address@hidden * k}}) that you last
-used, selecting the @kbd{C-x * c} interface by default.
-
-If @kbd{C-x *} doesn't work for you, you can always type explicit
-commands like @kbd{M-x calc} (for the standard user interface) or
address@hidden@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
-(that's Meta with the letter @kbd{x}), then, at the prompt,
-type the full command (like @kbd{calc-keypad}) and press Return.
-
-The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
-the Calculator also turn it off if it is already on.
-
address@hidden The Standard Interface, Quick Mode Overview, Starting Calc,
Using Calc
address@hidden The Standard Calc Interface
-
address@hidden
address@hidden Standard user interface
-Calc's standard interface acts like a traditional RPN calculator,
-operated by the normal Emacs keyboard. When you type @kbd{C-x * c}
-to start the Calculator, the Emacs screen splits into two windows
-with the file you were editing on top and Calc on the bottom.
-
address@hidden
address@hidden
-
-...
---**-Emacs: myfile (Fundamental)----All----------------------
---- Emacs Calculator Mode --- |Emacs Calculator Trail
-2: 17.3 | 17.3
-1: -5 | 3
- . | 2
- | 4
- | * 8
- | ->-5
- |
---%%-Calc: 12 Deg (Calculator)----All----- --%%-Emacs: *Calc Trail*
address@hidden group
address@hidden smallexample
-
-In this figure, the mode-line for @file{myfile} has moved up and the
-``Calculator'' window has appeared below it. As you can see, Calc
-actually makes two windows side-by-side. The lefthand one is
-called the @dfn{stack window} and the righthand one is called the
address@hidden window.} The stack holds the numbers involved in the
-calculation you are currently performing. The trail holds a complete
-record of all calculations you have done. In a desk calculator with
-a printer, the trail corresponds to the paper tape that records what
-you do.
-
-In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
-were first entered into the Calculator, then the 2 and 4 were
-multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
-(The @samp{>} symbol shows that this was the most recent calculation.)
-The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
-
-Most Calculator commands deal explicitly with the stack only, but
-there is a set of commands that allow you to search back through
-the trail and retrieve any previous result.
-
-Calc commands use the digits, letters, and punctuation keys.
-Shifted (i.e., upper-case) letters are different from lowercase
-letters. Some letters are @dfn{prefix} keys that begin two-letter
-commands. For example, @kbd{e} means ``enter exponent'' and shifted
address@hidden means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
-the letter ``e'' takes on very different meanings: @kbd{d e} means
-``engineering notation'' and @kbd{d E} means address@hidden language mode.''
-
-There is nothing stopping you from switching out of the Calc
-window and back into your editing window, say by using the Emacs
address@hidden@kbd{C-x o}} (@code{other-window}) command. When the cursor is
-inside a regular window, Emacs acts just like normal. When the
-cursor is in the Calc stack or trail windows, keys are interpreted
-as Calc commands.
-
-When you quit by pressing @kbd{C-x * c} a second time, the Calculator
-windows go away but the actual Stack and Trail are not gone, just
-hidden. When you press @kbd{C-x * c} once again you will get the
-same stack and trail contents you had when you last used the
-Calculator.
-
-The Calculator does not remember its state between Emacs sessions.
-Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
-a fresh stack and trail. There is a command (@kbd{m m}) that lets
-you save your favorite mode settings between sessions, though.
-One of the things it saves is which user interface (standard or
-Keypad) you last used; otherwise, a freshly started Emacs will
-always treat @kbd{C-x * *} the same as @kbd{C-x * c}.
-
-The @kbd{q} key is another equivalent way to turn the Calculator off.
-
-If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
-full-screen version of Calc (@code{full-calc}) in which the stack and
-trail windows are still side-by-side but are now as tall as the whole
-Emacs screen. When you press @kbd{q} or @kbd{C-x * c} again to quit,
-the file you were editing before reappears. The @kbd{C-x * b} key
-switches back and forth between ``big'' full-screen mode and the
-normal partial-screen mode.
-
-Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
-except that the Calc window is not selected. The buffer you were
-editing before remains selected instead. @kbd{C-x * o} is a handy
-way to switch out of Calc momentarily to edit your file; type
address@hidden * c} to switch back into Calc when you are done.
-
address@hidden Quick Mode Overview, Keypad Mode Overview, The Standard
Interface, Using Calc
address@hidden Quick Mode (Overview)
-
address@hidden
address@hidden mode} is a quick way to use Calc when you don't need the
-full complexity of the stack and trail. To use it, type @kbd{C-x * q}
-(@code{quick-calc}) in any regular editing buffer.
-
-Quick mode is very simple: It prompts you to type any formula in
-standard algebraic notation (like @samp{4 - 2/3}) and then displays
-the result at the bottom of the Emacs screen (@mathit{3.33333333333}
-in this case). You are then back in the same editing buffer you
-were in before, ready to continue editing or to type @kbd{C-x * q}
-again to do another quick calculation. The result of the calculation
-will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
-at this point will yank the result into your editing buffer.
-
-Calc mode settings affect Quick mode, too, though you will have to
-go into regular Calc (with @kbd{C-x * c}) to change the mode settings.
-
address@hidden [fix-ref Quick Calculator mode]
address@hidden Calculator}, for further information.
-
address@hidden Keypad Mode Overview, Standalone Operation, Quick Mode Overview,
Using Calc
address@hidden Keypad Mode (Overview)
-
address@hidden
address@hidden mode} is a mouse-based interface to the Calculator.
-It is designed for use with terminals that support a mouse. If you
-don't have a mouse, you will have to operate Keypad mode with your
-arrow keys (which is probably more trouble than it's worth).
-
-Type @kbd{C-x * k} to turn Keypad mode on or off. Once again you
-get two new windows, this time on the righthand side of the screen
-instead of at the bottom. The upper window is the familiar Calc
-Stack; the lower window is a picture of a typical calculator keypad.
-
address@hidden
-\dimen0=\pagetotal%
-\advance \dimen0 by 24\baselineskip%
-\ifdim \dimen0>\pagegoal \vfill\eject \fi%
-\medskip
address@hidden tex
address@hidden
address@hidden
-|--- Emacs Calculator Mode ---
-|2: 17.3
-|1: -5
-| .
-|--%%-Calc: 12 Deg (Calcul
-|----+-----Calc 2.1------+----1
-|FLR |CEIL|RND |TRNC|CLN2|FLT |
-|----+----+----+----+----+----|
-| LN |EXP | |ABS |IDIV|MOD |
-|----+----+----+----+----+----|
-|SIN |COS |TAN |SQRT|y^x |1/x |
-|----+----+----+----+----+----|
-| ENTER |+/- |EEX |UNDO| <- |
-|-----+---+-+--+--+-+---++----|
-| INV | 7 | 8 | 9 | / |
-|-----+-----+-----+-----+-----|
-| HYP | 4 | 5 | 6 | * |
-|-----+-----+-----+-----+-----|
-|EXEC | 1 | 2 | 3 | - |
-|-----+-----+-----+-----+-----|
-| OFF | 0 | . | PI | + |
-|-----+-----+-----+-----+-----+
address@hidden group
address@hidden smallexample
-
-Keypad mode is much easier for beginners to learn, because there
-is no need to memorize lots of obscure key sequences. But not all
-commands in regular Calc are available on the Keypad. You can
-always switch the cursor into the Calc stack window to use
-standard Calc commands if you need. Serious Calc users, though,
-often find they prefer the standard interface over Keypad mode.
-
-To operate the Calculator, just click on the ``buttons'' of the
-keypad using your left mouse button. To enter the two numbers
-shown here you would click @address@hidden 7 .@: 3 ENTER 5 +/- ENTER}}; to
-add them together you would then click @kbd{+} (to get 12.3 on
-the stack).
-
-If you click the right mouse button, the top three rows of the
-keypad change to show other sets of commands, such as advanced
-math functions, vector operations, and operations on binary
-numbers.
-
-Because Keypad mode doesn't use the regular keyboard, Calc leaves
-the cursor in your original editing buffer. You can type in
-this buffer in the usual way while also clicking on the Calculator
-keypad. One advantage of Keypad mode is that you don't need an
-explicit command to switch between editing and calculating.
-
-If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
-(@code{full-calc-keypad}) with three windows: The keypad in the lower
-left, the stack in the lower right, and the trail on top.
-
address@hidden [fix-ref Keypad Mode]
address@hidden Mode}, for further information.
-
address@hidden Standalone Operation, Embedded Mode Overview, Keypad Mode
Overview, Using Calc
address@hidden Standalone Operation
-
address@hidden
address@hidden Standalone Operation
-If you are not in Emacs at the moment but you wish to use Calc,
-you must start Emacs first. If all you want is to run Calc, you
-can give the commands:
-
address@hidden
-emacs -f full-calc
address@hidden example
-
address@hidden
-or
-
address@hidden
-emacs -f full-calc-keypad
address@hidden example
-
address@hidden
-which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
-a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
-In standalone operation, quitting the Calculator (by pressing
address@hidden or clicking on the keypad @key{EXIT} button) quits Emacs
-itself.
-
address@hidden Embedded Mode Overview, Other C-x * Commands, Standalone
Operation, Using Calc
address@hidden Embedded Mode (Overview)
-
address@hidden
address@hidden mode} is a way to use Calc directly from inside an
-editing buffer. Suppose you have a formula written as part of a
-document like this:
-
address@hidden
address@hidden
-The derivative of
-
- ln(ln(x))
-
-is
address@hidden group
address@hidden smallexample
-
address@hidden
-and you wish to have Calc compute and format the derivative for
-you and store this derivative in the buffer automatically. To
-do this with Embedded mode, first copy the formula down to where
-you want the result to be:
-
address@hidden
address@hidden
-The derivative of
-
- ln(ln(x))
-
-is
-
- ln(ln(x))
address@hidden group
address@hidden smallexample
-
-Now, move the cursor onto this new formula and press @kbd{C-x * e}.
-Calc will read the formula (using the surrounding blank lines to
-tell how much text to read), then push this formula (invisibly)
-onto the Calc stack. The cursor will stay on the formula in the
-editing buffer, but the buffer's mode line will change to look
-like the Calc mode line (with mode indicators like @samp{12 Deg}
-and so on). Even though you are still in your editing buffer,
-the keyboard now acts like the Calc keyboard, and any new result
-you get is copied from the stack back into the buffer. To take
-the derivative, you would type @kbd{a d x @key{RET}}.
-
address@hidden
address@hidden
-The derivative of
-
- ln(ln(x))
-
-is
-
-1 / ln(x) x
address@hidden group
address@hidden smallexample
-
-To make this look nicer, you might want to press @kbd{d =} to center
-the formula, and even @kbd{d B} to use Big display mode.
-
address@hidden
address@hidden
-The derivative of
-
- ln(ln(x))
-
-is
-% [calc-mode: justify: center]
-% [calc-mode: language: big]
-
- 1
- -------
- ln(x) x
address@hidden group
address@hidden smallexample
-
-Calc has added annotations to the file to help it remember the modes
-that were used for this formula. They are formatted like comments
-in the @TeX{} typesetting language, just in case you are using @TeX{} or
address@hidden (In this example @TeX{} is not being used, so you might want
-to move these comments up to the top of the file or otherwise put them
-out of the way.)
-
-As an extra flourish, we can add an equation number using a
-righthand label: Type @kbd{d @} (1) @key{RET}}.
-
address@hidden
address@hidden
-% [calc-mode: justify: center]
-% [calc-mode: language: big]
-% [calc-mode: right-label: " (1)"]
-
- 1
- ------- (1)
- ln(x) x
address@hidden group
address@hidden smallexample
-
-To leave Embedded mode, type @kbd{C-x * e} again. The mode line
-and keyboard will revert to the way they were before.
-
-The related command @kbd{C-x * w} operates on a single word, which
-generally means a single number, inside text. It uses any
-non-numeric characters rather than blank lines to delimit the
-formula it reads. Here's an example of its use:
-
address@hidden
-A slope of one-third corresponds to an angle of 1 degrees.
address@hidden smallexample
-
-Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
-Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
-and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
-then @address@hidden * w}} again to exit Embedded mode.
-
address@hidden
-A slope of one-third corresponds to an angle of 18.4349488229 degrees.
address@hidden smallexample
-
address@hidden [fix-ref Embedded Mode]
address@hidden Mode}, for full details.
-
address@hidden Other C-x * Commands, , Embedded Mode Overview, Using Calc
address@hidden Other @kbd{C-x *} Commands
-
address@hidden
-Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
-which ``grab'' data from a selected region of a buffer into the
-Calculator. The region is defined in the usual Emacs way, by
-a ``mark'' placed at one end of the region, and the Emacs
-cursor or ``point'' placed at the other.
-
-The @kbd{C-x * g} command reads the region in the usual left-to-right,
-top-to-bottom order. The result is packaged into a Calc vector
-of numbers and placed on the stack. Calc (in its standard
-user interface) is then started. Type @kbd{v u} if you want
-to unpack this vector into separate numbers on the stack. Also,
address@hidden C-x * g} interprets the region as a single number or
-formula.
-
-The @kbd{C-x * r} command reads a rectangle, with the point and
-mark defining opposite corners of the rectangle. The result
-is a matrix of numbers on the Calculator stack.
-
-Complementary to these is @kbd{C-x * y}, which ``yanks'' the
-value at the top of the Calc stack back into an editing buffer.
-If you type @address@hidden * y}} while in such a buffer, the value is
-yanked at the current position. If you type @kbd{C-x * y} while
-in the Calc buffer, Calc makes an educated guess as to which
-editing buffer you want to use. The Calc window does not have
-to be visible in order to use this command, as long as there
-is something on the Calc stack.
-
-Here, for reference, is the complete list of @kbd{C-x *} commands.
-The shift, control, and meta keys are ignored for the keystroke
-following @kbd{C-x *}.
-
address@hidden
-Commands for turning Calc on and off:
-
address@hidden @kbd
address@hidden *
-Turn Calc on or off, employing the same user interface as last time.
-
address@hidden =, +, -, /, \, &, #
-Alternatives for @kbd{*}.
-
address@hidden C
-Turn Calc on or off using its standard bottom-of-the-screen
-interface. If Calc is already turned on but the cursor is not
-in the Calc window, move the cursor into the window.
-
address@hidden O
-Same as @kbd{C}, but don't select the new Calc window. If
-Calc is already turned on and the cursor is in the Calc window,
-move it out of that window.
-
address@hidden B
-Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.
-
address@hidden Q
-Use Quick mode for a single short calculation.
-
address@hidden K
-Turn Calc Keypad mode on or off.
-
address@hidden E
-Turn Calc Embedded mode on or off at the current formula.
-
address@hidden J
-Turn Calc Embedded mode on or off, select the interesting part.
-
address@hidden W
-Turn Calc Embedded mode on or off at the current word (number).
-
address@hidden Z
-Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
-
address@hidden X
-Quit Calc; turn off standard, Keypad, or Embedded mode if on.
-(This is like @kbd{q} or @key{OFF} inside of Calc.)
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
-
address@hidden
-Commands for moving data into and out of the Calculator:
-
address@hidden @kbd
address@hidden G
-Grab the region into the Calculator as a vector.
-
address@hidden R
-Grab the rectangular region into the Calculator as a matrix.
-
address@hidden :
-Grab the rectangular region and compute the sums of its columns.
-
address@hidden _
-Grab the rectangular region and compute the sums of its rows.
-
address@hidden Y
-Yank a value from the Calculator into the current editing buffer.
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
-
address@hidden
-Commands for use with Embedded mode:
-
address@hidden @kbd
address@hidden A
-``Activate'' the current buffer. Locate all formulas that
-contain @samp{:=} or @samp{=>} symbols and record their locations
-so that they can be updated automatically as variables are changed.
-
address@hidden D
-Duplicate the current formula immediately below and select
-the duplicate.
-
address@hidden F
-Insert a new formula at the current point.
-
address@hidden N
-Move the cursor to the next active formula in the buffer.
-
address@hidden P
-Move the cursor to the previous active formula in the buffer.
-
address@hidden U
-Update (i.e., as if by the @kbd{=} key) the formula at the current point.
-
address@hidden `
-Edit (as if by @code{calc-edit}) the formula at the current point.
address@hidden table
address@hidden
address@hidden 2
address@hidden iftex
-
address@hidden
-Miscellaneous commands:
-
address@hidden @kbd
address@hidden I
-Run the Emacs Info system to read the Calc manual.
-(This is the same as @kbd{h i} inside of Calc.)
-
address@hidden T
-Run the Emacs Info system to read the Calc Tutorial.
-
address@hidden S
-Run the Emacs Info system to read the Calc Summary.
-
address@hidden L
-Load Calc entirely into memory. (Normally the various parts
-are loaded only as they are needed.)
-
address@hidden M
-Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
-and record them as the current keyboard macro.
-
address@hidden 0
-(This is the ``zero'' digit key.) Reset the Calculator to
-its initial state: Empty stack, and initial mode settings.
address@hidden table
-
address@hidden History and Acknowledgements, , Using Calc, Getting Started
address@hidden History and Acknowledgements
-
address@hidden
-Calc was originally started as a two-week project to occupy a lull
-in the author's schedule. Basically, a friend asked if I remembered
-the value of
address@hidden @math{2^{32}}.
address@hidden @expr{2^32}.
-I didn't offhand, but I said, ``that's easy, just call up an
address@hidden'' @code{Xcalc} duly reported that the answer to our
-question was @samp{4.294967e+09}---with no way to see the full ten
-digits even though we knew they were there in the program's memory! I
-was so annoyed, I vowed to write a calculator of my own, once and for
-all.
-
-I chose Emacs Lisp, a) because I had always been curious about it
-and b) because, being only a text editor extension language after
-all, Emacs Lisp would surely reach its limits long before the project
-got too far out of hand.
-
-To make a long story short, Emacs Lisp turned out to be a distressingly
-solid implementation of Lisp, and the humble task of calculating
-turned out to be more open-ended than one might have expected.
-
-Emacs Lisp didn't have built-in floating point math (now it does), so
-this had to be
-simulated in software. In fact, Emacs integers will only comfortably
-fit six decimal digits or so---not enough for a decent calculator. So
-I had to write my own high-precision integer code as well, and once I had
-this I figured that arbitrary-size integers were just as easy as large
-integers. Arbitrary floating-point precision was the logical next step.
-Also, since the large integer arithmetic was there anyway it seemed only
-fair to give the user direct access to it, which in turn made it practical
-to support fractions as well as floats. All these features inspired me
-to look around for other data types that might be worth having.
-
-Around this time, my friend Rick Koshi showed me his nifty new HP-28
-calculator. It allowed the user to manipulate formulas as well as
-numerical quantities, and it could also operate on matrices. I
-decided that these would be good for Calc to have, too. And once
-things had gone this far, I figured I might as well take a look at
-serious algebra systems for further ideas. Since these systems did
-far more than I could ever hope to implement, I decided to focus on
-rewrite rules and other programming features so that users could
-implement what they needed for themselves.
-
-Rick complained that matrices were hard to read, so I put in code to
-format them in a 2D style. Once these routines were in place, Big mode
-was obligatory. Gee, what other language modes would be useful?
-
-Scott Hemphill and Allen Knutson, two friends with a strong mathematical
-bent, contributed ideas and algorithms for a number of Calc features
-including modulo forms, primality testing, and float-to-fraction conversion.
-
-Units were added at the eager insistence of Mass Sivilotti. Later,
-Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
-expert assistance with the units table. As far as I can remember, the
-idea of using algebraic formulas and variables to represent units dates
-back to an ancient article in Byte magazine about muMath, an early
-algebra system for microcomputers.
-
-Many people have contributed to Calc by reporting bugs and suggesting
-features, large and small. A few deserve special mention: Tim Peters,
-who helped develop the ideas that led to the selection commands, rewrite
-rules, and many other algebra features;
address@hidden Fran\c{c}ois
address@hidden Francois
-Pinard, who contributed an early prototype of the Calc Summary appendix
-as well as providing valuable suggestions in many other areas of Calc;
-Carl Witty, whose eagle eyes discovered many typographical and factual
-errors in the Calc manual; Tim Kay, who drove the development of
-Embedded mode; Ove Ewerlid, who made many suggestions relating to the
-algebra commands and contributed some code for polynomial operations;
-Randal Schwartz, who suggested the @code{calc-eval} function; Robert
-J. Chassell, who suggested the Calc Tutorial and exercises; and Juha
-Sarlin, who first worked out how to split Calc into quickly-loading
-parts. Bob Weiner helped immensely with the Lucid Emacs port.
-
address@hidden Bibliography
address@hidden Knuth, Art of Computer Programming
address@hidden Numerical Recipes
address@hidden Should these be expanded into more complete references?
-Among the books used in the development of Calc were Knuth's @emph{Art
-of Computer Programming} (especially volume II, @emph{Seminumerical
-Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
-and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
-for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
-Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
address@hidden Standard Math Tables} (William H. Beyer, ed.); and
-Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
-Functions}. Also, of course, Calc could not have been written without
-the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
-Dan LaLiberte.
-
-Final thanks go to Richard Stallman, without whose fine implementations
-of the Emacs editor, language, and environment, Calc would have been
-finished in two weeks.
-
address@hidden [tutorial]
-
address@hidden
address@hidden This node is accessed by the `C-x * t' command.
address@hidden Interactive Tutorial, Tutorial, Getting Started, Top
address@hidden Tutorial
-
address@hidden
-Some brief instructions on using the Emacs Info system for this tutorial:
-
-Press the space bar and Delete keys to go forward and backward in a
-section by screenfuls (or use the regular Emacs scrolling commands
-for this).
-
-Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
-If the section has a @dfn{menu}, press a digit key like @kbd{1}
-or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
-go back up from a sub-section to the menu it is part of.
-
-Exercises in the tutorial all have cross-references to the
-appropriate page of the ``answers'' section. Press @kbd{f}, then
-the exercise number, to see the answer to an exercise. After
-you have followed a cross-reference, you can press the letter
address@hidden to return to where you were before.
-
-You can press @kbd{?} at any time for a brief summary of Info commands.
-
-Press @kbd{1} now to enter the first section of the Tutorial.
-
address@hidden
-* Tutorial::
address@hidden menu
-
address@hidden Tutorial, Introduction, Interactive Tutorial, Top
address@hidden ifinfo
address@hidden
address@hidden Tutorial, Introduction, Getting Started, Top
address@hidden ifnotinfo
address@hidden Tutorial
-
address@hidden
-This chapter explains how to use Calc and its many features, in
-a step-by-step, tutorial way. You are encouraged to run Calc and
-work along with the examples as you read (@pxref{Starting Calc}).
-If you are already familiar with advanced calculators, you may wish
address@hidden [not-split]
-to skip on to the rest of this manual.
address@hidden [when-split]
address@hidden to skip on to volume II of this manual, the @dfn{Calc Reference}.
-
address@hidden [fix-ref Embedded Mode]
-This tutorial describes the standard user interface of Calc only.
-The Quick mode and Keypad mode interfaces are fairly
-self-explanatory. @xref{Embedded Mode}, for a description of
-the Embedded mode interface.
-
-The easiest way to read this tutorial on-line is to have two windows on
-your Emacs screen, one with Calc and one with the Info system. (If you
-have a printed copy of the manual you can use that instead.) Press
address@hidden * c} to turn Calc on or to switch into the Calc window, and
-press @kbd{C-x * i} to start the Info system or to switch into its window.
-
-This tutorial is designed to be done in sequence. But the rest of this
-manual does not assume you have gone through the tutorial. The tutorial
-does not cover everything in the Calculator, but it touches on most
-general areas.
-
address@hidden
-You may wish to print out a copy of the Calc Summary and keep notes on
-it as you learn Calc. @xref{About This Manual}, to see how to make a
-printed summary. @xref{Summary}.
address@hidden ifnottex
address@hidden
-The Calc Summary at the end of the reference manual includes some blank
-space for your own use. You may wish to keep notes there as you learn
-Calc.
address@hidden iftex
-
address@hidden
-* Basic Tutorial::
-* Arithmetic Tutorial::
-* Vector/Matrix Tutorial::
-* Types Tutorial::
-* Algebra Tutorial::
-* Programming Tutorial::
-
-* Answers to Exercises::
address@hidden menu
-
address@hidden Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
address@hidden Basic Tutorial
-
address@hidden
-In this section, we learn how RPN and algebraic-style calculations
-work, how to undo and redo an operation done by mistake, and how
-to control various modes of the Calculator.
-
address@hidden
-* RPN Tutorial:: Basic operations with the stack.
-* Algebraic Tutorial:: Algebraic entry; variables.
-* Undo Tutorial:: If you make a mistake: Undo and the trail.
-* Modes Tutorial:: Common mode-setting commands.
address@hidden menu
-
address@hidden RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
address@hidden RPN Calculations and the Stack
-
address@hidden RPN notation
address@hidden
address@hidden
-Calc normally uses RPN notation. You may be familiar with the RPN
-system from Hewlett-Packard calculators, FORTH, or PostScript.
-(Reverse Polish Notation, RPN, is named after the Polish mathematician
-Jan Lukasiewicz.)
address@hidden ifnottex
address@hidden
-\noindent
-Calc normally uses RPN notation. You may be familiar with the RPN
-system from Hewlett-Packard calculators, FORTH, or PostScript.
-(Reverse Polish Notation, RPN, is named after the Polish mathematician
-Jan \L ukasiewicz.)
address@hidden tex
-
-The central component of an RPN calculator is the @dfn{stack}. A
-calculator stack is like a stack of dishes. New dishes (numbers) are
-added at the top of the stack, and numbers are normally only removed
-from the top of the stack.
-
address@hidden Operators
address@hidden Operands
-In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
-and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
-enter the operands first, then the operator. Each time you type a
-number, Calc adds or @dfn{pushes} it onto the top of the Stack.
-When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
-number of operands from the stack and pushes back the result.
-
-Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
address@hidden @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return,
corresponds to
-the @key{ENTER} key on traditional RPN calculators.) Try this now if
-you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
address@hidden * c} again or @kbd{C-x * o} to switch back to the Tutorial
window).
-The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
-The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
-and pushes the result (5) back onto the stack. Here's how the stack
-will look at various points throughout the calculation:
-
address@hidden
address@hidden
- . 1: 2 2: 2 1: 5 .
- . 1: 3 .
- .
-
- C-x * c 2 @key{RET} 3 @key{RET} +
@key{DEL}
address@hidden group
address@hidden smallexample
-
-The @samp{.} symbol is a marker that represents the top of the stack.
-Note that the ``top'' of the stack is really shown at the bottom of
-the Stack window. This may seem backwards, but it turns out to be
-less distracting in regular use.
-
address@hidden Stack levels
address@hidden Levels of stack
-The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
-numbers}. Old RPN calculators always had four stack levels called
address@hidden, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
-as large as you like, so it uses numbers instead of letters. Some
-stack-manipulation commands accept a numeric argument that says
-which stack level to work on. Normal commands like @kbd{+} always
-work on the top few levels of the stack.
-
address@hidden [fix-ref Truncating the Stack]
-The Stack buffer is just an Emacs buffer, and you can move around in
-it using the regular Emacs motion commands. But no matter where the
-cursor is, even if you have scrolled the @samp{.} marker out of
-view, most Calc commands always move the cursor back down to level 1
-before doing anything. It is possible to move the @samp{.} marker
-upwards through the stack, temporarily ``hiding'' some numbers from
-commands like @kbd{+}. This is called @dfn{stack truncation} and
-we will not cover it in this tutorial; @pxref{Truncating the Stack},
-if you are interested.
-
-You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
address@hidden +}. That's because if you type any operator name or
-other non-numeric key when you are entering a number, the Calculator
-automatically enters that number and then does the requested command.
-Thus @kbd{2 @key{RET} 3 +} will work just as well.
-
-Examples in this tutorial will often omit @key{RET} even when the
-stack displays shown would only happen if you did press @key{RET}:
-
address@hidden
address@hidden
-1: 2 2: 2 1: 5
- . 1: 3 .
- .
-
- 2 @key{RET} 3 +
address@hidden group
address@hidden smallexample
-
address@hidden
-Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
-with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
-press the optional @key{RET} to see the stack as the figure shows.
-
-(@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
-at various points. Try them if you wish. Answers to all the exercises
-are located at the end of the Tutorial chapter. Each exercise will
-include a cross-reference to its particular answer. If you are
-reading with the Emacs Info system, press @kbd{f} and the
-exercise number to go to the answer, then the letter @kbd{l} to
-return to where you were.)
-
address@hidden
-Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
address@hidden 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
-multiplication.) Figure it out by hand, then try it with Calc to see
-if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 2.} Compute
address@hidden @math{(2\times4) + (7\times9.4) + {5\over4}}
address@hidden @expr{2*4 + 7*9.5 + 5/4}
-using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
-
-The @key{DEL} key is called Backspace on some keyboards. It is
-whatever key you would use to correct a simple typing error when
-regularly using Emacs. The @key{DEL} key pops and throws away the
-top value on the stack. (You can still get that value back from
-the Trail if you should need it later on.) There are many places
-in this tutorial where we assume you have used @key{DEL} to erase the
-results of the previous example at the beginning of a new example.
-In the few places where it is really important to use @key{DEL} to
-clear away old results, the text will remind you to do so.
-
-(It won't hurt to let things accumulate on the stack, except that
-whenever you give a display-mode-changing command Calc will have to
-spend a long time reformatting such a large stack.)
-
-Since the @kbd{-} key is also an operator (it subtracts the top two
-stack elements), how does one enter a negative number? Calc uses
-the @kbd{_} (underscore) key to act like the minus sign in a number.
-So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
-will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
-
-You can also press @kbd{n}, which means ``change sign.'' It changes
-the number at the top of the stack (or the number being entered)
-from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
-
address@hidden Duplicating a stack entry
-If you press @key{RET} when you're not entering a number, the effect
-is to duplicate the top number on the stack. Consider this calculation:
-
address@hidden
address@hidden
-1: 3 2: 3 1: 9 2: 9 1: 81
- . 1: 3 . 1: 9 .
- . .
-
- 3 @key{RET} @key{RET} * @key{RET}
*
address@hidden group
address@hidden smallexample
-
address@hidden
-(Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
-to raise 3 to the fourth power.)
-
-The space-bar key (denoted @key{SPC} here) performs the same function
-as @key{RET}; you could replace all three occurrences of @key{RET} in
-the above example with @key{SPC} and the effect would be the same.
-
address@hidden Exchanging stack entries
-Another stack manipulation key is @key{TAB}. This exchanges the top
-two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
-to get 5, and then you realize what you really wanted to compute
-was @expr{20 / (2+3)}.
-
address@hidden
address@hidden
-1: 5 2: 5 2: 20 1: 4
- . 1: 20 1: 5 .
- . .
-
- 2 @key{RET} 3 + 20 @key{TAB} /
address@hidden group
address@hidden smallexample
-
address@hidden
-Planning ahead, the calculation would have gone like this:
-
address@hidden
address@hidden
-1: 20 2: 20 3: 20 2: 20 1: 4
- . 1: 2 2: 2 1: 5 .
- . 1: 3 .
- .
-
- 20 @key{RET} 2 @key{RET} 3 + /
address@hidden group
address@hidden smallexample
-
-A related stack command is @address@hidden (hold @key{META} and type
address@hidden). It rotates the top three elements of the stack upward,
-bringing the object in level 3 to the top.
-
address@hidden
address@hidden
-1: 10 2: 10 3: 10 3: 20 3: 30
- . 1: 20 2: 20 2: 30 2: 10
- . 1: 30 1: 10 1: 20
- . . .
-
- 10 @key{RET} 20 @key{RET} 30 @key{RET}
address@hidden address@hidden
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
-on the stack. Figure out how to add one to the number in level 2
-without affecting the rest of the stack. Also figure out how to add
-one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
-
-Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
-arguments from the stack and push a result. Operations like @kbd{n} and
address@hidden (square root) pop a single number and push the result. You can
-think of them as simply operating on the top element of the stack.
-
address@hidden
address@hidden
-1: 3 1: 9 2: 9 1: 25 1: 5
- . . 1: 16 . .
- .
-
- 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * +
Q
address@hidden group
address@hidden smallexample
-
address@hidden
-(Note that capital @kbd{Q} means to hold down the Shift key while
-typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
-
address@hidden Pythagorean Theorem
-Here we've used the Pythagorean Theorem to determine the hypotenuse of a
-right triangle. Calc actually has a built-in command for that called
address@hidden h}, but let's suppose we can't remember the necessary keystrokes.
-We can still enter it by its full name using @kbd{M-x} notation:
-
address@hidden
address@hidden
-1: 3 2: 3 1: 5
- . 1: 4 .
- .
-
- 3 @key{RET} 4 @key{RET} M-x calc-hypot
address@hidden group
address@hidden smallexample
-
-All Calculator commands begin with the word @samp{calc-}. Since it
-gets tiring to type this, Calc provides an @kbd{x} key which is just
-like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
-prefix for you:
-
address@hidden
address@hidden
-1: 3 2: 3 1: 5
- . 1: 4 .
- .
-
- 3 @key{RET} 4 @key{RET} x hypot
address@hidden group
address@hidden smallexample
-
-What happens if you take the square root of a negative number?
-
address@hidden
address@hidden
-1: 4 1: -4 1: (0, 2)
- . . .
-
- 4 @key{RET} n Q
address@hidden group
address@hidden smallexample
-
address@hidden
-The notation @expr{(a, b)} represents a complex number.
-Complex numbers are more traditionally written @expr{a + b i};
-Calc can display in this format, too, but for now we'll stick to the
address@hidden(a, b)} notation.
-
-If you don't know how complex numbers work, you can safely ignore this
-feature. Complex numbers only arise from operations that would be
-errors in a calculator that didn't have complex numbers. (For example,
-taking the square root or logarithm of a negative number produces a
-complex result.)
-
-Complex numbers are entered in the notation shown. The @kbd{(} and
address@hidden,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
-
address@hidden
address@hidden
-1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
- . 1: 2 . 3 .
- . .
-
- ( 2 , 3 )
address@hidden group
address@hidden smallexample
-
-You can perform calculations while entering parts of incomplete objects.
-However, an incomplete object cannot actually participate in a calculation:
-
address@hidden
address@hidden
-1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
- . 1: 2 2: 2 5 5
- . 1: 3 . .
- .
- (error)
- ( 2 @key{RET} 3 + +
address@hidden group
address@hidden smallexample
-
address@hidden
-Adding 5 to an incomplete object makes no sense, so the last command
-produces an error message and leaves the stack the same.
-
-Incomplete objects can't participate in arithmetic, but they can be
-moved around by the regular stack commands.
-
address@hidden
address@hidden
-2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
-1: 3 2: 3 2: ( ... 2 .
- . 1: ( ... 1: 2 3
- . . .
-
-2 @key{RET} 3 @key{RET} ( address@hidden
address@hidden )
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that the @kbd{,} (comma) key did not have to be used here.
-When you press @kbd{)} all the stack entries between the incomplete
-entry and the top are collected, so there's never really a reason
-to use the comma. It's up to you.
-
-(@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
-your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
-(Joe thought of a clever way to correct his mistake in only two
-keystrokes, but it didn't quite work. Try it to find out why.)
address@hidden Answer 4, 4}. (@bullet{})
-
-Vectors are entered the same way as complex numbers, but with square
-brackets in place of parentheses. We'll meet vectors again later in
-the tutorial.
-
-Any Emacs command can be given a @dfn{numeric prefix argument} by
-typing a series of @key{META}-digits beforehand. If @key{META} is
-awkward for you, you can instead type @kbd{C-u} followed by the
-necessary digits. Numeric prefix arguments can be negative, as in
address@hidden M-3 M-5} or @address@hidden - 3 5}}. Calc commands use numeric
-prefix arguments in a variety of ways. For example, a numeric prefix
-on the @kbd{+} operator adds any number of stack entries at once:
-
address@hidden
address@hidden
-1: 10 2: 10 3: 10 3: 10 1: 60
- . 1: 20 2: 20 2: 20 .
- . 1: 30 1: 30
- . .
-
- 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3
+
address@hidden group
address@hidden smallexample
-
-For stack manipulation commands like @key{RET}, a positive numeric
-prefix argument operates on the top @var{n} stack entries at once. A
-negative argument operates on the entry in level @var{n} only. An
-argument of zero operates on the entire stack. In this example, we copy
-the second-to-top element of the stack:
-
address@hidden
address@hidden
-1: 10 2: 10 3: 10 3: 10 4: 10
- . 1: 20 2: 20 2: 20 3: 20
- . 1: 30 1: 30 2: 30
- . . 1: 20
- .
-
- 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2
@key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden Clearing the stack
address@hidden Emptying the stack
-Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
-(The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
-entire stack.)
-
address@hidden Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
address@hidden Algebraic-Style Calculations
-
address@hidden
-If you are not used to RPN notation, you may prefer to operate the
-Calculator in Algebraic mode, which is closer to the way
-non-RPN calculators work. In Algebraic mode, you enter formulas
-in traditional @expr{2+3} notation.
-
address@hidden:} Note that @samp{/} has lower precedence than
address@hidden, so that @samp{a/b*c} is interpreted as @samp{a/(b*c)}. See
-below for details.
-
-You don't really need any special ``mode'' to enter algebraic formulas.
-You can enter a formula at any time by pressing the apostrophe (@kbd{'})
-key. Answer the prompt with the desired formula, then press @key{RET}.
-The formula is evaluated and the result is pushed onto the RPN stack.
-If you don't want to think in RPN at all, you can enter your whole
-computation as a formula, read the result from the stack, then press
address@hidden to delete it from the stack.
-
-Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
-The result should be the number 9.
-
-Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
address@hidden/}, and @samp{^}. You can use parentheses to make the order
-of evaluation clear. In the absence of parentheses, @samp{^} is
-evaluated first, then @samp{*}, then @samp{/}, then finally
address@hidden and @samp{-}. For example, the expression
-
address@hidden
-2 + 3*4*5 / 6*7^8 - 9
address@hidden example
-
address@hidden
-is equivalent to
-
address@hidden
-2 + ((3*4*5) / (6*(7^8)) - 9
address@hidden example
-
address@hidden
-or, in large mathematical notation,
-
address@hidden
address@hidden
address@hidden
- 3 * 4 * 5
-2 + --------- - 9
- 8
- 6 * 7
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
-\afterdisplay
address@hidden tex
-
address@hidden
-The result of this expression will be the number @mathit{-6.99999826533}.
-
-Calc's order of evaluation is the same as for most computer languages,
-except that @samp{*} binds more strongly than @samp{/}, as the above
-example shows. As in normal mathematical notation, the @samp{*} symbol
-can often be omitted: @samp{2 a} is the same as @samp{2*a}.
-
-Operators at the same level are evaluated from left to right, except
-that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
-equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
-to @samp{2^(3^4)} (a very large integer; try it!).
-
-If you tire of typing the apostrophe all the time, there is
-Algebraic mode, where Calc automatically senses
-when you are about to type an algebraic expression. To enter this
-mode, press the two letters @address@hidden a}}. (An @samp{Alg} indicator
-should appear in the Calc window's mode line.)
-
-Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
-
-In Algebraic mode, when you press any key that would normally begin
-entering a number (such as a digit, a decimal point, or the @kbd{_}
-key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
-an algebraic entry.
-
-Functions which do not have operator symbols like @samp{+} and @samp{*}
-must be entered in formulas using function-call notation. For example,
-the function name corresponding to the square-root key @kbd{Q} is
address@hidden To compute a square root in a formula, you would use
-the notation @samp{sqrt(@var{x})}.
-
-Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
-be @expr{0.16227766017}.
-
-Note that if the formula begins with a function name, you need to use
-the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
-out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
-command, and the @kbd{csin} will be taken as the name of the rewrite
-rule to use!
-
-Some people prefer to enter complex numbers and vectors in algebraic
-form because they find RPN entry with incomplete objects to be too
-distracting, even though they otherwise use Calc as an RPN calculator.
-
-Still in Algebraic mode, type:
-
address@hidden
address@hidden
-1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
- . 1: (1, -2) . 1: 1 .
- . .
-
- (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET}
+
address@hidden group
address@hidden smallexample
-
-Algebraic mode allows us to enter complex numbers without pressing
-an apostrophe first, but it also means we need to press @key{RET}
-after every entry, even for a simple number like @expr{1}.
-
-(You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
-mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
-though regular numeric keys still use RPN numeric entry. There is also
-Total Algebraic mode, started by typing @kbd{m t}, in which all
-normal keys begin algebraic entry. You must then use the @key{META} key
-to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
-mode, @kbd{M-q} to quit, etc.)
-
-If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
-
-Actual non-RPN calculators use a mixture of algebraic and RPN styles.
-In general, operators of two numbers (like @kbd{+} and @kbd{*})
-use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
-use RPN form. Also, a non-RPN calculator allows you to see the
-intermediate results of a calculation as you go along. You can
-accomplish this in Calc by performing your calculation as a series
-of algebraic entries, using the @kbd{$} sign to tie them together.
-In an algebraic formula, @kbd{$} represents the number on the top
-of the stack. Here, we perform the calculation
address@hidden @math{\sqrt{2\times4+1}},
address@hidden @expr{sqrt(2*4+1)},
-which on a traditional calculator would be done by pressing
address@hidden * 4 + 1 =} and then the square-root key.
-
address@hidden
address@hidden
-1: 8 1: 9 1: 3
- . . .
-
- ' 2*4 @key{RET} $+1 @key{RET} Q
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice that we didn't need to press an apostrophe for the @kbd{$+1},
-because the dollar sign always begins an algebraic entry.
-
-(@bullet{}) @strong{Exercise 1.} How could you get the same effect as
-pressing @kbd{Q} but using an algebraic entry instead? How about
-if the @kbd{Q} key on your keyboard were broken?
address@hidden Answer 1, 1}. (@bullet{})
-
-The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
-entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
-
-Algebraic formulas can include @dfn{variables}. To store in a
-variable, press @kbd{s s}, then type the variable name, then press
address@hidden (There are actually two flavors of store command:
address@hidden s} stores a number in a variable but also leaves the number
-on the stack, while @address@hidden t}} removes a number from the stack and
-stores it in the variable.) A variable name should consist of one
-or more letters or digits, beginning with a letter.
-
address@hidden
address@hidden
-1: 17 . 1: a + a^2 1: 306
- . . .
-
- 17 s t a @key{RET} ' a+a^2 @key{RET} =
address@hidden group
address@hidden smallexample
-
address@hidden
-The @kbd{=} key @dfn{evaluates} a formula by replacing all its
-variables by the values that were stored in them.
-
-For RPN calculations, you can recall a variable's value on the
-stack either by entering its name as a formula and pressing @kbd{=},
-or by using the @kbd{s r} command.
-
address@hidden
address@hidden
-1: 17 2: 17 3: 17 2: 17 1: 306
- . 1: 17 2: 17 1: 289 .
- . 1: 2 .
- .
-
- s r a @key{RET} ' a @key{RET} = 2 ^ +
address@hidden group
address@hidden smallexample
-
-If you press a single digit for a variable name (as in @kbd{s t 3}, you
-get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
-They are ``quick'' simply because you don't have to type the letter
address@hidden or the @key{RET} after their names. In fact, you can type
-simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
address@hidden 3} and @address@hidden 3}}.
-
-Any variables in an algebraic formula for which you have not stored
-values are left alone, even when you evaluate the formula.
-
address@hidden
address@hidden
-1: 2 a + 2 b 1: 34 + 2 b
- . .
-
- ' 2a+2b @key{RET} =
address@hidden group
address@hidden smallexample
-
-Calls to function names which are undefined in Calc are also left
-alone, as are calls for which the value is undefined.
-
address@hidden
address@hidden
-1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
- .
-
- ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-In this example, the first call to @code{log10} works, but the other
-calls are not evaluated. In the second call, the logarithm is
-undefined for that value of the argument; in the third, the argument
-is symbolic, and in the fourth, there are too many arguments. In the
-fifth case, there is no function called @code{foo}. You will see a
-``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
-Press the @kbd{w} (``why'') key to see any other messages that may
-have arisen from the last calculation. In this case you will get
-``logarithm of zero,'' then ``number expected: @code{x}''. Calc
-automatically displays the first message only if the message is
-sufficiently important; for example, Calc considers ``wrong number
-of arguments'' and ``logarithm of zero'' to be important enough to
-report automatically, while a message like ``number expected: @code{x}''
-will only show up if you explicitly press the @kbd{w} key.
-
-(@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
-stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
address@hidden y}. He then tried the same for the formula @samp{2 x (1+y)},
-expecting @samp{10 (1+y)}, but it didn't work. Why not?
address@hidden Answer 2, 2}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 3.} What result would you expect
address@hidden @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
address@hidden Answer 3, 3}. (@bullet{})
-
-One interesting way to work with variables is to use the
address@hidden (@samp{=>}) operator. It works like this:
-Enter a formula algebraically in the usual way, but follow
-the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
-command which builds an @samp{=>} formula using the stack.) On
-the stack, you will see two copies of the formula with an @samp{=>}
-between them. The lefthand formula is exactly like you typed it;
-the righthand formula has been evaluated as if by typing @kbd{=}.
-
address@hidden
address@hidden
-2: 2 + 3 => 5 2: 2 + 3 => 5
-1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
- . .
-
-' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice that the instant we stored a new value in @code{a}, all
address@hidden>} operators already on the stack that referred to @expr{a}
-were updated to use the new value. With @samp{=>}, you can push a
-set of formulas on the stack, then change the variables experimentally
-to see the effects on the formulas' values.
-
-You can also ``unstore'' a variable when you are through with it:
-
address@hidden
address@hidden
-2: 2 + 5 => 5
-1: 2 a + 2 b => 2 a + 2 b
- .
-
- s u a @key{RET}
address@hidden group
address@hidden smallexample
-
-We will encounter formulas involving variables and functions again
-when we discuss the algebra and calculus features of the Calculator.
-
address@hidden Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
address@hidden Undo and Redo
-
address@hidden
-If you make a mistake, you can usually correct it by pressing address@hidden,
-the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
-and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
-with a clean slate. Now:
-
address@hidden
address@hidden
-1: 2 2: 2 1: 8 2: 2 1: 6
- . 1: 3 . 1: 3 .
- . .
-
- 2 @key{RET} 3 ^ U *
address@hidden group
address@hidden smallexample
-
-You can undo any number of times. Calc keeps a complete record of
-all you have done since you last opened the Calc window. After the
-above example, you could type:
-
address@hidden
address@hidden
-1: 6 2: 2 1: 2 . .
- . 1: 3 .
- .
- (error)
- U U U U
address@hidden group
address@hidden smallexample
-
-You can also type @kbd{D} to ``redo'' a command that you have undone
-mistakenly.
-
address@hidden
address@hidden
- . 1: 2 2: 2 1: 6 1: 6
- . 1: 3 . .
- .
- (error)
- D D D D
address@hidden group
address@hidden smallexample
-
address@hidden
-It was not possible to redo past the @expr{6}, since that was placed there
-by something other than an undo command.
-
address@hidden Time travel
-You can think of undo and redo as a sort of ``time machine.'' Press
address@hidden to go backward in time, @kbd{D} to go forward. If you go
-backward and do something (like @kbd{*}) then, as any science fiction
-reader knows, you have changed your future and you cannot go forward
-again. Thus, the inability to redo past the @expr{6} even though there
-was an earlier undo command.
-
-You can always recall an earlier result using the Trail. We've ignored
-the trail so far, but it has been faithfully recording everything we
-did since we loaded the Calculator. If the Trail is not displayed,
-press @kbd{t d} now to turn it on.
-
-Let's try grabbing an earlier result. The @expr{8} we computed was
-undone by a @kbd{U} command, and was lost even to Redo when we pressed
address@hidden, but it's still there in the trail. There should be a little
address@hidden>} arrow (the @dfn{trail pointer}) resting on the last trail
-entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
-Now, press @address@hidden p}} to move the arrow onto the line containing
address@hidden, and press @address@hidden y}} to ``yank'' that number back onto
the
-stack.
-
-If you press @kbd{t ]} again, you will see that even our Yank command
-went into the trail.
-
-Let's go further back in time. Earlier in the tutorial we computed
-a huge integer using the formula @samp{2^3^4}. We don't remember
-what it was, but the first digits were ``241''. Press @kbd{t r}
-(which stands for trail-search-reverse), then type @kbd{241}.
-The trail cursor will jump back to the next previous occurrence of
-the string ``241'' in the trail. This is just a regular Emacs
-incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
-continue the search forwards or backwards as you like.
-
-To finish the search, press @key{RET}. This halts the incremental
-search and leaves the trail pointer at the thing we found. Now we
-can type @kbd{t y} to yank that number onto the stack. If we hadn't
-remembered the ``241'', we could simply have searched for @kbd{2^3^4},
-then pressed @address@hidden t n} to halt and then move to the next item.
-
-You may have noticed that all the trail-related commands begin with
-the letter @kbd{t}. (The store-and-recall commands, on the other hand,
-all began with @kbd{s}.) Calc has so many commands that there aren't
-enough keys for all of them, so various commands are grouped into
-two-letter sequences where the first letter is called the @dfn{prefix}
-key. If you type a prefix key by accident, you can press @kbd{C-g}
-to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
-anything in Emacs.) To get help on a prefix key, press that key
-followed by @kbd{?}. Some prefixes have several lines of help,
-so you need to press @kbd{?} repeatedly to see them all.
-You can also type @kbd{h h} to see all the help at once.
-
-Try pressing @kbd{t ?} now. You will see a line of the form,
-
address@hidden
-trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
address@hidden smallexample
-
address@hidden
-The word ``trail'' indicates that the @kbd{t} prefix key contains
-trail-related commands. Each entry on the line shows one command,
-with a single capital letter showing which letter you press to get
-that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
address@hidden y} so far. The @samp{[MORE]} means you can press @kbd{?}
-again to see more @kbd{t}-prefix commands. Notice that the commands
-are roughly divided (by semicolons) into related groups.
-
-When you are in the help display for a prefix key, the prefix is
-still active. If you press another key, like @kbd{y} for example,
-it will be interpreted as a @kbd{t y} command. If all you wanted
-was to look at the help messages, press @kbd{C-g} afterwards to cancel
-the prefix.
-
-One more way to correct an error is by editing the stack entries.
-The actual Stack buffer is marked read-only and must not be edited
-directly, but you can press @kbd{`} (the backquote or accent grave)
-to edit a stack entry.
-
-Try entering @samp{3.141439} now. If this is supposed to represent
address@hidden, it's got several errors. Press @kbd{`} to edit this number.
-Now use the normal Emacs cursor motion and editing keys to change
-the second 4 to a 5, and to transpose the 3 and the 9. When you
-press @key{RET}, the number on the stack will be replaced by your
-new number. This works for formulas, vectors, and all other types
-of values you can put on the stack. The @kbd{`} key also works
-during entry of a number or algebraic formula.
-
address@hidden Modes Tutorial, , Undo Tutorial, Basic Tutorial
address@hidden Mode-Setting Commands
-
address@hidden
-Calc has many types of @dfn{modes} that affect the way it interprets
-your commands or the way it displays data. We have already seen one
-mode, namely Algebraic mode. There are many others, too; we'll
-try some of the most common ones here.
-
-Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
-Notice the @samp{12} on the Calc window's mode line:
-
address@hidden
---%%-Calc: 12 Deg (Calculator)----All------
address@hidden smallexample
-
address@hidden
-Most of the symbols there are Emacs things you don't need to worry
-about, but the @samp{12} and the @samp{Deg} are mode indicators.
-The @samp{12} means that calculations should always be carried to
-12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
-we get @expr{0.142857142857} with exactly 12 digits, not counting
-leading and trailing zeros.
-
-You can set the precision to anything you like by pressing @kbd{p},
-then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
-then doing @kbd{1 @key{RET} 7 /} again:
-
address@hidden
address@hidden
-1: 0.142857142857
-2: 0.142857142857142857142857142857
- .
address@hidden group
address@hidden smallexample
-
-Although the precision can be set arbitrarily high, Calc always
-has to have @emph{some} value for the current precision. After
-all, the true value @expr{1/7} is an infinitely repeating decimal;
-Calc has to stop somewhere.
-
-Of course, calculations are slower the more digits you request.
-Press @address@hidden 12}} now to set the precision back down to the default.
-
-Calculations always use the current precision. For example, even
-though we have a 30-digit value for @expr{1/7} on the stack, if
-we use it in a calculation in 12-digit mode it will be rounded
-down to 12 digits before it is used. Try it; press @key{RET} to
-duplicate the number, then @address@hidden +}}. Notice that the @key{RET}
-key didn't round the number, because it doesn't do any calculation.
-But the instant we pressed @kbd{+}, the number was rounded down.
-
address@hidden
address@hidden
-1: 0.142857142857
-2: 0.142857142857142857142857142857
-3: 1.14285714286
- .
address@hidden group
address@hidden smallexample
-
address@hidden
-In fact, since we added a digit on the left, we had to lose one
-digit on the right from even the 12-digit value of @expr{1/7}.
-
-How did we get more than 12 digits when we computed @samp{2^3^4}? The
-answer is that Calc makes a distinction between @dfn{integers} and
address@hidden numbers, or @dfn{floats}. An integer is a number
-that does not contain a decimal point. There is no such thing as an
-``infinitely repeating fraction integer,'' so Calc doesn't have to limit
-itself. If you asked for @samp{2^10000} (don't try this!), you would
-have to wait a long time but you would eventually get an exact answer.
-If you ask for @samp{2.^10000}, you will quickly get an answer which is
-correct only to 12 places. The decimal point tells Calc that it should
-use floating-point arithmetic to get the answer, not exact integer
-arithmetic.
-
-You can use the @kbd{F} (@code{calc-floor}) command to convert a
-floating-point value to an integer, and @kbd{c f} (@code{calc-float})
-to convert an integer to floating-point form.
-
-Let's try entering that last calculation:
-
address@hidden
address@hidden
-1: 2. 2: 2. 1: 1.99506311689e3010
- . 1: 10000 .
- .
-
- 2.0 @key{RET} 10000 @key{RET} ^
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden Scientific notation, entry of
-Notice the letter @samp{e} in there. It represents ``times ten to the
-power of,'' and is used by Calc automatically whenever writing the
-number out fully would introduce more extra zeros than you probably
-want to see. You can enter numbers in this notation, too.
-
address@hidden
address@hidden
-1: 2. 2: 2. 1: 1.99506311678e3010
- . 1: 10000. .
- .
-
- 2.0 @key{RET} 1e4 @key{RET} ^
address@hidden group
address@hidden smallexample
-
address@hidden Round-off errors
address@hidden
-Hey, the answer is different! Look closely at the middle columns
-of the two examples. In the first, the stack contained the
-exact integer @expr{10000}, but in the second it contained
-a floating-point value with a decimal point. When you raise a
-number to an integer power, Calc uses repeated squaring and
-multiplication to get the answer. When you use a floating-point
-power, Calc uses logarithms and exponentials. As you can see,
-a slight error crept in during one of these methods. Which
-one should we trust? Let's raise the precision a bit and find
-out:
-
address@hidden
address@hidden
- . 1: 2. 2: 2. 1: 1.995063116880828e3010
- . 1: 10000. .
- .
-
- p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12
@key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden Guard digits
-Presumably, it doesn't matter whether we do this higher-precision
-calculation using an integer or floating-point power, since we
-have added enough ``guard digits'' to trust the first 12 digits
-no matter what. And the verdict address@hidden Integer powers were more
-accurate; in fact, the result was only off by one unit in the
-last place.
-
address@hidden Guard digits
-Calc does many of its internal calculations to a slightly higher
-precision, but it doesn't always bump the precision up enough.
-In each case, Calc added about two digits of precision during
-its calculation and then rounded back down to 12 digits
-afterward. In one case, it was enough; in the other, it
-wasn't. If you really need @var{x} digits of precision, it
-never hurts to do the calculation with a few extra guard digits.
-
-What if we want guard digits but don't want to look at them?
-We can set the @dfn{float format}. Calc supports four major
-formats for floating-point numbers, called @dfn{normal},
address@hidden, @dfn{scientific notation}, and @dfn{engineering
-notation}. You get them by pressing @address@hidden n}}, @kbd{d f},
address@hidden s}, and @kbd{d e}, respectively. In each case, you can
-supply a numeric prefix argument which says how many digits
-should be displayed. As an example, let's put a few numbers
-onto the stack and try some different display modes. First,
-use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
-numbers shown here:
-
address@hidden
address@hidden
-4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
-3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
-2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
-1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
- . . . . .
-
- d n M-3 d n d s M-3 d s M-3 d f
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
-to three significant digits, but then when we typed @kbd{d s} all
-five significant figures reappeared. The float format does not
-affect how numbers are stored, it only affects how they are
-displayed. Only the current precision governs the actual rounding
-of numbers in the Calculator's memory.
-
-Engineering notation, not shown here, is like scientific notation
-except the exponent (the power-of-ten part) is always adjusted to be
-a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
-there will be one, two, or three digits before the decimal point.
-
-Whenever you change a display-related mode, Calc redraws everything
-in the stack. This may be slow if there are many things on the stack,
-so Calc allows you to type address@hidden before any mode command to
-prevent it from updating the stack. Anything Calc displays after the
-mode-changing command will appear in the new format.
-
address@hidden
address@hidden
-4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
-3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
-2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
-1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
- . . . . .
-
- H d s @key{DEL} U @key{TAB} d @key{SPC}
d n
address@hidden group
address@hidden smallexample
-
address@hidden
-Here the @kbd{H d s} command changes to scientific notation but without
-updating the screen. Deleting the top stack entry and undoing it back
-causes it to show up in the new format; swapping the top two stack
-entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
-whole stack. The @kbd{d n} command changes back to the normal float
-format; since it doesn't have an @kbd{H} prefix, it also updates all
-the stack entries to be in @kbd{d n} format.
-
-Notice that the integer @expr{12345} was not affected by any
-of the float formats. Integers are integers, and are always
-displayed exactly.
-
address@hidden Large numbers, readability
-Large integers have their own problems. Let's look back at
-the result of @kbd{2^3^4}.
-
address@hidden
-2417851639229258349412352
address@hidden example
-
address@hidden
-Quick---how many digits does this have? Try typing @kbd{d g}:
-
address@hidden
-2,417,851,639,229,258,349,412,352
address@hidden example
-
address@hidden
-Now how many digits does this have? It's much easier to tell!
-We can actually group digits into clumps of any size. Some
-people prefer @kbd{M-5 d g}:
-
address@hidden
-24178,51639,22925,83494,12352
address@hidden example
-
-Let's see what happens to floating-point numbers when they are grouped.
-First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
-to get ourselves into trouble. Now, type @kbd{1e13 /}:
-
address@hidden
-24,17851,63922.9258349412352
address@hidden example
-
address@hidden
-The integer part is grouped but the fractional part isn't. Now try
address@hidden M-5 d g} (that's meta-minus-sign, meta-five):
-
address@hidden
-24,17851,63922.92583,49412,352
address@hidden example
-
-If you find it hard to tell the decimal point from the commas, try
-changing the grouping character to a space with @kbd{d , @key{SPC}}:
-
address@hidden
-24 17851 63922.92583 49412 352
address@hidden example
-
-Type @kbd{d , ,} to restore the normal grouping character, then
address@hidden g} again to turn grouping off. Also, press @kbd{p 12} to
-restore the default precision.
-
-Press @kbd{U} enough times to get the original big integer back.
-(Notice that @kbd{U} does not undo each mode-setting command; if
-you want to undo a mode-setting command, you have to do it yourself.)
-Now, type @kbd{d r 16 @key{RET}}:
-
address@hidden
-16#200000000000000000000
address@hidden example
-
address@hidden
-The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
-Suddenly it looks pretty simple; this should be no surprise, since we
-got this number by computing a power of two, and 16 is a power of 2.
-In fact, we can use @address@hidden r 2 @key{RET}}} to see it in actual binary
-form:
-
address@hidden
-2#1000000000000000000000000000000000000000000000000000000 @dots{}
address@hidden example
-
address@hidden
-We don't have enough space here to show all the zeros! They won't
-fit on a typical screen, either, so you will have to use horizontal
-scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
-stack window left and right by half its width. Another way to view
-something large is to press @kbd{`} (back-quote) to edit the top of
-stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
-
-You can enter non-decimal numbers using the @kbd{#} symbol, too.
-Let's see what the hexadecimal number @samp{5FE} looks like in
-binary. Type @kbd{16#5FE} (the letters can be typed in upper or
-lower case; they will always appear in upper case). It will also
-help to turn grouping on with @kbd{d g}:
-
address@hidden
-2#101,1111,1110
address@hidden example
-
-Notice that @kbd{d g} groups by fours by default if the display radix
-is binary or hexadecimal, but by threes if it is decimal, octal, or any
-other radix.
-
-Now let's see that number in decimal; type @kbd{d r 10}:
-
address@hidden
-1,534
address@hidden example
-
-Numbers are not @emph{stored} with any particular radix attached. They're
-just numbers; they can be entered in any radix, and are always displayed
-in whatever radix you've chosen with @kbd{d r}. The current radix applies
-to integers, fractions, and floats.
-
address@hidden Roundoff errors, in non-decimal numbers
-(@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
-as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
address@hidden (with 25 2's) in the display. When he multiplied
-that by three, he got @samp{3#0.222222...} instead of the expected
address@hidden Next, Joe entered @samp{3#0.2} and, to his great relief,
-saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
address@hidden (some zeros omitted). What's going on here?
address@hidden Answer 1, 1}. (@bullet{})
-
address@hidden Scientific notation, in non-decimal numbers
-(@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
-modes in the natural way (the exponent is a power of the radix instead of
-a power of ten, although the exponent itself is always written in decimal).
-Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
address@hidden times 16 to the 15th power: We write @samp{16#f.e8fe15}.
-What is wrong with this picture? What could we write instead that would
-work better? @xref{Modes Answer 2, 2}. (@bullet{})
-
-The @kbd{m} prefix key has another set of modes, relating to the way
-Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
-modes generally affect the way things look, @kbd{m}-prefix modes affect
-the way they are actually computed.
-
-The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
-the @samp{Deg} indicator in the mode line. This means that if you use
-a command that interprets a number as an angle, it will assume the
-angle is measured in degrees. For example,
-
address@hidden
address@hidden
-1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
- . . . .
-
- 45 S 2 ^ c 1
address@hidden group
address@hidden smallexample
-
address@hidden
-The address@hidden command computes the sine of an angle. The sine
-of 45 degrees is
address@hidden @math{\sqrt{2}/2};
address@hidden @expr{sqrt(2)/2};
-squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
-roundoff error because the representation of
address@hidden @math{\sqrt{2}/2}
address@hidden @expr{sqrt(2)/2}
-wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
-in this case; it temporarily reduces the precision by one digit while it
-re-rounds the number on the top of the stack.
-
address@hidden Roundoff errors, examples
-(@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
-of 45 degrees as shown above, then, hoping to avoid an inexact
-result, he increased the precision to 16 digits before squaring.
-What happened? @xref{Modes Answer 3, 3}. (@bullet{})
-
-To do this calculation in radians, we would type @kbd{m r} first.
-(The indicator changes to @samp{Rad}.) 45 degrees corresponds to
address@hidden radians. To get @cpi{}, press the @kbd{P} key. (Once
-again, this is a shifted capital @kbd{P}. Remember, unshifted
address@hidden sets the precision.)
-
address@hidden
address@hidden
-1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
- . . .
-
- P 4 / m r S
address@hidden group
address@hidden smallexample
-
-Likewise, inverse trigonometric functions generate results in
-either radians or degrees, depending on the current angular mode.
-
address@hidden
address@hidden
-1: 0.707106781187 1: 0.785398163398 1: 45.
- . . .
-
- .5 Q m r I S m d U I S
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we compute the Inverse Sine of
address@hidden @math{\sqrt{0.5}},
address@hidden @expr{sqrt(0.5)},
-first in radians, then in degrees.
-
-Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
-and vice-versa.
-
address@hidden
address@hidden
-1: 45 1: 0.785398163397 1: 45.
- . . .
-
- 45 c r c d
address@hidden group
address@hidden smallexample
-
-Another interesting mode is @dfn{Fraction mode}. Normally,
-dividing two integers produces a floating-point result if the
-quotient can't be expressed as an exact integer. Fraction mode
-causes integer division to produce a fraction, i.e., a rational
-number, instead.
-
address@hidden
address@hidden
-2: 12 1: 1.33333333333 1: 4:3
-1: 9 . .
- .
-
- 12 @key{RET} 9 / m f U / m f
address@hidden group
address@hidden smallexample
-
address@hidden
-In the first case, we get an approximate floating-point result.
-In the second case, we get an exact fractional result (four-thirds).
-
-You can enter a fraction at any time using @kbd{:} notation.
-(Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
-because @kbd{/} is already used to divide the top two stack
-elements.) Calculations involving fractions will always
-produce exact fractional results; Fraction mode only says
-what to do when dividing two integers.
-
address@hidden Fractions vs. floats
address@hidden Floats vs. fractions
-(@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
-why would you ever use floating-point numbers instead?
address@hidden Answer 4, 4}. (@bullet{})
-
-Typing @kbd{m f} doesn't change any existing values in the stack.
-In the above example, we had to Undo the division and do it over
-again when we changed to Fraction mode. But if you use the
-evaluates-to operator you can get commands like @kbd{m f} to
-recompute for you.
-
address@hidden
address@hidden
-1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
- . . .
-
- ' 12/9 => @key{RET} p 4 @key{RET} m f
address@hidden group
address@hidden smallexample
-
address@hidden
-In this example, the righthand side of the @samp{=>} operator
-on the stack is recomputed when we change the precision, then
-again when we change to Fraction mode. All @samp{=>} expressions
-on the stack are recomputed every time you change any mode that
-might affect their values.
-
address@hidden Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial,
Tutorial
address@hidden Arithmetic Tutorial
-
address@hidden
-In this section, we explore the arithmetic and scientific functions
-available in the Calculator.
-
-The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
-and @kbd{^}. Each normally takes two numbers from the top of the stack
-and pushes back a result. The @kbd{n} and @kbd{&} keys perform
-change-sign and reciprocal operations, respectively.
-
address@hidden
address@hidden
-1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
- . . . . .
-
- 5 & & n n
address@hidden group
address@hidden smallexample
-
address@hidden Binary operators
-You can apply a ``binary operator'' like @kbd{+} across any number of
-stack entries by giving it a numeric prefix. You can also apply it
-pairwise to several stack elements along with the top one if you use
-a negative prefix.
-
address@hidden
address@hidden
-3: 2 1: 9 3: 2 4: 2 3: 12
-2: 3 . 2: 3 3: 3 2: 13
-1: 4 1: 4 2: 4 1: 14
- . . 1: 10 .
- .
-
-2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M--
M-3 +
address@hidden group
address@hidden smallexample
-
address@hidden Unary operators
-You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
-stack entries with a numeric prefix, too.
-
address@hidden
address@hidden
-3: 2 3: 0.5 3: 0.5
-2: 3 2: 0.333333333333 2: 3.
-1: 4 1: 0.25 1: 4.
- . . .
-
-2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
address@hidden group
address@hidden smallexample
-
-Notice that the results here are left in floating-point form.
-We can convert them back to integers by pressing @kbd{F}, the
-``floor'' function. This function rounds down to the next lower
-integer. There is also @kbd{R}, which rounds to the nearest
-integer.
-
address@hidden
address@hidden
-7: 2. 7: 2 7: 2
-6: 2.4 6: 2 6: 2
-5: 2.5 5: 2 5: 3
-4: 2.6 4: 2 4: 3
-3: -2. 3: -2 3: -2
-2: -2.4 2: -3 2: -2
-1: -2.6 1: -3 1: -3
- . . .
-
- M-7 F U M-7 R
address@hidden group
address@hidden smallexample
-
-Since dividing-and-flooring (i.e., ``integer quotient'') is such a
-common operation, Calc provides a special command for that purpose, the
-backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
-computes the remainder that would arise from a @kbd{\} operation, i.e.,
-the ``modulo'' of two numbers. For example,
-
address@hidden
address@hidden
-2: 1234 1: 12 2: 1234 1: 34
-1: 100 . 1: 100 .
- . .
-
-1234 @key{RET} 100 \ U %
address@hidden group
address@hidden smallexample
-
-These commands actually work for any real numbers, not just integers.
-
address@hidden
address@hidden
-2: 3.1415 1: 3 2: 3.1415 1: 0.1415
-1: 1 . 1: 1 .
- . .
-
-3.1415 @key{RET} 1 \ U %
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
-frill, since you could always do the same thing with @kbd{/ F}. Think
-of a situation where this is not address@hidden/ F} would be inadequate.
-Now think of a way you could get around the problem if Calc didn't
-provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
-
-We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
-commands. Other commands along those lines are @kbd{C} (cosine),
address@hidden (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
-logarithm). These can be modified by the @kbd{I} (inverse) and
address@hidden (hyperbolic) prefix keys.
-
-Let's compute the sine and cosine of an angle, and verify the
-identity
address@hidden @math{\sin^2x + \cos^2x = 1}.
address@hidden @expr{sin(x)^2 + cos(x)^2 = 1}.
-We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
-With the angular mode set to degrees (type @address@hidden d}}), do:
-
address@hidden
address@hidden
-2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
-1: -64 1: -0.89879 1: -64 1: 0.43837 .
- . . . .
-
- 64 n @key{RET} @key{RET} S @key{TAB} C
f h
address@hidden group
address@hidden smallexample
-
address@hidden
-(For brevity, we're showing only five digits of the results here.
-You can of course do these calculations to any precision you like.)
-
-Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
-of squares, command.
-
-Another identity is
address@hidden @math{\displaystyle\tan x = {\sin x \over \cos x}}.
address@hidden @expr{tan(x) = sin(x) / cos(x)}.
address@hidden
address@hidden
-
-2: -0.89879 1: -2.0503 1: -64.
-1: 0.43837 . .
- .
-
- U / I T
address@hidden group
address@hidden smallexample
-
-A physical interpretation of this calculation is that if you move
address@hidden units downward and @expr{0.43837} units to the right,
-your direction of motion is @mathit{-64} degrees from horizontal. Suppose
-we move in the opposite direction, up and to the left:
-
address@hidden
address@hidden
-2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
-1: 0.43837 1: -0.43837 . .
- . .
-
- U U M-2 n / I T
address@hidden group
address@hidden smallexample
-
address@hidden
-How can the angle be the same? The answer is that the @kbd{/} operation
-loses information about the signs of its inputs. Because the quotient
-is negative, we know exactly one of the inputs was negative, but we
-can't tell which one. There is an @kbd{f T} address@hidden function which
-computes the inverse tangent of the quotient of a pair of numbers.
-Since you feed it the two original numbers, it has enough information
-to give you a full 360-degree answer.
-
address@hidden
address@hidden
-2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
-1: -0.43837 . 2: -0.89879 1: -64. .
- . 1: 0.43837 .
- .
-
- U U f T address@hidden M-2 n f T -
address@hidden group
address@hidden smallexample
-
address@hidden
-The resulting angles differ by 180 degrees; in other words, they
-point in opposite directions, just as we would expect.
-
-The @address@hidden we used in the third step is the
-``last-arguments'' command. It is sort of like Undo, except that it
-restores the arguments of the last command to the stack without removing
-the command's result. It is useful in situations like this one,
-where we need to do several operations on the same inputs. We could
-have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
-the top two stack elements right after the @kbd{U U}, then a pair of
address@hidden@key{TAB}} commands to cycle the 116 up around the duplicates.
-
-A similar identity is supposed to hold for hyperbolic sines and cosines,
-except that it is the @emph{difference}
address@hidden @math{\cosh^2x - \sinh^2x}
address@hidden @expr{cosh(x)^2 - sinh(x)^2}
-that always equals one. Let's try to verify this identity.
-
address@hidden
address@hidden
-2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
-1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
- . . . . .
-
- 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB}
H S 2 ^
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden Roundoff errors, examples
-Something's obviously wrong, because when we subtract these numbers
-the answer will clearly be zero! But if you think about it, if these
-numbers @emph{did} differ by one, it would be in the 55th decimal
-place. The difference we seek has been lost entirely to roundoff
-error.
-
-We could verify this hypothesis by doing the actual calculation with,
-say, 60 decimal places of precision. This will be slow, but not
-enormously so. Try it if you wish; sure enough, the answer is
-0.99999, reasonably close to 1.
-
-Of course, a more reasonable way to verify the identity is to use
-a more reasonable value for @expr{x}!
-
address@hidden Common logarithm
-Some Calculator commands use the Hyperbolic prefix for other purposes.
-The logarithm and exponential functions, for example, work to the base
address@hidden normally but use base-10 instead if you use the Hyperbolic
-prefix.
-
address@hidden
address@hidden
-1: 1000 1: 6.9077 1: 1000 1: 3
- . . . .
-
- 1000 L U H L
address@hidden group
address@hidden smallexample
-
address@hidden
-First, we mistakenly compute a natural logarithm. Then we undo
-and compute a common logarithm instead.
-
-The @kbd{B} key computes a general address@hidden logarithm for any
-value of @var{b}.
-
address@hidden
address@hidden
-2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
-1: 10 . . 1: 2.71828 .
- . .
-
- 1000 @key{RET} 10 B H E H P B
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we first use @kbd{B} to compute the base-10 logarithm, then use
-the ``hyperbolic'' exponential as a cheap hack to recover the number
-1000, then use @kbd{B} again to compute the natural logarithm. Note
-that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
-onto the stack.
-
-You may have noticed that both times we took the base-10 logarithm
-of 1000, we got an exact integer result. Calc always tries to give
-an exact rational result for calculations involving rational numbers
-where possible. But when we used @kbd{H E}, the result was a
-floating-point number for no apparent reason. In fact, if we had
-computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
-exact integer 1000. But the @kbd{H E} command is rigged to generate
-a floating-point result all of the time so that @kbd{1000 H E} will
-not waste time computing a thousand-digit integer when all you
-probably wanted was @samp{1e1000}.
-
-(@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
-the @kbd{B} command for which Calc could find an exact rational
-result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
-
-The Calculator also has a set of functions relating to combinatorics
-and statistics. You may be familiar with the @dfn{factorial} function,
-which computes the product of all the integers up to a given number.
-
address@hidden
address@hidden
-1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
- . . . .
-
- 100 ! U c f !
address@hidden group
address@hidden smallexample
-
address@hidden
-Recall, the @kbd{c f} command converts the integer or fraction at the
-top of the stack to floating-point format. If you take the factorial
-of a floating-point number, you get a floating-point result
-accurate to the current precision. But if you give @kbd{!} an
-exact integer, you get an exact integer result (158 digits long
-in this case).
-
-If you take the factorial of a non-integer, Calc uses a generalized
-factorial function defined in terms of Euler's Gamma function
address@hidden @math{\Gamma(n)}
address@hidden @expr{gamma(n)}
-(which is itself available as the @kbd{f g} command).
-
address@hidden
address@hidden
-3: 4. 3: 24. 1: 5.5 1: 52.342777847
-2: 4.5 2: 52.3427777847 . .
-1: 5. 1: 120.
- . .
-
- M-3 ! M-0 @key{DEL} 5.5 f g
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we verify the identity
address@hidden @math{n! = \Gamma(n+1)}.
address@hidden @address@hidden@: = gamma(@var{n}+1)}.
-
-The binomial coefficient @address@hidden
address@hidden or @math{\displaystyle {n \choose m}}
-is defined by
address@hidden @math{\displaystyle {n! \over m! \, (n-m)!}}
address@hidden @expr{n!@: / m!@: (n-m)!}
-for all reals @expr{n} and @expr{m}. The intermediate results in this
-formula can become quite large even if the final result is small; the
address@hidden c} command computes a binomial coefficient in a way that avoids
-large intermediate values.
-
-The @kbd{k} prefix key defines several common functions out of
-combinatorics and number theory. Here we compute the binomial
-coefficient 30-choose-20, then determine its prime factorization.
-
address@hidden
address@hidden
-2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
-1: 20 . .
- .
-
- 30 @key{RET} 20 k c k f
address@hidden group
address@hidden smallexample
-
address@hidden
-You can verify these prime factors by using @kbd{v u} to ``unpack''
-this vector into 8 separate stack entries, then @kbd{M-8 *} to
-multiply them back together. The result is the original number,
-30045015.
-
address@hidden Hash tables
-Suppose a program you are writing needs a hash table with at least
-10000 entries. It's best to use a prime number as the actual size
-of a hash table. Calc can compute the next prime number after 10000:
-
address@hidden
address@hidden
-1: 10000 1: 10007 1: 9973
- . . .
-
- 10000 k n I k n
address@hidden group
address@hidden smallexample
-
address@hidden
-Just for kicks we've also computed the next prime @emph{less} than
-10000.
-
address@hidden [fix-ref Financial Functions]
address@hidden Functions}, for a description of the Calculator
-commands that deal with business and financial calculations (functions
-like @code{pv}, @code{rate}, and @code{sln}).
-
address@hidden [fix-ref Binary Number Functions]
address@hidden Functions}, to read about the commands for operating
-on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
-
address@hidden Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial,
Tutorial
address@hidden Vector/Matrix Tutorial
-
address@hidden
-A @dfn{vector} is a list of numbers or other Calc data objects.
-Calc provides a large set of commands that operate on vectors. Some
-are familiar operations from vector analysis. Others simply treat
-a vector as a list of objects.
-
address@hidden
-* Vector Analysis Tutorial::
-* Matrix Tutorial::
-* List Tutorial::
address@hidden menu
-
address@hidden Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix
Tutorial, Vector/Matrix Tutorial
address@hidden Vector Analysis
-
address@hidden
-If you add two vectors, the result is a vector of the sums of the
-elements, taken pairwise.
-
address@hidden
address@hidden
-1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
- . 1: [7, 6, 0] .
- .
-
- [1,2,3] s 1 [7 6 0] s 2 +
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that we can separate the vector elements with either commas or
-spaces. This is true whether we are using incomplete vectors or
-algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
-vectors so we can easily reuse them later.
-
-If you multiply two vectors, the result is the sum of the products
-of the elements taken pairwise. This is called the @dfn{dot product}
-of the vectors.
-
address@hidden
address@hidden
-2: [1, 2, 3] 1: 19
-1: [7, 6, 0] .
- .
-
- r 1 r 2 *
address@hidden group
address@hidden smallexample
-
address@hidden Dot product
-The dot product of two vectors is equal to the product of their
-lengths times the cosine of the angle between them. (Here the vector
-is interpreted as a line from the origin @expr{(0,0,0)} to the
-specified point in three-dimensional space.) The @kbd{A}
-(absolute value) command can be used to compute the length of a
-vector.
-
address@hidden
address@hidden
-3: 19 3: 19 1: 0.550782 1: 56.579
-2: [1, 2, 3] 2: 3.741657 . .
-1: [7, 6, 0] 1: 9.219544
- . .
-
- address@hidden M-2 A * / I C
address@hidden group
address@hidden smallexample
-
address@hidden
-First we recall the arguments to the dot product command, then
-we compute the absolute values of the top two stack entries to
-obtain the lengths of the vectors, then we divide the dot product
-by the product of the lengths to get the cosine of the angle.
-The inverse cosine finds that the angle between the vectors
-is about 56 degrees.
-
address@hidden Cross product
address@hidden Perpendicular vectors
-The @dfn{cross product} of two vectors is a vector whose length
-is the product of the lengths of the inputs times the sine of the
-angle between them, and whose direction is perpendicular to both
-input vectors. Unlike the dot product, the cross product is
-defined only for three-dimensional vectors. Let's double-check
-our computation of the angle using the cross product.
-
address@hidden
address@hidden
-2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
-1: [7, 6, 0] 2: [1, 2, 3] . .
- . 1: [7, 6, 0]
- .
-
- r 1 r 2 V C s 3 address@hidden M-2 A * / A I S
address@hidden group
address@hidden smallexample
-
address@hidden
-First we recall the original vectors and compute their cross product,
-which we also store for later reference. Now we divide the vector
-by the product of the lengths of the original vectors. The length of
-this vector should be the sine of the angle; sure enough, it is!
-
address@hidden [fix-ref General Mode Commands]
-Vector-related commands generally begin with the @kbd{v} prefix key.
-Some are uppercase letters and some are lowercase. To make it easier
-to type these commands, the address@hidden prefix key acts the same as
-the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
-prefix keys have this property.)
-
-If we take the dot product of two perpendicular vectors we expect
-to get zero, since the cosine of 90 degrees is zero. Let's check
-that the cross product is indeed perpendicular to both inputs:
-
address@hidden
address@hidden
-2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
-1: [-18, 21, -8] . 1: [-18, 21, -8] .
- . .
-
- r 1 r 3 * @key{DEL} r 2 r 3 *
address@hidden group
address@hidden smallexample
-
address@hidden Normalizing a vector
address@hidden Unit vectors
-(@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
-stack, what keystrokes would you use to @dfn{normalize} the
-vector, i.e., to reduce its length to one without changing its
-direction? @xref{Vector Answer 1, 1}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
-at any of several positions along a ruler. You have a list of
-those positions in the form of a vector, and another list of the
-probabilities for the particle to be at the corresponding positions.
-Find the average position of the particle.
address@hidden Answer 2, 2}. (@bullet{})
-
address@hidden Matrix Tutorial, List Tutorial, Vector Analysis Tutorial,
Vector/Matrix Tutorial
address@hidden Matrices
-
address@hidden
-A @dfn{matrix} is just a vector of vectors, all the same length.
-This means you can enter a matrix using nested brackets. You can
-also use the semicolon character to enter a matrix. We'll show
-both methods here:
-
address@hidden
address@hidden
-1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
- [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
- . .
-
- [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-We'll be using this matrix again, so type @kbd{s 4} to save it now.
-
-Note that semicolons work with incomplete vectors, but they work
-better in algebraic entry. That's why we use the apostrophe in
-the second example.
-
-When two matrices are multiplied, the lefthand matrix must have
-the same number of columns as the righthand matrix has rows.
-Row @expr{i}, column @expr{j} of the result is effectively the
-dot product of row @expr{i} of the left matrix by column @expr{j}
-of the right matrix.
-
-If we try to duplicate this matrix and multiply it by itself,
-the dimensions are wrong and the multiplication cannot take place:
-
address@hidden
address@hidden
-1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
- [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
- .
-
- @key{RET} *
address@hidden group
address@hidden smallexample
-
address@hidden
-Though rather hard to read, this is a formula which shows the product
-of two matrices. The @samp{*} function, having invalid arguments, has
-been left in symbolic form.
-
-We can multiply the matrices if we @dfn{transpose} one of them first.
-
address@hidden
address@hidden
-2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
- [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
-1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
- [ 2, 5 ] .
- [ 3, 6 ] ]
- .
-
- U v t * U @key{TAB} *
address@hidden group
address@hidden smallexample
-
-Matrix multiplication is not commutative; indeed, switching the
-order of the operands can even change the dimensions of the result
-matrix, as happened here!
-
-If you multiply a plain vector by a matrix, it is treated as a
-single row or column depending on which side of the matrix it is
-on. The result is a plain vector which should also be interpreted
-as a row or column as appropriate.
-
address@hidden
address@hidden
-2: [ [ 1, 2, 3 ] 1: [14, 32]
- [ 4, 5, 6 ] ] .
-1: [1, 2, 3]
- .
-
- r 4 r 1 *
address@hidden group
address@hidden smallexample
-
-Multiplying in the other order wouldn't work because the number of
-rows in the matrix is different from the number of elements in the
-vector.
-
-(@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
-of the above
address@hidden @math{2\times3}
address@hidden 2x3
-matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
-to get @expr{[5, 7, 9]}.
address@hidden Answer 1, 1}. (@bullet{})
-
address@hidden Identity matrix
-An @dfn{identity matrix} is a square matrix with ones along the
-diagonal and zeros elsewhere. It has the property that multiplication
-by an identity matrix, on the left or on the right, always produces
-the original matrix.
-
address@hidden
address@hidden
-1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
- [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
- . 1: [ [ 1, 0, 0 ] .
- [ 0, 1, 0 ]
- [ 0, 0, 1 ] ]
- .
-
- r 4 v i 3 @key{RET} *
address@hidden group
address@hidden smallexample
-
-If a matrix is square, it is often possible to find its @dfn{inverse},
-that is, a matrix which, when multiplied by the original matrix, yields
-an identity matrix. The @kbd{&} (reciprocal) key also computes the
-inverse of a matrix.
-
address@hidden
address@hidden
-1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
- [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
- [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
- . .
-
- r 4 r 2 | s 5 &
address@hidden group
address@hidden smallexample
-
address@hidden
-The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
-matrices together. Here we have used it to add a new row onto
-our matrix to make it square.
-
-We can multiply these two matrices in either order to get an identity.
-
address@hidden
address@hidden
-1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
- [ 0., 1., 0. ] [ 0., 1., 0. ]
- [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
- . .
-
- address@hidden * U @key{TAB} *
address@hidden group
address@hidden smallexample
-
address@hidden Systems of linear equations
address@hidden Linear equations, systems of
-Matrix inverses are related to systems of linear equations in algebra.
-Suppose we had the following set of equations:
-
address@hidden
address@hidden
address@hidden
- a + 2b + 3c = 6
- 4a + 5b + 6c = 2
- 7a + 6b = 3
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplayh
-$$ \openup1\jot \tabskip=0pt plus1fil
-\halign to\displaywidth{\tabskip=0pt
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
- a&+&2b&+&3c&=6 \cr
- 4a&+&5b&+&6c&=2 \cr
- 7a&+&6b& & &=3 \cr}
-$$
-\afterdisplayh
address@hidden tex
-
address@hidden
-This can be cast into the matrix equation,
-
address@hidden
address@hidden
address@hidden
- [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
- [ 4, 5, 6 ] * [ b ] = [ 2 ]
- [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
- \times
- \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
-$$
-\afterdisplay
address@hidden tex
-
-We can solve this system of equations by multiplying both sides by the
-inverse of the matrix. Calc can do this all in one step:
-
address@hidden
address@hidden
-2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
-1: [ [ 1, 2, 3 ] .
- [ 4, 5, 6 ]
- [ 7, 6, 0 ] ]
- .
-
- [6,2,3] r 5 /
address@hidden group
address@hidden smallexample
-
address@hidden
-The result is the @expr{[a, b, c]} vector that solves the equations.
-(Dividing by a square matrix is equivalent to multiplying by its
-inverse.)
-
-Let's verify this solution:
-
address@hidden
address@hidden
-2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
- [ 4, 5, 6 ] .
- [ 7, 6, 0 ] ]
-1: [-12.6, 15.2, -3.93333]
- .
-
- r 5 @key{TAB} *
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that we had to be careful about the order in which we multiplied
-the matrix and vector. If we multiplied in the other order, Calc would
-assume the vector was a row vector in order to make the dimensions
-come out right, and the answer would be incorrect. If you
-don't feel safe letting Calc take either interpretation of your
-vectors, use explicit
address@hidden @math{N\times1}
address@hidden Nx1
-or
address@hidden @math{1\times N}
address@hidden 1xN
-matrices instead. In this case, you would enter the original column
-vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
-
-(@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
-vectors and matrices that include variables. Solve the following
-system of equations to get expressions for @expr{x} and @expr{y}
-in terms of @expr{a} and @expr{b}.
-
address@hidden
address@hidden
address@hidden
- x + a y = 6
- x + b y = 10
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \eqalign{ x &+ a y = 6 \cr
- x &+ b y = 10}
-$$
-\afterdisplay
address@hidden tex
-
address@hidden
address@hidden Answer 2, 2}. (@bullet{})
-
address@hidden Least-squares for over-determined systems
address@hidden Over-determined systems of equations
-(@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
-if it has more equations than variables. It is often the case that
-there are no values for the variables that will satisfy all the
-equations at once, but it is still useful to find a set of values
-which ``nearly'' satisfy all the equations. In terms of matrix equations,
-you can't solve @expr{A X = B} directly because the matrix @expr{A}
-is not square for an over-determined system. Matrix inversion works
-only for square matrices. One common trick is to multiply both sides
-on the left by the transpose of @expr{A}:
address@hidden
address@hidden(A)*A*X = trn(A)*B}.
address@hidden ifnottex
address@hidden
-\turnoffactive
-$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
address@hidden tex
-Now
address@hidden @math{A^T A}
address@hidden @expr{trn(A)*A}
-is a square matrix so a solution is possible. It turns out that the
address@hidden vector you compute in this way will be a ``least-squares''
-solution, which can be regarded as the ``closest'' solution to the set
-of equations. Use Calc to solve the following over-determined
-system:
-
address@hidden
address@hidden
address@hidden
- a + 2b + 3c = 6
- 4a + 5b + 6c = 2
- 7a + 6b = 3
- 2a + 4b + 6c = 11
address@hidden example
address@hidden group
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplayh
-$$ \openup1\jot \tabskip=0pt plus1fil
-\halign to\displaywidth{\tabskip=0pt
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
- a&+&2b&+&3c&=6 \cr
- 4a&+&5b&+&6c&=2 \cr
- 7a&+&6b& & &=3 \cr
- 2a&+&4b&+&6c&=11 \cr}
-$$
-\afterdisplayh
address@hidden tex
-
address@hidden
address@hidden Answer 3, 3}. (@bullet{})
-
address@hidden List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
address@hidden Vectors as Lists
-
address@hidden
address@hidden Lists
-Although Calc has a number of features for manipulating vectors and
-matrices as mathematical objects, you can also treat vectors as
-simple lists of values. For example, we saw that the @kbd{k f}
-command returns a vector which is a list of the prime factors of a
-number.
-
-You can pack and unpack stack entries into vectors:
-
address@hidden
address@hidden
-3: 10 1: [10, 20, 30] 3: 10
-2: 20 . 2: 20
-1: 30 1: 30
- . .
-
- M-3 v p v u
address@hidden group
address@hidden smallexample
-
-You can also build vectors out of consecutive integers, or out
-of many copies of a given value:
-
address@hidden
address@hidden
-1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
- . 1: 17 1: [17, 17, 17, 17]
- . .
-
- v x 4 @key{RET} 17 v b 4 @key{RET}
address@hidden group
address@hidden smallexample
-
-You can apply an operator to every element of a vector using the
address@hidden command.
-
address@hidden
address@hidden
-1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
- . . .
-
- V M * 2 V M ^ V M Q
address@hidden group
address@hidden smallexample
-
address@hidden
-In the first step, we multiply the vector of integers by the vector
-of 17's elementwise. In the second step, we raise each element to
-the power two. (The general rule is that both operands must be
-vectors of the same length, or else one must be a vector and the
-other a plain number.) In the final step, we take the square root
-of each element.
-
-(@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
-from
address@hidden @math{2^{-4}}
address@hidden @expr{2^-4}
-to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
-
-You can also @dfn{reduce} a binary operator across a vector.
-For example, reducing @samp{*} computes the product of all the
-elements in the vector:
-
address@hidden
address@hidden
-1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
- . . .
-
- 123123 k f V R *
address@hidden group
address@hidden smallexample
-
address@hidden
-In this example, we decompose 123123 into its prime factors, then
-multiply those factors together again to yield the original number.
-
-We could compute a dot product ``by hand'' using mapping and
-reduction:
-
address@hidden
address@hidden
-2: [1, 2, 3] 1: [7, 12, 0] 1: 19
-1: [7, 6, 0] . .
- .
-
- r 1 r 2 V M * V R +
address@hidden group
address@hidden smallexample
-
address@hidden
-Recalling two vectors from the previous section, we compute the
-sum of pairwise products of the elements to get the same answer
-for the dot product as before.
-
-A slight variant of vector reduction is the @dfn{accumulate} operation,
address@hidden U}. This produces a vector of the intermediate results from
-a corresponding reduction. Here we compute a table of factorials:
-
address@hidden
address@hidden
-1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
- . .
-
- v x 6 @key{RET} V U *
address@hidden group
address@hidden smallexample
-
-Calc allows vectors to grow as large as you like, although it gets
-rather slow if vectors have more than about a hundred elements.
-Actually, most of the time is spent formatting these large vectors
-for display, not calculating on them. Try the following experiment
-(if your computer is very fast you may need to substitute a larger
-vector size).
-
address@hidden
address@hidden
-1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
- . .
-
- v x 500 @key{RET} 1 V M +
address@hidden group
address@hidden smallexample
-
-Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
-experiment again. In @kbd{v .} mode, long vectors are displayed
-``abbreviated'' like this:
-
address@hidden
address@hidden
-1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
- . .
-
- v x 500 @key{RET} 1 V M +
address@hidden group
address@hidden smallexample
-
address@hidden
-(where now the @samp{...} is actually part of the Calc display).
-You will find both operations are now much faster. But notice that
-even in @address@hidden .}} mode, the full vectors are still shown in the
Trail.
-Type @address@hidden .}} to cause the trail to abbreviate as well, and try the
-experiment one more time. Operations on long vectors are now quite
-fast! (But of course if you use @kbd{t .} you will lose the ability
-to get old vectors back using the @kbd{t y} command.)
-
-An easy way to view a full vector when @kbd{v .} mode is active is
-to press @kbd{`} (back-quote) to edit the vector; editing always works
-with the full, unabbreviated value.
-
address@hidden Least-squares for fitting a straight line
address@hidden Fitting data to a line
address@hidden Line, fitting data to
address@hidden Data, extracting from buffers
address@hidden Columns of data, extracting
-As a larger example, let's try to fit a straight line to some data,
-using the method of least squares. (Calc has a built-in command for
-least-squares curve fitting, but we'll do it by hand here just to
-practice working with vectors.) Suppose we have the following list
-of values in a file we have loaded into Emacs:
-
address@hidden
- x y
- --- ---
- 1.34 0.234
- 1.41 0.298
- 1.49 0.402
- 1.56 0.412
- 1.64 0.466
- 1.73 0.473
- 1.82 0.601
- 1.91 0.519
- 2.01 0.603
- 2.11 0.637
- 2.22 0.645
- 2.33 0.705
- 2.45 0.917
- 2.58 1.009
- 2.71 0.971
- 2.85 1.062
- 3.00 1.148
- 3.15 1.157
- 3.32 1.354
address@hidden smallexample
-
address@hidden
-If you are reading this tutorial in printed form, you will find it
-easiest to press @kbd{C-x * i} to enter the on-line Info version of
-the manual and find this table there. (Press @kbd{g}, then type
address@hidden Tutorial}, to jump straight to this section.)
-
-Position the cursor at the upper-left corner of this table, just
-to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
-(On your system this may be @kbd{C-2}, @address@hidden, or @kbd{NUL}.)
-Now position the cursor to the lower-right, just after the @expr{1.354}.
-You have now defined this region as an Emacs ``rectangle.'' Still
-in the Info buffer, type @kbd{C-x * r}. This command
-(@code{calc-grab-rectangle}) will pop you back into the Calculator, with
-the contents of the rectangle you specified in the form of a matrix.
-
address@hidden
address@hidden
-1: [ [ 1.34, 0.234 ]
- [ 1.41, 0.298 ]
- @dots{}
address@hidden group
address@hidden smallexample
-
address@hidden
-(You may wish to use @kbd{v .} mode to abbreviate the display of this
-large matrix.)
-
-We want to treat this as a pair of lists. The first step is to
-transpose this matrix into a pair of rows. Remember, a matrix is
-just a vector of vectors. So we can unpack the matrix into a pair
-of row vectors on the stack.
-
address@hidden
address@hidden
-1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
- [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
- . .
-
- v t v u
address@hidden group
address@hidden smallexample
-
address@hidden
-Let's store these in quick variables 1 and 2, respectively.
-
address@hidden
address@hidden
-1: [1.34, 1.41, 1.49, ... ] .
- .
-
- t 2 t 1
address@hidden group
address@hidden smallexample
-
address@hidden
-(Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
-stored value from the stack.)
-
-In a least squares fit, the slope @expr{m} is given by the formula
-
address@hidden
address@hidden
-m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ m = {N \sum x y - \sum x \sum y \over
- N \sum x^2 - \left( \sum x \right)^2} $$
-\afterdisplay
address@hidden tex
-
address@hidden
-where
address@hidden @math{\sum x}
address@hidden @expr{sum(x)}
-represents the sum of all the values of @expr{x}. While there is an
-actual @code{sum} function in Calc, it's easier to sum a vector using a
-simple reduction. First, let's compute the four different sums that
-this formula uses.
-
address@hidden
address@hidden
-1: 41.63 1: 98.0003
- . .
-
- r 1 V R + t 3 r 1 2 V M ^ V R + t 4
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 13.613 1: 33.36554
- . .
-
- r 2 V R + t 5 r 1 r 2 V M * V R + t 6
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden
-These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
-respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
address@hidden(x y)}.)
address@hidden ifnottex
address@hidden
-\turnoffactive
-These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
-respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
-$\sum x y$.)
address@hidden tex
-
-Finally, we also need @expr{N}, the number of data points. This is just
-the length of either of our lists.
-
address@hidden
address@hidden
-1: 19
- .
-
- r 1 v l t 7
address@hidden group
address@hidden smallexample
-
address@hidden
-(That's @kbd{v} followed by a lower-case @kbd{l}.)
-
-Now we grind through the formula:
-
address@hidden
address@hidden
-1: 633.94526 2: 633.94526 1: 67.23607
- . 1: 566.70919 .
- .
-
- r 7 r 6 * r 3 r 5 * -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
-1: 1862.0057 2: 1862.0057 1: 128.9488 .
- . 1: 1733.0569 .
- .
-
- r 7 r 4 * r 3 2 ^ - / t 8
address@hidden group
address@hidden smallexample
-
-That gives us the slope @expr{m}. The y-intercept @expr{b} can now
-be found with the simple formula,
-
address@hidden
address@hidden
-b = (sum(y) - m sum(x)) / N
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ b = {\sum y - m \sum x \over N} $$
-\afterdisplay
-\vskip10pt
address@hidden tex
-
address@hidden
address@hidden
-1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
- . 1: 21.70658 . .
- .
-
- r 5 r 8 r 3 * - r 7 / t 9
address@hidden group
address@hidden smallexample
-
-Let's ``plot'' this straight line approximation,
address@hidden @math{y \approx m x + b},
address@hidden @expr{m x + b},
-and compare it with the original data.
-
address@hidden
address@hidden
-1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
- . .
-
- r 1 r 8 * r 9 + s 0
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice that multiplying a vector by a constant, and adding a constant
-to a vector, can be done without mapping commands since these are
-common operations from vector algebra. As far as Calc is concerned,
-we've just been doing geometry in 19-dimensional space!
-
-We can subtract this vector from our original @expr{y} vector to get
-a feel for the error of our fit. Let's find the maximum error:
-
address@hidden
address@hidden
-1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
- . . .
-
- r 2 - V M A V R X
address@hidden group
address@hidden smallexample
-
address@hidden
-First we compute a vector of differences, then we take the absolute
-values of these differences, then we reduce the @code{max} function
-across the vector. (The @code{max} function is on the two-key sequence
address@hidden x}; because it is so common to use @code{max} in a vector
-operation, the letters @kbd{X} and @kbd{N} are also accepted for
address@hidden and @code{min} in this context. In general, you answer
-the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
-invokes the function you want. You could have typed @kbd{V R f x} or
-even @kbd{V R x max @key{RET}} if you had preferred.)
-
-If your system has the GNUPLOT program, you can see graphs of your
-data and your straight line to see how well they match. (If you have
-GNUPLOT 3.0 or higher, the following instructions will work regardless
-of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
-may require additional steps to view the graphs.)
-
-Let's start by plotting the original data. Recall the address@hidden'' and
address@hidden''
-vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
-command does everything you need to do for simple, straightforward
-plotting of data.
-
address@hidden
address@hidden
-2: [1.34, 1.41, 1.49, ... ]
-1: [0.234, 0.298, 0.402, ... ]
- .
-
- r 1 r 2 g f
address@hidden group
address@hidden smallexample
-
-If all goes well, you will shortly get a new window containing a graph
-of the data. (If not, contact your GNUPLOT or Calc installer to find
-out what went wrong.) In the X window system, this will be a separate
-graphics window. For other kinds of displays, the default is to
-display the graph in Emacs itself using rough character graphics.
-Press @kbd{q} when you are done viewing the character graphics.
-
-Next, let's add the line we got from our least-squares fit.
address@hidden
-(If you are reading this tutorial on-line while running Calc, typing
address@hidden a} may cause the tutorial to disappear from its window and be
-replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
-will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
address@hidden ifinfo
-
address@hidden
address@hidden
-2: [1.34, 1.41, 1.49, ... ]
-1: [0.273, 0.309, 0.351, ... ]
- .
-
- @key{DEL} r 0 g a g p
address@hidden group
address@hidden smallexample
-
-It's not very useful to get symbols to mark the data points on this
-second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
-when you are done to remove the X graphics window and terminate GNUPLOT.
-
-(@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
-least squares fitting to a general system of equations. Our 19 data
-points are really 19 equations of the form @expr{y_i = m x_i + b} for
-different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
-to solve for @expr{m} and @expr{b}, duplicating the above result.
address@hidden Answer 2, 2}. (@bullet{})
-
address@hidden Geometric mean
-(@bullet{}) @strong{Exercise 3.} If the input data do not form a
-rectangle, you can use @address@hidden * g}} (@code{calc-grab-region})
-to grab the data the way Emacs normally works with regions---it reads
-left-to-right, top-to-bottom, treating line breaks the same as spaces.
-Use this command to find the geometric mean of the following numbers.
-(The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
-
address@hidden
-2.3 6 22 15.1 7
- 15 14 7.5
- 2.5
address@hidden example
-
address@hidden
-The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
-with or without surrounding vector brackets.
address@hidden Answer 3, 3}. (@bullet{})
-
address@hidden
-As another example, a theorem about binomial coefficients tells
-us that the alternating sum of binomial coefficients
address@hidden minus @var{n}-choose-1 plus @var{n}-choose-2, and so
-on up to @address@hidden,
-always comes out to zero. Let's verify this
-for @expr{n=6}.
address@hidden ifnottex
address@hidden
-As another example, a theorem about binomial coefficients tells
-us that the alternating sum of binomial coefficients
-${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
-always comes out to zero. Let's verify this
-for \cite{n=6}.
address@hidden tex
-
address@hidden
address@hidden
-1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
- . .
-
- v x 7 @key{RET} 1 -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [1, -6, 15, -20, 15, -6, 1] 1: 0
- . .
-
- V M ' (-1)^$ choose(6,$) @key{RET} V R +
address@hidden group
address@hidden smallexample
-
-The @kbd{V M '} command prompts you to enter any algebraic expression
-to define the function to map over the vector. The symbol @samp{$}
-inside this expression represents the argument to the function.
-The Calculator applies this formula to each element of the vector,
-substituting each element's value for the @samp{$} sign(s) in turn.
-
-To define a two-argument function, use @samp{$$} for the first
-argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
-equivalent to @kbd{V M -}. This is analogous to regular algebraic
-entry, where @samp{$$} would refer to the next-to-top stack entry
-and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
-would act exactly like @kbd{-}.
-
-Notice that the @kbd{V M '} command has recorded two things in the
-trail: The result, as usual, and also a funny-looking thing marked
address@hidden that represents the operator function you typed in.
-The function is enclosed in @samp{< >} brackets, and the argument is
-denoted by a @samp{#} sign. If there were several arguments, they
-would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
address@hidden M ' $$-$} will put the function @samp{<#1 - #2>} on the
-trail.) This object is a ``nameless function''; you can use nameless
address@hidden@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
-Nameless function notation has the interesting, occasionally useful
-property that a nameless function is not actually evaluated until
-it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
address@hidden(2.0)} once and adds that random number to all elements
-of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
address@hidden(2.0)} separately for each vector element.
-
-Another group of operators that are often useful with @kbd{V M} are
-the relational operators: @kbd{a =}, for example, compares two numbers
-and gives the result 1 if they are equal, or 0 if not. Similarly,
address@hidden@kbd{a <}} checks for one number being less than another.
-
-Other useful vector operations include @kbd{v v}, to reverse a
-vector end-for-end; @kbd{V S}, to sort the elements of a vector
-into increasing order; and @kbd{v r} and @address@hidden c}}, to extract
-one row or column of a matrix, or (in both cases) to extract one
-element of a plain vector. With a negative argument, @kbd{v r}
-and @kbd{v c} instead delete one row, column, or vector element.
-
address@hidden Divisor functions
-(@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
address@hidden
-$\sigma_k(n)$
address@hidden tex
-is the sum of the @expr{k}th powers of all the divisors of an
-integer @expr{n}. Figure out a method for computing the divisor
-function for reasonably small values of @expr{n}. As a test,
-the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
address@hidden Answer 4, 4}. (@bullet{})
-
address@hidden Square-free numbers
address@hidden Duplicate values in a list
-(@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
-list of prime factors for a number. Sometimes it is important to
-know that a number is @dfn{square-free}, i.e., that no prime occurs
-more than once in its list of prime factors. Find a sequence of
-keystrokes to tell if a number is square-free; your method should
-leave 1 on the stack if it is, or 0 if it isn't.
address@hidden Answer 5, 5}. (@bullet{})
-
address@hidden Triangular lists
-(@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
-like the following diagram. (You may wish to use the @kbd{v /}
-command to enable multi-line display of vectors.)
-
address@hidden
address@hidden
-1: [ [1],
- [1, 2],
- [1, 2, 3],
- [1, 2, 3, 4],
- [1, 2, 3, 4, 5],
- [1, 2, 3, 4, 5, 6] ]
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden Answer 6, 6}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 7.} Build the following list of lists.
-
address@hidden
address@hidden
-1: [ [0],
- [1, 2],
- [3, 4, 5],
- [6, 7, 8, 9],
- [10, 11, 12, 13, 14],
- [15, 16, 17, 18, 19, 20] ]
address@hidden group
address@hidden smallexample
-
address@hidden
address@hidden Answer 7, 7}. (@bullet{})
-
address@hidden Maximizing a function over a list of values
address@hidden [fix-ref Numerical Solutions]
-(@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
address@hidden @math{J_1(x)}
address@hidden @expr{J1}
-function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
-Find the value of @expr{x} (from among the above set of values) for
-which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
-i.e., just reading along the list by hand to find the largest value
-is not allowed! (There is an @kbd{a X} command which does this kind
-of thing automatically; @pxref{Numerical Solutions}.)
address@hidden Answer 8, 8}. (@bullet{})
-
address@hidden Digits, vectors of
-(@bullet{}) @strong{Exercise 9.} You are given an integer in the range
address@hidden @math{0 \le N < 10^m}
address@hidden @expr{0 <= N < 10^m}
-for @expr{m=12} (i.e., an integer of less than
-twelve digits). Convert this integer into a vector of @expr{m}
-digits, each in the range from 0 to 9. In vector-of-digits notation,
-add one to this integer to produce a vector of @expr{m+1} digits
-(since there could be a carry out of the most significant digit).
-Convert this vector back into a regular integer. A good integer
-to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
address@hidden R a =} to test if all numbers in a list were equal. What
-happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
-is @cpi{}. The area of the
address@hidden @math{2\times2}
address@hidden 2x2
-square that encloses that circle is 4. So if we throw @var{n} darts at
-random points in the square, about @cpiover{4} of them will land inside
-the circle. This gives us an entertaining way to estimate the value of
address@hidden The @address@hidden r}}
-command picks a random number between zero and the value on the stack.
-We could get a random floating-point number between @mathit{-1} and 1 by typing
address@hidden@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)}
points in
-this square, then use vector mapping and reduction to count how many
-points lie inside the unit circle. Hint: Use the @kbd{v b} command.
address@hidden Answer 11, 11}. (@bullet{})
-
address@hidden Matchstick problem
-(@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
-another way to calculate @cpi{}. Say you have an infinite field
-of vertical lines with a spacing of one inch. Toss a one-inch matchstick
-onto the field. The probability that the matchstick will land crossing
-a line turns out to be
address@hidden @math{2/\pi}.
address@hidden @expr{2/pi}.
-Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
-the probability that the GCD (@address@hidden g}}) of two large integers is
-one turns out to be
address@hidden @math{6/\pi^2}.
address@hidden @expr{6/pi^2}.
-That provides yet another way to estimate @cpi{}.)
address@hidden Answer 12, 12}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
-double-quote marks, @samp{"hello"}, creates a vector of the numerical
-(ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
-Sometimes it is convenient to compute a @dfn{hash code} of a string,
-which is just an integer that represents the value of that string.
-Two equal strings have the same hash code; two different strings
address@hidden have different hash codes. (For example, Calc has
-over 400 function names, but Emacs can quickly find the definition for
-any given name because it has sorted the functions into ``buckets'' by
-their hash codes. Sometimes a few names will hash into the same bucket,
-but it is easier to search among a few names than among all the names.)
-One popular hash function is computed as follows: First set @expr{h = 0}.
-Then, for each character from the string in turn, set @expr{h = 3h + c_i}
-where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
-we then take the hash code modulo 511 to get the bucket number. Develop a
-simple command or commands for converting string vectors into hash codes.
-The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
-511 is 121. @xref{List Answer 13, 13}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
-commands do nested function evaluations. @kbd{H V U} takes a starting
-value and a number of steps @var{n} from the stack; it then applies the
-function you give to the starting value 0, 1, 2, up to @var{n} times
-and returns a vector of the results. Use this command to create a
-``random walk'' of 50 steps. Start with the two-dimensional point
address@hidden(0,0)}; then take one step a random distance between @mathit{-1}
and 1
-in both @expr{x} and @expr{y}; then take another step, and so on. Use the
address@hidden f} command to display this random walk. Now modify your random
-walk to walk a unit distance, but in a random direction, at each step.
-(Hint: The @code{sincos} function returns a vector of the cosine and
-sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
-
address@hidden Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial,
Tutorial
address@hidden Types Tutorial
-
address@hidden
-Calc understands a variety of data types as well as simple numbers.
-In this section, we'll experiment with each of these types in turn.
-
-The numbers we've been using so far have mainly been either @dfn{integers}
-or @dfn{floats}. We saw that floats are usually a good approximation to
-the mathematical concept of real numbers, but they are only approximations
-and are susceptible to roundoff error. Calc also supports @dfn{fractions},
-which can exactly represent any rational number.
-
address@hidden
address@hidden
-1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
- . 1: 49 . . .
- .
-
- 10 ! 49 @key{RET} : 2 + &
address@hidden group
address@hidden smallexample
-
address@hidden
-The @kbd{:} command divides two integers to get a fraction; @kbd{/}
-would normally divide integers to get a floating-point result.
-Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
-since the @kbd{:} would otherwise be interpreted as part of a
-fraction beginning with 49.
-
-You can convert between floating-point and fractional format using
address@hidden f} and @kbd{c F}:
-
address@hidden
address@hidden
-1: 1.35027217629e-5 1: 7:518414
- . .
-
- c f c F
address@hidden group
address@hidden smallexample
-
-The @kbd{c F} command replaces a floating-point number with the
-``simplest'' fraction whose floating-point representation is the
-same, to within the current precision.
-
address@hidden
address@hidden
-1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
- . . . .
-
- P c F @key{DEL} p 5 @key{RET} P c F
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 1.} A calculation has produced the
-result 1.26508260337. You suspect it is the square root of the
-product of @cpi{} and some rational number. Is it? (Be sure
-to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
-
address@hidden numbers} can be stored in both rectangular and polar form.
-
address@hidden
address@hidden
-1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
- . . . . .
-
- 9 n Q c p 2 * Q
address@hidden group
address@hidden smallexample
-
address@hidden
-The square root of @mathit{-9} is by default rendered in rectangular form
-(@address@hidden + 3i}}), but we can convert it to polar form (3 with a
-phase angle of 90 degrees). All the usual arithmetic and scientific
-operations are defined on both types of complex numbers.
-
-Another generalized kind of number is @dfn{infinity}. Infinity
-isn't really a number, but it can sometimes be treated like one.
-Calc uses the symbol @code{inf} to represent positive infinity,
-i.e., a value greater than any real number. Naturally, you can
-also write @samp{-inf} for minus infinity, a value less than any
-real number. The word @code{inf} can only be input using
-algebraic entry.
-
address@hidden
address@hidden
-2: inf 2: -inf 2: -inf 2: -inf 1: nan
-1: -17 1: -inf 1: -inf 1: inf .
- . . . .
-
-' inf @key{RET} 17 n * @key{RET} 72 + A +
address@hidden group
address@hidden smallexample
-
address@hidden
-Since infinity is infinitely large, multiplying it by any finite
-number (like @mathit{-17}) has no effect, except that since @mathit{-17}
-is negative, it changes a plus infinity to a minus infinity.
-(``A huge positive number, multiplied by @mathit{-17}, yields a huge
-negative number.'') Adding any finite number to infinity also
-leaves it unchanged. Taking an absolute value gives us plus
-infinity again. Finally, we add this plus infinity to the minus
-infinity we had earlier. If you work it out, you might expect
-the answer to be @mathit{-72} for this. But the 72 has been completely
-lost next to the infinities; by the time we compute @address@hidden - inf}}
-the finite difference between them, if any, is undetectable.
-So we say the result is @dfn{indeterminate}, which Calc writes
-with the symbol @code{nan} (for Not A Number).
-
-Dividing by zero is normally treated as an error, but you can get
-Calc to write an answer in terms of infinity by pressing @kbd{m i}
-to turn on Infinite mode.
-
address@hidden
address@hidden
-3: nan 2: nan 2: nan 2: nan 1: nan
-2: 1 1: 1 / 0 1: uinf 1: uinf .
-1: 0 . . .
- .
-
- 1 @key{RET} 0 / m i U / 17 n * +
address@hidden group
address@hidden smallexample
-
address@hidden
-Dividing by zero normally is left unevaluated, but after @kbd{m i}
-it instead gives an infinite result. The answer is actually
address@hidden, ``undirected infinity.'' If you look at a graph of
address@hidden / x} around @address@hidden = 0}}, you'll see that it goes toward
-plus infinity as you approach zero from above, but toward minus
-infinity as you approach from below. Since we said only @expr{1 / 0},
-Calc knows that the answer is infinite but not in which direction.
-That's what @code{uinf} means. Notice that multiplying @code{uinf}
-by a negative number still leaves plain @code{uinf}; there's no
-point in saying @samp{-uinf} because the sign of @code{uinf} is
-unknown anyway. Finally, we add @code{uinf} to our @code{nan},
-yielding @code{nan} again. It's easy to see that, because
address@hidden means ``totally unknown'' while @code{uinf} means
-``unknown sign but known to be infinite,'' the more mysterious
address@hidden wins out when it is combined with @code{uinf}, or, for
-that matter, with anything else.
-
-(@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
-for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
address@hidden(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
address@hidden(uinf)}, @samp{ln(0)}.
address@hidden Answer 2, 2}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
-which stands for an unknown value. Can @code{nan} stand for
-a complex number? Can it stand for infinity?
address@hidden Answer 3, 3}. (@bullet{})
-
address@hidden forms} represent a value in terms of hours, minutes, and
-seconds.
-
address@hidden
address@hidden
-1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
- . . 1: 1@@ 45' 0." .
- .
-
- 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
address@hidden group
address@hidden smallexample
-
-HMS forms can also be used to hold angles in degrees, minutes, and
-seconds.
-
address@hidden
address@hidden
-1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
- . . . .
-
- 0.5 I T c h S
address@hidden group
address@hidden smallexample
-
address@hidden
-First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
-form, then we take the sine of that angle. Note that the trigonometric
-functions will accept HMS forms directly as input.
-
address@hidden Beatles
-(@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
-47 minutes and 26 seconds long, and contains 17 songs. What is the
-average length of a song on @emph{Abbey Road}? If the Extended Disco
-Version of @emph{Abbey Road} added 20 seconds to the length of each
-song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
-
-A @dfn{date form} represents a date, or a date and time. Dates must
-be entered using algebraic entry. Date forms are surrounded by
address@hidden< >} symbols; most standard formats for dates are recognized.
-
address@hidden
address@hidden
-2: <Sun Jan 13, 1991> 1: 2.25
-1: <6:00pm Thu Jan 10, 1991> .
- .
-
-' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
address@hidden group
address@hidden smallexample
-
address@hidden
-In this example, we enter two dates, then subtract to find the
-number of days between them. It is also possible to add an
-HMS form or a number (of days) to a date form to get another
-date form.
-
address@hidden
address@hidden
-1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
- . .
-
- t N 2 + 10@@ 5' +
address@hidden group
address@hidden smallexample
-
address@hidden [fix-ref Date Arithmetic]
address@hidden
-The @kbd{t N} (``now'') command pushes the current date and time on the
-stack; then we add two days, ten hours and five minutes to the date and
-time. Other date-and-time related commands include @kbd{t J}, which
-does Julian day conversions, @kbd{t W}, which finds the beginning of
-the week in which a date form lies, and @kbd{t I}, which increments a
-date by one or several months. @xref{Date Arithmetic}, for more.
-
-(@bullet{}) @strong{Exercise 5.} How many days until the next
-Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 6.} How many leap years will there be
-between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{})
-
address@hidden Slope and angle of a line
address@hidden Angle and slope of a line
-An @dfn{error form} represents a mean value with an attached standard
-deviation, or error estimate. Suppose our measurements indicate that
-a certain telephone pole is about 30 meters away, with an estimated
-error of 1 meter, and 8 meters tall, with an estimated error of 0.2
-meters. What is the slope of a line from here to the top of the
-pole, and what is the equivalent angle in degrees?
-
address@hidden
address@hidden
-1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
- . 1: 30 +/- 1 . .
- .
-
- 8 p .2 @key{RET} 30 p 1 / I T
address@hidden group
address@hidden smallexample
-
address@hidden
-This means that the angle is about 15 degrees, and, assuming our
-original error estimates were valid standard deviations, there is about
-a 60% chance that the result is correct within 0.59 degrees.
-
address@hidden Torus, volume of
-(@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
address@hidden @math{2 \pi^2 R r^2}
address@hidden @address@hidden pi^2 R r^2}}
-where @expr{R} is the radius of the circle that
-defines the center of the tube and @expr{r} is the radius of the tube
-itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
-within 5 percent. What is the volume and the relative uncertainty of
-the volume? @xref{Types Answer 7, 7}. (@bullet{})
-
-An @dfn{interval form} represents a range of values. While an
-error form is best for making statistical estimates, intervals give
-you exact bounds on an answer. Suppose we additionally know that
-our telephone pole is definitely between 28 and 31 meters away,
-and that it is between 7.7 and 8.1 meters tall.
-
address@hidden
address@hidden
-1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
- . 1: [28 .. 31] . .
- .
-
- [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
address@hidden group
address@hidden smallexample
-
address@hidden
-If our bounds were correct, then the angle to the top of the pole
-is sure to lie in the range shown.
-
-The square brackets around these intervals indicate that the endpoints
-themselves are allowable values. In other words, the distance to the
-telephone pole is between 28 and 31, @emph{inclusive}. You can also
-make an interval that is exclusive of its endpoints by writing
-parentheses instead of square brackets. You can even make an interval
-which is inclusive (``closed'') on one end and exclusive (``open'') on
-the other.
-
address@hidden
address@hidden
-1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
- . . 1: [2 .. 3) .
- .
-
- [ 1 .. 10 ) & [ 2 .. 3 ) *
address@hidden group
address@hidden smallexample
-
address@hidden
-The Calculator automatically keeps track of which end values should
-be open and which should be closed. You can also make infinite or
-semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
-or both endpoints.
-
-(@bullet{}) @strong{Exercise 8.} What answer would you expect from
address@hidden@w{1 /} @w{(0 .. 10)}}? What about @address@hidden /} @w{(-10 ..
0)}}? What
-about @address@hidden /} @w{[0 .. 10]}} (where the interval actually includes
-zero)? What about @address@hidden /} @w{(-10 .. 10)}}?
address@hidden Answer 8, 8}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
-are @address@hidden *} and @address@hidden ^}}. Normally these produce the
same
-answer. Would you expect this still to hold true for interval forms?
-If not, which of these will result in a larger interval?
address@hidden Answer 9, 9}. (@bullet{})
-
-A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
-For example, arithmetic involving time is generally done modulo 12
-or 24 hours.
-
address@hidden
address@hidden
-1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
- . . . .
-
- 17 M 24 @key{RET} 10 + n 5 /
address@hidden group
address@hidden smallexample
-
address@hidden
-In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
-new number which, when multiplied by 5 modulo 24, produces the original
-number, 21. If @var{m} is prime and the divisor is not a multiple of
address@hidden, it is always possible to find such a number. For non-prime
address@hidden like 24, it is only sometimes possible.
-
address@hidden
address@hidden
-1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
- . . . .
-
- 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
address@hidden group
address@hidden smallexample
-
address@hidden
-These two calculations get the same answer, but the first one is
-much more efficient because it avoids the huge intermediate value
-that arises in the second one.
-
address@hidden Fermat, primality test of
-(@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
-says that
address@hidden @address@hidden \bmod n = 1}}
address@hidden @expr{x^(n-1) mod n = 1}
-if @expr{n} is a prime number and @expr{x} is an integer less than
address@hidden If @expr{n} is @emph{not} a prime number, this will
address@hidden be true for most values of @expr{x}. Thus we can test
-informally if a number is prime by trying this formula for several
-values of @expr{x}. Use this test to tell whether the following numbers
-are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
-
-It is possible to use HMS forms as parts of error forms, intervals,
-modulo forms, or as the phase part of a polar complex number.
-For example, the @code{calc-time} command pushes the current time
-of day on the stack as an HMS/modulo form.
-
address@hidden
address@hidden
-1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
- . .
-
- x time @key{RET} n
address@hidden group
address@hidden smallexample
-
address@hidden
-This calculation tells me it is six hours and 22 minutes until midnight.
-
-(@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
-is about
address@hidden @math{\pi \times 10^7}
address@hidden @address@hidden * 10^7}}
-seconds. What time will it be that many seconds from right now?
address@hidden Answer 11, 11}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
-for the CD release of the Extended Disco Version of @emph{Abbey Road}.
-You are told that the songs will actually be anywhere from 20 to 60
-seconds longer than the originals. One CD can hold about 75 minutes
-of music. Should you order single or double packages?
address@hidden Answer 12, 12}. (@bullet{})
-
-Another kind of data the Calculator can manipulate is numbers with
address@hidden This isn't strictly a new data type; it's simply an
-application of algebraic expressions, where we use variables with
-suggestive names like @samp{cm} and @samp{in} to represent units
-like centimeters and inches.
-
address@hidden
address@hidden
-1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
- . . . .
-
- ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
address@hidden group
address@hidden smallexample
-
address@hidden
-We enter the quantity ``2 inches'' (actually an algebraic expression
-which means two times the variable @samp{in}), then we convert it
-first to centimeters, then to fathoms, then finally to ``base'' units,
-which in this case means meters.
-
address@hidden
address@hidden
-1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
- . . . .
-
- ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
- . . .
-
- u s 2 ^ u c cgs
address@hidden group
address@hidden smallexample
-
address@hidden
-Since units expressions are really just formulas, taking the square
-root of @samp{acre} is undefined. After all, @code{acre} might be an
-algebraic variable that you will someday assign a value. We use the
-``units-simplify'' command to simplify the expression with variables
-being interpreted as unit names.
-
-In the final step, we have converted not to a particular unit, but to a
-units system. The ``cgs'' system uses centimeters instead of meters
-as its standard unit of length.
-
-There is a wide variety of units defined in the Calculator.
-
address@hidden
address@hidden
-1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
- . . . .
-
- ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET}
u c c @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-We express a speed first in miles per hour, then in kilometers per
-hour, then again using a slightly more explicit notation, then
-finally in terms of fractions of the speed of light.
-
-Temperature conversions are a bit more tricky. There are two ways to
-interpret ``20 degrees Fahrenheit''---it could mean an actual
-temperature, or it could mean a change in temperature. For normal
-units there is no difference, but temperature units have an offset
-as well as a scale factor and so there must be two explicit commands
-for them.
-
address@hidden
address@hidden
-1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC
- . . . .
-
- ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET}
c f
address@hidden group
address@hidden smallexample
-
address@hidden
-First we convert a change of 20 degrees Fahrenheit into an equivalent
-change in degrees Celsius (or Centigrade). Then, we convert the
-absolute temperature 20 degrees Fahrenheit into Celsius. Since
-this comes out as an exact fraction, we then convert to floating-point
-for easier comparison with the other result.
-
-For simple unit conversions, you can put a plain number on the stack.
-Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
-When you use this method, you're responsible for remembering which
-numbers are in which units:
-
address@hidden
address@hidden
-1: 55 1: 88.5139 1: 8.201407e-8
- . . .
-
- 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c
@key{RET}
address@hidden group
address@hidden smallexample
-
-To see a complete list of built-in units, type @kbd{u v}. Press
address@hidden@kbd{C-x * c}} again to re-enter the Calculator when you're done
looking
-at the units table.
-
-(@bullet{}) @strong{Exercise 13.} How many seconds are there really
-in a year? @xref{Types Answer 13, 13}. (@bullet{})
-
address@hidden Speed of light
-(@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
-the speed of light (and of electricity, which is nearly as fast).
-Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
-cabinet is one meter across. Is speed of light going to be a
-significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
-five yards in an hour. He has obtained a supply of Power Pills; each
-Power Pill he eats doubles his speed. How many Power Pills can he
-swallow and still travel legally on most US highways?
address@hidden Answer 15, 15}. (@bullet{})
-
address@hidden Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
address@hidden Algebra and Calculus Tutorial
-
address@hidden
-This section shows how to use Calc's algebra facilities to solve
-equations, do simple calculus problems, and manipulate algebraic
-formulas.
-
address@hidden
-* Basic Algebra Tutorial::
-* Rewrites Tutorial::
address@hidden menu
-
address@hidden Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial,
Algebra Tutorial
address@hidden Basic Algebra
-
address@hidden
-If you enter a formula in Algebraic mode that refers to variables,
-the formula itself is pushed onto the stack. You can manipulate
-formulas as regular data objects.
-
address@hidden
address@hidden
-1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y)
- . . .
-
- ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
address@hidden' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
-Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
-
-There are also commands for doing common algebraic operations on
-formulas. Continuing with the formula from the last example,
-
address@hidden
address@hidden
-1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
- . .
-
- a x a c x @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-First we ``expand'' using the distributive law, then we ``collect''
-terms involving like powers of @expr{x}.
-
-Let's find the value of this expression when @expr{x} is 2 and @expr{y}
-is one-half.
-
address@hidden
address@hidden
-1: 17 x^2 - 6 x^4 + 3 1: -25
- . .
-
- 1:2 s l y @key{RET} 2 s l x @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-The @kbd{s l} command means ``let''; it takes a number from the top of
-the stack and temporarily assigns it as the value of the variable
-you specify. It then evaluates (as if by the @kbd{=} key) the
-next expression on the stack. After this command, the variable goes
-back to its original value, if any.
-
-(An earlier exercise in this tutorial involved storing a value in the
-variable @code{x}; if this value is still there, you will have to
-unstore it with @kbd{s u x @key{RET}} before the above example will work
-properly.)
-
address@hidden Maximum of a function using Calculus
-Let's find the maximum value of our original expression when @expr{y}
-is one-half and @expr{x} ranges over all possible values. We can
-do this by taking the derivative with respect to @expr{x} and examining
-values of @expr{x} for which the derivative is zero. If the second
-derivative of the function at that value of @expr{x} is negative,
-the function has a local maximum there.
-
address@hidden
address@hidden
-1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
- . .
-
- U @key{DEL} s 1 a d x @key{RET} s 2
address@hidden group
address@hidden smallexample
-
address@hidden
-Well, the derivative is clearly zero when @expr{x} is zero. To find
-the other root(s), let's divide through by @expr{x} and then solve:
-
address@hidden
address@hidden
-1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2
- . . .
-
- ' x @key{RET} / a x a s
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 34 - 24 x^2 = 0 1: x = 1.19023
- . .
-
- 0 a = s 3 a S x @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice the use of @kbd{a s} to ``simplify'' the formula. When the
-default algebraic simplifications don't do enough, you can use
address@hidden s} to tell Calc to spend more time on the job.
-
-Now we compute the second derivative and plug in our values of @expr{x}:
-
address@hidden
address@hidden
-1: 1.19023 2: 1.19023 2: 1.19023
- . 1: 34 x - 24 x^3 1: 34 - 72 x^2
- . .
-
- a . r 2 a d x @key{RET} s 4
address@hidden group
address@hidden smallexample
-
address@hidden
-(The @kbd{a .} command extracts just the righthand side of an equation.
-Another method would have been to use @kbd{v u} to unpack the equation
address@hidden@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M--
M-2 @key{DEL}}
-to delete the @samp{x}.)
-
address@hidden
address@hidden
-2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
-1: 1.19023 . 1: 0 .
- . .
-
- @key{TAB} s l x @key{RET} U @key{DEL} 0 s
l x @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-The first of these second derivatives is negative, so we know the function
-has a maximum value at @expr{x = 1.19023}. (The function also has a
-local @emph{minimum} at @expr{x = 0}.)
-
-When we solved for @expr{x}, we got only one value even though
address@hidden - 24 x^2 = 0} is a quadratic equation that ought to have
-two solutions. The reason is that @address@hidden S}} normally returns a
-single ``principal'' solution. If it needs to come up with an
-arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
-If it needs an arbitrary integer, it picks zero. We can get a full
-solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
-
address@hidden
address@hidden
-1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023
- . . .
-
- r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Calc has invented the variable @samp{s1} to represent an unknown sign;
-it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
-the ``let'' command to evaluate the expression when the sign is negative.
-If we plugged this into our second derivative we would get the same,
-negative, answer, so @expr{x = -1.19023} is also a maximum.
-
-To find the actual maximum value, we must plug our two values of @expr{x}
-into the original formula.
-
address@hidden
address@hidden
-2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
-1: x = 1.19023 s1 .
- .
-
- r 1 r 5 s l @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-(Here we see another way to use @kbd{s l}; if its input is an equation
-with a variable on the lefthand side, then @kbd{s l} treats the equation
-like an assignment to that variable if you don't give a variable name.)
-
-It's clear that this will have the same value for either sign of
address@hidden, but let's work it out anyway, just for the exercise:
-
address@hidden
address@hidden
-2: [-1, 1] 1: [15.04166, 15.04166]
-1: 24.08333 s1^2 ... .
- .
-
- [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we have used a vector mapping operation to evaluate the function
-at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
-except that it takes the formula from the top of the stack. The
-formula is interpreted as a function to apply across the vector at the
-next-to-top stack level. Since a formula on the stack can't contain
address@hidden signs, Calc assumes the variables in the formula stand for
-different arguments. It prompts you for an @dfn{argument list}, giving
-the list of all variables in the formula in alphabetical order as the
-default list. In this case the default is @samp{(s1)}, which is just
-what we want so we simply press @key{RET} at the prompt.
-
-If there had been several different values, we could have used
address@hidden@kbd{V R X}} to find the global maximum.
-
-Calc has a built-in @kbd{a P} command that solves an equation using
address@hidden@kbd{H a S}} and returns a vector of all the solutions. It simply
-automates the job we just did by hand. Applied to our original
-cubic polynomial, it would produce the vector of solutions
address@hidden, -1.19023, 0]}. (There is also an @kbd{a X} command
-which finds a local maximum of a function. It uses a numerical search
-method rather than examining the derivatives, and thus requires you
-to provide some kind of initial guess to show it where to look.)
-
-(@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
-polynomial (such as the output of an @kbd{a P} command), what
-sequence of commands would you use to reconstruct the original
-polynomial? (The answer will be unique to within a constant
-multiple; choose the solution where the leading coefficient is one.)
address@hidden Answer 2, 2}. (@bullet{})
-
-The @kbd{m s} command enables Symbolic mode, in which formulas
-like @samp{sqrt(5)} that can't be evaluated exactly are left in
-symbolic form rather than giving a floating-point approximate answer.
-Fraction mode (@kbd{m f}) is also useful when doing algebra.
-
address@hidden
address@hidden
-2: 34 x - 24 x^3 2: 34 x - 24 x^3
-1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
- . .
-
- r 2 @key{RET} m s m f a P x @key{RET}
address@hidden group
address@hidden smallexample
-
-One more mode that makes reading formulas easier is Big mode.
-
address@hidden
address@hidden
- 3
-2: 34 x - 24 x
-
- ____ ____
- V 51 V 51
-1: [-----, -----, 0]
- 6 -6
-
- .
-
- d B
address@hidden group
address@hidden smallexample
-
-Here things like powers, square roots, and quotients and fractions
-are displayed in a two-dimensional pictorial form. Calc has other
-language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
-and address@hidden mode.
-
address@hidden
address@hidden
-2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
-1: @{sqrt(51) / 6, sqrt(51) / -6, address@hidden 1: /sqrt(51) / 6, sqrt(51)
/ -6, 0/
- . .
-
- d C d F
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-3: 34 x - 24 x^3
-2: address@hidden@address@hidden \over address@hidden, @address@hidden@}
\over address@hidden, 0]
-1: @{2 \over address@hidden address@hidden@}
- .
-
- d T ' 2 address@hidden@} \over 3 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-As you can see, language modes affect both entry and display of
-formulas. They affect such things as the names used for built-in
-functions, the set of arithmetic operators and their precedences,
-and notations for vectors and matrices.
-
-Notice that @samp{sqrt(51)} may cause problems with older
-implementations of C and FORTRAN, which would require something more
-like @samp{sqrt(51.0)}. It is always wise to check over the formulas
-produced by the various language modes to make sure they are fully
-correct.
-
-Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
-may prefer to remain in Big mode, but all the examples in the tutorial
-are shown in normal mode.)
-
address@hidden Area under a curve
-What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
-This is simply the integral of the function:
-
address@hidden
address@hidden
-1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
- . .
-
- r 1 a i x
address@hidden group
address@hidden smallexample
-
address@hidden
-We want to evaluate this at our two values for @expr{x} and subtract.
-One way to do it is again with vector mapping and reduction:
-
address@hidden
address@hidden
-2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
-1: 5.6666 x^3 ... . .
-
- [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
-of
address@hidden @math{x \sin \pi x}
address@hidden @address@hidden sin(pi x)}}
-(where the sine is calculated in radians). Find the values of the
-integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
-3}. (@bullet{})
-
-Calc's integrator can do many simple integrals symbolically, but many
-others are beyond its capabilities. Suppose we wish to find the area
-under the curve
address@hidden @math{\sin x \ln x}
address@hidden @expr{sin(x) ln(x)}
-over the same range of @expr{x}. If you entered this formula and typed
address@hidden i x @key{RET}} (don't bother to try this), Calc would work for a
-long time but would be unable to find a solution. In fact, there is no
-closed-form solution to this integral. Now what do we do?
-
address@hidden Integration, numerical
address@hidden Numerical integration
-One approach would be to do the integral numerically. It is not hard
-to do this by hand using vector mapping and reduction. It is rather
-slow, though, since the sine and logarithm functions take a long time.
-We can save some time by reducing the working precision.
-
address@hidden
address@hidden
-3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
-2: 1 .
-1: 0.1
- .
-
- 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
address@hidden group
address@hidden smallexample
-
address@hidden
-(Note that we have used the extended version of @kbd{v x}; we could
-also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
-
address@hidden
address@hidden
-2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
-1: sin(x) ln(x) .
- .
-
- ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 3.4195 0.34195
- . .
-
- V R + 0.1 *
address@hidden group
address@hidden smallexample
-
address@hidden
-(If you got wildly different results, did you remember to switch
-to Radians mode?)
-
-Here we have divided the curve into ten segments of equal width;
-approximating these segments as rectangular boxes (i.e., assuming
-the curve is nearly flat at that resolution), we compute the areas
-of the boxes (height times width), then sum the areas. (It is
-faster to sum first, then multiply by the width, since the width
-is the same for every box.)
-
-The true value of this integral turns out to be about 0.374, so
-we're not doing too well. Let's try another approach.
-
address@hidden
address@hidden
-1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
- . .
-
- r 1 a t x=1 @key{RET} 4 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we have computed the Taylor series expansion of the function
-about the point @expr{x=1}. We can now integrate this polynomial
-approximation, since polynomials are easy to integrate.
-
address@hidden
address@hidden
-1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
- . . .
-
- a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V
R -
address@hidden group
address@hidden smallexample
-
address@hidden
-Better! By increasing the precision and/or asking for more terms
-in the Taylor series, we can get a result as accurate as we like.
-(Taylor series converge better away from singularities in the
-function such as the one at @code{ln(0)}, so it would also help to
-expand the series about the points @expr{x=2} or @expr{x=1.5} instead
-of @expr{x=1}.)
-
address@hidden Simpson's rule
address@hidden Integration by Simpson's rule
-(@bullet{}) @strong{Exercise 4.} Our first method approximated the
-curve by stairsteps of width 0.1; the total area was then the sum
-of the areas of the rectangles under these stairsteps. Our second
-method approximated the function by a polynomial, which turned out
-to be a better approximation than stairsteps. A third method is
address@hidden's rule}, which is like the stairstep method except
-that the steps are not required to be flat. Simpson's rule boils
-down to the formula,
-
address@hidden
address@hidden
-(h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
- + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \displaylines{
- \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
- \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
-} $$
-\afterdisplay
address@hidden tex
-
address@hidden
-where @expr{n} (which must be even) is the number of slices and @expr{h}
-is the width of each slice. These are 10 and 0.1 in our example.
-For reference, here is the corresponding formula for the stairstep
-method:
-
address@hidden
address@hidden
-h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
- + f(a+(n-2)*h) + f(a+(n-1)*h))
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
- + f(a+(n-2)h) + f(a+(n-1)h)) $$
-\afterdisplay
address@hidden tex
-
-Compute the integral from 1 to 2 of
address@hidden @math{\sin x \ln x}
address@hidden @expr{sin(x) ln(x)}
-using Simpson's rule with 10 slices.
address@hidden Answer 4, 4}. (@bullet{})
-
-Calc has a built-in @kbd{a I} command for doing numerical integration.
-It uses @dfn{Romberg's method}, which is a more sophisticated cousin
-of Simpson's rule. In particular, it knows how to keep refining the
-result until the current precision is satisfied.
-
address@hidden [fix-ref Selecting Sub-Formulas]
-Aside from the commands we've seen so far, Calc also provides a
-large set of commands for operating on parts of formulas. You
-indicate the desired sub-formula by placing the cursor on any part
-of the formula before giving a @dfn{selection} command. Selections won't
-be covered in the tutorial; @pxref{Selecting Subformulas}, for
-details and examples.
-
address@hidden hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1)
2^(n - 1)
address@hidden to 2^((n-1)*(r-1)).
-
address@hidden Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
address@hidden Rewrite Rules
-
address@hidden
-No matter how many built-in commands Calc provided for doing algebra,
-there would always be something you wanted to do that Calc didn't have
-in its repertoire. So Calc also provides a @dfn{rewrite rule} system
-that you can use to define your own algebraic manipulations.
-
-Suppose we want to simplify this trigonometric formula:
-
address@hidden
address@hidden
-1: 1 / cos(x) - sin(x) tan(x)
- .
-
- ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
address@hidden group
address@hidden smallexample
-
address@hidden
-If we were simplifying this by hand, we'd probably replace the
address@hidden with a @samp{sin/cos} first, then combine over a common
-denominator. There is no Calc command to do the former; the @kbd{a n}
-algebra command will do the latter but we'll do both with rewrite
-rules just for practice.
-
-Rewrite rules are written with the @samp{:=} symbol.
-
address@hidden
address@hidden
-1: 1 / cos(x) - sin(x)^2 / cos(x)
- .
-
- a r tan(a) := sin(a)/cos(a) @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-(The ``assignment operator'' @samp{:=} has several uses in Calc. All
-by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
-but when it is given to the @kbd{a r} command, that command interprets
-it as a rewrite rule.)
-
-The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
-rewrite rule. Calc searches the formula on the stack for parts that
-match the pattern. Variables in a rewrite pattern are called
address@hidden, and when matching the pattern each meta-variable
-can match any sub-formula. Here, the meta-variable @samp{a} matched
-the actual variable @samp{x}.
-
-When the pattern part of a rewrite rule matches a part of the formula,
-that part is replaced by the righthand side with all the meta-variables
-substituted with the things they matched. So the result is
address@hidden(x) / cos(x)}. Calc's normal algebraic simplifications then
-mix this in with the rest of the original formula.
-
-To merge over a common denominator, we can use another simple rule:
-
address@hidden
address@hidden
-1: (1 - sin(x)^2) / cos(x)
- .
-
- a r a/x + b/x := (a+b)/x @key{RET}
address@hidden group
address@hidden smallexample
-
-This rule points out several interesting features of rewrite patterns.
-First, if a meta-variable appears several times in a pattern, it must
-match the same thing everywhere. This rule detects common denominators
-because the same meta-variable @samp{x} is used in both of the
-denominators.
-
-Second, meta-variable names are independent from variables in the
-target formula. Notice that the meta-variable @samp{x} here matches
-the subformula @samp{cos(x)}; Calc never confuses the two meanings of
address@hidden
-
-And third, rewrite patterns know a little bit about the algebraic
-properties of formulas. The pattern called for a sum of two quotients;
-Calc was able to match a difference of two quotients by matching
address@hidden = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
-
address@hidden [fix-ref Algebraic Properties of Rewrite Rules]
-We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
-the rule. It would have worked just the same in all cases. (If we
-really wanted the rule to apply only to @samp{+} or only to @samp{-},
-we could have used the @code{plain} symbol. @xref{Algebraic Properties
-of Rewrite Rules}, for some examples of this.)
-
-One more rewrite will complete the job. We want to use the identity
address@hidden(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
-the identity in a way that matches our formula. The obvious rule
-would be @address@hidden - sin(x)^2} := cos(x)^2}, but a little thought shows
-that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
-latter rule has a more general pattern so it will work in many other
-situations, too.
-
address@hidden
address@hidden
-1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
- . .
-
- a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
address@hidden group
address@hidden smallexample
-
-You may ask, what's the point of using the most general rule if you
-have to type it in every time anyway? The answer is that Calc allows
-you to store a rewrite rule in a variable, then give the variable
-name in the @kbd{a r} command. In fact, this is the preferred way to
-use rewrites. For one, if you need a rule once you'll most likely
-need it again later. Also, if the rule doesn't work quite right you
-can simply Undo, edit the variable, and run the rule again without
-having to retype it.
-
address@hidden
address@hidden
-' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET}
-' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
-' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
-
-1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
- . .
-
- r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr
@key{RET} a s
address@hidden group
address@hidden smallexample
-
-To edit a variable, type @kbd{s e} and the variable name, use regular
-Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
-the edited value back into the variable.
-You can also use @address@hidden e}} to create a new variable if you wish.
-
-Notice that the first time you use each rule, Calc puts up a ``compiling''
-message briefly. The pattern matcher converts rules into a special
-optimized pattern-matching language rather than using them directly.
-This allows @kbd{a r} to apply even rather complicated rules very
-efficiently. If the rule is stored in a variable, Calc compiles it
-only once and stores the compiled form along with the variable. That's
-another good reason to store your rules in variables rather than
-entering them on the fly.
-
-(@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
-mode, then enter the formula @address@hidden(2 + sqrt(2))} / @w{(1 +
sqrt(2))}}.
-Using a rewrite rule, simplify this formula by multiplying the top and
-bottom by the conjugate @address@hidden - sqrt(2)}}. The result will have
-to be expanded by the distributive law; do this with another
-rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
-
-The @kbd{a r} command can also accept a vector of rewrite rules, or
-a variable containing a vector of rules.
-
address@hidden
address@hidden
-1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ]
- . .
-
- ' [tsc,merge,sinsqr] @key{RET} =
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
- . .
-
- s t trig @key{RET} r 1 a r trig @key{RET} a s
address@hidden group
address@hidden smallexample
-
address@hidden [fix-ref Nested Formulas with Rewrite Rules]
-Calc tries all the rules you give against all parts of the formula,
-repeating until no further change is possible. (The exact order in
-which things are tried is rather complex, but for simple rules like
-the ones we've used here the order doesn't really matter.
address@hidden Formulas with Rewrite Rules}.)
-
-Calc actually repeats only up to 100 times, just in case your rule set
-has gotten into an infinite loop. You can give a numeric prefix argument
-to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
-only one rewrite at a time.
-
address@hidden
address@hidden
-1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x)
- . .
-
- r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
address@hidden group
address@hidden smallexample
-
-You can type @kbd{M-0 a r} if you want no limit at all on the number
-of rewrites that occur.
-
-Rewrite rules can also be @dfn{conditional}. Simply follow the rule
-with a @samp{::} symbol and the desired condition. For example,
-
address@hidden
address@hidden
-1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
- .
-
- ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 1 + exp(3 pi i) + 1
- .
-
- a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-(Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
-which will be zero only when @samp{k} is an even integer.)
-
-An interesting point is that the variables @samp{pi} and @samp{i}
-were matched literally rather than acting as meta-variables.
-This is because they are special-constant variables. The special
-constants @samp{e}, @samp{phi}, and so on also match literally.
-A common error with rewrite
-rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
-to match any @samp{f} with five arguments but in fact matching
-only when the fifth argument is literally @samp{e}!
-
address@hidden Fibonacci numbers
address@hidden
address@hidden
address@hidden ignore
address@hidden fib
-Rewrite rules provide an interesting way to define your own functions.
-Suppose we want to define @samp{fib(n)} to produce the @var{n}th
-Fibonacci number. The first two Fibonacci numbers are each 1;
-later numbers are formed by summing the two preceding numbers in
-the sequence. This is easy to express in a set of three rules:
-
address@hidden
address@hidden
-' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
-
-1: fib(7) 1: 13
- . .
-
- ' fib(7) @key{RET} a r fib @key{RET}
address@hidden group
address@hidden smallexample
-
-One thing that is guaranteed about the order that rewrites are tried
-is that, for any given subformula, earlier rules in the rule set will
-be tried for that subformula before later ones. So even though the
-first and third rules both match @samp{fib(1)}, we know the first will
-be used preferentially.
-
-This rule set has one dangerous bug: Suppose we apply it to the
-formula @samp{fib(x)}? (Don't actually try this.) The third rule
-will match @samp{fib(x)} and replace it with @address@hidden(x-1) + fib(x-2)}}.
-Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
-fib(x-4)}, and so on, expanding forever. What we really want is to apply
-the third rule only when @samp{n} is an integer greater than two. Type
address@hidden@kbd{s e fib @key{RET}}}, then edit the third rule to:
-
address@hidden
-fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
address@hidden smallexample
-
address@hidden
-Now:
-
address@hidden
address@hidden
-1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0)
- . .
-
- ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-We've created a new function, @code{fib}, and a new command,
address@hidden@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib}
calls in
-this formula.'' To make things easier still, we can tell Calc to
-apply these rules automatically by storing them in the special
-variable @code{EvalRules}.
-
address@hidden
address@hidden
-1: [fib(1) := ...] . 1: [8, 13]
- . .
-
- s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)]
@key{RET}
address@hidden group
address@hidden smallexample
-
-It turns out that this rule set has the problem that it does far
-more work than it needs to when @samp{n} is large. Consider the
-first few steps of the computation of @samp{fib(6)}:
-
address@hidden
address@hidden
-fib(6) =
-fib(5) + fib(4) =
-fib(4) + fib(3) + fib(3) + fib(2) =
-fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that @samp{fib(3)} appears three times here. Unless Calc's
-algebraic simplifier notices the multiple @samp{fib(3)}s and combines
-them (and, as it happens, it doesn't), this rule set does lots of
-needless recomputation. To cure the problem, type @code{s e EvalRules}
-to edit the rules (or just @kbd{s E}, a shorthand command for editing
address@hidden) and add another condition:
-
address@hidden
-fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
address@hidden smallexample
-
address@hidden
-If a @samp{:: remember} condition appears anywhere in a rule, then if
-that rule succeeds Calc will add another rule that describes that match
-to the front of the rule set. (Remembering works in any rule set, but
-for technical reasons it is most effective in @code{EvalRules}.) For
-example, if the rule rewrites @samp{fib(7)} to something that evaluates
-to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
-
-Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
-type @kbd{s E} again to see what has happened to the rule set.
-
-With the @code{remember} feature, our rule set can now compute
address@hidden(@var{n})} in just @var{n} steps. In the process it builds
-up a table of all Fibonacci numbers up to @var{n}. After we have
-computed the result for a particular @var{n}, we can get it back
-(and the results for all smaller @var{n}) later in just one step.
-
-All Calc operations will run somewhat slower whenever @code{EvalRules}
-contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
-un-store the variable.
-
-(@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
-a problem to reduce the amount of recursion necessary to solve it.
-Create a rule that, in about @var{n} simple steps and without recourse
-to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
address@hidden(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
address@hidden and @var{n+1}st Fibonacci numbers, respectively. This rule is
-rather clunky to use, so add a couple more rules to make the ``user
-interface'' the same as for our first version: enter @samp{fib(@var{n})},
-get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
-
-There are many more things that rewrites can do. For example, there
-are @samp{&&&} and @samp{|||} pattern operators that create ``and''
-and ``or'' combinations of rules. As one really simple example, we
-could combine our first two Fibonacci rules thusly:
-
address@hidden
-[fib(1 ||| 2) := 1, fib(n) := ... ]
address@hidden example
-
address@hidden
-That means address@hidden of something matching either 1 or 2 rewrites
-to 1.''
-
-You can also make meta-variables optional by enclosing them in @code{opt}.
-For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
address@hidden + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) +
opt(b) x}
-matches all of these forms, filling in a default of zero for @samp{a}
-and one for @samp{b}.
-
-(@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
-on the stack and tried to use the rule
address@hidden(a) + opt(b) x := f(a, b, x)}. What happened?
address@hidden Answer 3, 3}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
-divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
-Now repeat this step over and over. A famous unproved conjecture
-is that for any starting @expr{a}, the sequence always eventually
-reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
-rules that convert this into @samp{seq(1, @var{n})} where @var{n}
-is the number of steps it took the sequence to reach the value 1.
-Now enhance the rules to accept @samp{seq(@var{a})} as a starting
-configuration, and to stop with just the number @var{n} by itself.
-Now make the result be a vector of values in the sequence, from @var{a}
-to 1. (The formula @address@hidden|@var{y}} appends the vectors @var{x}
-and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
-vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
address@hidden Answer 4, 4}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
address@hidden(@var{x})} that returns the number of terms in the sum
address@hidden, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
-is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
-so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
address@hidden Answer 5, 5}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
-infinite series that exactly equals the value of that function at
-values of @expr{x} near zero.
-
address@hidden
address@hidden
-cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
-\afterdisplay
address@hidden tex
-
-The @kbd{a t} command produces a @dfn{truncated Taylor series} which
-is obtained by dropping all the terms higher than, say, @expr{x^2}.
-Calc represents the truncated Taylor series as a polynomial in @expr{x}.
-Mathematicians often write a truncated series using a ``big-O'' notation
-that records what was the lowest term that was truncated.
-
address@hidden
address@hidden
-cos(x) = 1 - x^2 / 2! + O(x^3)
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
-\afterdisplay
address@hidden tex
-
address@hidden
-The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
-if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
-
-The exercise is to create rewrite rules that simplify sums and products of
-power series represented as @address@hidden + O(@address@hidden)}.
-For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
-on the stack, we want to be able to type @kbd{*} and get the result
address@hidden - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
-rearranged or if @kbd{a s} needs to be typed after rewriting. (This one
-is rather tricky; the solution at the end of this chapter uses 6 rewrite
-rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is
-a number.) @xref{Rewrites Answer 6, 6}. (@bullet{})
-
-Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
-What happens? (Be sure to remove this rule afterward, or you might get
-a nasty surprise when you use Calc to balance your checkbook!)
-
address@hidden Rules}, for the whole story on rewrite rules.
-
address@hidden Programming Tutorial, Answers to Exercises, Algebra Tutorial,
Tutorial
address@hidden Programming Tutorial
-
address@hidden
-The Calculator is written entirely in Emacs Lisp, a highly extensible
-language. If you know Lisp, you can program the Calculator to do
-anything you like. Rewrite rules also work as a powerful programming
-system. But Lisp and rewrite rules take a while to master, and often
-all you want to do is define a new function or repeat a command a few
-times. Calc has features that allow you to do these things easily.
-
-One very limited form of programming is defining your own functions.
-Calc's @kbd{Z F} command allows you to define a function name and
-key sequence to correspond to any formula. Programming commands use
-the address@hidden prefix; the user commands they create use the lower
-case @kbd{z} prefix.
-
address@hidden
address@hidden
-1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6
- . .
-
- ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET}
@key{RET} @key{RET} y
address@hidden group
address@hidden smallexample
-
-This polynomial is a Taylor series approximation to @samp{exp(x)}.
-The @kbd{Z F} command asks a number of questions. The above answers
-say that the key sequence for our function should be @kbd{z e}; the
address@hidden equivalent should be @code{calc-myexp}; the name of the
-function in algebraic formulas should also be @code{myexp}; the
-default argument list @samp{(x)} is acceptable; and finally @kbd{y}
-answers the question ``leave it in symbolic form for non-constant
-arguments?''
-
address@hidden
address@hidden
-1: 1.3495 2: 1.3495 3: 1.3495
- . 1: 1.34986 2: 1.34986
- . 1: myexp(a + 1)
- .
-
- .3 z e .3 E ' a+1 @key{RET} z e
address@hidden group
address@hidden smallexample
-
address@hidden
-First we call our new @code{exp} approximation with 0.3 as an
-argument, and compare it with the true @code{exp} function. Then
-we note that, as requested, if we try to give @kbd{z e} an
-argument that isn't a plain number, it leaves the @code{myexp}
-function call in symbolic form. If we had answered @kbd{n} to the
-final question, @samp{myexp(a + 1)} would have evaluated by plugging
-in @samp{a + 1} for @samp{x} in the defining formula.
-
address@hidden Sine integral Si(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden Si
-(@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
address@hidden @math{{\rm Si}(x)}
address@hidden @expr{Si(x)}
-is defined as the integral of @samp{sin(t)/t} for
address@hidden = 0} to @expr{x} in radians. (It was invented because this
-integral has no solution in terms of basic functions; if you give it
-to Calc's @kbd{a i} command, it will ponder it for a long time and then
-give up.) We can use the numerical integration command, however,
-which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
-with any integrand @samp{f(t)}. Define a @kbd{z s} command and
address@hidden function that implement this. You will need to edit the
-default argument list a bit. As a test, @samp{Si(1)} should return
-0.946083. (If you don't get this answer, you might want to check that
-Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
-you reduce the precision to, say, six digits beforehand.)
address@hidden Answer 1, 1}. (@bullet{})
-
-The simplest way to do real ``programming'' of Emacs is to define a
address@hidden macro}. A keyboard macro is simply a sequence of
-keystrokes which Emacs has stored away and can play back on demand.
-For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
-you may wish to program a keyboard macro to type this for you.
-
address@hidden
address@hidden
-1: y = sqrt(x) 1: x = y^2
- . .
-
- ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
-
-1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1
- . .
-
- ' y=cos(x) @key{RET} X
address@hidden group
address@hidden smallexample
-
address@hidden
-When you type @kbd{C-x (}, Emacs begins recording. But it is also
-still ready to execute your keystrokes, so you're really ``training''
-Emacs by walking it through the procedure once. When you type
address@hidden@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
-re-execute the same keystrokes.
-
-You can give a name to your macro by typing @kbd{Z K}.
-
address@hidden
address@hidden
-1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
- . .
-
- Z K x @key{RET} ' y=x^4 @key{RET} z x
address@hidden group
address@hidden smallexample
-
address@hidden
-Notice that we use address@hidden to define the command, and lower-case
address@hidden to call it up.
-
-Keyboard macros can call other macros.
-
address@hidden
address@hidden
-1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
- . . . .
-
- ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET}
X
address@hidden group
address@hidden smallexample
-
-(@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
-the item in level 3 of the stack, without disturbing the rest of
-the stack. @xref{Programming Answer 2, 2}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
-the following functions:
-
address@hidden
address@hidden
-Compute
address@hidden @math{\displaystyle{\sin x \over x}},
address@hidden @expr{sin(x) / x},
-where @expr{x} is the number on the top of the stack.
-
address@hidden
-Compute the address@hidden logarithm, just like the @kbd{B} key except
-the arguments are taken in the opposite order.
-
address@hidden
-Produce a vector of integers from 1 to the integer on the top of
-the stack.
address@hidden enumerate
address@hidden
address@hidden Answer 3, 3}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
-the average (mean) value of a list of numbers.
address@hidden Answer 4, 4}. (@bullet{})
-
-In many programs, some of the steps must execute several times.
-Calc has @dfn{looping} commands that allow this. Loops are useful
-inside keyboard macros, but actually work at any time.
-
address@hidden
address@hidden
-1: x^6 2: x^6 1: 360 x^2
- . 1: 4 .
- .
-
- ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we have computed the fourth derivative of @expr{x^6} by
-enclosing a derivative command in a ``repeat loop'' structure.
-This structure pops a repeat count from the stack, then
-executes the body of the loop that many times.
-
-If you make a mistake while entering the body of the loop,
-type @address@hidden C-g}} to cancel the loop command.
-
address@hidden Fibonacci numbers
-Here's another example:
-
address@hidden
address@hidden
-3: 1 2: 10946
-2: 1 1: 17711
-1: 20 .
- .
-
-1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
address@hidden group
address@hidden smallexample
-
address@hidden
-The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
-numbers, respectively. (To see what's going on, try a few repetitions
-of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
-key if you have one, makes a copy of the number in level 2.)
-
address@hidden Golden ratio
address@hidden Phi, golden ratio
-A fascinating property of the Fibonacci numbers is that the @expr{n}th
-Fibonacci number can be found directly by computing
address@hidden @math{\phi^n / \sqrt{5}}
address@hidden @expr{phi^n / sqrt(5)}
-and then rounding to the nearest integer, where
address@hidden @math{\phi} (``phi''),
address@hidden @expr{phi},
-the ``golden ratio,'' is
address@hidden @math{(1 + \sqrt{5}) / 2}.
address@hidden @expr{(1 + sqrt(5)) / 2}.
-(For convenience, this constant is available from the @code{phi}
-variable, or the @kbd{I H P} command.)
-
address@hidden
address@hidden
-1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
- . . . .
-
- I H P 21 ^ 5 Q / R
address@hidden group
address@hidden smallexample
-
address@hidden Continued fractions
-(@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
-representation of
address@hidden @math{\phi}
address@hidden @expr{phi}
-is
address@hidden @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
address@hidden @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
-We can compute an approximate value by carrying this however far
-and then replacing the innermost
address@hidden @math{1/( \ldots )}
address@hidden @expr{1/( ...@: )}
-by 1. Approximate
address@hidden @math{\phi}
address@hidden @expr{phi}
-using a twenty-term continued fraction.
address@hidden Answer 5, 5}. (@bullet{})
-
-(@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
-Fibonacci numbers can be expressed in terms of matrices. Given a
-vector @address@hidden, b]}} determine a matrix which, when multiplied by this
-vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
address@hidden are three successive Fibonacci numbers. Now write a program
-that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
-using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
-
address@hidden Harmonic numbers
-A more sophisticated kind of loop is the @dfn{for} loop. Suppose
-we wish to compute the 20th ``harmonic'' number, which is equal to
-the sum of the reciprocals of the integers from 1 to 20.
-
address@hidden
address@hidden
-3: 0 1: 3.597739
-2: 1 .
-1: 20
- .
-
-0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
address@hidden group
address@hidden smallexample
-
address@hidden
-The ``for'' loop pops two numbers, the lower and upper limits, then
-repeats the body of the loop as an internal counter increases from
-the lower limit to the upper one. Just before executing the loop
-body, it pushes the current loop counter. When the loop body
-finishes, it pops the ``step,'' i.e., the amount by which to
-increment the loop counter. As you can see, our loop always
-uses a step of one.
-
-This harmonic number function uses the stack to hold the running
-total as well as for the various loop housekeeping functions. If
-you find this disorienting, you can sum in a variable instead:
-
address@hidden
address@hidden
-1: 0 2: 1 . 1: 3.597739
- . 1: 20 .
- .
-
- 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
address@hidden group
address@hidden smallexample
-
address@hidden
-The @kbd{s +} command adds the top-of-stack into the value in a
-variable (and removes that value from the stack).
-
-It's worth noting that many jobs that call for a ``for'' loop can
-also be done more easily by Calc's high-level operations. Two
-other ways to compute harmonic numbers are to use vector mapping
-and reduction (@kbd{v x 20}, then @address@hidden M &}}, then @kbd{V R +}),
-or to use the summation command @kbd{a +}. Both of these are
-probably easier than using loops. However, there are some
-situations where loops really are the way to go:
-
-(@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
-harmonic number which is greater than 4.0.
address@hidden Answer 7, 7}. (@bullet{})
-
-Of course, if we're going to be using variables in our programs,
-we have to worry about the programs clobbering values that the
-caller was keeping in those same variables. This is easy to
-fix, though:
-
address@hidden
address@hidden
- . 1: 0.6667 1: 0.6667 3: 0.6667
- . . 2: 3.597739
- 1: 0.6667
- .
-
- Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r
a @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-When we type @kbd{Z `} (that's a back-quote character), Calc saves
-its mode settings and the contents of the ten ``quick variables''
-for later reference. When we type @kbd{Z '} (that's an apostrophe
-now), Calc restores those saved values. Thus the @kbd{p 4} and
address@hidden 7} commands have no effect outside this sequence. Wrapping
-this around the body of a keyboard macro ensures that it doesn't
-interfere with what the user of the macro was doing. Notice that
-the contents of the stack, and the values of named variables,
-survive past the @kbd{Z '} command.
-
address@hidden Bernoulli numbers, approximate
-The @dfn{Bernoulli numbers} are a sequence with the interesting
-property that all of the odd Bernoulli numbers are zero, and the
-even ones, while difficult to compute, can be roughly approximated
-by the formula
address@hidden @math{\displaystyle{2 n! \over (2 \pi)^n}}.
address@hidden @expr{2 n!@: / (2 pi)^n}.
-Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
-(Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
-this command is very slow for large @expr{n} since the higher Bernoulli
-numbers are very large fractions.)
-
address@hidden
address@hidden
-1: 10 1: 0.0756823
- . .
-
- 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET}
= Z ] C-x )
address@hidden group
address@hidden smallexample
-
address@hidden
-You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
address@hidden ]} as ``end-if.'' There is no need for an explicit ``if''
-command. For the purposes of @address@hidden [}}, the condition is ``true''
-if the value it pops from the stack is a nonzero number, or ``false''
-if it pops zero or something that is not a number (like a formula).
-Here we take our integer argument modulo 2; this will be nonzero
-if we're asking for an odd Bernoulli number.
-
-The actual tenth Bernoulli number is @expr{5/66}.
-
address@hidden
address@hidden
-3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
-2: 5:66 . . . .
-1: 0.0757575
- .
-
-10 k b @key{RET} c f M-0 @key{DEL} 11 X @key{DEL} 12 X @key{DEL} 13
X @key{DEL} 14 X
address@hidden group
address@hidden smallexample
-
-Just to exercise loops a bit more, let's compute a table of even
-Bernoulli numbers.
-
address@hidden
address@hidden
-3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
-2: 2 .
-1: 30
- .
-
- [ ] 2 @key{RET} 30 Z ( X | 2 Z )
address@hidden group
address@hidden smallexample
-
address@hidden
-The vertical-bar @kbd{|} is the vector-concatenation command. When
-we execute it, the list we are building will be in stack level 2
-(initially this is an empty list), and the next Bernoulli number
-will be in level 1. The effect is to append the Bernoulli number
-onto the end of the list. (To create a table of exact fractional
-Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
-sequence of keystrokes.)
-
-With loops and conditionals, you can program essentially anything
-in Calc. One other command that makes looping easier is @kbd{Z /},
-which takes a condition from the stack and breaks out of the enclosing
-loop if the condition is true (non-zero). You can use this to make
-``while'' and ``until'' style loops.
-
-If you make a mistake when entering a keyboard macro, you can edit
-it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
-One technique is to enter a throwaway dummy definition for the macro,
-then enter the real one in the edit command.
-
address@hidden
address@hidden
-1: 3 1: 3 Calc Macro Edit Mode.
- . . Original keys: 1 <return> 2 +
-
- 1 ;; calc
digits
- RET ;;
calc-enter
- 2 ;; calc
digits
- + ;; calc-plus
-
-C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
address@hidden group
address@hidden smallexample
-
address@hidden
-A keyboard macro is stored as a pure keystroke sequence. The
address@hidden package (invoked by @kbd{Z E}) scans along the
-macro and tries to decode it back into human-readable steps.
-Descriptions of the keystrokes are given as comments, which begin with
address@hidden;;}, and which are ignored when the edited macro is saved.
-Spaces and line breaks are also ignored when the edited macro is saved.
-To enter a space into the macro, type @code{SPC}. All the special
-characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
-and @code{NUL} must be written in all uppercase, as must the prefixes
address@hidden and @code{M-}.
-
-Let's edit in a new definition, for computing harmonic numbers.
-First, erase the four lines of the old definition. Then, type
-in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
-to copy it from this page of the Info file; you can of course skip
-typing the comments, which begin with @samp{;;}).
-
address@hidden
-Z` ;; calc-kbd-push (Save local values)
-0 ;; calc digits (Push a zero onto the stack)
-st ;; calc-store-into (Store it in the following
variable)
-1 ;; calc quick variable (Quick variable q1)
-1 ;; calc digits (Initial value for the loop)
-TAB ;; calc-roll-down (Swap initial and final)
-Z( ;; calc-kbd-for (Begin the "for" loop)
-& ;; calc-inv (Take the reciprocal)
-s+ ;; calc-store-plus (Add to the following variable)
-1 ;; calc quick variable (Quick variable q1)
-1 ;; calc digits (The loop step is 1)
-Z) ;; calc-kbd-end-for (End the "for" loop)
-sr ;; calc-recall (Recall the final accumulated
value)
-1 ;; calc quick variable (Quick variable q1)
-Z' ;; calc-kbd-pop (Restore values)
address@hidden smallexample
-
address@hidden
-Press @kbd{C-c C-c} to finish editing and return to the Calculator.
-
address@hidden
address@hidden
-1: 20 1: 3.597739
- . .
-
- 20 z h
address@hidden group
address@hidden smallexample
-
-The @file{edmacro} package defines a handy @code{read-kbd-macro} command
-which reads the current region of the current buffer as a sequence of
-keystroke names, and defines that sequence on the @kbd{X}
-(and @kbd{C-x e}) key. Because this is so useful, Calc puts this
-command on the @kbd{C-x * m} key. Try reading in this macro in the
-following form: Press @kbd{C-@@} (or @address@hidden) at
-one end of the text below, then type @kbd{C-x * m} at the other.
-
address@hidden
address@hidden
-Z ` 0 t 1
- 1 TAB
- Z ( & s + 1 1 Z )
- r 1
-Z '
address@hidden group
address@hidden example
-
-(@bullet{}) @strong{Exercise 8.} A general algorithm for solving
-equations numerically is @dfn{Newton's Method}. Given the equation
address@hidden(x) = 0} for any function @expr{f}, and an initial guess
address@hidden which is reasonably close to the desired solution, apply
-this formula over and over:
-
address@hidden
address@hidden
-new_x = x - f(x)/f'(x)
address@hidden example
address@hidden ifnottex
address@hidden
-\beforedisplay
-$$ x_{\rm new} = x - {f(x) \over f'(x)} $$
-\afterdisplay
address@hidden tex
-
address@hidden
-where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
-values will quickly converge to a solution, i.e., eventually
address@hidden @math{x_{\rm new}}
address@hidden @expr{new_x}
-and @expr{x} will be equal to within the limits
-of the current precision. Write a program which takes a formula
-involving the variable @expr{x}, and an initial guess @expr{x_0},
-on the stack, and produces a value of @expr{x} for which the formula
-is zero. Use it to find a solution of
address@hidden @math{\sin(\cos x) = 0.5}
address@hidden @expr{sin(cos(x)) = 0.5}
-near @expr{x = 4.5}. (Use angles measured in radians.) Note that
-the built-in @address@hidden R}} (@code{calc-find-root}) command uses Newton's
-method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
-
address@hidden Digamma function
address@hidden Gamma constant, Euler's
address@hidden Euler's gamma constant
-(@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
address@hidden @math{\psi(z) (``psi'')}
address@hidden @expr{psi(z)}
-is defined as the derivative of
address@hidden @math{\ln \Gamma(z)}.
address@hidden @expr{ln(gamma(z))}.
-For large values of @expr{z}, it can be approximated by the infinite sum
-
address@hidden
address@hidden
-psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
address@hidden example
address@hidden ifnottex
address@hidden
-\beforedisplay
-$$ \psi(z) \approx \ln z - {1\over2z} -
- \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
-$$
-\afterdisplay
address@hidden tex
-
address@hidden
-where
address@hidden @math{\sum}
address@hidden @expr{sum}
-represents the sum over @expr{n} from 1 to infinity
-(or to some limit high enough to give the desired accuracy), and
-the @code{bern} function produces (exact) Bernoulli numbers.
-While this sum is not guaranteed to converge, in practice it is safe.
-An interesting mathematical constant is Euler's gamma, which is equal
-to about 0.5772. One way to compute it is by the formula,
address@hidden @math{\gamma = -\psi(1)}.
address@hidden @expr{gamma = -psi(1)}.
-Unfortunately, 1 isn't a large enough argument
-for the above formula to work (5 is a much safer value for @expr{z}).
-Fortunately, we can compute
address@hidden @math{\psi(1)}
address@hidden @expr{psi(1)}
-from
address@hidden @math{\psi(5)}
address@hidden @expr{psi(5)}
-using the recurrence
address@hidden @math{\psi(z+1) = \psi(z) + {1 \over z}}.
address@hidden @expr{psi(z+1) = psi(z) + 1/z}.
-Your task: Develop a program to compute
address@hidden @math{\psi(z)};
address@hidden @expr{psi(z)};
-it should ``pump up'' @expr{z}
-if necessary to be greater than 5, then use the above summation
-formula. Use looping commands to compute the sum. Use your function
-to compute
address@hidden @math{\gamma}
address@hidden @expr{gamma}
-to twelve decimal places. (Calc has a built-in command
-for Euler's constant, @kbd{I P}, which you can use to check your answer.)
address@hidden Answer 9, 9}. (@bullet{})
-
address@hidden Polynomial, list of coefficients
-(@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
-a number @expr{m} on the stack, where the polynomial is of degree
address@hidden or less (i.e., does not have any terms higher than @expr{x^m}),
-write a program to convert the polynomial into a list-of-coefficients
-notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
-should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
-a way to convert from this form back to the standard algebraic form.
address@hidden Answer 10, 10}. (@bullet{})
-
address@hidden Recursion
-(@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
-first kind} are defined by the recurrences,
-
address@hidden
address@hidden
-s(n,n) = 1 for n >= 0,
-s(n,0) = 0 for n > 0,
-s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
- s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
- s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
- \hbox{for } n \ge m \ge 1.}
-$$
-\afterdisplay
-\vskip5pt
-(These numbers are also sometimes written $\displaystyle{n \brack m}$.)
address@hidden tex
-
-This can be implemented using a @dfn{recursive} program in Calc; the
-program must invoke itself in order to calculate the two righthand
-terms in the general formula. Since it always invokes itself with
-``simpler'' arguments, it's easy to see that it must eventually finish
-the computation. Recursion is a little difficult with Emacs keyboard
-macros since the macro is executed before its definition is complete.
-So here's the recommended strategy: Create a ``dummy macro'' and assign
-it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
-using the @kbd{z s} command to call itself recursively, then assign it
-to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
-the complete recursive program. (Another way is to use @address@hidden E}}
-or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
-thus avoiding the ``training'' phase.) The task: Write a program
-that computes Stirling numbers of the first kind, given @expr{n} and
address@hidden on the stack. Test it with @emph{small} inputs like
address@hidden(4,2)}. (There is a built-in command for Stirling numbers,
address@hidden s}, which you can use to check your answers.)
address@hidden Answer 11, 11}. (@bullet{})
-
-The programming commands we've seen in this part of the tutorial
-are low-level, general-purpose operations. Often you will find
-that a higher-level function, such as vector mapping or rewrite
-rules, will do the job much more easily than a detailed, step-by-step
-program can:
-
-(@bullet{}) @strong{Exercise 12.} Write another program for
-computing Stirling numbers of the first kind, this time using
-rewrite rules. Once again, @expr{n} and @expr{m} should be taken
-from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
-
address@hidden
-
address@hidden example
-This ends the tutorial section of the Calc manual. Now you know enough
-about Calc to use it effectively for many kinds of calculations. But
-Calc has many features that were not even touched upon in this tutorial.
address@hidden [not-split]
-The rest of this manual tells the whole story.
address@hidden [when-split]
address@hidden Volume II of this manual, the @dfn{Calc Reference}, tells the
whole story.
-
address@hidden
address@hidden Answers to Exercises, , Programming Tutorial, Tutorial
address@hidden Answers to Exercises
-
address@hidden
-This section includes answers to all the exercises in the Calc tutorial.
-
address@hidden
-* RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
-* RPN Answer 2:: 2*4 + 7*9.5 + 5/4
-* RPN Answer 3:: Operating on levels 2 and 3
-* RPN Answer 4:: Joe's complex problems
-* Algebraic Answer 1:: Simulating Q command
-* Algebraic Answer 2:: Joe's algebraic woes
-* Algebraic Answer 3:: 1 / 0
-* Modes Answer 1:: 3#0.1 = 3#0.0222222?
-* Modes Answer 2:: 16#f.e8fe15
-* Modes Answer 3:: Joe's rounding bug
-* Modes Answer 4:: Why floating point?
-* Arithmetic Answer 1:: Why the \ command?
-* Arithmetic Answer 2:: Tripping up the B command
-* Vector Answer 1:: Normalizing a vector
-* Vector Answer 2:: Average position
-* Matrix Answer 1:: Row and column sums
-* Matrix Answer 2:: Symbolic system of equations
-* Matrix Answer 3:: Over-determined system
-* List Answer 1:: Powers of two
-* List Answer 2:: Least-squares fit with matrices
-* List Answer 3:: Geometric mean
-* List Answer 4:: Divisor function
-* List Answer 5:: Duplicate factors
-* List Answer 6:: Triangular list
-* List Answer 7:: Another triangular list
-* List Answer 8:: Maximum of Bessel function
-* List Answer 9:: Integers the hard way
-* List Answer 10:: All elements equal
-* List Answer 11:: Estimating pi with darts
-* List Answer 12:: Estimating pi with matchsticks
-* List Answer 13:: Hash codes
-* List Answer 14:: Random walk
-* Types Answer 1:: Square root of pi times rational
-* Types Answer 2:: Infinities
-* Types Answer 3:: What can "nan" be?
-* Types Answer 4:: Abbey Road
-* Types Answer 5:: Friday the 13th
-* Types Answer 6:: Leap years
-* Types Answer 7:: Erroneous donut
-* Types Answer 8:: Dividing intervals
-* Types Answer 9:: Squaring intervals
-* Types Answer 10:: Fermat's primality test
-* Types Answer 11:: pi * 10^7 seconds
-* Types Answer 12:: Abbey Road on CD
-* Types Answer 13:: Not quite pi * 10^7 seconds
-* Types Answer 14:: Supercomputers and c
-* Types Answer 15:: Sam the Slug
-* Algebra Answer 1:: Squares and square roots
-* Algebra Answer 2:: Building polynomial from roots
-* Algebra Answer 3:: Integral of x sin(pi x)
-* Algebra Answer 4:: Simpson's rule
-* Rewrites Answer 1:: Multiplying by conjugate
-* Rewrites Answer 2:: Alternative fib rule
-* Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
-* Rewrites Answer 4:: Sequence of integers
-* Rewrites Answer 5:: Number of terms in sum
-* Rewrites Answer 6:: Truncated Taylor series
-* Programming Answer 1:: Fresnel's C(x)
-* Programming Answer 2:: Negate third stack element
-* Programming Answer 3:: Compute sin(x) / x, etc.
-* Programming Answer 4:: Average value of a list
-* Programming Answer 5:: Continued fraction phi
-* Programming Answer 6:: Matrix Fibonacci numbers
-* Programming Answer 7:: Harmonic number greater than 4
-* Programming Answer 8:: Newton's method
-* Programming Answer 9:: Digamma function
-* Programming Answer 10:: Unpacking a polynomial
-* Programming Answer 11:: Recursive Stirling numbers
-* Programming Answer 12:: Stirling numbers with rewrites
address@hidden menu
-
address@hidden The following kludgery prevents the individual answers from
address@hidden being entered on the table of contents.
address@hidden
-\global\let\oldwrite=\write
-\gdef\skipwrite#1#2{\let\write=\oldwrite}
-\global\let\oldchapternofonts=\chapternofonts
-\gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
address@hidden tex
-
address@hidden RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to
Exercises
address@hidden RPN Tutorial Exercise 1
-
address@hidden
address@hidden @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
-
-The result is
address@hidden @math{1 - (2 \times (3 + 4)) = -13}.
address@hidden @expr{1 - (2 * (3 + 4)) = -13}.
-
address@hidden RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
address@hidden RPN Tutorial Exercise 2
-
address@hidden
address@hidden @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
address@hidden @expr{2*4 + 7*9.5 + 5/4 = 75.75}
-
-After computing the intermediate term
address@hidden @math{2\times4 = 8},
address@hidden @expr{2*4 = 8},
-you can leave that result on the stack while you compute the second
-term. With both of these results waiting on the stack you can then
-compute the final term, then press @kbd{+ +} to add everything up.
-
address@hidden
address@hidden
-2: 2 1: 8 3: 8 2: 8
-1: 4 . 2: 7 1: 66.5
- . 1: 9.5 .
- .
-
- 2 @key{RET} 4 * 7 @key{RET} 9.5 *
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-4: 8 3: 8 2: 8 1: 75.75
-3: 66.5 2: 66.5 1: 67.75 .
-2: 5 1: 1.25 .
-1: 4 .
- .
-
- 5 @key{RET} 4 / + +
address@hidden group
address@hidden smallexample
-
-Alternatively, you could add the first two terms before going on
-with the third term.
-
address@hidden
address@hidden
-2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
-1: 66.5 . 2: 5 1: 1.25 .
- . 1: 4 .
- .
-
- ... + 5 @key{RET} 4 / +
address@hidden group
address@hidden smallexample
-
-On an old-style RPN calculator this second method would have the
-advantage of using only three stack levels. But since Calc's stack
-can grow arbitrarily large this isn't really an issue. Which method
-you choose is purely a matter of taste.
-
address@hidden RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
address@hidden RPN Tutorial Exercise 3
-
address@hidden
-The @key{TAB} key provides a way to operate on the number in level 2.
-
address@hidden
address@hidden
-3: 10 3: 10 4: 10 3: 10 3: 10
-2: 20 2: 30 3: 30 2: 30 2: 21
-1: 30 1: 20 2: 20 1: 21 1: 30
- . . 1: 1 . .
- .
-
- @key{TAB} 1 + @key{TAB}
address@hidden group
address@hidden smallexample
-
-Similarly, @address@hidden gives you access to the number in level 3.
-
address@hidden
address@hidden
-3: 10 3: 21 3: 21 3: 30 3: 11
-2: 21 2: 30 2: 30 2: 11 2: 21
-1: 30 1: 10 1: 11 1: 21 1: 30
- . . . . .
-
- address@hidden 1 + address@hidden
address@hidden
address@hidden group
address@hidden smallexample
-
address@hidden RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to
Exercises
address@hidden RPN Tutorial Exercise 4
-
address@hidden
-Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
-but using both the comma and the space at once yields:
-
address@hidden
address@hidden
-1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
- . 1: 2 . 1: (2, ... 1: (2, 3)
- . . .
-
- ( 2 , @key{SPC} 3 )
address@hidden group
address@hidden smallexample
-
-Joe probably tried to type @address@hidden @key{DEL}} to swap the
-extra incomplete object to the top of the stack and delete it.
-But a feature of Calc is that @key{DEL} on an incomplete object
-deletes just one component out of that object, so he had to press
address@hidden twice to finish the job.
-
address@hidden
address@hidden
-2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
-1: (2, 3) 1: (2, ... 1: ( ... .
- . . .
-
- @key{TAB} @key{DEL} @key{DEL}
address@hidden group
address@hidden smallexample
-
-(As it turns out, deleting the second-to-top stack entry happens often
-enough that Calc provides a special key, @address@hidden, to do just that.
address@hidden@key{DEL}} is just like @address@hidden @key{DEL}}, except that
it doesn't exhibit
-the ``feature'' that tripped poor Joe.)
-
address@hidden Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to
Exercises
address@hidden Algebraic Entry Tutorial Exercise 1
-
address@hidden
-Type @kbd{' sqrt($) @key{RET}}.
-
-If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
-Or, RPN style, @kbd{0.5 ^}.
-
-(Actually, @samp{$^1:2}, using the fraction one-half as the power, is
-a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
address@hidden(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
-
address@hidden Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1,
Answers to Exercises
address@hidden Algebraic Entry Tutorial Exercise 2
-
address@hidden
-In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
-name with @samp{1+y} as its argument. Assigning a value to a variable
-has no relation to a function by the same name. Joe needed to use an
-explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
-
address@hidden Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers
to Exercises
address@hidden Algebraic Entry Tutorial Exercise 3
-
address@hidden
-The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
-The ``function'' @samp{/} cannot be evaluated when its second argument
-is zero, so it is left in symbolic form. When you now type @kbd{0 *},
-the result will be zero because Calc uses the general rule that ``zero
-times anything is zero.''
-
address@hidden [fix-ref Infinities]
-The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
-results in a special symbol that represents ``infinity.'' If you
-multiply infinity by zero, Calc uses another special new symbol to
-show that the answer is ``indeterminate.'' @xref{Infinities}, for
-further discussion of infinite and indeterminate values.
-
address@hidden Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to
Exercises
address@hidden Modes Tutorial Exercise 1
-
address@hidden
-Calc always stores its numbers in decimal, so even though one-third has
-an exact base-3 representation (@samp{3#0.1}), it is still stored as
-0.3333333 (chopped off after 12 or however many decimal digits) inside
-the calculator's memory. When this inexact number is converted back
-to base 3 for display, it may still be slightly inexact. When we
-multiply this number by 3, we get 0.999999, also an inexact value.
-
-When Calc displays a number in base 3, it has to decide how many digits
-to show. If the current precision is 12 (decimal) digits, that corresponds
-to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
-exact integer, Calc shows only 25 digits, with the result that stored
-numbers carry a little bit of extra information that may not show up on
-the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
-happened to round to a pleasing value when it lost that last 0.15 of a
-digit, but it was still inexact in Calc's memory. When he divided by 2,
-he still got the dreaded inexact value 0.333333. (Actually, he divided
-0.666667 by 2 to get 0.333334, which is why he got something a little
-higher than @code{3#0.1} instead of a little lower.)
-
-If Joe didn't want to be bothered with all this, he could have typed
address@hidden d n} to display with one less digit than the default. (If
-you give @kbd{d n} a negative argument, it uses default-minus-that,
-so @kbd{M-- d n} would be an easier way to get the same effect.) Those
-inexact results would still be lurking there, but they would now be
-rounded to nice, natural-looking values for display purposes. (Remember,
address@hidden in base 3 is like @samp{0.099999} in base 10; rounding
-off one digit will round the number up to @samp{0.1}.) Depending on the
-nature of your work, this hiding of the inexactness may be a benefit or
-a danger. With the @kbd{d n} command, Calc gives you the choice.
-
-Incidentally, another consequence of all this is that if you type
address@hidden d n} to display more digits than are ``really there,''
-you'll see garbage digits at the end of the number. (In decimal
-display mode, with decimally-stored numbers, these garbage digits are
-always zero so they vanish and you don't notice them.) Because Calc
-rounds off that 0.15 digit, there is the danger that two numbers could
-be slightly different internally but still look the same. If you feel
-uneasy about this, set the @kbd{d n} precision to be a little higher
-than normal; you'll get ugly garbage digits, but you'll always be able
-to tell two distinct numbers apart.
-
-An interesting side note is that most computers store their
-floating-point numbers in binary, and convert to decimal for display.
-Thus everyday programs have the same problem: Decimal 0.1 cannot be
-represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
-comes out as an inexact approximation to 1 on some machines (though
-they generally arrange to hide it from you by rounding off one digit as
-we did above). Because Calc works in decimal instead of binary, you can
-be sure that numbers that look exact @emph{are} exact as long as you stay
-in decimal display mode.
-
-It's not hard to show that any number that can be represented exactly
-in binary, octal, or hexadecimal is also exact in decimal, so the kinds
-of problems we saw in this exercise are likely to be severe only when
-you use a relatively unusual radix like 3.
-
address@hidden Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to
Exercises
address@hidden Modes Tutorial Exercise 2
-
-If the radix is 15 or higher, we can't use the letter @samp{e} to mark
-the exponent because @samp{e} is interpreted as a digit. When Calc
-needs to display scientific notation in a high radix, it writes
address@hidden You can enter a number like this as an
-algebraic entry. Also, pressing @kbd{e} without any digits before it
-normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
-puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is
another
-way to enter this number.
-
-The reason Calc puts a decimal point in the @samp{16.^} is to prevent
-huge integers from being generated if the exponent is large (consider
address@hidden, where we compute @samp{16^1000} as a giant
-exact integer and then throw away most of the digits when we multiply
-it by the floating-point @samp{16#1.23}). While this wouldn't normally
-matter for display purposes, it could give you a nasty surprise if you
-copied that number into a file and later moved it back into Calc.
-
address@hidden Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to
Exercises
address@hidden Modes Tutorial Exercise 3
-
address@hidden
-The answer he got was @expr{0.5000000000006399}.
-
-The problem is not that the square operation is inexact, but that the
-sine of 45 that was already on the stack was accurate to only 12 places.
-Arbitrary-precision calculations still only give answers as good as
-their inputs.
-
-The real problem is that there is no 12-digit number which, when
-squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
-commands decrease or increase a number by one unit in the last
-place (according to the current precision). They are useful for
-determining facts like this.
-
address@hidden
address@hidden
-1: 0.707106781187 1: 0.500000000001
- . .
-
- 45 S 2 ^
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
- . . .
-
- U @key{DEL} f [ 2 ^
address@hidden group
address@hidden smallexample
-
-A high-precision calculation must be carried out in high precision
-all the way. The only number in the original problem which was known
-exactly was the quantity 45 degrees, so the precision must be raised
-before anything is done after the number 45 has been entered in order
-for the higher precision to be meaningful.
-
address@hidden Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to
Exercises
address@hidden Modes Tutorial Exercise 4
-
address@hidden
-Many calculations involve real-world quantities, like the width and
-height of a piece of wood or the volume of a jar. Such quantities
-can't be measured exactly anyway, and if the data that is input to
-a calculation is inexact, doing exact arithmetic on it is a waste
-of time.
-
-Fractions become unwieldy after too many calculations have been
-done with them. For example, the sum of the reciprocals of the
-integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
-9304682830147:2329089562800. After a point it will take a long
-time to add even one more term to this sum, but a floating-point
-calculation of the sum will not have this problem.
-
-Also, rational numbers cannot express the results of all calculations.
-There is no fractional form for the square root of two, so if you type
address@hidden@kbd{2 Q}}, Calc has no choice but to give you a floating-point
answer.
-
address@hidden Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4,
Answers to Exercises
address@hidden Arithmetic Tutorial Exercise 1
-
address@hidden
-Dividing two integers that are larger than the current precision may
-give a floating-point result that is inaccurate even when rounded
-down to an integer. Consider @expr{123456789 / 2} when the current
-precision is 6 digits. The true answer is @expr{61728394.5}, but
-with a precision of 6 this will be rounded to
address@hidden @math{12345700.0/2.0 = 61728500.0}.
address@hidden @expr{12345700.@: / 2.@: = 61728500.}.
-The result, when converted to an integer, will be off by 106.
-
-Here are two solutions: Raise the precision enough that the
-floating-point round-off error is strictly to the right of the
-decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
-produces the exact fraction @expr{123456789:2}, which can be rounded
-down by the @kbd{F} command without ever switching to floating-point
-format.
-
address@hidden Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1,
Answers to Exercises
address@hidden Arithmetic Tutorial Exercise 2
-
address@hidden
address@hidden @key{RET} 9 B} could give the exact result @expr{3:2}, but it
-does a floating-point calculation instead and produces @expr{1.5}.
-
-Calc will find an exact result for a logarithm if the result is an integer
-or (when in Fraction mode) the reciprocal of an integer. But there is
-no efficient way to search the space of all possible rational numbers
-for an exact answer, so Calc doesn't try.
-
address@hidden Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers
to Exercises
address@hidden Vector Tutorial Exercise 1
-
address@hidden
-Duplicate the vector, compute its length, then divide the vector
-by its length: @address@hidden A /}.
-
address@hidden
address@hidden
-1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
- . 1: 3.74165738677 . .
- .
-
- r 1 @key{RET} A / A
address@hidden group
address@hidden smallexample
-
-The final @kbd{A} command shows that the normalized vector does
-indeed have unit length.
-
address@hidden Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to
Exercises
address@hidden Vector Tutorial Exercise 2
-
address@hidden
-The average position is equal to the sum of the products of the
-positions times their corresponding probabilities. This is the
-definition of the dot product operation. So all you need to do
-is to put the two vectors on the stack and press @kbd{*}.
-
address@hidden Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 1
-
address@hidden
-The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
-get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
-
address@hidden Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 2
-
address@hidden
address@hidden
address@hidden
- x + a y = 6
- x + b y = 10
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \eqalign{ x &+ a y = 6 \cr
- x &+ b y = 10}
-$$
-\afterdisplay
address@hidden tex
-
-Just enter the righthand side vector, then divide by the lefthand side
-matrix as usual.
-
address@hidden
address@hidden
-1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ]
- . 1: [ [ 1, a ] .
- [ 1, b ] ]
- .
-
-' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
address@hidden group
address@hidden smallexample
-
-This can be made more readable using @kbd{d B} to enable Big display
-mode:
-
address@hidden
address@hidden
- 4 a 4
-1: [6 - -----, -----]
- b - a b - a
address@hidden group
address@hidden smallexample
-
-Type @kbd{d N} to return to Normal display mode afterwards.
-
address@hidden Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to
Exercises
address@hidden Matrix Tutorial Exercise 3
-
address@hidden
-To solve
address@hidden @math{A^T A \, X = A^T B},
address@hidden @expr{trn(A) * A * X = trn(A) * B},
-first we compute
address@hidden @math{A' = A^T A}
address@hidden @expr{A2 = trn(A) * A}
-and
address@hidden @math{B' = A^T B};
address@hidden @expr{B2 = trn(A) * B};
-now, we have a system
address@hidden @math{A' X = B'}
address@hidden @expr{A2 * X = B2}
-which we can solve using Calc's @samp{/} command.
-
address@hidden
address@hidden
address@hidden
- a + 2b + 3c = 6
- 4a + 5b + 6c = 2
- 7a + 6b = 3
- 2a + 4b + 6c = 11
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplayh
-$$ \openup1\jot \tabskip=0pt plus1fil
-\halign to\displaywidth{\tabskip=0pt
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&$\hfil{}#{}$&
- $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
- a&+&2b&+&3c&=6 \cr
- 4a&+&5b&+&6c&=2 \cr
- 7a&+&6b& & &=3 \cr
- 2a&+&4b&+&6c&=11 \cr}
-$$
-\afterdisplayh
address@hidden tex
-
-The first step is to enter the coefficient matrix. We'll store it in
-quick variable number 7 for later reference. Next, we compute the
address@hidden @math{B'}
address@hidden @expr{B2}
-vector.
-
address@hidden
address@hidden
-1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
- [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
- [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
- [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
- . .
-
-' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
address@hidden group
address@hidden smallexample
-
address@hidden
-Now we compute the matrix
address@hidden @math{A'}
address@hidden @expr{A2}
-and divide.
-
address@hidden
address@hidden
-2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
-1: [ [ 70, 72, 39 ] .
- [ 72, 81, 60 ]
- [ 39, 60, 81 ] ]
- .
-
- r 7 v t r 7 * /
address@hidden group
address@hidden smallexample
-
address@hidden
-(The actual computed answer will be slightly inexact due to
-round-off error.)
-
-Notice that the answers are similar to those for the
address@hidden @math{3\times3}
address@hidden 3x3
-system solved in the text. That's because the fourth equation that was
-added to the system is almost identical to the first one multiplied
-by two. (If it were identical, we would have gotten the exact same
-answer since the
address@hidden @math{4\times3}
address@hidden 4x3
-system would be equivalent to the original
address@hidden @math{3\times3}
address@hidden 3x3
-system.)
-
-Since the first and fourth equations aren't quite equivalent, they
-can't both be satisfied at once. Let's plug our answers back into
-the original system of equations to see how well they match.
-
address@hidden
address@hidden
-2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
-1: [ [ 1, 2, 3 ] .
- [ 4, 5, 6 ]
- [ 7, 6, 0 ]
- [ 2, 4, 6 ] ]
- .
-
- r 7 @key{TAB} *
address@hidden group
address@hidden smallexample
-
address@hidden
-This is reasonably close to our original @expr{B} vector,
address@hidden, 2, 3, 11]}.
-
address@hidden List Answer 1, List Answer 2, Matrix Answer 3, Answers to
Exercises
address@hidden List Tutorial Exercise 1
-
address@hidden
-We can use @kbd{v x} to build a vector of integers. This needs to be
-adjusted to get the range of integers we desire. Mapping @samp{-}
-across the vector will accomplish this, although it turns out the
-plain @samp{-} key will work just as well.
-
address@hidden
address@hidden
-2: 2 2: 2
-1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
- . .
-
- 2 v x 9 @key{RET} 5 V M - or 5 -
address@hidden group
address@hidden smallexample
-
address@hidden
-Now we use @kbd{V M ^} to map the exponentiation operator across the
-vector.
-
address@hidden
address@hidden
-1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
- .
-
- V M ^
address@hidden group
address@hidden smallexample
-
address@hidden List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
address@hidden List Tutorial Exercise 2
-
address@hidden
-Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
-the first job is to form the matrix that describes the problem.
-
address@hidden
address@hidden
- m*x + b*1 = y
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ m \times x + b \times 1 = y $$
-\afterdisplay
address@hidden tex
-
-Thus we want a
address@hidden @math{19\times2}
address@hidden 19x2
-matrix with our @expr{x} vector as one column and
-ones as the other column. So, first we build the column of ones, then
-we combine the two columns to form our @expr{A} matrix.
-
address@hidden
address@hidden
-2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
-1: [1, 1, 1, ...] [ 1.41, 1 ]
- . [ 1.49, 1 ]
- @dots{}
-
- r 1 1 v b 19 @key{RET} M-2 v p v t s 3
address@hidden group
address@hidden smallexample
-
address@hidden
-Now we compute
address@hidden @math{A^T y}
address@hidden @expr{trn(A) * y}
-and
address@hidden @math{A^T A}
address@hidden @expr{trn(A) * A}
-and divide.
-
address@hidden
address@hidden
-1: [33.36554, 13.613] 2: [33.36554, 13.613]
- . 1: [ [ 98.0003, 41.63 ]
- [ 41.63, 19 ] ]
- .
-
- v t r 2 * r 3 v t r 3 *
address@hidden group
address@hidden smallexample
-
address@hidden
-(Hey, those numbers look familiar!)
-
address@hidden
address@hidden
-1: [0.52141679, -0.425978]
- .
-
- /
address@hidden group
address@hidden smallexample
-
-Since we were solving equations of the form
address@hidden @math{m \times x + b \times 1 = y},
address@hidden @expr{m*x + b*1 = y},
-these numbers should be @expr{m} and @expr{b}, respectively. Sure
-enough, they agree exactly with the result computed using @kbd{V M} and
address@hidden R}!
-
-The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
-your problem, but there is often an easier way using the higher-level
-arithmetic functions!
-
address@hidden [fix-ref Curve Fitting]
-In fact, there is a built-in @kbd{a F} command that does least-squares
-fits. @xref{Curve Fitting}.
-
address@hidden List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
address@hidden List Tutorial Exercise 3
-
address@hidden
-Move to one end of the list and press @kbd{C-@@} (or @address@hidden or
-whatever) to set the mark, then move to the other end of the list
-and type @address@hidden * g}}.
-
address@hidden
address@hidden
-1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
- .
address@hidden group
address@hidden smallexample
-
-To make things interesting, let's assume we don't know at a glance
-how many numbers are in this list. Then we could type:
-
address@hidden
address@hidden
-2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
-1: [2.3, 6, 22, ... ] 1: 126356422.5
- . .
-
- @key{RET} V R *
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: 126356422.5 2: 126356422.5 1: 7.94652913734
-1: [2.3, 6, 22, ... ] 1: 9 .
- . .
-
- @key{TAB} v l I ^
address@hidden group
address@hidden smallexample
-
address@hidden
-(The @kbd{I ^} command computes the @var{n}th root of a number.
-You could also type @kbd{& ^} to take the reciprocal of 9 and
-then raise the number to that power.)
-
address@hidden List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
address@hidden List Tutorial Exercise 4
-
address@hidden
-A number @expr{j} is a divisor of @expr{n} if
address@hidden @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
address@hidden @samp{n % j = 0}.
-The first step is to get a vector that identifies the divisors.
-
address@hidden
address@hidden
-2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
-1: [1, 2, 3, 4, ...] 1: 0 .
- . .
-
- 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
address@hidden group
address@hidden smallexample
-
address@hidden
-This vector has 1's marking divisors of 30 and 0's marking non-divisors.
-
-The zeroth divisor function is just the total number of divisors.
-The first divisor function is the sum of the divisors.
-
address@hidden
address@hidden
-1: 8 3: 8 2: 8 2: 8
- 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
- 1: [1, 1, 1, 0, ...] . .
- .
-
- V R + r 1 r 2 V M * V R +
address@hidden group
address@hidden smallexample
-
address@hidden
-Once again, the last two steps just compute a dot product for which
-a simple @kbd{*} would have worked equally well.
-
address@hidden List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
address@hidden List Tutorial Exercise 5
-
address@hidden
-The obvious first step is to obtain the list of factors with @kbd{k f}.
-This list will always be in sorted order, so if there are duplicates
-they will be right next to each other. A suitable method is to compare
-the list with a copy of itself shifted over by one.
-
address@hidden
address@hidden
-1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
- . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
- . .
-
- 19551 k f @key{RET} 0 | @key{TAB} 0
@key{TAB} |
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
- . . .
-
- V M a = V R + 0 a =
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that we have to arrange for both vectors to have the same length
-so that the mapping operation works; no prime factor will ever be
-zero, so adding zeros on the left and right is safe. From then on
-the job is pretty straightforward.
-
-Incidentally, Calc provides the
address@hidden @dfn{M@"obius} @math{\mu}
address@hidden @dfn{Moebius mu}
-function which is zero if and only if its argument is square-free. It
-would be a much more convenient way to do the above test in practice.
-
address@hidden List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
address@hidden List Tutorial Exercise 6
-
address@hidden
-First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
-to get a list of lists of integers!
-
address@hidden List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
address@hidden List Tutorial Exercise 7
-
address@hidden
-Here's one solution. First, compute the triangular list from the previous
-exercise and type @kbd{1 -} to subtract one from all the elements.
-
address@hidden
address@hidden
-1: [ [0],
- [0, 1],
- [0, 1, 2],
- @dots{}
-
- 1 -
address@hidden group
address@hidden smallexample
-
-The numbers down the lefthand edge of the list we desire are called
-the ``triangular numbers'' (now you know why!). The @expr{n}th
-triangular number is the sum of the integers from 1 to @expr{n}, and
-can be computed directly by the formula
address@hidden @math{n (n+1) \over 2}.
address@hidden @expr{n * (n+1) / 2}.
-
address@hidden
address@hidden
-2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
-1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
- . .
-
- v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Adding this list to the above list of lists produces the desired
-result:
-
address@hidden
address@hidden
-1: [ [0],
- [1, 2],
- [3, 4, 5],
- [6, 7, 8, 9],
- [10, 11, 12, 13, 14],
- [15, 16, 17, 18, 19, 20] ]
- .
-
- V M +
address@hidden group
address@hidden smallexample
-
-If we did not know the formula for triangular numbers, we could have
-computed them using a @kbd{V U +} command. We could also have
-gotten them the hard way by mapping a reduction across the original
-triangular list.
-
address@hidden
address@hidden
-2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
-1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
- . .
-
- @key{RET} V M V R +
address@hidden group
address@hidden smallexample
-
address@hidden
-(This means ``map a @kbd{V R +} command across the vector,'' and
-since each element of the main vector is itself a small vector,
address@hidden R +} computes the sum of its elements.)
-
address@hidden List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
address@hidden List Tutorial Exercise 8
-
address@hidden
-The first step is to build a list of values of @expr{x}.
-
address@hidden
address@hidden
-1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
- . . .
-
- v x 21 @key{RET} 1 - 4 / s 1
address@hidden group
address@hidden smallexample
-
-Next, we compute the Bessel function values.
-
address@hidden
address@hidden
-1: [0., 0.124, 0.242, ..., -0.328]
- .
-
- V M ' besJ(1,$) @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-(Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
-
-A way to isolate the maximum value is to compute the maximum using
address@hidden R X}, then compare all the Bessel values with that maximum.
-
address@hidden
address@hidden
-2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
-1: 0.5801562 . 1: 1
- . .
-
- @key{RET} V R X V M a = @key{RET} V R
+ @key{DEL}
address@hidden group
address@hidden smallexample
-
address@hidden
-It's a good idea to verify, as in the last step above, that only
-one value is equal to the maximum. (After all, a plot of
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
-might have many points all equal to the maximum value, 1.)
-
-The vector we have now has a single 1 in the position that indicates
-the maximum value of @expr{x}. Now it is a simple matter to convert
-this back into the corresponding value itself.
-
address@hidden
address@hidden
-2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
-1: [0, 0.25, 0.5, ... ] . .
- .
-
- r 1 V M * V R +
address@hidden group
address@hidden smallexample
-
-If @kbd{a =} had produced more than one @expr{1} value, this method
-would have given the sum of all maximum @expr{x} values; not very
-useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
-instead. This command deletes all elements of a ``data'' vector that
-correspond to zeros in a ``mask'' vector, leaving us with, in this
-example, a vector of maximum @expr{x} values.
-
-The built-in @kbd{a X} command maximizes a function using more
-efficient methods. Just for illustration, let's use @kbd{a X}
-to maximize @samp{besJ(1,x)} over this same interval.
-
address@hidden
address@hidden
-2: besJ(1, x) 1: [1.84115, 0.581865]
-1: [0 .. 5] .
- .
-
-' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-The output from @kbd{a X} is a vector containing the value of @expr{x}
-that maximizes the function, and the function's value at that maximum.
-As you can see, our simple search got quite close to the right answer.
-
address@hidden List Answer 9, List Answer 10, List Answer 8, Answers to
Exercises
address@hidden List Tutorial Exercise 9
-
address@hidden
-Step one is to convert our integer into vector notation.
-
address@hidden
address@hidden
-1: 25129925999 3: 25129925999
- . 2: 10
- 1: [11, 10, 9, ..., 1, 0]
- .
-
- 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET}
-
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
-2: [100000000000, ... ] .
- .
-
- V M ^ s 1 V M \
address@hidden group
address@hidden smallexample
-
address@hidden
-(Recall, the @kbd{\} command computes an integer quotient.)
-
address@hidden
address@hidden
-1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
- .
-
- 10 V M % s 2
address@hidden group
address@hidden smallexample
-
-Next we must increment this number. This involves adding one to
-the last digit, plus handling carries. There is a carry to the
-left out of a digit if that digit is a nine and all the digits to
-the right of it are nines.
-
address@hidden
address@hidden
-1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
- . .
-
- 9 V M a = v v
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
- . .
-
- V U * v v 1 |
address@hidden group
address@hidden smallexample
-
address@hidden
-Accumulating @kbd{*} across a vector of ones and zeros will preserve
-only the initial run of ones. These are the carries into all digits
-except the rightmost digit. Concatenating a one on the right takes
-care of aligning the carries properly, and also adding one to the
-rightmost digit.
-
address@hidden
address@hidden
-2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
-1: [0, 0, 2, 5, ... ] .
- .
-
- 0 r 2 | V M + 10 V M %
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we have concatenated 0 to the @emph{left} of the original number;
-this takes care of shifting the carries by one with respect to the
-digits that generated them.
-
-Finally, we must convert this list back into an integer.
-
address@hidden
address@hidden
-3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
-2: 1000000000000 1: [1000000000000, 100000000000, ... ]
-1: [100000000000, ... ] .
- .
-
- 10 @key{RET} 12 ^ r 1 |
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
- . .
-
- V M * V R +
address@hidden group
address@hidden smallexample
-
address@hidden
-Another way to do this final step would be to reduce the formula
address@hidden@samp{10 $$ + $}} across the vector of digits.
-
address@hidden
address@hidden
-1: [0, 0, 2, 5, ... ] 1: 25129926000
- . .
-
- V R ' 10 $$ + $ @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden List Answer 10, List Answer 11, List Answer 9, Answers to
Exercises
address@hidden List Tutorial Exercise 10
-
address@hidden
-For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
-which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
-then compared with @expr{c} to produce another 1 or 0, which is then
-compared with @expr{d}. This is not at all what Joe wanted.
-
-Here's a more correct method:
-
address@hidden
address@hidden
-1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
- . 1: 7
- .
-
- ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [1, 1, 1, 0, 1] 1: 0
- . .
-
- V M a = V R *
address@hidden group
address@hidden smallexample
-
address@hidden List Answer 11, List Answer 12, List Answer 10, Answers to
Exercises
address@hidden List Tutorial Exercise 11
-
address@hidden
-The circle of unit radius consists of those points @expr{(x,y)} for which
address@hidden + y^2 < 1}. We start by generating a vector of @expr{x^2}
-and a vector of @expr{y^2}.
-
-We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
-commands.
-
address@hidden
address@hidden
-2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
-1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
- . .
-
- v . t . 2. v b 100 @key{RET} @key{RET} V M k r
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
-1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
- . .
-
- 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
address@hidden group
address@hidden smallexample
-
-Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
-get a vector of 1/0 truth values, then sum the truth values.
-
address@hidden
address@hidden
-1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
- . . .
-
- + 1 V M a < V R +
address@hidden group
address@hidden smallexample
-
address@hidden
-The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
-
address@hidden
address@hidden
-1: 0.84 1: 3.36 2: 3.36 1: 1.0695
- . . 1: 3.14159 .
-
- 100 / 4 * P /
address@hidden group
address@hidden smallexample
-
address@hidden
-Our estimate, 3.36, is off by about 7%. We could get a better estimate
-by taking more points (say, 1000), but it's clear that this method is
-not very efficient!
-
-(Naturally, since this example uses random numbers your own answer
-will be slightly different from the one shown here!)
-
-If you typed @kbd{v .} and @kbd{t .} before, type them again to
-return to full-sized display of vectors.
-
address@hidden List Answer 12, List Answer 13, List Answer 11, Answers to
Exercises
address@hidden List Tutorial Exercise 12
-
address@hidden
-This problem can be made a lot easier by taking advantage of some
-symmetries. First of all, after some thought it's clear that the
address@hidden axis can be ignored altogether. Just pick a random @expr{x}
-component for one end of the match, pick a random direction
address@hidden @math{\theta},
address@hidden @expr{theta},
-and see if @expr{x} and
address@hidden @math{x + \cos \theta}
address@hidden @expr{x + cos(theta)}
-(which is the @expr{x} coordinate of the other endpoint) cross a line.
-The lines are at integer coordinates, so this happens when the two
-numbers surround an integer.
-
-Since the two endpoints are equivalent, we may as well choose the leftmost
-of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
-to the right, in the range -90 to 90 degrees. (We could use radians, but
-it would feel like cheating to refer to @cpiover{2} radians while trying
-to estimate @cpi{}!)
-
-In fact, since the field of lines is infinite we can choose the
-coordinates 0 and 1 for the lines on either side of the leftmost
-endpoint. The rightmost endpoint will be between 0 and 1 if the
-match does not cross a line, or between 1 and 2 if it does. So:
-Pick random @expr{x} and
address@hidden @math{\theta},
address@hidden @expr{theta},
-compute
address@hidden @math{x + \cos \theta},
address@hidden @expr{x + cos(theta)},
-and count how many of the results are greater than one. Simple!
-
-We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
-commands.
-
address@hidden
address@hidden
-1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
- . 1: [78.4, 64.5, ..., -42.9]
- .
-
-v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
address@hidden group
address@hidden smallexample
-
address@hidden
-(The next step may be slow, depending on the speed of your computer.)
-
address@hidden
address@hidden
-2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
-1: [0.20, 0.43, ..., 0.73] .
- .
-
- m d V M C +
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [0, 1, ..., 1] 1: 0.64 1: 3.125
- . . .
-
- 1 V M a > V R + 100 / 2 @key{TAB} /
address@hidden group
address@hidden smallexample
-
-Let's try the third method, too. We'll use random integers up to
-one million. The @kbd{k r} command with an integer argument picks
-a random integer.
-
address@hidden
address@hidden
-2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
-1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
- . .
-
- 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M
k r
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
- . . .
-
- V M k g 1 V M a = V R + 100 /
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: 10.714 1: 3.273
- . .
-
- 6 @key{TAB} / Q
address@hidden group
address@hidden smallexample
-
-For a proof of this property of the GCD function, see section 4.5.2,
-exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
-
-If you typed @kbd{v .} and @kbd{t .} before, type them again to
-return to full-sized display of vectors.
-
address@hidden List Answer 13, List Answer 14, List Answer 12, Answers to
Exercises
address@hidden List Tutorial Exercise 13
-
address@hidden
-First, we put the string on the stack as a vector of ASCII codes.
-
address@hidden
address@hidden
-1: [84, 101, 115, ..., 51]
- .
-
- "Testing, 1, 2, 3 @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
-there was no need to type an apostrophe. Also, Calc didn't mind that
-we omitted the closing @kbd{"}. (The same goes for all closing delimiters
-like @kbd{)} and @kbd{]} at the end of a formula.
-
-We'll show two different approaches here. In the first, we note that
-if the input vector is @expr{[a, b, c, d]}, then the hash code is
address@hidden (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
-it's a sum of descending powers of three times the ASCII codes.
-
address@hidden
address@hidden
-2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
-1: 16 1: [15, 14, 13, ..., 0]
- . .
-
- @key{RET} v l v x 16 @key{RET} -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
-1: [14348907, ..., 1] . .
- .
-
- 3 @key{TAB} V M ^ * 511 %
address@hidden group
address@hidden smallexample
-
address@hidden
-Once again, @kbd{*} elegantly summarizes most of the computation.
-But there's an even more elegant approach: Reduce the formula
address@hidden $$ + $} across the vector. Recall that this represents a
-function of two arguments that computes its first argument times three
-plus its second argument.
-
address@hidden
address@hidden
-1: [84, 101, 115, ..., 51] 1: 1960915098
- . .
-
- "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-If you did the decimal arithmetic exercise, this will be familiar.
-Basically, we're turning a base-3 vector of digits into an integer,
-except that our ``digits'' are much larger than real digits.
-
-Instead of typing @kbd{511 %} again to reduce the result, we can be
-cleverer still and notice that rather than computing a huge integer
-and taking the modulo at the end, we can take the modulo at each step
-without affecting the result. While this means there are more
-arithmetic operations, the numbers we operate on remain small so
-the operations are faster.
-
address@hidden
address@hidden
-1: [84, 101, 115, ..., 51] 1: 121
- . .
-
- "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
address@hidden group
address@hidden smallexample
-
-Why does this work? Think about a two-step computation:
address@hidden@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically
means
-subtracting off enough 511's to put the result in the desired range.
-So the result when we take the modulo after every step is,
-
address@hidden
address@hidden
-3 (3 a + b - 511 m) + c - 511 n
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ 3 (3 a + b - 511 m) + c - 511 n $$
-\afterdisplay
address@hidden tex
-
address@hidden
-for some suitable integers @expr{m} and @expr{n}. Expanding out by
-the distributive law yields
-
address@hidden
address@hidden
-9 a + 3 b + c - 511*3 m - 511 n
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ 9 a + 3 b + c - 511\times3 m - 511 n $$
-\afterdisplay
address@hidden tex
-
address@hidden
-The @expr{m} term in the latter formula is redundant because any
-contribution it makes could just as easily be made by the @expr{n}
-term. So we can take it out to get an equivalent formula with
address@hidden' = 3m + n},
-
address@hidden
address@hidden
-9 a + 3 b + c - 511 n'
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ 9 a + 3 b + c - 511 n' $$
-\afterdisplay
address@hidden tex
-
address@hidden
-which is just the formula for taking the modulo only at the end of
-the calculation. Therefore the two methods are essentially the same.
-
-Later in the tutorial we will encounter @dfn{modulo forms}, which
-basically automate the idea of reducing every intermediate result
-modulo some value @var{m}.
-
address@hidden List Answer 14, Types Answer 1, List Answer 13, Answers to
Exercises
address@hidden List Tutorial Exercise 14
-
-We want to use @kbd{H V U} to nest a function which adds a random
-step to an @expr{(x,y)} coordinate. The function is a bit long, but
-otherwise the problem is quite straightforward.
-
address@hidden
address@hidden
-2: [0, 0] 1: [ [ 0, 0 ]
-1: 50 [ 0.4288, -0.1695 ]
- . [ -0.4787, -0.9027 ]
- ...
-
- [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
address@hidden group
address@hidden smallexample
-
-Just as the text recommended, we used @samp{< >} nameless function
-notation to keep the two @code{random} calls from being evaluated
-before nesting even begins.
-
-We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
-rules acts like a matrix. We can transpose this matrix and unpack
-to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
-
address@hidden
address@hidden
-2: [ 0, 0.4288, -0.4787, ... ]
-1: [ 0, -0.1696, -0.9027, ... ]
- .
-
- v t v u g f
address@hidden group
address@hidden smallexample
-
-Incidentally, because the @expr{x} and @expr{y} are completely
-independent in this case, we could have done two separate commands
-to create our @expr{x} and @expr{y} vectors of numbers directly.
-
-To make a random walk of unit steps, we note that @code{sincos} of
-a random direction exactly gives us an @expr{[x, y]} step of unit
-length; in fact, the new nesting function is even briefer, though
-we might want to lower the precision a bit for it.
-
address@hidden
address@hidden
-2: [0, 0] 1: [ [ 0, 0 ]
-1: 50 [ 0.1318, 0.9912 ]
- . [ -0.5965, 0.3061 ]
- ...
-
- [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))>
@key{RET}
address@hidden group
address@hidden smallexample
-
-Another @kbd{v t v u g f} sequence will graph this new random walk.
-
-An interesting twist on these random walk functions would be to use
-complex numbers instead of 2-vectors to represent points on the plane.
-In the first example, we'd use something like @samp{random + random*(0,1)},
-and in the second we could use polar complex numbers with random phase
-angles. (This exercise was first suggested in this form by Randal
-Schwartz.)
-
address@hidden Types Answer 1, Types Answer 2, List Answer 14, Answers to
Exercises
address@hidden Types Tutorial Exercise 1
-
address@hidden
-If the number is the square root of @cpi{} times a rational number,
-then its square, divided by @cpi{}, should be a rational number.
-
address@hidden
address@hidden
-1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
- . . .
-
- 2 ^ P / c F
address@hidden group
address@hidden smallexample
-
address@hidden
-Technically speaking this is a rational number, but not one that is
-likely to have arisen in the original problem. More likely, it just
-happens to be the fraction which most closely represents some
-irrational number to within 12 digits.
-
-But perhaps our result was not quite exact. Let's reduce the
-precision slightly and try again:
-
address@hidden
address@hidden
-1: 0.509433962268 1: 27:53
- . .
-
- U p 10 @key{RET} c F
address@hidden group
address@hidden smallexample
-
address@hidden
-Aha! It's unlikely that an irrational number would equal a fraction
-this simple to within ten digits, so our original number was probably
address@hidden @math{\sqrt{27 \pi / 53}}.
address@hidden @expr{sqrt(27 pi / 53)}.
-
-Notice that we didn't need to re-round the number when we reduced the
-precision. Remember, arithmetic operations always round their inputs
-to the current precision before they begin.
-
address@hidden Types Answer 2, Types Answer 3, Types Answer 1, Answers to
Exercises
address@hidden Types Tutorial Exercise 2
-
address@hidden
address@hidden / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
-But if @address@hidden inf = inf}}, then @samp{17 inf / inf = inf / inf = 17},
too.
-
address@hidden(inf) = inf}. It's tempting to say that the exponential
-of infinity must be ``bigger'' than ``regular'' infinity, but as
-far as Calc is concerned all infinities are as just as big.
-In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
-to infinity, but the fact the @expr{e^x} grows much faster than
address@hidden is not relevant here.
-
address@hidden(-inf) = 0}. Here we have a finite answer even though
-the input is infinite.
-
address@hidden(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
-represents the imaginary number @expr{i}. Here's a derivation:
address@hidden(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
-The first part is, by definition, @expr{i}; the second is @code{inf}
-because, once again, all infinities are the same size.
-
address@hidden(uinf) = uinf}. In fact, we do know something about the
-direction because @code{sqrt} is defined to return a value in the
-right half of the complex plane. But Calc has no notation for this,
-so it settles for the conservative answer @code{uinf}.
-
address@hidden(uinf) = inf}. No matter which direction @expr{x} points,
address@hidden(x)} always points along the positive real axis.
-
address@hidden(0) = -inf}. Here we have an infinite answer to a finite
-input. As in the @expr{1 / 0} case, Calc will only use infinities
-here if you have turned on Infinite mode. Otherwise, it will
-treat @samp{ln(0)} as an error.
-
address@hidden Types Answer 3, Types Answer 4, Types Answer 2, Answers to
Exercises
address@hidden Types Tutorial Exercise 3
-
address@hidden
-We can make @samp{inf - inf} be any real number we like, say,
address@hidden, just by claiming that we added @expr{a} to the first
-infinity but not to the second. This is just as true for complex
-values of @expr{a}, so @code{nan} can stand for a complex number.
-(And, similarly, @code{uinf} can stand for an infinity that points
-in any direction in the complex plane, such as @samp{(0, 1) inf}).
-
-In fact, we can multiply the first @code{inf} by two. Surely
address@hidden@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf
= nan}.
-So @code{nan} can even stand for infinity. Obviously it's just
-as easy to make it stand for minus infinity as for plus infinity.
-
-The moral of this story is that ``infinity'' is a slippery fish
-indeed, and Calc tries to handle it by having a very simple model
-for infinities (only the direction counts, not the ``size''); but
-Calc is careful to write @code{nan} any time this simple model is
-unable to tell what the true answer is.
-
address@hidden Types Answer 4, Types Answer 5, Types Answer 3, Answers to
Exercises
address@hidden Types Tutorial Exercise 4
-
address@hidden
address@hidden
-2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
-1: 17 .
- .
-
- 0@@ 47' 26" @key{RET} 17 /
address@hidden group
address@hidden smallexample
-
address@hidden
-The average song length is two minutes and 47.4 seconds.
-
address@hidden
address@hidden
-2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
-1: 0@@ 0' 20" . .
- .
-
- 20" + 17 *
address@hidden group
address@hidden smallexample
-
address@hidden
-The album would be 53 minutes and 6 seconds long.
-
address@hidden Types Answer 5, Types Answer 6, Types Answer 4, Answers to
Exercises
address@hidden Types Tutorial Exercise 5
-
address@hidden
-Let's suppose it's January 14, 1991. The easiest thing to do is
-to keep trying 13ths of months until Calc reports a Friday.
-We can do this by manually entering dates, or by using @kbd{t I}:
-
address@hidden
address@hidden
-1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
- . . .
-
- ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
address@hidden group
address@hidden smallexample
-
address@hidden
-(Calc assumes the current year if you don't say otherwise.)
-
-This is getting tedious---we can keep advancing the date by typing
address@hidden I} over and over again, but let's automate the job by using
-vector mapping. The @kbd{t I} command actually takes a second
-``how-many-months'' argument, which defaults to one. This
-argument is exactly what we want to map over:
-
address@hidden
address@hidden
-2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
-1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
- . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
- .
-
- v x 6 @key{RET} V M t I
address@hidden group
address@hidden smallexample
-
address@hidden
-Et address@hidden, September 13, 1991 is a Friday.
-
address@hidden
address@hidden
-1: 242
- .
-
-' <sep 13> - <jan 14> @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-And the answer to our original question: 242 days to go.
-
address@hidden Types Answer 6, Types Answer 7, Types Answer 5, Answers to
Exercises
address@hidden Types Tutorial Exercise 6
-
address@hidden
-The full rule for leap years is that they occur in every year divisible
-by four, except that they don't occur in years divisible by 100, except
-that they @emph{do} in years divisible by 400. We could work out the
-answer by carefully counting the years divisible by four and the
-exceptions, but there is a much simpler way that works even if we
-don't know the leap year rule.
-
-Let's assume the present year is 1991. Years have 365 days, except
-that leap years (whenever they occur) have 366 days. So let's count
-the number of days between now and then, and compare that to the
-number of years times 365. The number of extra days we find must be
-equal to the number of leap years there were.
-
address@hidden
address@hidden
-1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
- . 1: <Tue Jan 1, 1991> .
- .
-
- ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-3: 2925593 2: 2925593 2: 2925593 1: 1943
-2: 10001 1: 8010 1: 2923650 .
-1: 1991 . .
- .
-
- 10001 @key{RET} 1991 - 365 * -
address@hidden group
address@hidden smallexample
-
address@hidden [fix-ref Date Forms]
address@hidden
-There will be 1943 leap years before the year 10001. (Assuming,
-of course, that the algorithm for computing leap years remains
-unchanged for that long. @xref{Date Forms}, for some interesting
-background information in that regard.)
-
address@hidden Types Answer 7, Types Answer 8, Types Answer 6, Answers to
Exercises
address@hidden Types Tutorial Exercise 7
-
address@hidden
-The relative errors must be converted to absolute errors so that
address@hidden/-} notation may be used.
-
address@hidden
address@hidden
-1: 1. 2: 1.
- . 1: 0.2
- .
-
- 20 @key{RET} .05 * 4 @key{RET} .05 *
address@hidden group
address@hidden smallexample
-
-Now we simply chug through the formula.
-
address@hidden
address@hidden
-1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
- . . .
-
- 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
address@hidden group
address@hidden smallexample
-
-It turns out the @kbd{v u} command will unpack an error form as
-well as a vector. This saves us some retyping of numbers.
-
address@hidden
address@hidden
-3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
-2: 6316.5 1: 0.1118
-1: 706.21 .
- .
-
- @key{RET} v u @key{TAB} /
address@hidden group
address@hidden smallexample
-
address@hidden
-Thus the volume is 6316 cubic centimeters, within about 11 percent.
-
address@hidden Types Answer 8, Types Answer 9, Types Answer 7, Answers to
Exercises
address@hidden Types Tutorial Exercise 8
-
address@hidden
-The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
-Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
-close to zero, its reciprocal can get arbitrarily large, so the answer
-is an interval that effectively means, ``any number greater than 0.1''
-but with no upper bound.
-
-The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
-
-Calc normally treats division by zero as an error, so that the formula
address@hidden@samp{1 / 0}} is left unsimplified. Our third problem,
address@hidden@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because
zero
-is now a member of the interval. So Calc leaves this one unevaluated, too.
-
-If you turn on Infinite mode by pressing @kbd{m i}, you will
-instead get the answer @samp{[0.1 .. inf]}, which includes infinity
-as a possible value.
-
-The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
-Zero is buried inside the interval, but it's still a possible value.
-It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
-will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
-the interval goes from minus infinity to plus infinity, with a ``hole''
-in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
-represent this, so it just reports @samp{[-inf .. inf]} as the answer.
-It may be disappointing to hear ``the answer lies somewhere between
-minus infinity and plus infinity, inclusive,'' but that's the best
-that interval arithmetic can do in this case.
-
address@hidden Types Answer 9, Types Answer 10, Types Answer 8, Answers to
Exercises
address@hidden Types Tutorial Exercise 9
-
address@hidden
address@hidden
-1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
- . 1: [0 .. 9] 1: [-9 .. 9]
- . .
-
- [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
address@hidden group
address@hidden smallexample
-
address@hidden
-In the first case the result says, ``if a number is between @mathit{-3} and
-3, its square is between 0 and 9.'' The second case says, ``the product
-of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
-
-An interval form is not a number; it is a symbol that can stand for
-many different numbers. Two identical-looking interval forms can stand
-for different numbers.
-
-The same issue arises when you try to square an error form.
-
address@hidden Types Answer 10, Types Answer 11, Types Answer 9, Answers to
Exercises
address@hidden Types Tutorial Exercise 10
-
address@hidden
-Testing the first number, we might arbitrarily choose 17 for @expr{x}.
-
address@hidden
address@hidden
-1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
- . 811749612 .
- .
-
- 17 M 811749613 @key{RET} 811749612 ^
address@hidden group
address@hidden smallexample
-
address@hidden
-Since 533694123 is (considerably) different from 1, the number 811749613
-must not be prime.
-
-It's awkward to type the number in twice as we did above. There are
-various ways to avoid this, and algebraic entry is one. In fact, using
-a vector mapping operation we can perform several tests at once. Let's
-use this method to test the second number.
-
address@hidden
address@hidden
-2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
-1: 15485863 .
- .
-
- [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-The result is three ones (modulo @expr{n}), so it's very probable that
-15485863 is prime. (In fact, this number is the millionth prime.)
-
-Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
-would have been hopelessly inefficient, since they would have calculated
-the power using full integer arithmetic.
-
-Calc has a @kbd{k p} command that does primality testing. For small
-numbers it does an exact test; for large numbers it uses a variant
-of the Fermat test we used here. You can use @kbd{k p} repeatedly
-to prove that a large integer is prime with any desired probability.
-
address@hidden Types Answer 11, Types Answer 12, Types Answer 10, Answers to
Exercises
address@hidden Types Tutorial Exercise 11
-
address@hidden
-There are several ways to insert a calculated number into an HMS form.
-One way to convert a number of seconds to an HMS form is simply to
-multiply the number by an HMS form representing one second:
-
address@hidden
address@hidden
-1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
- . 1: 0@@ 0' 1" .
- .
-
- P 1e7 * 0@@ 0' 1" *
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
-1: 15@@ 27' 16" mod 24@@ 0' 0" .
- .
-
- x time @key{RET} +
address@hidden group
address@hidden smallexample
-
address@hidden
-It will be just after six in the morning.
-
-The algebraic @code{hms} function can also be used to build an
-HMS form:
-
address@hidden
address@hidden
-1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
- . .
-
- ' hms(0, 0, 1e7 pi) @key{RET} =
address@hidden group
address@hidden smallexample
-
address@hidden
-The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
-the actual number 3.14159...
-
address@hidden Types Answer 12, Types Answer 13, Types Answer 11, Answers to
Exercises
address@hidden Types Tutorial Exercise 12
-
address@hidden
-As we recall, there are 17 songs of about 2 minutes and 47 seconds
-each.
-
address@hidden
address@hidden
-2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
-1: [0@@ 0' 20" .. 0@@ 1' 0"] .
- .
-
- [ 0@@ 20" .. 0@@ 1' ] +
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [0@@ 52' 59." .. 1@@ 4' 19."]
- .
-
- 17 *
address@hidden group
address@hidden smallexample
-
address@hidden
-No matter how long it is, the album will fit nicely on one CD.
-
address@hidden Types Answer 13, Types Answer 14, Types Answer 12, Answers to
Exercises
address@hidden Types Tutorial Exercise 13
-
address@hidden
-Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
-
address@hidden Types Answer 14, Types Answer 15, Types Answer 13, Answers to
Exercises
address@hidden Types Tutorial Exercise 14
-
address@hidden
-How long will it take for a signal to get from one end of the computer
-to the other?
-
address@hidden
address@hidden
-1: m / c 1: 3.3356 ns
- . .
-
- ' 1 m / c @key{RET} u c ns @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-(Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
-
address@hidden
address@hidden
-1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356
-2: 4.1 ns . .
- .
-
- ' 4.1 ns @key{RET} / u s
address@hidden group
address@hidden smallexample
-
address@hidden
-Thus a signal could take up to 81 percent of a clock cycle just to
-go from one place to another inside the computer, assuming the signal
-could actually attain the full speed of light. Pretty tight!
-
address@hidden Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to
Exercises
address@hidden Types Tutorial Exercise 15
-
address@hidden
-The speed limit is 55 miles per hour on most highways. We want to
-find the ratio of Sam's speed to the US speed limit.
-
address@hidden
address@hidden
-1: 55 mph 2: 55 mph 3: 11 hr mph / yd
- . 1: 5 yd / hr .
- .
-
- ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
address@hidden group
address@hidden smallexample
-
-The @kbd{u s} command cancels out these units to get a plain
-number. Now we take the logarithm base two to find the final
-answer, assuming that each successive pill doubles his speed.
-
address@hidden
address@hidden
-1: 19360. 2: 19360. 1: 14.24
- . 1: 2 .
- .
-
- u s 2 B
address@hidden group
address@hidden smallexample
-
address@hidden
-Thus Sam can take up to 14 pills without a worry.
-
address@hidden Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 1
-
address@hidden
address@hidden [fix-ref Declarations]
-The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
-Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
-if @address@hidden = -4}}.) If @expr{x} is real, this formula could be
-simplified to @samp{abs(x)}, but for general complex arguments even
-that is not safe. (@xref{Declarations}, for a way to tell Calc
-that @expr{x} is known to be real.)
-
address@hidden Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 2
-
address@hidden
-Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
-is zero when @expr{x} is any of these values. The trivial polynomial
address@hidden is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
-will do the job. We can use @kbd{a c x} to write this in a more
-familiar form.
-
address@hidden
address@hidden
-1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
- . .
-
- r 2 a P x @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x
- . .
-
- V M ' x-$ @key{RET} V R *
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: x^3 - 1.41666 x 1: 34 x - 24 x^3
- . .
-
- a c x @key{RET} 24 n * a x
address@hidden group
address@hidden smallexample
-
address@hidden
-Sure enough, our answer (multiplied by a suitable constant) is the
-same as the original polynomial.
-
address@hidden Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to
Exercises
address@hidden Algebra Tutorial Exercise 3
-
address@hidden
address@hidden
-1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2
- . .
-
- ' x sin(pi x) @key{RET} m r a i x @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [y, 1]
-2: (sin(pi x) - pi x cos(pi x)) / pi^2
- .
-
- ' [y,1] @key{RET} @key{TAB}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
- .
-
- V M $ @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
- .
-
- V R -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
- .
-
- =
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
- .
-
- v x 5 @key{RET} @key{TAB} V M $ @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers
to Exercises
address@hidden Algebra Tutorial Exercise 4
-
address@hidden
-The hard part is that @kbd{V R +} is no longer sufficient to add up all
-the contributions from the slices, since the slices have varying
-coefficients. So first we must come up with a vector of these
-coefficients. Here's one way:
-
address@hidden
address@hidden
-2: -1 2: 3 1: [4, 2, ..., 4]
-1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
- . .
-
- 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
- . .
-
- 1 | 1 @key{TAB} |
address@hidden group
address@hidden smallexample
-
address@hidden
-Now we compute the function values. Note that for this method we need
-eleven values, including both endpoints of the desired interval.
-
address@hidden
address@hidden
-2: [1, 4, 2, ..., 4, 1]
-1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
- .
-
- 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: [1, 4, 2, ..., 4, 1]
-1: [0., 0.084941, 0.16993, ... ]
- .
-
- ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
address@hidden group
address@hidden smallexample
-
address@hidden
-Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
-same thing.
-
address@hidden
address@hidden
-1: 11.22 1: 1.122 1: 0.374
- . . .
-
- * .1 * 3 /
address@hidden group
address@hidden smallexample
-
address@hidden
-Wow! That's even better than the result from the Taylor series method.
-
address@hidden Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 1
-
address@hidden
-We'll use Big mode to make the formulas more readable.
-
address@hidden
address@hidden
- ___
- 2 + V 2
-1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------
- . ___
- 1 + V 2
-
- .
-
- ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
address@hidden group
address@hidden smallexample
-
address@hidden
-Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
-
address@hidden
address@hidden
- ___ ___
-1: (2 + V 2 ) (V 2 - 1)
- .
-
- a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
- ___ ___
-1: 2 + V 2 - 2 1: V 2
- . .
-
- a r a*(b+c) := a*b + a*c a s
address@hidden group
address@hidden smallexample
-
address@hidden
-(We could have used @kbd{a x} instead of a rewrite rule for the
-second step.)
-
-The multiply-by-conjugate rule turns out to be useful in many
-different circumstances, such as when the denominator involves
-sines and cosines or the imaginary constant @code{i}.
-
address@hidden Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 2
-
address@hidden
-Here is the rule set:
-
address@hidden
address@hidden
-[ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
- fib(1, x, y) := x,
- fib(n, x, y) := fib(n-1, y, x+y) ]
address@hidden group
address@hidden smallexample
-
address@hidden
-The first rule turns a one-argument @code{fib} that people like to write
-into a three-argument @code{fib} that makes computation easier. The
-second rule converts back from three-argument form once the computation
-is done. The third rule does the computation itself. It basically
-says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
-then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
-numbers.
-
-Notice that because the number @expr{n} was ``validated'' by the
-conditions on the first rule, there is no need to put conditions on
-the other rules because the rule set would never get that far unless
-the input were valid. That further speeds computation, since no
-extra conditions need to be checked at every step.
-
-Actually, a user with a nasty sense of humor could enter a bad
-three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
-which would get the rules into an infinite loop. One thing that would
-help keep this from happening by accident would be to use something like
address@hidden instead of @code{fib} as the name of the three-argument
-function.
-
address@hidden Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 3
-
address@hidden
-He got an infinite loop. First, Calc did as expected and rewrote
address@hidden@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
-apply the rule again, and found that @samp{f(2, 3, x)} looks like
address@hidden + b x} with @address@hidden = 0}} and @samp{b = 1}, so it
rewrote to
address@hidden(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
-around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
-to make sure the rule applied only once.
-
-(Actually, even the first step didn't work as he expected. What Calc
-really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
-treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
-to it. While this may seem odd, it's just as valid a solution as the
-``obvious'' one. One way to fix this would be to add the condition
address@hidden:: variable(x)} to the rule, to make sure the thing that matches
address@hidden is indeed a variable, or to change @samp{x} to @samp{quote(x)}
-on the lefthand side, so that the rule matches the actual variable
address@hidden rather than letting @samp{x} stand for something else.)
-
address@hidden Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 4
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden seq
-Here is a suitable set of rules to solve the first part of the problem:
-
address@hidden
address@hidden
-[ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
- seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
address@hidden group
address@hidden smallexample
-
-Given the initial formula @samp{seq(6, 0)}, application of these
-rules produces the following sequence of formulas:
-
address@hidden
-seq( 3, 1)
-seq(10, 2)
-seq( 5, 3)
-seq(16, 4)
-seq( 8, 5)
-seq( 4, 6)
-seq( 2, 7)
-seq( 1, 8)
address@hidden example
-
address@hidden
-whereupon neither of the rules match, and rewriting stops.
-
-We can pretty this up a bit with a couple more rules:
-
address@hidden
address@hidden
-[ seq(n) := seq(n, 0),
- seq(1, c) := c,
- ... ]
address@hidden group
address@hidden smallexample
-
address@hidden
-Now, given @samp{seq(6)} as the starting configuration, we get 8
-as the result.
-
-The change to return a vector is quite simple:
-
address@hidden
address@hidden
-[ seq(n) := seq(n, []) :: integer(n) :: n > 0,
- seq(1, v) := v | 1,
- seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
- seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
address@hidden group
address@hidden smallexample
-
address@hidden
-Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
-
-Notice that the @expr{n > 1} guard is no longer necessary on the last
-rule since the @expr{n = 1} case is now detected by another rule.
-But a guard has been added to the initial rule to make sure the
-initial value is suitable before the computation begins.
-
-While still a good idea, this guard is not as vitally important as it
-was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
-will not get into an infinite loop. Calc will not be able to prove
-the symbol @samp{x} is either even or odd, so none of the rules will
-apply and the rewrites will stop right away.
-
address@hidden Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers
to Exercises
address@hidden Rewrites Tutorial Exercise 5
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden nterms
-If @expr{x} is the sum @expr{a + b}, then
address@hidden(address@hidden@tfn{)}' must
-be address@hidden(address@hidden@tfn{)}' plus
address@hidden(address@hidden@tfn{)}'. If @expr{x}
-is not a sum, then address@hidden(address@hidden@tfn{)}' = 1.
-
address@hidden
address@hidden
-[ nterms(a + b) := nterms(a) + nterms(b),
- nterms(x) := 1 ]
address@hidden group
address@hidden smallexample
-
address@hidden
-Here we have taken advantage of the fact that earlier rules always
-match before later rules; @samp{nterms(x)} will only be tried if we
-already know that @samp{x} is not a sum.
-
address@hidden Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5,
Answers to Exercises
address@hidden Rewrites Tutorial Exercise 6
-
address@hidden
-Here is a rule set that will do the job:
-
address@hidden
address@hidden
-[ a*(b + c) := a*b + a*c,
- opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
- :: constant(a) :: constant(b),
- opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
- :: constant(a) :: constant(b),
- a O(x^n) := O(x^n) :: constant(a),
- x^opt(m) O(x^n) := O(x^(n+m)),
- O(x^n) O(x^m) := O(x^(n+m)) ]
address@hidden group
address@hidden smallexample
-
-If we really want the @kbd{+} and @kbd{*} keys to operate naturally
-on power series, we should put these rules in @code{EvalRules}. For
-testing purposes, it is better to put them in a different variable,
-say, @code{O}, first.
-
-The first rule just expands products of sums so that the rest of the
-rules can assume they have an expanded-out polynomial to work with.
-Note that this rule does not mention @samp{O} at all, so it will
-apply to any product-of-sum it encounters---this rule may surprise
-you if you put it into @code{EvalRules}!
-
-In the second rule, the sum of two O's is changed to the smaller O.
-The optional constant coefficients are there mostly so that
address@hidden(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
-as well as @samp{O(x^2) + O(x^3)}.
-
-The third rule absorbs higher powers of @samp{x} into O's.
-
-The fourth rule says that a constant times a negligible quantity
-is still negligible. (This rule will also match @samp{O(x^3) / 4},
-with @samp{a = 1/4}.)
-
-The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
-(It is easy to see that if one of these forms is negligible, the other
-is, too.) Notice the @samp{x^opt(m)} to pick up terms like
address@hidden@samp{x O(x^3)}}. Optional powers will match @samp{x} as
@samp{x^1}
-but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
-
-The sixth rule is the corresponding rule for products of two O's.
-
-Another way to solve this problem would be to create a new ``data type''
-that represents truncated power series. We might represent these as
-function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
-a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
-on. Rules would exist for sums and products of such @code{series}
-objects, and as an optional convenience could also know how to combine a
address@hidden object with a normal polynomial. (With this, and with a
-rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
-you could still enter power series in exactly the same notation as
-before.) Operations on such objects would probably be more efficient,
-although the objects would be a bit harder to read.
-
address@hidden [fix-ref Compositions]
-Some other symbolic math programs provide a power series data type
-similar to this. Mathematica, for example, has an object that looks
-like @address@hidden, @var{x0}, @var{coefs}, @var{nmin},
address@hidden, @var{den}]}, where @var{x0} is the point about which the
-power series is taken (we've been assuming this was always zero),
-and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
-with fractional or negative powers. Also, the @code{PowerSeries}
-objects have a special display format that makes them look like
address@hidden x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
-for a way to do this in Calc, although for something as involved as
-this it would probably be better to write the formatting routine
-in Lisp.)
-
address@hidden Programming Answer 1, Programming Answer 2, Rewrites Answer 6,
Answers to Exercises
address@hidden Programming Tutorial Exercise 1
-
address@hidden
-Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
address@hidden F}, and answer the questions. Since this formula contains two
-variables, the default argument list will be @samp{(t x)}. We want to
-change this to @samp{(x)} since @expr{t} is really a dummy variable
-to be used within @code{ninteg}.
-
-The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL}
@key{DEL} @key{RET} y}.
-(The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
-
address@hidden Programming Answer 2, Programming Answer 3, Programming Answer
1, Answers to Exercises
address@hidden Programming Tutorial Exercise 2
-
address@hidden
-One way is to move the number to the top of the stack, operate on
-it, then move it back: @kbd{C-x ( address@hidden n address@hidden
address@hidden C-x )}.
-
-Another way is to negate the top three stack entries, then negate
-again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
-
-Finally, it turns out that a negative prefix argument causes a
-command like @kbd{n} to operate on the specified stack entry only,
-which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
-
-Just for kicks, let's also do it algebraically:
address@hidden@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
-
address@hidden Programming Answer 3, Programming Answer 4, Programming Answer
2, Answers to Exercises
address@hidden Programming Tutorial Exercise 3
-
address@hidden
-Each of these functions can be computed using the stack, or using
-algebraic entry, whichever way you prefer:
-
address@hidden
-Computing
address@hidden @math{\displaystyle{\sin x \over x}}:
address@hidden @expr{sin(x) / x}:
-
-Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
-
-Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
-
address@hidden
-Computing the logarithm:
-
-Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
-
-Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
-
address@hidden
-Computing the vector of integers:
-
-Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
address@hidden v x} takes the vector size, starting value, and increment
-from the stack.)
-
-Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
-number from the stack and uses it as the prefix argument for the
-next command.)
-
-Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
-
address@hidden Programming Answer 4, Programming Answer 5, Programming Answer
3, Answers to Exercises
address@hidden Programming Tutorial Exercise 4
-
address@hidden
-Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
-
address@hidden Programming Answer 5, Programming Answer 6, Programming Answer
4, Answers to Exercises
address@hidden Programming Tutorial Exercise 5
-
address@hidden
address@hidden
-2: 1 1: 1.61803398502 2: 1.61803398502
-1: 20 . 1: 1.61803398875
- . .
-
- 1 @key{RET} 20 Z < & 1 + Z > I H P
address@hidden group
address@hidden smallexample
-
address@hidden
-This answer is quite accurate.
-
address@hidden Programming Answer 6, Programming Answer 7, Programming Answer
5, Answers to Exercises
address@hidden Programming Tutorial Exercise 6
-
address@hidden
-Here is the matrix:
-
address@hidden
-[ [ 0, 1 ] * [a, b] = [b, a + b]
- [ 1, 1 ] ]
address@hidden example
-
address@hidden
-Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
-and @expr{n+2}. Here's one program that does the job:
-
address@hidden
-C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
address@hidden example
-
address@hidden
-This program is quite efficient because Calc knows how to raise a
-matrix (or other value) to the power @expr{n} in only
address@hidden @math{\log_2 n}
address@hidden @expr{log(n,2)}
-steps. For example, this program can compute the 1000th Fibonacci
-number (a 209-digit integer!) in about 10 steps; even though the
address@hidden < ... Z >} solution had much simpler steps, it would have
-required so many steps that it would not have been practical.
-
address@hidden Programming Answer 7, Programming Answer 8, Programming Answer
6, Answers to Exercises
address@hidden Programming Tutorial Exercise 7
-
address@hidden
-The trick here is to compute the harmonic numbers differently, so that
-the loop counter itself accumulates the sum of reciprocals. We use
-a separate variable to hold the integer counter.
-
address@hidden
address@hidden
-1: 1 2: 1 1: .
- . 1: 4
- .
-
- 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
address@hidden group
address@hidden smallexample
-
address@hidden
-The body of the loop goes as follows: First save the harmonic sum
-so far in variable 2. Then delete it from the stack; the for loop
-itself will take care of remembering it for us. Next, recall the
-count from variable 1, add one to it, and feed its reciprocal to
-the for loop to use as the step value. The for loop will increase
-the ``loop counter'' by that amount and keep going until the
-loop counter exceeds 4.
-
address@hidden
address@hidden
-2: 31 3: 31
-1: 3.99498713092 2: 3.99498713092
- . 1: 4.02724519544
- .
-
- r 1 r 2 @key{RET} 31 & +
address@hidden group
address@hidden smallexample
-
-Thus we find that the 30th harmonic number is 3.99, and the 31st
-harmonic number is 4.02.
-
address@hidden Programming Answer 8, Programming Answer 9, Programming Answer
7, Answers to Exercises
address@hidden Programming Tutorial Exercise 8
-
address@hidden
-The first step is to compute the derivative @expr{f'(x)} and thus
-the formula
address@hidden @math{\displaystyle{x - {f(x) \over f'(x)}}}.
address@hidden @expr{x - f(x)/f'(x)}.
-
-(Because this definition is long, it will be repeated in concise form
-below. You can use @address@hidden * m}} to load it from there. While you are
-entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
-keystrokes without executing them. In the following diagrams we'll
-pretend Calc actually executed the keystrokes as you typed them,
-just for purposes of illustration.)
-
address@hidden
address@hidden
-2: sin(cos(x)) - 0.5 3: 4.5
-1: 4.5 2: sin(cos(x)) - 0.5
- . 1: -(sin(x) cos(cos(x)))
- .
-
-' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x
@key{RET}
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: 4.5
-1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
- .
-
- / ' x @key{RET} @key{TAB} - t 1
address@hidden group
address@hidden smallexample
-
-Now, we enter the loop. We'll use a repeat loop with a 20-repetition
-limit just in case the method fails to converge for some reason.
-(Normally, the @address@hidden /}} command will stop the loop before all 20
-repetitions are done.)
-
address@hidden
address@hidden
-1: 4.5 3: 4.5 2: 4.5
- . 2: x + (sin(cos(x)) ... 1: 5.24196456928
- 1: 4.5 .
- .
-
- 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
address@hidden group
address@hidden smallexample
-
-This is the new guess for @expr{x}. Now we compare it with the
-old one to see if we've converged.
-
address@hidden
address@hidden
-3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
-2: 5.24196 1: 0 . .
-1: 4.5 .
- .
-
- @key{RET} address@hidden a = Z / Z > Z ' C-x
)
address@hidden group
address@hidden smallexample
-
-The loop converges in just a few steps to this value. To check
-the result, we can simply substitute it back into the equation.
-
address@hidden
address@hidden
-2: 5.26345856348
-1: 0.499999999997
- .
-
- @key{RET} ' sin(cos($)) @key{RET}
address@hidden group
address@hidden smallexample
-
-Let's test the new definition again:
-
address@hidden
address@hidden
-2: x^2 - 9 1: 3.
-1: 1 .
- .
-
- ' x^2-9 @key{RET} 1 X
address@hidden group
address@hidden smallexample
-
-Once again, here's the full Newton's Method definition:
-
address@hidden
address@hidden
-C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} -
t 1
- 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
- @key{RET} address@hidden a = Z /
- Z >
- Z '
-C-x )
address@hidden group
address@hidden example
-
address@hidden [fix-ref Nesting and Fixed Points]
-It turns out that Calc has a built-in command for applying a formula
-repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
-to see how to use it.
-
address@hidden [fix-ref Root Finding]
-Also, of course, @kbd{a R} is a built-in command that uses Newton's
-method (among others) to look for numerical solutions to any equation.
address@hidden Finding}.
-
address@hidden Programming Answer 9, Programming Answer 10, Programming Answer
8, Answers to Exercises
address@hidden Programming Tutorial Exercise 9
-
address@hidden
-The first step is to adjust @expr{z} to be greater than 5. A simple
-``for'' loop will do the job here. If @expr{z} is less than 5, we
-reduce the problem using
address@hidden @math{\psi(z) = \psi(z+1) - 1/z}.
address@hidden @expr{psi(z) = psi(z+1) - 1/z}. We go
-on to compute
address@hidden @math{\psi(z+1)},
address@hidden @expr{psi(z+1)},
-and remember to add back a factor of @expr{-1/z} when we're done. This
-step is repeated until @expr{z > 5}.
-
-(Because this definition is long, it will be repeated in concise form
-below. You can use @address@hidden * m}} to load it from there. While you are
-entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
-keystrokes without executing them. In the following diagrams we'll
-pretend Calc actually executed the keystrokes as you typed them,
-just for purposes of illustration.)
-
address@hidden
address@hidden
-1: 1. 1: 1.
- . .
-
- 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
address@hidden group
address@hidden smallexample
-
-Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
-factor. If @expr{z < 5}, we use a loop to increase it.
-
-(By the way, we started with @samp{1.0} instead of the integer 1 because
-otherwise the calculation below will try to do exact fractional arithmetic,
-and will never converge because fractions compare equal only if they
-are exactly equal, not just equal to within the current precision.)
-
address@hidden
address@hidden
-3: 1. 2: 1. 1: 6.
-2: 1. 1: 1 .
-1: 5 .
- .
-
- @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
address@hidden group
address@hidden smallexample
-
-Now we compute the initial part of the sum:
address@hidden @math{\ln z - {1 \over 2z}}
address@hidden @expr{ln(z) - 1/2z}
-minus the adjustment factor.
-
address@hidden
address@hidden
-2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
-1: 0.0833333333333 1: 2.28333333333 .
- . .
-
- L r 1 2 * & - r 2 -
address@hidden group
address@hidden smallexample
-
-Now we evaluate the series. We'll use another ``for'' loop counting
-up the value of @expr{2 n}. (Calc does have a summation command,
address@hidden +}, but we'll use loops just to get more practice with them.)
-
address@hidden
address@hidden
-3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
-2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
-1: 40 1: 2 2: 2 .
- . . 1: 36.
- .
-
- 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1
@key{TAB} ^ * /
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
-2: -0.5749 2: -0.5772 1: 0 .
-1: 2.3148e-3 1: -0.5749 .
- . .
-
- @key{TAB} @key{RET} address@hidden - @key{RET} address@hidden a =
Z / 2 Z ) Z ' C-x )
address@hidden group
address@hidden smallexample
-
-This is the value of
address@hidden @math{-\gamma},
address@hidden @expr{- gamma},
-with a slight bit of roundoff error. To get a full 12 digits, let's use
-a higher precision:
-
address@hidden
address@hidden
-2: -0.577215664892 2: -0.577215664892
-1: 1. 1: -0.577215664901532
-
- 1. @key{RET} p 16 @key{RET} X
address@hidden group
address@hidden smallexample
-
-Here's the complete sequence of keystrokes:
-
address@hidden
address@hidden
-C-x ( Z ` s 1 0 t 2
- @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
- L r 1 2 * & - r 2 -
- 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1
@key{TAB} ^ * /
- @key{TAB} @key{RET} address@hidden - @key{RET}
address@hidden a = Z /
- 2 Z )
- Z '
-C-x )
address@hidden group
address@hidden example
-
address@hidden Programming Answer 10, Programming Answer 11, Programming Answer
9, Answers to Exercises
address@hidden Programming Tutorial Exercise 10
-
address@hidden
-Taking the derivative of a term of the form @expr{x^n} will produce
-a term like
address@hidden @math{n x^{n-1}}.
address@hidden @expr{n x^(n-1)}.
-Taking the derivative of a constant
-produces zero. From this it is easy to see that the @expr{n}th
-derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
-coefficient on the @expr{x^n} term times @expr{n!}.
-
-(Because this definition is long, it will be repeated in concise form
-below. You can use @address@hidden * m}} to load it from there. While you are
-entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
-keystrokes without executing them. In the following diagrams we'll
-pretend Calc actually executed the keystrokes as you typed them,
-just for purposes of illustration.)
-
address@hidden
address@hidden
-2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
-1: 6 2: 0
- . 1: 6
- .
-
- ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
address@hidden group
address@hidden smallexample
-
address@hidden
-Variable 1 will accumulate the vector of coefficients.
-
address@hidden
address@hidden
-2: 0 3: 0 2: 5 x^4 + ...
-1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
- . 1: 1 .
- .
-
- Z ( @key{TAB} @key{RET} 0 s l x @key{RET} address@hidden
! / s | 1
address@hidden group
address@hidden smallexample
-
address@hidden
-Note that @kbd{s | 1} appends the top-of-stack value to the vector
-in a variable; it is completely analogous to @kbd{s + 1}. We could
-have written instead, @kbd{r 1 @key{TAB} | t 1}.
-
address@hidden
address@hidden
-1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
- . . .
-
- a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
address@hidden group
address@hidden smallexample
-
-To convert back, a simple method is just to map the coefficients
-against a table of powers of @expr{x}.
-
address@hidden
address@hidden
-2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
-1: 6 1: [0, 1, 2, 3, 4, 5, 6]
- . .
-
- 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
-1: [1, x, x^2, x^3, ... ] .
- .
-
- ' x @key{RET} @key{TAB} V M ^ *
address@hidden group
address@hidden smallexample
-
-Once again, here are the whole polynomial to/from vector programs:
-
address@hidden
address@hidden
-C-x ( Z ` [ ] t 1 0 @key{TAB}
- Z ( @key{TAB} @key{RET} 0 s l x @key{RET} address@hidden ! / s | 1
- a d x @key{RET}
- 1 Z ) r 1
- Z '
-C-x )
-
-C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
address@hidden group
address@hidden example
-
address@hidden Programming Answer 11, Programming Answer 12, Programming Answer
10, Answers to Exercises
address@hidden Programming Tutorial Exercise 11
-
address@hidden
-First we define a dummy program to go on the @kbd{z s} key. The true
address@hidden@kbd{z s}} key is supposed to take two numbers from the stack and
-return one number, so @key{DEL} as a dummy definition will make
-sure the stack comes out right.
-
address@hidden
address@hidden
-2: 4 1: 4 2: 4
-1: 2 . 1: 2
- . .
-
- 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
address@hidden group
address@hidden smallexample
-
-The last step replaces the 2 that was eaten during the creation
-of the dummy @kbd{z s} command. Now we move on to the real
-definition. The recurrence needs to be rewritten slightly,
-to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
-
-(Because this definition is long, it will be repeated in concise form
-below. You can use @kbd{C-x * m} to load it from there.)
-
address@hidden
address@hidden
-2: 4 4: 4 3: 4 2: 4
-1: 2 3: 2 2: 2 1: 2
- . 2: 4 1: 0 .
- 1: 2 .
- .
-
- C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
-3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
-2: 2 . . 2: 3 2: 3 1: 3
-1: 0 1: 2 1: 1 .
- . . .
-
- @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB}
M-2 @key{RET} 1 - z s
address@hidden group
address@hidden smallexample
-
address@hidden
-(Note that the value 3 that our dummy @kbd{z s} produces is not correct;
-it is merely a placeholder that will do just as well for now.)
-
address@hidden
address@hidden
-3: 3 4: 3 3: 3 2: 3 1: -6
-2: 3 3: 3 2: 3 1: 9 .
-1: 2 2: 3 1: 3 .
- . 1: 2 .
- .
-
- address@hidden address@hidden @key{TAB} @key{RET} address@hidden
z s * -
-
address@hidden group
address@hidden smallexample
address@hidden
address@hidden
address@hidden
-1: -6 2: 4 1: 11 2: 11
- . 1: 2 . 1: 11
- . .
-
- Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s
address@hidden k s
address@hidden group
address@hidden smallexample
-
-Even though the result that we got during the definition was highly
-bogus, once the definition is complete the @kbd{z s} command gets
-the right answers.
-
-Here's the full program once again:
-
address@hidden
address@hidden
-C-x ( M-2 @key{RET} a =
- Z [ @key{DEL} @key{DEL} 1
- Z : @key{RET} 0 a =
- Z [ @key{DEL} @key{DEL} 0
- Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
- address@hidden address@hidden @key{TAB} @key{RET}
address@hidden z s * -
- Z ]
- Z ]
-C-x )
address@hidden group
address@hidden example
-
-You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
-followed by @kbd{Z K s}, without having to make a dummy definition
-first, because @code{read-kbd-macro} doesn't need to execute the
-definition as it reads it in. For this reason, @code{C-x * m} is often
-the easiest way to create recursive programs in Calc.
-
address@hidden Programming Answer 12, , Programming Answer 11, Answers to
Exercises
address@hidden Programming Tutorial Exercise 12
-
address@hidden
-This turns out to be a much easier way to solve the problem. Let's
-denote Stirling numbers as calls of the function @samp{s}.
-
-First, we store the rewrite rules corresponding to the definition of
-Stirling numbers in a convenient variable:
-
address@hidden
-s e StirlingRules @key{RET}
-[ s(n,n) := 1 :: n >= 0,
- s(n,0) := 0 :: n > 0,
- s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
-C-c C-c
address@hidden smallexample
-
-Now, it's just a matter of applying the rules:
-
address@hidden
address@hidden
-2: 4 1: s(4, 2) 1: 11
-1: 2 . .
- .
-
- 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules
@key{RET} C-x )
address@hidden group
address@hidden smallexample
-
-As in the case of the @code{fib} rules, it would be useful to put these
-rules in @code{EvalRules} and to add a @samp{:: remember} condition to
-the last rule.
-
address@hidden This ends the table-of-contents kludge from above:
address@hidden
-\global\let\chapternofonts=\oldchapternofonts
address@hidden tex
-
address@hidden [reference]
-
address@hidden Introduction, Data Types, Tutorial, Top
address@hidden Introduction
-
address@hidden
-This chapter is the beginning of the Calc reference manual.
-It covers basic concepts such as the stack, algebraic and
-numeric entry, undo, numeric prefix arguments, etc.
-
address@hidden [when-split]
address@hidden (Chapter 2, the Tutorial, has been printed in a separate volume.)
-
address@hidden
-* Basic Commands::
-* Help Commands::
-* Stack Basics::
-* Numeric Entry::
-* Algebraic Entry::
-* Quick Calculator::
-* Prefix Arguments::
-* Undo::
-* Error Messages::
-* Multiple Calculators::
-* Troubleshooting Commands::
address@hidden menu
-
address@hidden Basic Commands, Help Commands, Introduction, Introduction
address@hidden Basic Commands
-
address@hidden
address@hidden calc
address@hidden calc-mode
address@hidden Starting the Calculator
address@hidden Running the Calculator
-To start the Calculator in its standard interface, type @kbd{M-x calc}.
-By default this creates a pair of small windows, @samp{*Calculator*}
-and @samp{*Calc Trail*}. The former displays the contents of the
-Calculator stack and is manipulated exclusively through Calc commands.
-It is possible (though not usually necessary) to create several Calc
-mode buffers each of which has an independent stack, undo list, and
-mode settings. There is exactly one Calc Trail buffer; it records a
-list of the results of all calculations that have been done. The
-Calc Trail buffer uses a variant of Calc mode, so Calculator commands
-still work when the trail buffer's window is selected. It is possible
-to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
-still exists and is updated silently. @xref{Trail Commands}.
-
address@hidden C-x * c
address@hidden C-x * *
address@hidden
address@hidden @null
address@hidden ignore
-In most installations, the @kbd{C-x * c} key sequence is a more
-convenient way to start the Calculator. Also, @kbd{C-x * *}
-is a synonym for @kbd{C-x * c} unless you last used Calc
-in its Keypad mode.
-
address@hidden x
address@hidden M-x
address@hidden calc-execute-extended-command
-Most Calc commands use one or two keystrokes. Lower- and upper-case
-letters are distinct. Commands may also be entered in full @kbd{M-x} form;
-for some commands this is the only form. As a convenience, the @kbd{x}
-key (@code{calc-execute-extended-command})
-is like @kbd{M-x} except that it enters the initial string @samp{calc-}
-for you. For example, the following key sequences are equivalent:
address@hidden, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
-
address@hidden Extensions module
address@hidden @file{calc-ext} module
-The Calculator exists in many parts. When you type @kbd{C-x * c}, the
-Emacs ``auto-load'' mechanism will bring in only the first part, which
-contains the basic arithmetic functions. The other parts will be
-auto-loaded the first time you use the more advanced commands like trig
-functions or matrix operations. This is done to improve the response time
-of the Calculator in the common case when all you need to do is a
-little arithmetic. If for some reason the Calculator fails to load an
-extension module automatically, you can force it to load all the
-extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
-command. @xref{Mode Settings}.
-
-If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
-the Calculator is loaded if necessary, but it is not actually started.
-If the argument is positive, the @file{calc-ext} extensions are also
-loaded if necessary. User-written Lisp code that wishes to make use
-of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
-to auto-load the Calculator.
-
address@hidden C-x * b
address@hidden full-calc
-If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
-will get a Calculator that uses the full height of the Emacs screen.
-When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
-command instead of @code{calc}. From the Unix shell you can type
address@hidden -f full-calc} to start a new Emacs specifically for use
-as a calculator. When Calc is started from the Emacs command line
-like this, Calc's normal ``quit'' commands actually quit Emacs itself.
-
address@hidden C-x * o
address@hidden calc-other-window
-The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
-window is not actually selected. If you are already in the Calc
-window, @kbd{C-x * o} switches you out of it. (The regular Emacs
address@hidden o} command would also work for this, but it has a
-tendency to drop you into the Calc Trail window instead, which
address@hidden * o} takes care not to do.)
-
address@hidden
address@hidden C-x * q
address@hidden ignore
-For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
-which prompts you for a formula (like @samp{2+3/4}). The result is
-displayed at the bottom of the Emacs screen without ever creating
-any special Calculator windows. @xref{Quick Calculator}.
-
address@hidden
address@hidden C-x * k
address@hidden ignore
-Finally, if you are using the X window system you may want to try
address@hidden * k} (@code{calc-keypad}) which runs Calc with a
-``calculator keypad'' picture as well as a stack display. Click on
-the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
-
address@hidden q
address@hidden calc-quit
address@hidden Quitting the Calculator
address@hidden Exiting the Calculator
-The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
-Calculator's window(s). It does not delete the Calculator buffers.
-If you type @kbd{M-x calc} again, the Calculator will reappear with the
-contents of the stack intact. Typing @kbd{C-x * c} or @kbd{C-x * *}
-again from inside the Calculator buffer is equivalent to executing
address@hidden; you can think of @kbd{C-x * *} as toggling the
-Calculator on and off.
-
address@hidden C-x * x
-The @kbd{C-x * x} command also turns the Calculator off, no matter which
-user interface (standard, Keypad, or Embedded) is currently active.
-It also cancels @code{calc-edit} mode if used from there.
-
address@hidden d @key{SPC}
address@hidden calc-refresh
address@hidden Refreshing a garbled display
address@hidden Garbled displays, refreshing
-The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
-of the Calculator buffer from memory. Use this if the contents of the
-buffer have been damaged somehow.
-
address@hidden
address@hidden o
address@hidden ignore
-The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
-``home'' position at the bottom of the Calculator buffer.
-
address@hidden <
address@hidden >
address@hidden calc-scroll-left
address@hidden calc-scroll-right
address@hidden Horizontal scrolling
address@hidden Scrolling
address@hidden Wide text, scrolling
-The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
address@hidden These are just like the normal horizontal
-scrolling commands except that they scroll one half-screen at a time by
-default. (Calc formats its output to fit within the bounds of the
-window whenever it can.)
-
address@hidden @{
address@hidden @}
address@hidden calc-scroll-down
address@hidden calc-scroll-up
address@hidden Vertical scrolling
-The @address@hidden and @address@hidden keys are bound to
@code{calc-scroll-down}
-and @code{calc-scroll-up}. They scroll up or down by one-half the
-height of the Calc window.
-
address@hidden C-x * 0
address@hidden calc-reset
-The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
-by a zero) resets the Calculator to its initial state. This clears
-the stack, resets all the modes to their initial values (the values
-that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
-caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
-values of any variables.) With an argument of 0, Calc will be reset to
-its default state; namely, the modes will be given their default values.
-With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
-the stack but resets everything else to its initial state; with a
-negative prefix argument, @kbd{C-x * 0} preserves the contents of the
-stack but resets everything else to its default state.
-
address@hidden calc-version
-The @kbd{M-x calc-version} command displays the current version number
-of Calc and the name of the person who installed it on your system.
-(This information is also present in the @samp{*Calc Trail*} buffer,
-and in the output of the @kbd{h h} command.)
-
address@hidden Help Commands, Stack Basics, Basic Commands, Introduction
address@hidden Help Commands
-
address@hidden
address@hidden Help commands
address@hidden ?
address@hidden calc-help
-The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
-Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
address@hidden and @kbd{C-x} prefixes. You can type
address@hidden after a prefix to see a list of commands beginning with that
-prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
-to see additional commands for that prefix.)
-
address@hidden h h
address@hidden calc-full-help
-The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
-responses at once. When printed, this makes a nice, compact (three pages)
-summary of Calc keystrokes.
-
-In general, the @kbd{h} key prefix introduces various commands that
-provide help within Calc. Many of the @kbd{h} key functions are
-Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
-
address@hidden h i
address@hidden C-x * i
address@hidden i
address@hidden calc-info
-The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
-to read this manual on-line. This is basically the same as typing
address@hidden i} (the regular way to run the Info system), then, if Info
-is not already in the Calc manual, selecting the beginning of the
-manual. The @kbd{C-x * i} command is another way to read the Calc
-manual; it is different from @kbd{h i} in that it works any time,
-not just inside Calc. The plain @kbd{i} key is also equivalent to
address@hidden i}, though this key is obsolete and may be replaced with a
-different command in a future version of Calc.
-
address@hidden h t
address@hidden C-x * t
address@hidden calc-tutorial
-The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
-the Tutorial section of the Calc manual. It is like @kbd{h i},
-except that it selects the starting node of the tutorial rather
-than the beginning of the whole manual. (It actually selects the
-node ``Interactive Tutorial'' which tells a few things about
-using the Info system before going on to the actual tutorial.)
-The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
-all times).
-
address@hidden h s
address@hidden C-x * s
address@hidden calc-info-summary
-The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
-on the Summary node of the Calc manual. @xref{Summary}. The @kbd{C-x * s}
-key is equivalent to @kbd{h s}.
-
address@hidden h k
address@hidden calc-describe-key
-The @kbd{h k} (@code{calc-describe-key}) command looks up a key
-sequence in the Calc manual. For example, @kbd{h k H a S} looks
-up the documentation on the @kbd{H a S} (@code{calc-solve-for})
-command. This works by looking up the textual description of
-the key(s) in the Key Index of the manual, then jumping to the
-node indicated by the index.
-
-Most Calc commands do not have traditional Emacs documentation
-strings, since the @kbd{h k} command is both more convenient and
-more instructive. This means the regular Emacs @kbd{C-h k}
-(@code{describe-key}) command will not be useful for Calc keystrokes.
-
address@hidden h c
address@hidden calc-describe-key-briefly
-The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
-key sequence and displays a brief one-line description of it at
-the bottom of the screen. It looks for the key sequence in the
-Summary node of the Calc manual; if it doesn't find the sequence
-there, it acts just like its regular Emacs counterpart @kbd{C-h c}
-(@code{describe-key-briefly}). For example, @kbd{h c H a S}
-gives the description:
-
address@hidden
-H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
address@hidden smallexample
-
address@hidden
-which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
-takes a value @expr{a} from the stack, prompts for a value @expr{v},
-then applies the algebraic function @code{fsolve} to these values.
-The @samp{?=notes} message means you can now type @kbd{?} to see
-additional notes from the summary that apply to this command.
-
address@hidden h f
address@hidden calc-describe-function
-The @kbd{h f} (@code{calc-describe-function}) command looks up an
-algebraic function or a command name in the Calc manual. Enter an
-algebraic function name to look up that function in the Function
-Index or enter a command name beginning with @samp{calc-} to look it
-up in the Command Index. This command will also look up operator
-symbols that can appear in algebraic formulas, like @samp{%} and
address@hidden>}.
-
address@hidden h v
address@hidden calc-describe-variable
-The @kbd{h v} (@code{calc-describe-variable}) command looks up a
-variable in the Calc manual. Enter a variable name like @code{pi} or
address@hidden
-
address@hidden h b
address@hidden describe-bindings
-The @kbd{h b} (@code{calc-describe-bindings}) command is just like
address@hidden b}, except that only local (Calc-related) key bindings are
-listed.
-
address@hidden h n
-The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
-the ``news'' or change history of Calc. This is kept in the file
address@hidden, which Calc looks for in the same directory as the Calc
-source files.
-
address@hidden h C-c
address@hidden h C-d
address@hidden h C-w
-The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
-distribution, and warranty information about Calc. These work by
-pulling up the appropriate parts of the ``Copying'' or ``Reporting
-Bugs'' sections of the manual.
-
address@hidden Stack Basics, Numeric Entry, Help Commands, Introduction
address@hidden Stack Basics
-
address@hidden
address@hidden Stack basics
address@hidden [fix-tut RPN Calculations and the Stack]
-Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
-Tutorial}.
-
-To add the numbers 1 and 2 in Calc you would type the keys:
address@hidden @key{RET} 2 +}.
-(@key{RET} corresponds to the @key{ENTER} key on most calculators.)
-The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
address@hidden key always ``pops'' the top two numbers from the stack, adds
them,
-and pushes the result (3) back onto the stack. This number is ready for
-further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
-3 and 5, subtracts them, and pushes the result (@mathit{-2}).
-
-Note that the ``top'' of the stack actually appears at the @emph{bottom}
-of the buffer. A line containing a single @samp{.} character signifies
-the end of the buffer; Calculator commands operate on the number(s)
-directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
-command allows you to move the @samp{.} marker up and down in the stack;
address@hidden the Stack}.
-
address@hidden d l
address@hidden calc-line-numbering
-Stack elements are numbered consecutively, with number 1 being the top of
-the stack. These line numbers are ordinarily displayed on the lefthand side
-of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
-whether these numbers appear. (Line numbers may be turned off since they
-slow the Calculator down a bit and also clutter the display.)
-
address@hidden o
address@hidden calc-realign
-The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
-the cursor to its top-of-stack ``home'' position. It also undoes any
-horizontal scrolling in the window. If you give it a numeric prefix
-argument, it instead moves the cursor to the specified stack element.
-
-The @key{RET} (or equivalent @key{SPC}) key is only required to separate
-two consecutive numbers.
-(After all, if you typed @kbd{1 2} by themselves the Calculator
-would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
-right after typing a number, the key duplicates the number on the top of
-the stack. @address@hidden *} is thus a handy way to square a number.
-
-The @key{DEL} key pops and throws away the top number on the stack.
-The @key{TAB} key swaps the top two objects on the stack.
address@hidden and Trail}, for descriptions of these and other stack-related
-commands.
-
address@hidden Numeric Entry, Algebraic Entry, Stack Basics, Introduction
address@hidden Numeric Entry
-
address@hidden
address@hidden 0-9
address@hidden .
address@hidden e
address@hidden Numeric entry
address@hidden Entering numbers
-Pressing a digit or other numeric key begins numeric entry using the
-minibuffer. The number is pushed on the stack when you press the @key{RET}
-or @key{SPC} keys. If you press any other non-numeric key, the number is
-pushed onto the stack and the appropriate operation is performed. If
-you press a numeric key which is not valid, the key is ignored.
-
address@hidden Minus signs
address@hidden Negative numbers, entering
address@hidden _
-There are three different concepts corresponding to the word ``minus,''
-typified by @expr{a-b} (subtraction), @expr{-x}
-(change-sign), and @expr{-5} (negative number). Calc uses three
-different keys for these operations, respectively:
address@hidden, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key
subtracts
-the two numbers on the top of the stack. The @kbd{n} key changes the sign
-of the number on the top of the stack or the number currently being entered.
-The @kbd{_} key begins entry of a negative number or changes the sign of
-the number currently being entered. The following sequences all enter the
-number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
address@hidden @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
-
-Some other keys are active during numeric entry, such as @kbd{#} for
-non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
-These notations are described later in this manual with the corresponding
-data types. @xref{Data Types}.
-
-During numeric entry, the only editing key available is @key{DEL}.
-
address@hidden Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
address@hidden Algebraic Entry
-
address@hidden
address@hidden '
address@hidden calc-algebraic-entry
address@hidden Algebraic notation
address@hidden Formulas, entering
-Calculations can also be entered in algebraic form. This is accomplished
-by typing the apostrophe key, ', followed by the expression in
-standard format:
-
address@hidden
-' 2+3*4 @key{RET}.
address@hidden example
-
address@hidden
-This will compute
address@hidden @math{2+(3\times4) = 14}
address@hidden @expr{2+(3*4) = 14}
-and push it on the stack. If you wish you can
-ignore the RPN aspect of Calc altogether and simply enter algebraic
-expressions in this way. You may want to use @key{DEL} every so often to
-clear previous results off the stack.
-
-You can press the apostrophe key during normal numeric entry to switch
-the half-entered number into Algebraic entry mode. One reason to do this
-would be to use the full Emacs cursor motion and editing keys, which are
-available during algebraic entry but not during numeric entry.
-
-In the same vein, during either numeric or algebraic entry you can
-press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
-you complete your half-finished entry in a separate buffer.
address@hidden Stack Entries}.
-
address@hidden m a
address@hidden calc-algebraic-mode
address@hidden Algebraic Mode
-If you prefer algebraic entry, you can use the command @kbd{m a}
-(@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
-digits and other keys that would normally start numeric entry instead
-start full algebraic entry; as long as your formula begins with a digit
-you can omit the apostrophe. Open parentheses and square brackets also
-begin algebraic entry. You can still do RPN calculations in this mode,
-but you will have to press @key{RET} to terminate every number:
address@hidden @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
-thing as @kbd{2*3+4 @key{RET}}.
-
address@hidden Incomplete Algebraic Mode
-If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
-command, it enables Incomplete Algebraic mode; this is like regular
-Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
-only. Numeric keys still begin a numeric entry in this mode.
-
address@hidden m t
address@hidden calc-total-algebraic-mode
address@hidden Total Algebraic Mode
-The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
-stronger algebraic-entry mode, in which @emph{all} regular letter and
-punctuation keys begin algebraic entry. Use this if you prefer typing
address@hidden@kbd{sqrt( )}} instead of @kbd{Q}, @address@hidden( )}} instead of
address@hidden f}, and so on. To type regular Calc commands when you are in
-Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
-is the command to quit Calc, @kbd{M-p} sets the precision, and
address@hidden t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
-mode back off again. Meta keys also terminate algebraic entry, so
-that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
address@hidden will appear in the mode line whenever you are in this mode.
-
-Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
-algebraic formula. You can then use the normal Emacs editing keys to
-modify this formula to your liking before pressing @key{RET}.
-
address@hidden $
address@hidden Formulas, referring to stack
-Within a formula entered from the keyboard, the symbol @kbd{$}
-represents the number on the top of the stack. If an entered formula
-contains any @kbd{$} characters, the Calculator replaces the top of
-stack with that formula rather than simply pushing the formula onto the
-stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
address@hidden replaces it with 6. Note that the @kbd{$} key always
-initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
-first character in the new formula.
-
-Higher stack elements can be accessed from an entered formula with the
-symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
-removed (to be replaced by the entered values) equals the number of dollar
-signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
-adds the second and third stack elements, replacing the top three elements
-with the answer. (All information about the top stack element is thus lost
-since no single @samp{$} appears in this formula.)
-
-A slightly different way to refer to stack elements is with a dollar
-sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
-like @samp{$}, @samp{$$}, etc., except that stack entries referred
-to numerically are not replaced by the algebraic entry. That is, while
address@hidden replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
-on the stack and pushes an additional 6.
-
-If a sequence of formulas are entered separated by commas, each formula
-is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
-those three numbers onto the stack (leaving the 3 at the top), and
address@hidden,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
address@hidden,$$} exchanges the top two elements of the stack, just like the
address@hidden key.
-
-You can finish an algebraic entry with @kbd{M-=} or @address@hidden instead
-of @key{RET}. This uses @kbd{=} to evaluate the variables in each
-formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
-the variable @samp{pi}, but @kbd{' pi address@hidden pushes 3.1415.)
-
-If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
-instead of @key{RET}, Calc disables the default simplifications
-(as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
-is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
-on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
-you might then press @kbd{=} when it is time to evaluate this formula.
-
address@hidden Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
address@hidden ``Quick Calculator'' Mode
-
address@hidden
address@hidden C-x * q
address@hidden quick-calc
address@hidden Quick Calculator
-There is another way to invoke the Calculator if all you need to do
-is make one or two quick calculations. Type @kbd{C-x * q} (or
address@hidden quick-calc}), then type any formula as an algebraic entry.
-The Calculator will compute the result and display it in the echo
-area, without ever actually putting up a Calc window.
-
-You can use the @kbd{$} character in a Quick Calculator formula to
-refer to the previous Quick Calculator result. Older results are
-not retained; the Quick Calculator has no effect on the full
-Calculator's stack or trail. If you compute a result and then
-forget what it was, just run @code{C-x * q} again and enter
address@hidden as the formula.
-
-If this is the first time you have used the Calculator in this Emacs
-session, the @kbd{C-x * q} command will create the @code{*Calculator*}
-buffer and perform all the usual initializations; it simply will
-refrain from putting that buffer up in a new window. The Quick
-Calculator refers to the @code{*Calculator*} buffer for all mode
-settings. Thus, for example, to set the precision that the Quick
-Calculator uses, simply run the full Calculator momentarily and use
-the regular @kbd{p} command.
-
-If you use @code{C-x * q} from inside the Calculator buffer, the
-effect is the same as pressing the apostrophe key (algebraic entry).
-
-The result of a Quick calculation is placed in the Emacs ``kill ring''
-as well as being displayed. A subsequent @kbd{C-y} command will
-yank the result into the editing buffer. You can also use this
-to yank the result into the next @kbd{C-x * q} input line as a more
-explicit alternative to @kbd{$} notation, or to yank the result
-into the Calculator stack after typing @kbd{C-x * c}.
-
-If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
-of @key{RET}, the result is inserted immediately into the current
-buffer rather than going into the kill ring.
-
-Quick Calculator results are actually evaluated as if by the @kbd{=}
-key (which replaces variable names by their stored values, if any).
-If the formula you enter is an assignment to a variable using the
address@hidden:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
-then the result of the evaluation is stored in that Calc variable.
address@hidden and Recall}.
-
-If the result is an integer and the current display radix is decimal,
-the number will also be displayed in hex, octal and binary formats. If
-the integer is in the range from 1 to 126, it will also be displayed as
-an ASCII character.
-
-For example, the quoted character @samp{"x"} produces the vector
-result @samp{[120]} (because 120 is the ASCII code of the lower-case
-`x'; @pxref{Strings}). Since this is a vector, not an integer, it
-is displayed only according to the current mode settings. But
-running Quick Calc again and entering @samp{120} will produce the
-result @samp{120 (16#78, 8#170, x)} which shows the number in its
-decimal, hexadecimal, octal, and ASCII forms.
-
-Please note that the Quick Calculator is not any faster at loading
-or computing the answer than the full Calculator; the name ``quick''
-merely refers to the fact that it's much less hassle to use for
-small calculations.
-
address@hidden Prefix Arguments, Undo, Quick Calculator, Introduction
address@hidden Numeric Prefix Arguments
-
address@hidden
-Many Calculator commands use numeric prefix arguments. Some, such as
address@hidden s} (@code{calc-sci-notation}), set a parameter to the value of
-the prefix argument or use a default if you don't use a prefix.
-Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
-and prompt for a number if you don't give one as a prefix.
-
-As a rule, stack-manipulation commands accept a numeric prefix argument
-which is interpreted as an index into the stack. A positive argument
-operates on the top @var{n} stack entries; a negative argument operates
-on the @var{n}th stack entry in isolation; and a zero argument operates
-on the entire stack.
-
-Most commands that perform computations (such as the arithmetic and
-scientific functions) accept a numeric prefix argument that allows the
-operation to be applied across many stack elements. For unary operations
-(that is, functions of one argument like absolute value or complex
-conjugate), a positive prefix argument applies that function to the top
address@hidden stack entries simultaneously, and a negative argument applies it
-to the @var{n}th stack entry only. For binary operations (functions of
-two arguments like addition, GCD, and vector concatenation), a positive
-prefix argument ``reduces'' the function across the top @var{n}
-stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
address@hidden and Mapping}), and a negative argument maps the next-to-top
address@hidden stack elements with the top stack element as a second argument
-(for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
-This feature is not available for operations which use the numeric prefix
-argument for some other purpose.
-
-Numeric prefixes are specified the same way as always in Emacs: Press
-a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
-or press @kbd{C-u} followed by digits. Some commands treat plain
address@hidden (without any actual digits) specially.
-
address@hidden ~
address@hidden calc-num-prefix
-You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
-top of the stack and enter it as the numeric prefix for the next command.
-For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
-(silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
-to the fourth power and set the precision to that value.
-
-Conversely, if you have typed a numeric prefix argument the @kbd{~} key
-pushes it onto the stack in the form of an integer.
-
address@hidden Undo, Error Messages, Prefix Arguments, Introduction
address@hidden Undoing Mistakes
-
address@hidden
address@hidden U
address@hidden C-_
address@hidden calc-undo
address@hidden Mistakes, undoing
address@hidden Undoing mistakes
address@hidden Errors, undoing
-The address@hidden key (@code{calc-undo}) undoes the most recent operation.
-If that operation added or dropped objects from the stack, those objects
-are removed or restored. If it was a ``store'' operation, you are
-queried whether or not to restore the variable to its original value.
-The @kbd{U} key may be pressed any number of times to undo successively
-farther back in time; with a numeric prefix argument it undoes a
-specified number of operations. The undo history is cleared only by the
address@hidden (@code{calc-quit}) command. (Recall that @kbd{C-x * c} is
-synonymous with @code{calc-quit} while inside the Calculator; this
-also clears the undo history.)
-
-Currently the mode-setting commands (like @code{calc-precision}) are not
-undoable. You can undo past a point where you changed a mode, but you
-will need to reset the mode yourself.
-
address@hidden D
address@hidden calc-redo
address@hidden Redoing after an Undo
-The address@hidden key (@code{calc-redo}) redoes an operation that was
-mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
-equivalent to executing @code{calc-redo}. You can redo any number of
-times, up to the number of recent consecutive undo commands. Redo
-information is cleared whenever you give any command that adds new undo
-information, i.e., if you undo, then enter a number on the stack or make
-any other change, then it will be too late to redo.
-
address@hidden address@hidden
address@hidden calc-last-args
address@hidden Last-arguments feature
address@hidden Arguments, restoring
-The @address@hidden key (@code{calc-last-args}) is like undo in that
-it restores the arguments of the most recent command onto the stack;
-however, it does not remove the result of that command. Given a numeric
-prefix argument, this command applies to the @expr{n}th most recent
-command which removed items from the stack; it pushes those items back
-onto the stack.
-
-The @kbd{K} (@code{calc-keep-args}) command provides a related function
-to @address@hidden @xref{Stack and Trail}.
-
-It is also possible to recall previous results or inputs using the trail.
address@hidden Commands}.
-
-The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
-
address@hidden Error Messages, Multiple Calculators, Undo, Introduction
address@hidden Error Messages
-
address@hidden
address@hidden w
address@hidden calc-why
address@hidden Errors, messages
address@hidden Why did an error occur?
-Many situations that would produce an error message in other calculators
-simply create unsimplified formulas in the Emacs Calculator. For example,
address@hidden @key{RET} 0 /} pushes the formula @expr{1 / 0}; @address@hidden
L}} pushes
-the formula @samp{ln(0)}. Floating-point overflow and underflow are also
-reasons for this to happen.
-
-When a function call must be left in symbolic form, Calc usually
-produces a message explaining why. Messages that are probably
-surprising or indicative of user errors are displayed automatically.
-Other messages are simply kept in Calc's memory and are displayed only
-if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
-the same computation results in several messages. (The first message
-will end with @samp{[w=more]} in this case.)
-
address@hidden d w
address@hidden calc-auto-why
-The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
-are displayed automatically. (Calc effectively presses @kbd{w} for you
-after your computation finishes.) By default, this occurs only for
-``important'' messages. The other possible modes are to report
address@hidden messages automatically, or to report none automatically (so
-that you must always press @kbd{w} yourself to see the messages).
-
address@hidden Multiple Calculators, Troubleshooting Commands, Error Messages,
Introduction
address@hidden Multiple Calculators
-
address@hidden
address@hidden another-calc
-It is possible to have any number of Calc mode buffers at once.
-Usually this is done by executing @kbd{M-x another-calc}, which
-is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
-buffer already exists, a new, independent one with a name of the
-form @samp{*Calculator*<@var{n}>} is created. You can also use the
-command @code{calc-mode} to put any buffer into Calculator mode, but
-this would ordinarily never be done.
-
-The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
-it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
-Calculator buffer.
-
-Each Calculator buffer keeps its own stack, undo list, and mode settings
-such as precision, angular mode, and display formats. In Emacs terms,
-variables such as @code{calc-stack} are buffer-local variables. The
-global default values of these variables are used only when a new
-Calculator buffer is created. The @code{calc-quit} command saves
-the stack and mode settings of the buffer being quit as the new defaults.
-
-There is only one trail buffer, @samp{*Calc Trail*}, used by all
-Calculator buffers.
-
address@hidden Troubleshooting Commands, , Multiple Calculators, Introduction
address@hidden Troubleshooting Commands
-
address@hidden
-This section describes commands you can use in case a computation
-incorrectly fails or gives the wrong answer.
-
address@hidden Bugs}, if you find a problem that appears to be due
-to a bug or deficiency in Calc.
-
address@hidden
-* Autoloading Problems::
-* Recursion Depth::
-* Caches::
-* Debugging Calc::
address@hidden menu
-
address@hidden Autoloading Problems, Recursion Depth, Troubleshooting Commands,
Troubleshooting Commands
address@hidden Autoloading Problems
-
address@hidden
-The Calc program is split into many component files; components are
-loaded automatically as you use various commands that require them.
-Occasionally Calc may lose track of when a certain component is
-necessary; typically this means you will type a command and it won't
-work because some function you've never heard of was undefined.
-
address@hidden C-x * L
address@hidden calc-load-everything
-If this happens, the easiest workaround is to type @kbd{C-x * L}
-(@code{calc-load-everything}) to force all the parts of Calc to be
-loaded right away. This will cause Emacs to take up a lot more
-memory than it would otherwise, but it's guaranteed to fix the problem.
-
address@hidden Recursion Depth, Caches, Autoloading Problems, Troubleshooting
Commands
address@hidden Recursion Depth
-
address@hidden
address@hidden M
address@hidden I M
address@hidden calc-more-recursion-depth
address@hidden calc-less-recursion-depth
address@hidden Recursion depth
address@hidden ``Computation got stuck'' message
address@hidden @code{max-lisp-eval-depth}
address@hidden @code{max-specpdl-size}
-Calc uses recursion in many of its calculations. Emacs Lisp keeps a
-variable @code{max-lisp-eval-depth} which limits the amount of recursion
-possible in an attempt to recover from program bugs. If a calculation
-ever halts incorrectly with the message ``Computation got stuck or
-ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
-to increase this limit. (Of course, this will not help if the
-calculation really did get stuck due to some problem inside Calc.)
-
-The limit is always increased (multiplied) by a factor of two. There
-is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
-decreases this limit by a factor of two, down to a minimum value of 200.
-The default value is 1000.
-
-These commands also double or halve @code{max-specpdl-size}, another
-internal Lisp recursion limit. The minimum value for this limit is 600.
-
address@hidden Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
address@hidden Caches
-
address@hidden
address@hidden Caches
address@hidden Flushing caches
-Calc saves certain values after they have been computed once. For
-example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
-constant @cpi{} to about 20 decimal places; if the current precision
-is greater than this, it will recompute @cpi{} using a series
-approximation. This value will not need to be recomputed ever again
-unless you raise the precision still further. Many operations such as
-logarithms and sines make use of similarly cached values such as
address@hidden and
address@hidden @math{\ln 2}.
address@hidden @expr{ln(2)}.
-The visible effect of caching is that
-high-precision computations may seem to do extra work the first time.
-Other things cached include powers of two (for the binary arithmetic
-functions), matrix inverses and determinants, symbolic integrals, and
-data points computed by the graphing commands.
-
address@hidden calc-flush-caches
-If you suspect a Calculator cache has become corrupt, you can use the
address@hidden command to reset all caches to the empty state.
-(This should only be necessary in the event of bugs in the Calculator.)
-The @kbd{C-x * 0} (with the zero key) command also resets caches along
-with all other aspects of the Calculator's state.
-
address@hidden Debugging Calc, , Caches, Troubleshooting Commands
address@hidden Debugging Calc
-
address@hidden
-A few commands exist to help in the debugging of Calc commands.
address@hidden, to see the various ways that you can write
-your own Calc commands.
-
address@hidden Z T
address@hidden calc-timing
-The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
-in which the timing of slow commands is reported in the Trail.
-Any Calc command that takes two seconds or longer writes a line
-to the Trail showing how many seconds it took. This value is
-accurate only to within one second.
-
-All steps of executing a command are included; in particular, time
-taken to format the result for display in the stack and trail is
-counted. Some prompts also count time taken waiting for them to
-be answered, while others do not; this depends on the exact
-implementation of the command. For best results, if you are timing
-a sequence that includes prompts or multiple commands, define a
-keyboard macro to run the whole sequence at once. Calc's @kbd{X}
-command (@pxref{Keyboard Macros}) will then report the time taken
-to execute the whole macro.
-
-Another advantage of the @kbd{X} command is that while it is
-executing, the stack and trail are not updated from step to step.
-So if you expect the output of your test sequence to leave a result
-that may take a long time to format and you don't wish to count
-this formatting time, end your sequence with a @key{DEL} keystroke
-to clear the result from the stack. When you run the sequence with
address@hidden, Calc will never bother to format the large result.
-
-Another thing @kbd{Z T} does is to increase the Emacs variable
address@hidden to a much higher value (two million; the
-usual default in Calc is 250,000) for the duration of each command.
-This generally prevents garbage collection during the timing of
-the command, though it may cause your Emacs process to grow
-abnormally large. (Garbage collection time is a major unpredictable
-factor in the timing of Emacs operations.)
-
-Another command that is useful when debugging your own Lisp
-extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
-the error handler that changes the address@hidden
-exceeded'' message to the much more friendly ``Computation got
-stuck or ran too long.'' This handler interferes with the Emacs
-Lisp debugger's @code{debug-on-error} mode. Errors are reported
-in the handler itself rather than at the true location of the
-error. After you have executed @code{calc-pass-errors}, Lisp
-errors will be reported correctly but the user-friendly message
-will be lost.
-
address@hidden Data Types, Stack and Trail, Introduction, Top
address@hidden Data Types
-
address@hidden
-This chapter discusses the various types of objects that can be placed
-on the Calculator stack, how they are displayed, and how they are
-entered. (@xref{Data Type Formats}, for information on how these data
-types are represented as underlying Lisp objects.)
-
-Integers, fractions, and floats are various ways of describing real
-numbers. HMS forms also for many purposes act as real numbers. These
-types can be combined to form complex numbers, modulo forms, error forms,
-or interval forms. (But these last four types cannot be combined
-arbitrarily:@: error forms may not contain modulo forms, for example.)
-Finally, all these types of numbers may be combined into vectors,
-matrices, or algebraic formulas.
-
address@hidden
-* Integers:: The most basic data type.
-* Fractions:: This and above are called @dfn{rationals}.
-* Floats:: This and above are called @dfn{reals}.
-* Complex Numbers:: This and above are called @dfn{numbers}.
-* Infinities::
-* Vectors and Matrices::
-* Strings::
-* HMS Forms::
-* Date Forms::
-* Modulo Forms::
-* Error Forms::
-* Interval Forms::
-* Incomplete Objects::
-* Variables::
-* Formulas::
address@hidden menu
-
address@hidden Integers, Fractions, Data Types, Data Types
address@hidden Integers
-
address@hidden
address@hidden Integers
-The Calculator stores integers to arbitrary precision. Addition,
-subtraction, and multiplication of integers always yields an exact
-integer result. (If the result of a division or exponentiation of
-integers is not an integer, it is expressed in fractional or
-floating-point form according to the current Fraction mode.
address@hidden Mode}.)
-
-A decimal integer is represented as an optional sign followed by a
-sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
-insert a comma at every third digit for display purposes, but you
-must not type commas during the entry of numbers.
-
address@hidden #
-A non-decimal integer is represented as an optional sign, a radix
-between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
-and above, the letters A through Z (upper- or lower-case) count as
-digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
-to set the default radix for display of integers. Numbers of any radix
-may be entered at any time. If you press @kbd{#} at the beginning of a
-number, the current display radix is used.
-
address@hidden Fractions, Floats, Integers, Data Types
address@hidden Fractions
-
address@hidden
address@hidden Fractions
-A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
-written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
-performs RPN division; the following two sequences push the number
address@hidden:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3
/}
-assuming Fraction mode has been enabled.)
-When the Calculator produces a fractional result it always reduces it to
-simplest form, which may in fact be an integer.
-
-Fractions may also be entered in a three-part form, where @samp{2:3:4}
-represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
-display formats.
-
-Non-decimal fractions are entered and displayed as
address@hidden@address@hidden:@var{denom}} (or in the analogous three-part
-form). The numerator and denominator always use the same radix.
-
address@hidden Floats, Complex Numbers, Fractions, Data Types
address@hidden Floats
-
address@hidden
address@hidden Floating-point numbers
-A floating-point number or @dfn{float} is a number stored in scientific
-notation. The number of significant digits in the fractional part is
-governed by the current floating precision (@pxref{Precision}). The
-range of acceptable values is from
address@hidden @math{10^{-3999999}}
address@hidden @expr{10^-3999999}
-(inclusive) to
address@hidden @math{10^{4000000}}
address@hidden @expr{10^4000000}
-(exclusive), plus the corresponding negative values and zero.
-
-Calculations that would exceed the allowable range of values (such
-as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
-messages ``floating-point overflow'' or ``floating-point underflow''
-indicate that during the calculation a number would have been produced
-that was too large or too close to zero, respectively, to be represented
-by Calc. This does not necessarily mean the final result would have
-overflowed, just that an overflow occurred while computing the result.
-(In fact, it could report an underflow even though the final result
-would have overflowed!)
-
-If a rational number and a float are mixed in a calculation, the result
-will in general be expressed as a float. Commands that require an integer
-value (such as @kbd{k g} address@hidden) will also accept integer-valued
-floats, i.e., floating-point numbers with nothing after the decimal point.
-
-Floats are identified by the presence of a decimal point and/or an
-exponent. In general a float consists of an optional sign, digits
-including an optional decimal point, and an optional exponent consisting
-of an @samp{e}, an optional sign, and up to seven exponent digits.
-For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
-or 0.235.
-
-Floating-point numbers are normally displayed in decimal notation with
-all significant figures shown. Exceedingly large or small numbers are
-displayed in scientific notation. Various other display options are
-available. @xref{Float Formats}.
-
address@hidden Accuracy of calculations
-Floating-point numbers are stored in decimal, not binary. The result
-of each operation is rounded to the nearest value representable in the
-number of significant digits specified by the current precision,
-rounding away from zero in the case of a tie. Thus (in the default
-display mode) what you see is exactly what you get. Some operations such
-as square roots and transcendental functions are performed with several
-digits of extra precision and then rounded down, in an effort to make the
-final result accurate to the full requested precision. However,
-accuracy is not rigorously guaranteed. If you suspect the validity of a
-result, try doing the same calculation in a higher precision. The
-Calculator's arithmetic is not intended to be IEEE-conformant in any
-way.
-
-While floats are always @emph{stored} in decimal, they can be entered
-and displayed in any radix just like integers and fractions. Since a
-float that is entered in a radix other that 10 will be converted to
-decimal, the number that Calc stores may not be exactly the number that
-was entered, it will be the closest decimal approximation given the
-current precison. The notation @address@hidden@address@hidden
-is a floating-point number whose digits are in the specified radix.
-Note that the @samp{.} is more aptly referred to as a ``radix point''
-than as a decimal point in this case. The number @samp{8#123.4567} is
-defined as @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can
-use @samp{e} notation to write a non-decimal number in scientific
-notation. The exponent is written in decimal, and is considered to be a
-power of the radix: @samp{8#1234567e-4}. If the radix is 15 or above,
-the letter @samp{e} is a digit, so scientific notation must be written
-out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
-Modes Tutorial explore some of the properties of non-decimal floats.
-
address@hidden Complex Numbers, Infinities, Floats, Data Types
address@hidden Complex Numbers
-
address@hidden
address@hidden Complex numbers
-There are two supported formats for complex numbers: rectangular and
-polar. The default format is rectangular, displayed in the form
address@hidden(@var{real},@var{imag})} where @var{real} is the real part and
address@hidden is the imaginary part, each of which may be any real number.
-Rectangular complex numbers can also be displayed in @address@hidden@var{b}i}
-notation; @pxref{Complex Formats}.
-
-Polar complex numbers are displayed in the form
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'
-where @var{r} is the nonnegative magnitude and
address@hidden @math{\theta}
address@hidden @var{theta}
-is the argument or phase angle. The range of
address@hidden @math{\theta}
address@hidden @var{theta}
-depends on the current angular mode (@pxref{Angular Modes}); it is
-generally between @mathit{-180} and @mathit{+180} degrees or the equivalent
range
-in radians.
-
-Complex numbers are entered in stages using incomplete objects.
address@hidden Objects}.
-
-Operations on rectangular complex numbers yield rectangular complex
-results, and similarly for polar complex numbers. Where the two types
-are mixed, or where new complex numbers arise (as for the square root of
-a negative real), the current @dfn{Polar mode} is used to determine the
-type. @xref{Polar Mode}.
-
-A complex result in which the imaginary part is zero (or the phase angle
-is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
-number.
-
address@hidden Infinities, Vectors and Matrices, Complex Numbers, Data Types
address@hidden Infinities
-
address@hidden
address@hidden Infinity
address@hidden @code{inf} variable
address@hidden @code{uinf} variable
address@hidden @code{nan} variable
address@hidden inf
address@hidden uinf
address@hidden nan
-The word @code{inf} represents the mathematical concept of @dfn{infinity}.
-Calc actually has three slightly different infinity-like values:
address@hidden, @code{uinf}, and @code{nan}. These are just regular
-variable names (@pxref{Variables}); you should avoid using these
-names for your own variables because Calc gives them special
-treatment. Infinities, like all variable names, are normally
-entered using algebraic entry.
-
-Mathematically speaking, it is not rigorously correct to treat
-``infinity'' as if it were a number, but mathematicians often do
-so informally. When they say that @samp{1 / inf = 0}, what they
-really mean is that @expr{1 / x}, as @expr{x} becomes larger and
-larger, becomes arbitrarily close to zero. So you can imagine
-that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
-would go all the way to zero. Similarly, when they say that
address@hidden(inf) = inf}, they mean that
address@hidden @math{e^x}
address@hidden @expr{exp(x)}
-grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
-stands for an infinitely negative real value; for example, we say that
address@hidden(-inf) = 0}. You can have an infinity pointing in any
-direction on the complex plane: @samp{sqrt(-inf) = i inf}.
-
-The same concept of limits can be used to define @expr{1 / 0}. We
-really want the value that @expr{1 / x} approaches as @expr{x}
-approaches zero. But if all we have is @expr{1 / 0}, we can't
-tell which direction @expr{x} was coming from. If @expr{x} was
-positive and decreasing toward zero, then we should say that
address@hidden / 0 = inf}. But if @expr{x} was negative and increasing
-toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
-could be an imaginary number, giving the answer @samp{i inf} or
address@hidden inf}. Calc uses the special symbol @samp{uinf} to mean
address@hidden infinity}, i.e., a value which is infinitely
-large but with an unknown sign (or direction on the complex plane).
-
-Calc actually has three modes that say how infinities are handled.
-Normally, infinities never arise from calculations that didn't
-already have them. Thus, @expr{1 / 0} is treated simply as an
-error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
-command (@pxref{Infinite Mode}) enables a mode in which
address@hidden / 0} evaluates to @code{uinf} instead. There is also
-an alternative type of infinite mode which says to treat zeros
-as if they were positive, so that @samp{1 / 0 = inf}. While this
-is less mathematically correct, it may be the answer you want in
-some cases.
-
-Since all infinities are ``as large'' as all others, Calc simplifies,
-e.g., @samp{5 inf} to @samp{inf}. Another example is
address@hidden - inf = -inf}, where the @samp{-inf} is so large that
-adding a finite number like five to it does not affect it.
-Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
-that variables like @code{a} always stand for finite quantities.
-Just to show that infinities really are all the same size,
-note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
-notation.
-
-It's not so easy to define certain formulas like @samp{0 * inf} and
address@hidden / inf}. Depending on where these zeros and infinities
-came from, the answer could be literally anything. The latter
-formula could be the limit of @expr{x / x} (giving a result of one),
-or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
-or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
-to represent such an @dfn{indeterminate} value. (The name ``nan''
-comes from analogy with the ``NAN'' concept of IEEE standard
-arithmetic; it stands for ``Not A Number.'' This is somewhat of a
-misnomer, since @code{nan} @emph{does} stand for some number or
-infinity, it's just that @emph{which} number it stands for
-cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
-and @samp{inf / inf = nan}. A few other common indeterminate
-expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
address@hidden / 0 = nan} if you have turned on Infinite mode
-(as described above).
-
-Infinities are especially useful as parts of @dfn{intervals}.
address@hidden Forms}.
-
address@hidden Vectors and Matrices, Strings, Infinities, Data Types
address@hidden Vectors and Matrices
-
address@hidden
address@hidden Vectors
address@hidden Plain vectors
address@hidden Matrices
-The @dfn{vector} data type is flexible and general. A vector is simply a
-list of zero or more data objects. When these objects are numbers, the
-whole is a vector in the mathematical sense. When these objects are
-themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
-A vector which is not a matrix is referred to here as a @dfn{plain vector}.
-
-A vector is displayed as a list of values separated by commas and enclosed
-in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
-3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
-numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
-During algebraic entry, vectors are entered all at once in the usual
-brackets-and-commas form. Matrices may be entered algebraically as nested
-vectors, or using the shortcut notation @address@hidden, 2, 3; 4, 5, 6]}},
-with rows separated by semicolons. The commas may usually be omitted
-when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
-place of brackets: @address@hidden, 2, address@hidden, but the commas are
required in
-this case.
-
-Traditional vector and matrix arithmetic is also supported;
address@hidden Arithmetic} and @pxref{Matrix Functions}.
-Many other operations are applied to vectors element-wise. For example,
-the complex conjugate of a vector is a vector of the complex conjugates
-of its elements.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden vec
-Algebraic functions for building vectors include @samp{vec(a, b, c)}
-to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
address@hidden @math{n\times m}
address@hidden @address@hidden
-matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
-from 1 to @samp{n}.
-
address@hidden Strings, HMS Forms, Vectors and Matrices, Data Types
address@hidden Strings
-
address@hidden
address@hidden "
address@hidden Strings
address@hidden Character strings
-Character strings are not a special data type in the Calculator.
-Rather, a string is represented simply as a vector all of whose
-elements are integers in the range 0 to 255 (ASCII codes). You can
-enter a string at any time by pressing the @kbd{"} key. Quotation
-marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
-inside strings. Other notations introduced by backslashes are:
-
address@hidden
address@hidden
-\a 7 \^@@ 0
-\b 8 \^a-z 1-26
-\e 27 \^[ 27
-\f 12 \^\\ 28
-\n 10 \^] 29
-\r 13 \^^ 30
-\t 9 \^_ 31
- \^? 127
address@hidden group
address@hidden example
-
address@hidden
-Finally, a backslash followed by three octal digits produces any
-character from its ASCII code.
-
address@hidden d "
address@hidden calc-display-strings
-Strings are normally displayed in vector-of-integers form. The
address@hidden@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
-which any vectors of small integers are displayed as quoted strings
-instead.
-
-The backslash notations shown above are also used for displaying
-strings. Characters 128 and above are not translated by Calc; unless
-you have an Emacs modified for 8-bit fonts, these will show up in
-backslash-octal-digits notation. For characters below 32, and
-for character 127, Calc uses the backslash-letter combination if
-there is one, or otherwise uses a @samp{\^} sequence.
-
-The only Calc feature that uses strings is @dfn{compositions};
address@hidden Strings also provide a convenient
-way to do conversions between ASCII characters and integers.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden string
-There is a @code{string} function which provides a different display
-format for strings. Basically, @samp{string(@var{s})}, where @var{s}
-is a vector of integers in the proper range, is displayed as the
-corresponding string of characters with no surrounding quotation
-marks or other modifications. Thus @samp{string("ABC")} (or
address@hidden([65 66 67])}) will look like @samp{ABC} on the stack.
-This happens regardless of whether @address@hidden "}} has been used. The
-only way to turn it off is to use @kbd{d U} (unformatted language
-mode) which will display @samp{string("ABC")} instead.
-
-Control characters are displayed somewhat differently by @code{string}.
-Characters below 32, and character 127, are shown using @samp{^} notation
-(same as shown above, but without the backslash). The quote and
-backslash characters are left alone, as are characters 128 and above.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden bstring
-The @code{bstring} function is just like @code{string} except that
-the resulting string is breakable across multiple lines if it doesn't
-fit all on one line. Potential break points occur at every space
-character in the string.
-
address@hidden HMS Forms, Date Forms, Strings, Data Types
address@hidden HMS Forms
-
address@hidden
address@hidden Hours-minutes-seconds forms
address@hidden Degrees-minutes-seconds forms
address@hidden stands for Hours-Minutes-Seconds; when used as an angular
-argument, the interpretation is Degrees-Minutes-Seconds. All functions
-that operate on angles accept HMS forms. These are interpreted as
-degrees regardless of the current angular mode. It is also possible to
-use HMS as the angular mode so that calculated angles are expressed in
-degrees, minutes, and seconds.
-
address@hidden @@
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ' (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden " (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden h (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden o (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden m (HMS forms)
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s (HMS forms)
-The default format for HMS values is
address@hidden@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
address@hidden (for ``hours'') or
address@hidden (approximating the ``degrees'' symbol) are accepted as well as
address@hidden@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
-accepted in place of @samp{"}.
-The @var{hours} value is an integer (or integer-valued float).
-The @var{mins} value is an integer or integer-valued float between 0 and 59.
-The @var{secs} value is a real number between 0 (inclusive) and 60
-(exclusive). A positive HMS form is interpreted as @var{hours} +
address@hidden/60 + @var{secs}/3600. A negative HMS form is interpreted
-as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
-Display format for HMS forms is quite flexible. @xref{HMS Formats}.
-
-HMS forms can be added and subtracted. When they are added to numbers,
-the numbers are interpreted according to the current angular mode. HMS
-forms can also be multiplied and divided by real numbers. Dividing
-two HMS forms produces a real-valued ratio of the two angles.
-
address@hidden calc-time
address@hidden Time of day
-Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
-the stack as an HMS form.
-
address@hidden Date Forms, Modulo Forms, HMS Forms, Data Types
address@hidden Date Forms
-
address@hidden
address@hidden Date forms
-A @dfn{date form} represents a date and possibly an associated time.
-Simple date arithmetic is supported: Adding a number to a date
-produces a new date shifted by that many days; adding an HMS form to
-a date shifts it by that many hours. Subtracting two date forms
-computes the number of days between them (represented as a simple
-number). Many other operations, such as multiplying two date forms,
-are nonsensical and are not allowed by Calc.
-
-Date forms are entered and displayed enclosed in @samp{< >} brackets.
-The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
-or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
-Input is flexible; date forms can be entered in any of the usual
-notations for dates and times. @xref{Date Formats}.
-
-Date forms are stored internally as numbers, specifically the number
-of days since midnight on the morning of January 1 of the year 1 AD.
-If the internal number is an integer, the form represents a date only;
-if the internal number is a fraction or float, the form represents
-a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>}
-is represented by the number 726842.25. The standard precision of
-12 decimal digits is enough to ensure that a (reasonable) date and
-time can be stored without roundoff error.
-
-If the current precision is greater than 12, date forms will keep
-additional digits in the seconds position. For example, if the
-precision is 15, the seconds will keep three digits after the
-decimal point. Decreasing the precision below 12 may cause the
-time part of a date form to become inaccurate. This can also happen
-if astronomically high years are used, though this will not be an
-issue in everyday (or even everymillennium) use. Note that date
-forms without times are stored as exact integers, so roundoff is
-never an issue for them.
-
-You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
-(@code{calc-unpack}) commands to get at the numerical representation
-of a date form. @xref{Packing and Unpacking}.
-
-Date forms can go arbitrarily far into the future or past. Negative
-year numbers represent years BC. Calc uses a combination of the
-Gregorian and Julian calendars, following the history of Great
-Britain and the British colonies. This is the same calendar that
-is used by the @code{cal} program in most Unix implementations.
-
address@hidden Julian calendar
address@hidden Gregorian calendar
-Some historical background: The Julian calendar was created by
-Julius Caesar in the year 46 BC as an attempt to fix the gradual
-drift caused by the lack of leap years in the calendar used
-until that time. The Julian calendar introduced an extra day in
-all years divisible by four. After some initial confusion, the
-calendar was adopted around the year we call 8 AD. Some centuries
-later it became apparent that the Julian year of 365.25 days was
-itself not quite right. In 1582 Pope Gregory XIII introduced the
-Gregorian calendar, which added the new rule that years divisible
-by 100, but not by 400, were not to be considered leap years
-despite being divisible by four. Many countries delayed adoption
-of the Gregorian calendar because of religious differences;
-in Britain it was put off until the year 1752, by which time
-the Julian calendar had fallen eleven days behind the true
-seasons. So the switch to the Gregorian calendar in early
-September 1752 introduced a discontinuity: The day after
-Sep 2, 1752 is Sep 14, 1752. Calc follows this convention.
-To take another example, Russia waited until 1918 before
-adopting the new calendar, and thus needed to remove thirteen
-days (between Feb 1, 1918 and Feb 14, 1918). This means that
-Calc's reckoning will be inconsistent with Russian history between
-1752 and 1918, and similarly for various other countries.
-
-Today's timekeepers introduce an occasional ``leap second'' as
-well, but Calc does not take these minor effects into account.
-(If it did, it would have to report a non-integer number of days
-between, say, @samp{<12:00am Mon Jan 1, 1900>} and
address@hidden<12:00am Sat Jan 1, 2000>}.)
-
-Calc uses the Julian calendar for all dates before the year 1752,
-including dates BC when the Julian calendar technically had not
-yet been invented. Thus the claim that day number @mathit{-10000} is
-called ``August 16, 28 BC'' should be taken with a grain of salt.
-
-Please note that there is no ``year 0''; the day before
address@hidden<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are
-days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.
-
address@hidden Julian day counting
-Another day counting system in common use is, confusingly, also called
-``Julian.'' The Julian day number is the numbers of days since
-12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT)
-is @mathit{-1721423.5} (recall that Calc starts at midnight instead
-of noon). Thus to convert a Calc date code obtained by unpacking a
-date form into a Julian day number, simply add 1721423.5 after
-compensating for the time zone difference. The built-in @kbd{t J}
-command performs this conversion for you.
-
-The Julian day number is based on the Julian cycle, which was invented
-in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
-since it is involves the Julian calendar, but some have suggested that
-Scaliger named it in honor of his father, Julius Caesar Scaliger. The
-Julian cycle is based it on three other cycles: the indiction cycle,
-the Metonic cycle, and the solar cycle. The indiction cycle is a 15
-year cycle originally used by the Romans for tax purposes but later
-used to date medieval documents. The Metonic cycle is a 19 year
-cycle; 19 years is close to being a common multiple of a solar year
-and a lunar month, and so every 19 years the phases of the moon will
-occur on the same days of the year. The solar cycle is a 28 year
-cycle; the Julian calendar repeats itself every 28 years. The
-smallest time period which contains multiples of all three cycles is
-the least common multiple of 15 years, 19 years and 28 years, which
-(since they're pairwise relatively prime) is
address@hidden @math{15\times 19\times 28 = 7980} years.
address@hidden 15*19*28 = 7980 years.
-This is the length of a Julian cycle. Working backwards, the previous
-year in which all three cycles began was 4713 BC, and so Scalinger
-chose that year as the beginning of a Julian cycle. Since at the time
-there were no historical records from before 4713 BC, using this year
-as a starting point had the advantage of avoiding negative year
-numbers. In 1849, the astronomer John Herschel (son of William
-Herschel) suggested using the number of days since the beginning of
-the Julian cycle as an astronomical dating system; this idea was taken
-up by other astronomers. (At the time, noon was the start of the
-astronomical day. Herschel originally suggested counting the days
-since Jan 1, 4713 BC at noon Alexandria time; this was later amended to
-noon GMT.) Julian day numbering is largely used in astronomy.
-
address@hidden Unix time format
-The Unix operating system measures time as an integer number of
-seconds since midnight, Jan 1, 1970. To convert a Calc date
-value into a Unix time stamp, first subtract 719164 (the code
-for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
-seconds in a day) and press @kbd{R} to round to the nearest
-integer. If you have a date form, you can simply subtract the
-day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
-719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
-to convert from Unix time to a Calc date form. (Note that
-Unix normally maintains the time in the GMT time zone; you may
-need to subtract five hours to get New York time, or eight hours
-for California time. The same is usually true of Julian day
-counts.) The built-in @kbd{t U} command performs these
-conversions.
-
address@hidden Modulo Forms, Error Forms, Date Forms, Data Types
address@hidden Modulo Forms
-
address@hidden
address@hidden Modulo forms
-A @dfn{modulo form} is a real number which is taken modulo (i.e., within
-an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
-often arises in number theory. Modulo forms are written
address@hidden @tfn{mod} @var{M}',
-where @var{a} and @var{M} are real numbers or HMS forms, and
address@hidden @math{0 \le a < M}.
address@hidden @expr{0 <= a < @var{M}}.
-In many applications @expr{a} and @expr{M} will be
-integers but this is not required.
-
address@hidden
address@hidden M
address@hidden ignore
address@hidden M (modulo forms)
address@hidden
address@hidden mod
address@hidden ignore
address@hidden mod (operator)
-To create a modulo form during numeric entry, press the address@hidden
-key to enter the word @samp{mod}. As a special convenience, pressing
address@hidden a second time automatically enters the value of @expr{M}
-that was most recently used before. During algebraic entry, either
-type @samp{mod} by hand or press @kbd{M-m} (that's @address@hidden).
-Once again, pressing this a second time enters the current modulo.
-
-Modulo forms are not to be confused with the modulo operator @samp{%}.
-The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
-the result 7. Further computations treat this 7 as just a regular integer.
-The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
-further computations with this value are again reduced modulo 10 so that
-the result always lies in the desired range.
-
-When two modulo forms with identical @expr{M}'s are added or multiplied,
-the Calculator simply adds or multiplies the values, then reduces modulo
address@hidden If one argument is a modulo form and the other a plain number,
-the plain number is treated like a compatible modulo form. It is also
-possible to raise modulo forms to powers; the result is the value raised
-to the power, then reduced modulo @expr{M}. (When all values involved
-are integers, this calculation is done much more efficiently than
-actually computing the power and then reducing.)
-
address@hidden Modulo division
-Two modulo forms address@hidden @tfn{mod} @var{M}' and address@hidden
@tfn{mod} @var{M}'
-can be divided if @expr{a}, @expr{b}, and @expr{M} are all
-integers. The result is the modulo form which, when multiplied by
address@hidden @tfn{mod} @var{M}', produces address@hidden @tfn{mod} @var{M}'.
If
-there is no solution to this equation (which can happen only when
address@hidden is non-prime), or if any of the arguments are non-integers, the
-division is left in symbolic form. Other operations, such as square
-roots, are not yet supported for modulo forms. (Note that, although
address@hidden@tfn{(address@hidden @tfn{mod} @address@hidden)^.5}'} will
compute a ``modulo square root''
-in the sense of reducing
address@hidden @math{\sqrt a}
address@hidden @expr{sqrt(a)}
-modulo @expr{M}, this is not a useful definition from the
-number-theoretical point of view.)
-
-It is possible to mix HMS forms and modulo forms. For example, an
-HMS form modulo 24 could be used to manipulate clock times; an HMS
-form modulo 360 would be suitable for angles. Making the modulo @expr{M}
-also be an HMS form eliminates troubles that would arise if the angular
-mode were inadvertently set to Radians, in which case
address@hidden@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees
modulo
-24 radians!
-
-Modulo forms cannot have variables or formulas for components. If you
-enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
-to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
-
-You can use @kbd{v p} and @kbd{%} to modify modulo forms.
address@hidden and Unpacking}. @xref{Basic Arithmetic}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden makemod
-The algebraic function @samp{makemod(a, m)} builds the modulo form
address@hidden@samp{a mod m}}.
-
address@hidden Error Forms, Interval Forms, Modulo Forms, Data Types
address@hidden Error Forms
-
address@hidden
address@hidden Error forms
address@hidden Standard deviations
-An @dfn{error form} is a number with an associated standard
-deviation, as in @samp{2.3 +/- 0.12}. The notation
address@hidden address@hidden @tfn{+/-} @math{\sigma}'
address@hidden address@hidden @tfn{+/-} sigma'
-stands for an uncertain value which follows
-a normal or Gaussian distribution of mean @expr{x} and standard
-deviation or ``error''
address@hidden @math{\sigma}.
address@hidden @expr{sigma}.
-Both the mean and the error can be either numbers or
-formulas. Generally these are real numbers but the mean may also be
-complex. If the error is negative or complex, it is changed to its
-absolute value. An error form with zero error is converted to a
-regular number by the Calculator.
-
-All arithmetic and transcendental functions accept error forms as input.
-Operations on the mean-value part work just like operations on regular
-numbers. The error part for any function @expr{f(x)} (such as
address@hidden @math{\sin x}
address@hidden @expr{sin(x)})
-is defined by the error of @expr{x} times the derivative of @expr{f}
-evaluated at the mean value of @expr{x}. For a two-argument function
address@hidden(x,y)} (such as addition) the error is the square root of the sum
-of the squares of the errors due to @expr{x} and @expr{y}.
address@hidden
-$$ \eqalign{
- f(x \hbox{\code{ +/- }} \sigma)
- &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
- f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
- &= f(x,y) \hbox{\code{ +/- }}
- \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
- \right| \right)^2
- +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
- \right| \right)^2 } \cr
-} $$
address@hidden tex
-Note that this
-definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
-A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
-is not the same as @samp{(2 +/- 1)^2}; the former represents the product
-of two independent values which happen to have the same probability
-distributions, and the latter is the product of one random value with itself.
-The former will produce an answer with less error, since on the average
-the two independent errors can be expected to cancel out.
-
-Consult a good text on error analysis for a discussion of the proper use
-of standard deviations. Actual errors often are neither Gaussian-distributed
-nor uncorrelated, and the above formulas are valid only when errors
-are small. As an example, the error arising from
address@hidden address@hidden(address@hidden @tfn{+/-} @address@hidden)}'
address@hidden address@hidden(address@hidden @tfn{+/-} @address@hidden)}'
-is
address@hidden address@hidden @tfn{abs(cos(address@hidden@tfn{))}'.
address@hidden address@hidden @tfn{abs(cos(address@hidden@tfn{))}'.
-When @expr{x} is close to zero,
address@hidden @math{\cos x}
address@hidden @expr{cos(x)}
-is close to one so the error in the sine is close to
address@hidden @math{\sigma};
address@hidden @expr{sigma};
-this makes sense, since
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
-is approximately @expr{x} near zero, so a given error in @expr{x} will
-produce about the same error in the sine. Likewise, near 90 degrees
address@hidden @math{\cos x}
address@hidden @expr{cos(x)}
-is nearly zero and so the computed error is
-small: The sine curve is nearly flat in that region, so an error in @expr{x}
-has relatively little effect on the value of
address@hidden @math{\sin x}.
address@hidden @expr{sin(x)}.
-However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
-Calc will report zero error! We get an obviously wrong result because
-we have violated the small-error approximation underlying the error
-analysis. If the error in @expr{x} had been small, the error in
address@hidden @math{\sin x}
address@hidden @expr{sin(x)}
-would indeed have been negligible.
-
address@hidden
address@hidden p
address@hidden ignore
address@hidden p (error forms)
address@hidden +/-
-To enter an error form during regular numeric entry, use the @kbd{p}
-(``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
-typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
address@hidden command!) Within an algebraic formula, you can press @kbd{M-+}
to
-type the @samp{+/-} symbol, or type it out by hand.
-
-Error forms and complex numbers can be mixed; the formulas shown above
-are used for complex numbers, too; note that if the error part evaluates
-to a complex number its absolute value (or the square root of the sum of
-the squares of the absolute values of the two error contributions) is
-used. Mathematically, this corresponds to a radially symmetric Gaussian
-distribution of numbers on the complex plane. However, note that Calc
-considers an error form with real components to represent a real number,
-not a complex distribution around a real mean.
-
-Error forms may also be composed of HMS forms. For best results, both
-the mean and the error should be HMS forms if either one is.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden sdev
-The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
-
address@hidden Interval Forms, Incomplete Objects, Error Forms, Data Types
address@hidden Interval Forms
-
address@hidden
address@hidden Interval forms
-An @dfn{interval} is a subset of consecutive real numbers. For example,
-the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
-inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
-obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
-you multiply some number in the range @samp{[2 ..@: 4]} by some other
-number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
-from 1 to 8. Interval arithmetic is used to get a worst-case estimate
-of the possible range of values a computation will produce, given the
-set of possible values of the input.
-
address@hidden
-Calc supports several varieties of intervals, including @dfn{closed}
-intervals of the type shown above, @dfn{open} intervals such as
address@hidden(2 ..@: 4)}, which represents the range of numbers from 2 to 4
address@hidden, and @dfn{semi-open} intervals in which one end
-uses a round parenthesis and the other a square bracket. In mathematical
-terms,
address@hidden ..@: 4]} means @expr{2 <= x <= 4}, whereas
address@hidden ..@: 4)} represents @expr{2 <= x < 4},
address@hidden(2 ..@: 4]} represents @expr{2 < x <= 4}, and
address@hidden(2 ..@: 4)} represents @expr{2 < x < 4}.
address@hidden ifnottex
address@hidden
-Calc supports several varieties of intervals, including \dfn{closed}
-intervals of the type shown above, \dfn{open} intervals such as
-\samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
-\emph{exclusive}, and \dfn{semi-open} intervals in which one end
-uses a round parenthesis and the other a square bracket. In mathematical
-terms,
-$$ \eqalign{
- [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
- [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
- (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
- (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
-} $$
address@hidden tex
-
-The lower and upper limits of an interval must be either real numbers
-(or HMS or date forms), or symbolic expressions which are assumed to be
-real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
-must be less than the upper limit. A closed interval containing only
-one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
-automatically. An interval containing no values at all (such as
address@hidden ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
-guaranteed to behave well when used in arithmetic. Note that the
-interval @samp{[3 .. inf)} represents all real numbers greater than
-or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
-In fact, @samp{[-inf .. inf]} represents all real numbers including
-the real infinities.
-
-Intervals are entered in the notation shown here, either as algebraic
-formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
-In algebraic formulas, multiple periods in a row are collected from
-left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
-rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
-get the other interpretation. If you omit the lower or upper limit,
-a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
-
-Infinite mode also affects operations on intervals
-(@pxref{Infinities}). Calc will always introduce an open infinity,
-as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
address@hidden@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
-otherwise they are left unevaluated. Note that the ``direction'' of
-a zero is not an issue in this case since the zero is always assumed
-to be continuous with the rest of the interval. For intervals that
-contain zero inside them Calc is forced to give the result,
address@hidden / (-2 .. 2) = [-inf .. inf]}.
-
-While it may seem that intervals and error forms are similar, they are
-based on entirely different concepts of inexact quantities. An error
-form
address@hidden address@hidden @tfn{+/-} @math{\sigma}'
address@hidden address@hidden @tfn{+/-} @var{sigma}'
-means a variable is random, and its value could
-be anything but is ``probably'' within one
address@hidden @math{\sigma}
address@hidden @var{sigma}
-of the mean value @expr{x}. An interval
address@hidden@var{a} @tfn{..@:} @address@hidden' means a
-variable's value is unknown, but guaranteed to lie in the specified
-range. Error forms are statistical or ``average case'' approximations;
-interval arithmetic tends to produce ``worst case'' bounds on an
-answer.
-
-Intervals may not contain complex numbers, but they may contain
-HMS forms or date forms.
-
address@hidden Operations}, for commands that interpret interval forms
-as subsets of the set of real numbers.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden intv
-The algebraic function @samp{intv(n, a, b)} builds an interval form
-from @samp{a} to @samp{b}; @samp{n} is an integer code which must
-be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
-3 for @samp{[..]}.
-
-Please note that in fully rigorous interval arithmetic, care would be
-taken to make sure that the computation of the lower bound rounds toward
-minus infinity, while upper bound computations round toward plus
-infinity. Calc's arithmetic always uses a round-to-nearest mode,
-which means that roundoff errors could creep into an interval
-calculation to produce intervals slightly smaller than they ought to
-be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
-should yield the interval @samp{[1..2]} again, but in fact it yields the
-(slightly too small) interval @samp{[1..1.9999999]} due to roundoff
-error.
-
address@hidden Incomplete Objects, Variables, Interval Forms, Data Types
address@hidden Incomplete Objects
-
address@hidden
address@hidden
address@hidden [ ]
address@hidden ignore
address@hidden [
address@hidden
address@hidden ( )
address@hidden ignore
address@hidden (
address@hidden ,
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ]
address@hidden
address@hidden @null
address@hidden ignore
address@hidden )
address@hidden Incomplete vectors
address@hidden Incomplete complex numbers
address@hidden Incomplete interval forms
-When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
-vector, respectively, the effect is to push an @dfn{incomplete} complex
-number or vector onto the stack. The @kbd{,} key adds the value(s) at
-the top of the stack onto the current incomplete object. The @kbd{)}
-and @kbd{]} keys ``close'' the incomplete object after adding any values
-on the top of the stack in front of the incomplete object.
-
-As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
-pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
-pushes the complex number @samp{(1, 1.414)} (approximately).
-
-If several values lie on the stack in front of the incomplete object,
-all are collected and appended to the object. Thus the @kbd{,} key
-is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
-prefer the equivalent @key{SPC} key to @key{RET}.
-
-As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
address@hidden,} adds a zero or duplicates the preceding value in the list being
-formed. Typing @key{DEL} during incomplete entry removes the last item
-from the list.
-
address@hidden ;
-The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
-numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
-creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
-equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
-
address@hidden ..
address@hidden calc-dots
-Incomplete entry is also used to enter intervals. For example,
address@hidden 2 ..@: 4 )} enters a semi-open interval. Note that when you type
-the first period, it will be interpreted as a decimal point, but when
-you type a second period immediately afterward, it is re-interpreted as
-part of the interval symbol. Typing @kbd{..} corresponds to executing
-the @code{calc-dots} command.
-
-If you find incomplete entry distracting, you may wish to enter vectors
-and complex numbers as algebraic formulas by pressing the apostrophe key.
-
address@hidden Variables, Formulas, Incomplete Objects, Data Types
address@hidden Variables
-
address@hidden
address@hidden Variables, in formulas
-A @dfn{variable} is somewhere between a storage register on a conventional
-calculator, and a variable in a programming language. (In fact, a Calc
-variable is really just an Emacs Lisp variable that contains a Calc number
-or formula.) A variable's name is normally composed of letters and digits.
-Calc also allows apostrophes and @code{#} signs in variable names.
-(The Calc variable @code{foo} corresponds to the Emacs Lisp variable
address@hidden, but unless you access the variable from within Emacs
-Lisp, you don't need to worry about it. Variable names in algebraic
-formulas implicitly have @samp{var-} prefixed to their names. The
address@hidden character in variable names used in algebraic formulas
-corresponds to a dash @samp{-} in the Lisp variable name. If the name
-contains any dashes, the prefix @samp{var-} is @emph{not} automatically
-added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
-refer to the same variable.)
-
-In a command that takes a variable name, you can either type the full
-name of a variable, or type a single digit to use one of the special
-convenience variables @code{q0} through @code{q9}. For example,
address@hidden s s 2} stores the number 3 in variable @code{q2}, and
address@hidden@kbd{3 s s foo @key{RET}}} stores that number in variable
address@hidden
-
-To push a variable itself (as opposed to the variable's value) on the
-stack, enter its name as an algebraic expression using the apostrophe
-(@key{'}) key.
-
address@hidden =
address@hidden calc-evaluate
address@hidden Evaluation of variables in a formula
address@hidden Variables, evaluation
address@hidden Formulas, evaluation
-The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
-replacing all variables in the formula which have been given values by a
address@hidden or @code{calc-let} command by their stored values.
-Other variables are left alone. Thus a variable that has not been
-stored acts like an abstract variable in algebra; a variable that has
-been stored acts more like a register in a traditional calculator.
-With a positive numeric prefix argument, @kbd{=} evaluates the top
address@hidden stack entries; with a negative argument, @kbd{=} evaluates
-the @var{n}th stack entry.
-
address@hidden @code{e} variable
address@hidden @code{pi} variable
address@hidden @code{i} variable
address@hidden @code{phi} variable
address@hidden @code{gamma} variable
address@hidden e
address@hidden pi
address@hidden i
address@hidden phi
address@hidden gamma
-A few variables are called @dfn{special constants}. Their names are
address@hidden, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
-(@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
-their values are calculated if necessary according to the current precision
-or complex polar mode. If you wish to use these symbols for other purposes,
-simply undefine or redefine them using @code{calc-store}.
-
-The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
-infinite or indeterminate values. It's best not to use them as
-regular variables, since Calc uses special algebraic rules when
-it manipulates them. Calc displays a warning message if you store
-a value into any of these special variables.
-
address@hidden and Recall}, for a discussion of commands dealing with variables.
-
address@hidden Formulas, , Variables, Data Types
address@hidden Formulas
-
address@hidden
address@hidden Formulas
address@hidden Expressions
address@hidden Operators in formulas
address@hidden Precedence of operators
-When you press the apostrophe key you may enter any expression or formula
-in algebraic form. (Calc uses the terms ``expression'' and ``formula''
-interchangeably.) An expression is built up of numbers, variable names,
-and function calls, combined with various arithmetic operators.
-Parentheses may
-be used to indicate grouping. Spaces are ignored within formulas, except
-that spaces are not permitted within variable names or numbers.
-Arithmetic operators, in order from highest to lowest precedence, and
-with their equivalent function names, are:
-
address@hidden address@hidden (subscripts);
-
-postfix @samp{%} address@hidden (as in @samp{25% = 0.25});
-
-prefix @samp{+} and @samp{-} address@hidden (as in @samp{-x})
-and prefix @samp{!} address@hidden (logical ``not,'' as in @samp{!x});
-
address@hidden/-} address@hidden (the standard deviation symbol) and
address@hidden address@hidden (the symbol for modulo forms);
-
-postfix @samp{!} address@hidden (factorial, as in @samp{n!})
-and postfix @samp{!!} address@hidden (double factorial);
-
address@hidden address@hidden (raised-to-the-power-of);
-
address@hidden address@hidden;
-
address@hidden/} address@hidden, @samp{%} address@hidden (modulo), and
address@hidden address@hidden (integer division);
-
-infix @samp{+} address@hidden and @samp{-} address@hidden (as in @samp{x-y});
-
address@hidden|} address@hidden (vector concatenation);
-
-relations @samp{=} address@hidden, @samp{!=} address@hidden, @samp{<}
address@hidden,
address@hidden>} address@hidden, @samp{<=} address@hidden, and @samp{>=}
address@hidden;
-
address@hidden&&} address@hidden (logical ``and'');
-
address@hidden||} address@hidden (logical ``or'');
-
-the C-style ``if'' operator @samp{a?b:c} address@hidden;
-
address@hidden address@hidden (rewrite pattern ``not'');
-
address@hidden&&&} address@hidden (rewrite pattern ``and'');
-
address@hidden|||} address@hidden (rewrite pattern ``or'');
-
address@hidden:=} address@hidden (for assignments and rewrite rules);
-
address@hidden::} address@hidden (rewrite pattern condition);
-
address@hidden>} address@hidden
-
-Note that, unlike in usual computer notation, multiplication binds more
-strongly than division: @samp{a*b/c*d} is equivalent to
address@hidden @math{a b \over c d}.
address@hidden @expr{(a*b)/(c*d)}.
-
address@hidden Multiplication, implicit
address@hidden Implicit multiplication
-The multiplication sign @samp{*} may be omitted in many cases. In particular,
-if the righthand side is a number, variable name, or parenthesized
-expression, the @samp{*} may be omitted. Implicit multiplication has the
-same precedence as the explicit @samp{*} operator. The one exception to
-the rule is that a variable name followed by a parenthesized expression,
-as in @samp{f(x)},
-is interpreted as a function call, not an implicit @samp{*}. In many
-cases you must use a space if you omit the @samp{*}: @samp{2a} is the
-same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
-is a variable called @code{ab}, @emph{not} the product of @samp{a} and
address@hidden Also note that @samp{f (x)} is still a function call.
-
address@hidden Implicit comma in vectors
-The rules are slightly different for vectors written with square brackets.
-In vectors, the space character is interpreted (like the comma) as a
-separator of elements of the vector. Thus @address@hidden 2a b+c d ]}} is
-equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
-to @samp{2*a*b + c*d}.
-Note that spaces around the brackets, and around explicit commas, are
-ignored. To force spaces to be interpreted as multiplication you can
-enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
-interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
-between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
-
-Vectors that contain commas (not embedded within nested parentheses or
-brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
-of two elements. Also, if it would be an error to treat spaces as
-separators, but not otherwise, then Calc will ignore spaces:
address@hidden@samp{[a - b]}} is a vector of one element, but @address@hidden
-b]}} is
-a vector of two elements. Finally, vectors entered with curly braces
-instead of square brackets do not give spaces any special treatment.
-When Calc displays a vector that does not contain any commas, it will
-insert parentheses if necessary to make the meaning clear:
address@hidden@samp{[(a b)]}}.
-
-The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
-or five modulo minus-two? Calc always interprets the leftmost symbol as
-an infix operator preferentially (modulo, in this case), so you would
-need to write @samp{(5%)-2} to get the former interpretation.
-
address@hidden Function call notation
-A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
address@hidden corresponds to the Emacs Lisp function @code{calcFunc-foo},
-but unless you access the function from within Emacs Lisp, you don't
-need to worry about it.) Most mathematical Calculator commands like
address@hidden have function equivalents like @code{sin}.
-If no Lisp function is defined for a function called by a formula, the
-call is left as it is during algebraic manipulation: @samp{f(x+y)} is
-left alone. Beware that many innocent-looking short names like @code{in}
-and @code{re} have predefined meanings which could surprise you; however,
-single letters or single letters followed by digits are always safe to
-use for your own function names. @xref{Function Index}.
-
-In the documentation for particular commands, the notation @kbd{H S}
-(@code{calc-sinh}) address@hidden means that the key sequence @kbd{H S}, the
-command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
-represent the same operation.
-
-Commands that interpret (``parse'') text as algebraic formulas include
-algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
-the contents of the editing buffer when you finish, the @kbd{C-x * g}
-and @address@hidden * r}} commands, the @kbd{C-y} command, the X window system
-``paste'' mouse operation, and Embedded mode. All of these operations
-use the same rules for parsing formulas; in particular, language modes
-(@pxref{Language Modes}) affect them all in the same way.
-
-When you read a large amount of text into the Calculator (say a vector
-which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
-you may wish to include comments in the text. Calc's formula parser
-ignores the symbol @samp{%%} and anything following it on a line:
-
address@hidden
-[ a + b, %% the sum of "a" and "b"
- c + d,
- %% last line is coming up:
- e + f ]
address@hidden example
-
address@hidden
-This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
-
address@hidden Tables}, for a way to create your own operators and other
-input notations. @xref{Compositions}, for a way to create new display
-formats.
-
address@hidden, for commands for manipulating formulas symbolically.
-
address@hidden Stack and Trail, Mode Settings, Data Types, Top
address@hidden Stack and Trail Commands
-
address@hidden
-This chapter describes the Calc commands for manipulating objects on the
-stack and in the trail buffer. (These commands operate on objects of any
-type, such as numbers, vectors, formulas, and incomplete objects.)
-
address@hidden
-* Stack Manipulation::
-* Editing Stack Entries::
-* Trail Commands::
-* Keep Arguments::
address@hidden menu
-
address@hidden Stack Manipulation, Editing Stack Entries, Stack and Trail,
Stack and Trail
address@hidden Stack Manipulation Commands
-
address@hidden
address@hidden @key{RET}
address@hidden @key{SPC}
address@hidden calc-enter
address@hidden Duplicating stack entries
-To duplicate the top object on the stack, press @key{RET} or @key{SPC}
-(two equivalent keys for the @code{calc-enter} command).
-Given a positive numeric prefix argument, these commands duplicate
-several elements at the top of the stack.
-Given a negative argument,
-these commands duplicate the specified element of the stack.
-Given an argument of zero, they duplicate the entire stack.
-For example, with @samp{10 20 30} on the stack,
address@hidden creates @samp{10 20 30 30},
address@hidden 2 @key{RET}} creates @samp{10 20 30 20 30},
address@hidden - 2 @key{RET}} creates @samp{10 20 30 20}, and
address@hidden 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
-
address@hidden @key{LFD}
address@hidden calc-over
-The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
-have it, else on @kbd{C-j}) is like @code{calc-enter}
-except that the sign of the numeric prefix argument is interpreted
-oppositely. Also, with no prefix argument the default argument is 2.
-Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
-are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
address@hidden 20 30 20}.
-
address@hidden @key{DEL}
address@hidden C-d
address@hidden calc-pop
address@hidden Removing stack entries
address@hidden Deleting stack entries
-To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
-The @kbd{C-d} key is a synonym for @key{DEL}.
-(If the top element is an incomplete object with at least one element, the
-last element is removed from it.) Given a positive numeric prefix argument,
-several elements are removed. Given a negative argument, the specified
-element of the stack is deleted. Given an argument of zero, the entire
-stack is emptied.
-For example, with @samp{10 20 30} on the stack,
address@hidden leaves @samp{10 20},
address@hidden 2 @key{DEL}} leaves @samp{10},
address@hidden - 2 @key{DEL}} leaves @samp{10 30}, and
address@hidden 0 @key{DEL}} leaves an empty stack.
-
address@hidden address@hidden
address@hidden calc-pop-above
-The @address@hidden (@code{calc-pop-above}) command is to @key{DEL} what
address@hidden is to @key{RET}: It interprets the sign of the numeric
-prefix argument in the opposite way, and the default argument is 2.
-Thus @address@hidden by itself removes the second-from-top stack element,
-leaving the first, third, fourth, and so on; @kbd{M-3 address@hidden deletes
-the third stack element.
-
address@hidden @key{TAB}
address@hidden calc-roll-down
-To exchange the top two elements of the stack, press @key{TAB}
-(@code{calc-roll-down}). Given a positive numeric prefix argument, the
-specified number of elements at the top of the stack are rotated downward.
-Given a negative argument, the entire stack is rotated downward the specified
-number of times. Given an argument of zero, the entire stack is reversed
-top-for-bottom.
-For example, with @samp{10 20 30 40 50} on the stack,
address@hidden creates @samp{10 20 30 50 40},
address@hidden 3 @key{TAB}} creates @samp{10 20 50 30 40},
address@hidden - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
address@hidden 0 @key{TAB}} creates @samp{50 40 30 20 10}.
-
address@hidden address@hidden
address@hidden calc-roll-up
-The command @address@hidden (@code{calc-roll-up}) is analogous to @key{TAB}
-except that it rotates upward instead of downward. Also, the default
-with no prefix argument is to rotate the top 3 elements.
-For example, with @samp{10 20 30 40 50} on the stack,
address@hidden@key{TAB}} creates @samp{10 20 40 50 30},
address@hidden 4 address@hidden creates @samp{10 30 40 50 20},
address@hidden - 2 address@hidden creates @samp{30 40 50 10 20}, and
address@hidden 0 address@hidden creates @samp{50 40 30 20 10}.
-
-A good way to view the operation of @key{TAB} and @address@hidden is in
-terms of moving a particular element to a new position in the stack.
-With a positive argument @var{n}, @key{TAB} moves the top stack
-element down to level @var{n}, making room for it by pulling all the
-intervening stack elements toward the top. @address@hidden moves the
-element at level @var{n} up to the top. (Compare with @key{LFD},
-which copies instead of moving the element in level @var{n}.)
-
-With a negative argument @address@hidden, @key{TAB} rotates the stack
-to move the object in level @var{n} to the deepest place in the
-stack, and the object in level @address@hidden to the top. @address@hidden
-rotates the deepest stack element to be in level @mathit{n}, also
-putting the top stack element in level @address@hidden
-
address@hidden Subformulas}, for a way to apply these commands to
-any portion of a vector or formula on the stack.
-
address@hidden Editing Stack Entries, Trail Commands, Stack Manipulation, Stack
and Trail
address@hidden Editing Stack Entries
-
address@hidden
address@hidden `
address@hidden calc-edit
address@hidden calc-edit-finish
address@hidden Editing the stack with Emacs
-The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary
-buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using
-regular Emacs commands. With a numeric prefix argument, it edits the
-specified number of stack entries at once. (An argument of zero edits
-the entire stack; a negative argument edits one specific stack entry.)
-
-When you are done editing, press @kbd{C-c C-c} to finish and return
-to Calc. The @key{RET} and @key{LFD} keys also work to finish most
-sorts of editing, though in some cases Calc leaves @key{RET} with its
-usual meaning (``insert a newline'') if it's a situation where you
-might want to insert new lines into the editing buffer.
-
-When you finish editing, the Calculator parses the lines of text in
-the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
-original stack elements in the original buffer with these new values,
-then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
-continues to exist during editing, but for best results you should be
-careful not to change it until you have finished the edit. You can
-also cancel the edit by killing the buffer with @kbd{C-x k}.
-
-The formula is normally reevaluated as it is put onto the stack.
-For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
address@hidden C-c} will push 5 on the stack. If you use @key{LFD} to
-finish, Calc will put the result on the stack without evaluating it.
-
-If you give a prefix argument to @kbd{C-c C-c},
-Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
-back to that buffer and continue editing if you wish. However, you
-should understand that if you initiated the edit with @kbd{`}, the
address@hidden C-c} operation will be programmed to replace the top of the
-stack with the new edited value, and it will do this even if you have
-rearranged the stack in the meanwhile. This is not so much of a problem
-with other editing commands, though, such as @kbd{s e}
-(@code{calc-edit-variable}; @pxref{Operations on Variables}).
-
-If the @code{calc-edit} command involves more than one stack entry,
-each line of the @samp{*Calc Edit*} buffer is interpreted as a
-separate formula. Otherwise, the entire buffer is interpreted as
-one formula, with line breaks ignored. (You can use @kbd{C-o} or
address@hidden C-j} to insert a newline in the buffer without pressing
@key{RET}.)
-
-The @kbd{`} key also works during numeric or algebraic entry. The
-text entered so far is moved to the @code{*Calc Edit*} buffer for
-more extensive editing than is convenient in the minibuffer.
-
address@hidden Trail Commands, Keep Arguments, Editing Stack Entries, Stack and
Trail
address@hidden Trail Commands
-
address@hidden
address@hidden Trail buffer
-The commands for manipulating the Calc Trail buffer are two-key sequences
-beginning with the @kbd{t} prefix.
-
address@hidden t d
address@hidden calc-trail-display
-The @kbd{t d} (@code{calc-trail-display}) command turns display of the
-trail on and off. Normally the trail display is toggled on if it was off,
-off if it was on. With a numeric prefix of zero, this command always
-turns the trail off; with a prefix of one, it always turns the trail on.
-The other trail-manipulation commands described here automatically turn
-the trail on. Note that when the trail is off values are still recorded
-there; they are simply not displayed. To set Emacs to turn the trail
-off by default, type @kbd{t d} and then save the mode settings with
address@hidden m} (@code{calc-save-modes}).
-
address@hidden t i
address@hidden calc-trail-in
address@hidden t o
address@hidden calc-trail-out
-The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
-(@code{calc-trail-out}) commands switch the cursor into and out of the
-Calc Trail window. In practice they are rarely used, since the commands
-shown below are a more convenient way to move around in the
-trail, and they work ``by remote control'' when the cursor is still
-in the Calculator window.
-
address@hidden Trail pointer
-There is a @dfn{trail pointer} which selects some entry of the trail at
-any given time. The trail pointer looks like a @samp{>} symbol right
-before the selected number. The following commands operate on the
-trail pointer in various ways.
-
address@hidden t y
address@hidden calc-trail-yank
address@hidden Retrieving previous results
-The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
-the trail and pushes it onto the Calculator stack. It allows you to
-re-use any previously computed value without retyping. With a numeric
-prefix argument @var{n}, it yanks the value @var{n} lines above the current
-trail pointer.
-
address@hidden t <
address@hidden calc-trail-scroll-left
address@hidden t >
address@hidden calc-trail-scroll-right
-The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
-(@code{calc-trail-scroll-right}) commands horizontally scroll the trail
-window left or right by one half of its width.
-
address@hidden t n
address@hidden calc-trail-next
address@hidden t p
address@hidden calc-trail-previous
address@hidden t f
address@hidden calc-trail-forward
address@hidden t b
address@hidden calc-trail-backward
-The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
-(@code{calc-trail-previous)} commands move the trail pointer down or up
-one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
-(@code{calc-trail-backward}) commands move the trail pointer down or up
-one screenful at a time. All of these commands accept numeric prefix
-arguments to move several lines or screenfuls at a time.
-
address@hidden t [
address@hidden calc-trail-first
address@hidden t ]
address@hidden calc-trail-last
address@hidden t h
address@hidden calc-trail-here
-The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
-(@code{calc-trail-last}) commands move the trail pointer to the first or
-last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
-moves the trail pointer to the cursor position; unlike the other trail
-commands, @kbd{t h} works only when Calc Trail is the selected window.
-
address@hidden t s
address@hidden calc-trail-isearch-forward
address@hidden t r
address@hidden calc-trail-isearch-backward
address@hidden
-The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
-(@code{calc-trail-isearch-backward}) commands perform an incremental
-search forward or backward through the trail. You can press @key{RET}
-to terminate the search; the trail pointer moves to the current line.
-If you cancel the search with @kbd{C-g}, the trail pointer stays where
-it was when the search began.
address@hidden ifnottex
address@hidden
-The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
-(@code{calc-trail-isearch-backward}) com\-mands perform an incremental
-search forward or backward through the trail. You can press @key{RET}
-to terminate the search; the trail pointer moves to the current line.
-If you cancel the search with @kbd{C-g}, the trail pointer stays where
-it was when the search began.
address@hidden tex
-
address@hidden t m
address@hidden calc-trail-marker
-The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
-line of text of your own choosing into the trail. The text is inserted
-after the line containing the trail pointer; this usually means it is
-added to the end of the trail. Trail markers are useful mainly as the
-targets for later incremental searches in the trail.
-
address@hidden t k
address@hidden calc-trail-kill
-The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
-from the trail. The line is saved in the Emacs kill ring suitable for
-yanking into another buffer, but it is not easy to yank the text back
-into the trail buffer. With a numeric prefix argument, this command
-kills the @var{n} lines below or above the selected one.
-
-The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
-elsewhere; @pxref{Vector and Matrix Formats}.
-
address@hidden Keep Arguments, , Trail Commands, Stack and Trail
address@hidden Keep Arguments
-
address@hidden
address@hidden K
address@hidden calc-keep-args
-The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
-the following command. It prevents that command from removing its
-arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
-the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
-the stack contains the arguments and the result: @samp{2 3 5}.
-
-With the exception of keyboard macros, this works for all commands that
-take arguments off the stack. (To avoid potentially unpleasant behavior,
-a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
-prefix called @emph{within} the keyboard macro will still take effect.)
-As another example, @kbd{K a s} simplifies a formula, pushing the
-simplified version of the formula onto the stack after the original
-formula (rather than replacing the original formula). Note that you
-could get the same effect by typing @address@hidden a s}, copying the
-formula and then simplifying the copy. One difference is that for a very
-large formula the time taken to format the intermediate copy in
address@hidden@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
-extra work.
-
-Even stack manipulation commands are affected. @key{TAB} works by
-popping two values and pushing them back in the opposite order,
-so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
-
-A few Calc commands provide other ways of doing the same thing.
-For example, @kbd{' sin($)} replaces the number on the stack with
-its sine using algebraic entry; to push the sine and keep the
-original argument you could use either @kbd{' sin($1)} or
address@hidden ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
-command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
-
-If you execute a command and then decide you really wanted to keep
-the argument, you can press @address@hidden (@code{calc-last-args}).
-This command pushes the last arguments that were popped by any command
-onto the stack. Note that the order of things on the stack will be
-different than with @kbd{K}: @kbd{2 @key{RET} 3 + address@hidden leaves
address@hidden 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
-
address@hidden Mode Settings, Arithmetic, Stack and Trail, Top
address@hidden Mode Settings
-
address@hidden
-This chapter describes commands that set modes in the Calculator.
-They do not affect the contents of the stack, although they may change
-the @emph{appearance} or @emph{interpretation} of the stack's contents.
-
address@hidden
-* General Mode Commands::
-* Precision::
-* Inverse and Hyperbolic::
-* Calculation Modes::
-* Simplification Modes::
-* Declarations::
-* Display Modes::
-* Language Modes::
-* Modes Variable::
-* Calc Mode Line::
address@hidden menu
-
address@hidden General Mode Commands, Precision, Mode Settings, Mode Settings
address@hidden General Mode Commands
-
address@hidden
address@hidden m m
address@hidden calc-save-modes
address@hidden Continuous memory
address@hidden Saving mode settings
address@hidden Permanent mode settings
address@hidden Calc init file, mode settings
-You can save all of the current mode settings in your Calc init file
-(the file given by the variable @code{calc-settings-file}, typically
address@hidden/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
-This will cause Emacs to reestablish these modes each time it starts up.
-The modes saved in the file include everything controlled by the @kbd{m}
-and @kbd{d} prefix keys, the current precision and binary word size,
-whether or not the trail is displayed, the current height of the Calc
-window, and more. The current interface (used when you type @kbd{C-x * *})
-is also saved. If there were already saved mode settings in the
-file, they are replaced. Otherwise, the new mode information is
-appended to the end of the file.
-
address@hidden m R
address@hidden calc-mode-record-mode
-The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
-record all the mode settings (as if by pressing @kbd{m m}) every
-time a mode setting changes. If the modes are saved this way, then this
-``automatic mode recording'' mode is also saved.
-Type @kbd{m R} again to disable this method of recording the mode
-settings. To turn it off permanently, the @kbd{m m} command will also be
-necessary. (If Embedded mode is enabled, other options for recording
-the modes are available; @pxref{Mode Settings in Embedded Mode}.)
-
address@hidden m F
address@hidden calc-settings-file-name
-The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
-choose a different file than the current value of @code{calc-settings-file}
-for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
-You are prompted for a file name. All Calc modes are then reset to
-their default values, then settings from the file you named are loaded
-if this file exists, and this file becomes the one that Calc will
-use in the future for commands like @kbd{m m}. The default settings
-file name is @file{~/.calc.el}. You can see the current file name by
-giving a blank response to the @kbd{m F} prompt. See also the
-discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
-
-If the file name you give is your user init file (typically
address@hidden/.emacs}), @kbd{m F} will not automatically load the new file.
This
-is because your user init file may contain other things you don't want
-to reread. You can give
-a numeric prefix argument of 1 to @kbd{m F} to force it to read the
-file no matter what. Conversely, an argument of @mathit{-1} tells
address@hidden F} @emph{not} to read the new file. An argument of 2 or
@mathit{-2}
-tells @kbd{m F} not to reset the modes to their defaults beforehand,
-which is useful if you intend your new file to have a variant of the
-modes present in the file you were using before.
-
address@hidden m x
address@hidden calc-always-load-extensions
-The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
-in which the first use of Calc loads the entire program, including all
-extensions modules. Otherwise, the extensions modules will not be loaded
-until the various advanced Calc features are used. Since this mode only
-has effect when Calc is first loaded, @kbd{m x} is usually followed by
address@hidden m} to make the mode-setting permanent. To load all of Calc just
-once, rather than always in the future, you can press @kbd{C-x * L}.
-
address@hidden m S
address@hidden calc-shift-prefix
-The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
-all of Calc's letter prefix keys may be typed shifted as well as unshifted.
-If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
-you might find it easier to turn this mode on so that you can type
address@hidden S} instead. When this mode is enabled, the commands that used to
-be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
-now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
-that the @kbd{v} prefix key always works both shifted and unshifted, and
-the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
-prefix is not affected by this mode. Press @kbd{m S} again to disable
-shifted-prefix mode.
-
address@hidden Precision, Inverse and Hyperbolic, General Mode Commands, Mode
Settings
address@hidden Precision
-
address@hidden
address@hidden p
address@hidden calc-precision
address@hidden Precision of calculations
-The @kbd{p} (@code{calc-precision}) command controls the precision to
-which floating-point calculations are carried. The precision must be
-at least 3 digits and may be arbitrarily high, within the limits of
-memory and time. This affects only floats: Integer and rational
-calculations are always carried out with as many digits as necessary.
-
-The @kbd{p} key prompts for the current precision. If you wish you
-can instead give the precision as a numeric prefix argument.
-
-Many internal calculations are carried to one or two digits higher
-precision than normal. Results are rounded down afterward to the
-current precision. Unless a special display mode has been selected,
-floats are always displayed with their full stored precision, i.e.,
-what you see is what you get. Reducing the current precision does not
-round values already on the stack, but those values will be rounded
-down before being used in any calculation. The @kbd{c 0} through
address@hidden 9} commands (@pxref{Conversions}) can be used to round an
-existing value to a new precision.
-
address@hidden Accuracy of calculations
-It is important to distinguish the concepts of @dfn{precision} and
address@hidden In the normal usage of these words, the number
-123.4567 has a precision of 7 digits but an accuracy of 4 digits.
-The precision is the total number of digits not counting leading
-or trailing zeros (regardless of the position of the decimal point).
-The accuracy is simply the number of digits after the decimal point
-(again not counting trailing zeros). In Calc you control the precision,
-not the accuracy of computations. If you were to set the accuracy
-instead, then calculations like @samp{exp(100)} would generate many
-more digits than you would typically need, while @samp{exp(-100)} would
-probably round to zero! In Calc, both these computations give you
-exactly 12 (or the requested number of) significant digits.
-
-The only Calc features that deal with accuracy instead of precision
-are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
-and the rounding functions like @code{floor} and @code{round}
-(@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
-deal with both precision and accuracy depending on the magnitudes
-of the numbers involved.
-
-If you need to work with a particular fixed accuracy (say, dollars and
-cents with two digits after the decimal point), one solution is to work
-with integers and an ``implied'' decimal point. For example, $8.99
-divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
-(actually $1.49833 with our implied decimal point); pressing @kbd{R}
-would round this to 150 cents, i.e., $1.50.
-
address@hidden, for still more on floating-point precision and related
-issues.
-
address@hidden Inverse and Hyperbolic, Calculation Modes, Precision, Mode
Settings
address@hidden Inverse and Hyperbolic Flags
-
address@hidden
address@hidden I
address@hidden calc-inverse
-There is no single-key equivalent to the @code{calc-arcsin} function.
-Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
-the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
-The @kbd{I} key actually toggles the Inverse Flag. When this flag
-is set, the word @samp{Inv} appears in the mode line.
-
address@hidden H
address@hidden calc-hyperbolic
-Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
-Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
-If both of these flags are set at once, the effect will be
address@hidden (The Hyperbolic flag is also used by some
-non-trigonometric commands; for example @kbd{H L} computes a base-10,
-instead of address@hidden, logarithm.)
-
-Command names like @code{calc-arcsin} are provided for completeness, and
-may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
-toggle the Inverse and/or Hyperbolic flags and then execute the
-corresponding base command (@code{calc-sin} in this case).
-
-The Inverse and Hyperbolic flags apply only to the next Calculator
-command, after which they are automatically cleared. (They are also
-cleared if the next keystroke is not a Calc command.) Digits you
-type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
-arguments for the next command, not as numeric entries. The same
-is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
-subtract and keep arguments).
-
-The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
-elsewhere. @xref{Keep Arguments}.
-
address@hidden Calculation Modes, Simplification Modes, Inverse and Hyperbolic,
Mode Settings
address@hidden Calculation Modes
-
address@hidden
-The commands in this section are two-key sequences beginning with
-the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
-The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
-(@pxref{Algebraic Entry}).
-
address@hidden
-* Angular Modes::
-* Polar Mode::
-* Fraction Mode::
-* Infinite Mode::
-* Symbolic Mode::
-* Matrix Mode::
-* Automatic Recomputation::
-* Working Message::
address@hidden menu
-
address@hidden Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
address@hidden Angular Modes
-
address@hidden
address@hidden Angular mode
-The Calculator supports three notations for angles: radians, degrees,
-and degrees-minutes-seconds. When a number is presented to a function
-like @code{sin} that requires an angle, the current angular mode is
-used to interpret the number as either radians or degrees. If an HMS
-form is presented to @code{sin}, it is always interpreted as
-degrees-minutes-seconds.
-
-Functions that compute angles produce a number in radians, a number in
-degrees, or an HMS form depending on the current angular mode. If the
-result is a complex number and the current mode is HMS, the number is
-instead expressed in degrees. (Complex-number calculations would
-normally be done in Radians mode, though. Complex numbers are converted
-to degrees by calculating the complex result in radians and then
-multiplying by 180 over @cpi{}.)
-
address@hidden m r
address@hidden calc-radians-mode
address@hidden m d
address@hidden calc-degrees-mode
address@hidden m h
address@hidden calc-hms-mode
-The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
-and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
-The current angular mode is displayed on the Emacs mode line.
-The default angular mode is Degrees.
-
address@hidden Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
address@hidden Polar Mode
-
address@hidden
address@hidden Polar mode
-The Calculator normally ``prefers'' rectangular complex numbers in the
-sense that rectangular form is used when the proper form can not be
-decided from the input. This might happen by multiplying a rectangular
-number by a polar one, by taking the square root of a negative real
-number, or by entering @kbd{( 2 @key{SPC} 3 )}.
-
address@hidden m p
address@hidden calc-polar-mode
-The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
-preference between rectangular and polar forms. In Polar mode, all
-of the above example situations would produce polar complex numbers.
-
address@hidden Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
address@hidden Fraction Mode
-
address@hidden
address@hidden Fraction mode
address@hidden Division of integers
-Division of two integers normally yields a floating-point number if the
-result cannot be expressed as an integer. In some cases you would
-rather get an exact fractional answer. One way to accomplish this is
-to use the @kbd{:} (@code{calc-fdiv}) address@hidden command, which
-divides the two integers on the top of the stack to produce a fraction:
address@hidden @key{RET} 4 :} produces @expr{3:2} even though
address@hidden @key{RET} 4 /} produces @expr{1.5}.
-
address@hidden m f
address@hidden calc-frac-mode
-To set the Calculator to produce fractional results for normal integer
-divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
-For example, @expr{8/4} produces @expr{2} in either mode,
-but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
-Float mode.
-
-At any time you can use @kbd{c f} (@code{calc-float}) to convert a
-fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
-float to a fraction. @xref{Conversions}.
-
address@hidden Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
address@hidden Infinite Mode
-
address@hidden
address@hidden Infinite mode
-The Calculator normally treats results like @expr{1 / 0} as errors;
-formulas like this are left in unsimplified form. But Calc can be
-put into a mode where such calculations instead produce ``infinite''
-results.
-
address@hidden m i
address@hidden calc-infinite-mode
-The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
-on and off. When the mode is off, infinities do not arise except
-in calculations that already had infinities as inputs. (One exception
-is that infinite open intervals like @samp{[0 .. inf)} can be
-generated; however, intervals closed at infinity (@samp{[0 .. inf]})
-will not be generated when Infinite mode is off.)
-
-With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
-an undirected infinity. @xref{Infinities}, for a discussion of the
-difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
-evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
-functions can also return infinities in this mode; for example,
address@hidden(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
-note that @samp{exp(inf) = inf} regardless of Infinite mode because
-this calculation has infinity as an input.
-
address@hidden Positive Infinite mode
-The @kbd{m i} command with a numeric prefix argument of zero,
-i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
-which zero is treated as positive instead of being directionless.
-Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
-Note that zero never actually has a sign in Calc; there are no
-separate representations for @mathit{+0} and @mathit{-0}. Positive
-Infinite mode merely changes the interpretation given to the
-single symbol, @samp{0}. One consequence of this is that, while
-you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
-is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
-
address@hidden Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
address@hidden Symbolic Mode
-
address@hidden
address@hidden Symbolic mode
address@hidden Inexact results
-Calculations are normally performed numerically wherever possible.
-For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
-algebraic expression, produces a numeric answer if the argument is a
-number or a symbolic expression if the argument is an expression:
address@hidden Q} pushes 1.4142 but @address@hidden'} x+1 @key{RET} Q} pushes
@samp{sqrt(x+1)}.
-
address@hidden m s
address@hidden calc-symbolic-mode
-In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
-command, functions which would produce inexact, irrational results are
-left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
address@hidden(2)}.
-
address@hidden N
address@hidden calc-eval-num
-The address@hidden (@code{calc-eval-num}) command evaluates numerically
-the expression at the top of the stack, by temporarily disabling
address@hidden and executing @kbd{=} (@code{calc-evaluate}).
-Given a numeric prefix argument, it also
-sets the floating-point precision to the specified value for the duration
-of the command.
-
-To evaluate a formula numerically without expanding the variables it
-contains, you can use the key sequence @kbd{m s a v m s} (this uses
address@hidden, which resimplifies but doesn't evaluate
-variables.)
-
address@hidden Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation
Modes
address@hidden Matrix and Scalar Modes
-
address@hidden
address@hidden Matrix mode
address@hidden Scalar mode
-Calc sometimes makes assumptions during algebraic manipulation that
-are awkward or incorrect when vectors and matrices are involved.
-Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
-modify its behavior around vectors in useful ways.
-
address@hidden m v
address@hidden calc-matrix-mode
-Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
-In this mode, all objects are assumed to be matrices unless provably
-otherwise. One major effect is that Calc will no longer consider
-multiplication to be commutative. (Recall that in matrix arithmetic,
address@hidden is not the same as @samp{B*A}.) This assumption affects
-rewrite rules and algebraic simplification. Another effect of this
-mode is that calculations that would normally produce constants like
-0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
-produce function calls that represent ``generic'' zero or identity
-matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
address@hidden(@var{a},@var{n})} returns @var{a} times an @address@hidden
-identity matrix; if @var{n} is omitted, it doesn't know what
-dimension to use and so the @code{idn} call remains in symbolic
-form. However, if this generic identity matrix is later combined
-with a matrix whose size is known, it will be converted into
-a true identity matrix of the appropriate size. On the other hand,
-if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
-will assume it really was a scalar after all and produce, e.g., 3.
-
-Press @kbd{m v} a second time to get Scalar mode. Here, objects are
-assumed @emph{not} to be vectors or matrices unless provably so.
-For example, normally adding a variable to a vector, as in
address@hidden, y, z] + a}, will leave the sum in symbolic form because
-as far as Calc knows, @samp{a} could represent either a number or
-another 3-vector. In Scalar mode, @samp{a} is assumed to be a
-non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
-
-Press @kbd{m v} a third time to return to the normal mode of operation.
-
-If you press @kbd{m v} with a numeric prefix argument @var{n}, you
-get a special ``dimensioned'' Matrix mode in which matrices of
-unknown size are assumed to be @address@hidden square matrices.
-Then, the function call @samp{idn(1)} will expand into an actual
-matrix rather than representing a ``generic'' matrix. Simply typing
address@hidden m v} will get you a square Matrix mode, in which matrices of
-unknown size are assumed to be square matrices of unspecified size.
-
address@hidden Declaring scalar variables
-Of course these modes are approximations to the true state of
-affairs, which is probably that some quantities will be matrices
-and others will be scalars. One solution is to ``declare''
-certain variables or functions to be scalar-valued.
address@hidden, to see how to make declarations in Calc.
-
-There is nothing stopping you from declaring a variable to be
-scalar and then storing a matrix in it; however, if you do, the
-results you get from Calc may not be valid. Suppose you let Calc
-get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
address@hidden, 2, 3]} in @samp{a}. The result would not be the same as
-for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
-your earlier promise to Calc that @samp{a} would be scalar.
-
-Another way to mix scalars and matrices is to use selections
-(@pxref{Selecting Subformulas}). Use Matrix mode when operating on
-your formula normally; then, to apply Scalar mode to a certain part
-of the formula without affecting the rest just select that part,
-change into Scalar mode and press @kbd{=} to resimplify the part
-under this mode, then change back to Matrix mode before deselecting.
-
address@hidden Automatic Recomputation, Working Message, Matrix Mode,
Calculation Modes
address@hidden Automatic Recomputation
-
address@hidden
-The @dfn{evaluates-to} operator, @samp{=>}, has the special
-property that any @samp{=>} formulas on the stack are recomputed
-whenever variable values or mode settings that might affect them
-are changed. @xref{Evaluates-To Operator}.
-
address@hidden m C
address@hidden calc-auto-recompute
-The @kbd{m C} (@code{calc-auto-recompute}) command turns this
-automatic recomputation on and off. If you turn it off, Calc will
-not update @samp{=>} operators on the stack (nor those in the
-attached Embedded mode buffer, if there is one). They will not
-be updated unless you explicitly do so by pressing @kbd{=} or until
-you press @kbd{m C} to turn recomputation back on. (While automatic
-recomputation is off, you can think of @kbd{m C m C} as a command
-to update all @samp{=>} operators while leaving recomputation off.)
-
-To update @samp{=>} operators in an Embedded buffer while
-automatic recomputation is off, use @address@hidden * u}}.
address@hidden Mode}.
-
address@hidden Working Message, , Automatic Recomputation, Calculation Modes
address@hidden Working Messages
-
address@hidden
address@hidden Performance
address@hidden Working messages
-Since the Calculator is written entirely in Emacs Lisp, which is not
-designed for heavy numerical work, many operations are quite slow.
-The Calculator normally displays the message @samp{Working...} in the
-echo area during any command that may be slow. In addition, iterative
-operations such as square roots and trigonometric functions display the
-intermediate result at each step. Both of these types of messages can
-be disabled if you find them distracting.
-
address@hidden m w
address@hidden calc-working
-Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
-disable all ``working'' messages. Use a numeric prefix of 1 to enable
-only the plain @samp{Working...} message. Use a numeric prefix of 2 to
-see intermediate results as well. With no numeric prefix this displays
-the current mode.
-
-While it may seem that the ``working'' messages will slow Calc down
-considerably, experiments have shown that their impact is actually
-quite small. But if your terminal is slow you may find that it helps
-to turn the messages off.
-
address@hidden Simplification Modes, Declarations, Calculation Modes, Mode
Settings
address@hidden Simplification Modes
-
address@hidden
-The current @dfn{simplification mode} controls how numbers and formulas
-are ``normalized'' when being taken from or pushed onto the stack.
-Some normalizations are unavoidable, such as rounding floating-point
-results to the current precision, and reducing fractions to simplest
-form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
-are done by default but can be turned off when necessary.
-
-When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
-stack, Calc pops these numbers, normalizes them, creates the formula
address@hidden, normalizes it, and pushes the result. Of course the standard
-rules for normalizing @expr{2+3} will produce the result @expr{5}.
-
-Simplification mode commands consist of the lower-case @kbd{m} prefix key
-followed by a shifted letter.
-
address@hidden m O
address@hidden calc-no-simplify-mode
-The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
-simplifications. These would leave a formula like @expr{2+3} alone. In
-fact, nothing except simple numbers are ever affected by normalization
-in this mode.
-
address@hidden m N
address@hidden calc-num-simplify-mode
-The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
-of any formulas except those for which all arguments are constants. For
-example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
-simplified to @expr{a+0} but no further, since one argument of the sum
-is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
-because the top-level @samp{-} operator's arguments are not both
-constant numbers (one of them is the formula @expr{a+2}).
-A constant is a number or other numeric object (such as a constant
-error form or modulo form), or a vector all of whose
-elements are constant.
-
address@hidden m D
address@hidden calc-default-simplify-mode
-The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
-default simplifications for all formulas. This includes many easy and
-fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
address@hidden + 2 a} to @expr{3 a}, as well as evaluating functions like
address@hidden@tfn{deriv}(x^2, x)} to @expr{2 x}.
-
address@hidden m B
address@hidden calc-bin-simplify-mode
-The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
-simplifications to a result and then, if the result is an integer,
-uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
-to the current binary word size. @xref{Binary Functions}. Real numbers
-are rounded to the nearest integer and then clipped; other kinds of
-results (after the default simplifications) are left alone.
-
address@hidden m A
address@hidden calc-alg-simplify-mode
-The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
-simplification; it applies all the default simplifications, and also
-the more powerful (and slower) simplifications made by @kbd{a s}
-(@code{calc-simplify}). @xref{Algebraic Simplifications}.
-
address@hidden m E
address@hidden calc-ext-simplify-mode
-The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
-algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
-command. @xref{Unsafe Simplifications}.
-
address@hidden m U
address@hidden calc-units-simplify-mode
-The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
-simplification; it applies the command @kbd{u s}
-(@code{calc-simplify-units}), which in turn
-is a superset of @kbd{a s}. In this mode, variable names which
-are identifiable as unit names (like @samp{mm} for ``millimeters'')
-are simplified with their unit definitions in mind.
-
-A common technique is to set the simplification mode down to the lowest
-amount of simplification you will allow to be applied automatically, then
-use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
-perform higher types of simplifications on demand. @xref{Algebraic
-Definitions}, for another sample use of No-Simplification mode.
-
address@hidden Declarations, Display Modes, Simplification Modes, Mode Settings
address@hidden Declarations
-
address@hidden
-A @dfn{declaration} is a statement you make that promises you will
-use a certain variable or function in a restricted way. This may
-give Calc the freedom to do things that it couldn't do if it had to
-take the fully general situation into account.
-
address@hidden
-* Declaration Basics::
-* Kinds of Declarations::
-* Functions for Declarations::
address@hidden menu
-
address@hidden Declaration Basics, Kinds of Declarations, Declarations,
Declarations
address@hidden Declaration Basics
-
address@hidden
address@hidden s d
address@hidden calc-declare-variable
-The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
-way to make a declaration for a variable. This command prompts for
-the variable name, then prompts for the declaration. The default
-at the declaration prompt is the previous declaration, if any.
-You can edit this declaration, or press @kbd{C-k} to erase it and
-type a new declaration. (Or, erase it and press @key{RET} to clear
-the declaration, effectively ``undeclaring'' the variable.)
-
-A declaration is in general a vector of @dfn{type symbols} and
address@hidden values. If there is only one type symbol or range value,
-you can write it directly rather than enclosing it in a vector.
-For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
-be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
-declares @code{bar} to be a constant integer between 1 and 6.
-(Actually, you can omit the outermost brackets and Calc will
-provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
-
address@hidden @code{Decls} variable
address@hidden Decls
-Declarations in Calc are kept in a special variable called @code{Decls}.
-This variable encodes the set of all outstanding declarations in
-the form of a matrix. Each row has two elements: A variable or
-vector of variables declared by that row, and the declaration
-specifier as described above. You can use the @kbd{s D} command to
-edit this variable if you wish to see all the declarations at once.
address@hidden on Variables}, for a description of this command
-and the @kbd{s p} command that allows you to save your declarations
-permanently if you wish.
-
-Items being declared can also be function calls. The arguments in
-the call are ignored; the effect is to say that this function returns
-values of the declared type for any valid arguments. The @kbd{s d}
-command declares only variables, so if you wish to make a function
-declaration you will have to edit the @code{Decls} matrix yourself.
-
-For example, the declaration matrix
-
address@hidden
address@hidden
-[ [ foo, real ]
- [ [j, k, n], int ]
- [ f(1,2,3), [0 .. inf) ] ]
address@hidden group
address@hidden smallexample
-
address@hidden
-declares that @code{foo} represents a real number, @code{j}, @code{k}
-and @code{n} represent integers, and the function @code{f} always
-returns a real number in the interval shown.
-
address@hidden All
-If there is a declaration for the variable @code{All}, then that
-declaration applies to all variables that are not otherwise declared.
-It does not apply to function names. For example, using the row
address@hidden, real]} says that all your variables are real unless they
-are explicitly declared without @code{real} in some other row.
-The @kbd{s d} command declares @code{All} if you give a blank
-response to the variable-name prompt.
-
address@hidden Kinds of Declarations, Functions for Declarations, Declaration
Basics, Declarations
address@hidden Kinds of Declarations
-
address@hidden
-The type-specifier part of a declaration (that is, the second prompt
-in the @kbd{s d} command) can be a type symbol, an interval, or a
-vector consisting of zero or more type symbols followed by zero or
-more intervals or numbers that represent the set of possible values
-for the variable.
-
address@hidden
address@hidden
-[ [ a, [1, 2, 3, 4, 5] ]
- [ b, [1 .. 5] ]
- [ c, [int, 1 .. 5] ] ]
address@hidden group
address@hidden smallexample
-
-Here @code{a} is declared to contain one of the five integers shown;
address@hidden is any number in the interval from 1 to 5 (any real number
-since we haven't specified), and @code{c} is any integer in that
-interval. Thus the declarations for @code{a} and @code{c} are
-nearly equivalent (see below).
-
-The type-specifier can be the empty vector @samp{[]} to say that
-nothing is known about a given variable's value. This is the same
-as not declaring the variable at all except that it overrides any
address@hidden declaration which would otherwise apply.
-
-The initial value of @code{Decls} is the empty vector @samp{[]}.
-If @code{Decls} has no stored value or if the value stored in it
-is not valid, it is ignored and there are no declarations as far
-as Calc is concerned. (The @kbd{s d} command will replace such a
-malformed value with a fresh empty matrix, @samp{[]}, before recording
-the new declaration.) Unrecognized type symbols are ignored.
-
-The following type symbols describe what sorts of numbers will be
-stored in a variable:
-
address@hidden @code
address@hidden int
-Integers.
address@hidden numint
-Numerical integers. (Integers or integer-valued floats.)
address@hidden frac
-Fractions. (Rational numbers which are not integers.)
address@hidden rat
-Rational numbers. (Either integers or fractions.)
address@hidden float
-Floating-point numbers.
address@hidden real
-Real numbers. (Integers, fractions, or floats. Actually,
-intervals and error forms with real components also count as
-reals here.)
address@hidden pos
-Positive real numbers. (Strictly greater than zero.)
address@hidden nonneg
-Nonnegative real numbers. (Greater than or equal to zero.)
address@hidden number
-Numbers. (Real or complex.)
address@hidden table
-
-Calc uses this information to determine when certain simplifications
-of formulas are safe. For example, @samp{(x^y)^z} cannot be
-simplified to @samp{x^(y z)} in general; for example,
address@hidden((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is
@mathit{-3}.
-However, this simplification @emph{is} safe if @code{z} is known
-to be an integer, or if @code{x} is known to be a nonnegative
-real number. If you have given declarations that allow Calc to
-deduce either of these facts, Calc will perform this simplification
-of the formula.
-
-Calc can apply a certain amount of logic when using declarations.
-For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
-has been declared @code{int}; Calc knows that an integer times an
-integer, plus an integer, must always be an integer. (In fact,
-Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
-it is able to determine that @samp{2n+1} must be an odd integer.)
-
-Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
-because Calc knows that the @code{abs} function always returns a
-nonnegative real. If you had a @code{myabs} function that also had
-this property, you could get Calc to recognize it by adding the row
address@hidden(), nonneg]} to the @code{Decls} matrix.
-
-One instance of this simplification is @samp{sqrt(x^2)} (since the
address@hidden function is effectively a one-half power). Normally
-Calc leaves this formula alone. After the command
address@hidden d x @key{RET} real @key{RET}}, however, it can simplify the
formula to
address@hidden(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
-simplify this formula all the way to @samp{x}.
-
-If there are any intervals or real numbers in the type specifier,
-they comprise the set of possible values that the variable or
-function being declared can have. In particular, the type symbol
address@hidden is effectively the same as the range @samp{[-inf .. inf]}
-(note that infinity is included in the range of possible values);
address@hidden is the same as @samp{(0 .. inf]}, and @code{nonneg} is
-the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
-redundant because the fact that the variable is real can be
-deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
address@hidden, [-5 .. 5]]} are useful combinations.
-
-Note that the vector of intervals or numbers is in the same format
-used by Calc's set-manipulation commands. @xref{Set Operations}.
-
-The type specifier @samp{[1, 2, 3]} is equivalent to
address@hidden, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
-In other words, the range of possible values means only that
-the variable's value must be numerically equal to a number in
-that range, but not that it must be equal in type as well.
-Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
-and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
-
-If you use a conflicting combination of type specifiers, the
-results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
-where the interval does not lie in the range described by the
-type symbol.
-
-``Real'' declarations mostly affect simplifications involving powers
-like the one described above. Another case where they are used
-is in the @kbd{a P} command which returns a list of all roots of a
-polynomial; if the variable has been declared real, only the real
-roots (if any) will be included in the list.
-
-``Integer'' declarations are used for simplifications which are valid
-only when certain values are integers (such as @samp{(x^y)^z}
-shown above).
-
-Another command that makes use of declarations is @kbd{a s}, when
-simplifying equations and inequalities. It will cancel @code{x}
-from both sides of @samp{a x = b x} only if it is sure @code{x}
-is non-zero, say, because it has a @code{pos} declaration.
-To declare specifically that @code{x} is real and non-zero,
-use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
-current notation to say that @code{x} is nonzero but not necessarily
-real.) The @kbd{a e} command does ``unsafe'' simplifications,
-including cancelling @samp{x} from the equation when @samp{x} is
-not known to be nonzero.
-
-Another set of type symbols distinguish between scalars and vectors.
-
address@hidden @code
address@hidden scalar
-The value is not a vector.
address@hidden vector
-The value is a vector.
address@hidden matrix
-The value is a matrix (a rectangular vector of vectors).
address@hidden sqmatrix
-The value is a square matrix.
address@hidden table
-
-These type symbols can be combined with the other type symbols
-described above; @samp{[int, matrix]} describes an object which
-is a matrix of integers.
-
-Scalar/vector declarations are used to determine whether certain
-algebraic operations are safe. For example, @samp{[a, b, c] + x}
-is normally not simplified to @samp{[a + x, b + x, c + x]}, but
-it will be if @code{x} has been declared @code{scalar}. On the
-other hand, multiplication is usually assumed to be commutative,
-but the terms in @samp{x y} will never be exchanged if both @code{x}
-and @code{y} are known to be vectors or matrices. (Calc currently
-never distinguishes between @code{vector} and @code{matrix}
-declarations.)
-
address@hidden Mode}, for a discussion of Matrix mode and
-Scalar mode, which are similar to declaring @samp{[All, matrix]}
-or @samp{[All, scalar]} but much more convenient.
-
-One more type symbol that is recognized is used with the @kbd{H a d}
-command for taking total derivatives of a formula. @xref{Calculus}.
-
address@hidden @code
address@hidden const
-The value is a constant with respect to other variables.
address@hidden table
-
-Calc does not check the declarations for a variable when you store
-a value in it. However, storing @mathit{-3.5} in a variable that has
-been declared @code{pos}, @code{int}, or @code{matrix} may have
-unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
-if it substitutes the value first, or to @expr{-3.5} if @code{x}
-was declared @code{pos} and the formula @samp{sqrt(x^2)} is
-simplified to @samp{x} before the value is substituted. Before
-using a variable for a new purpose, it is best to use @kbd{s d}
-or @kbd{s D} to check to make sure you don't still have an old
-declaration for the variable that will conflict with its new meaning.
-
address@hidden Functions for Declarations, , Kinds of Declarations,
Declarations
address@hidden Functions for Declarations
-
address@hidden
-Calc has a set of functions for accessing the current declarations
-in a convenient manner. These functions return 1 if the argument
-can be shown to have the specified property, or 0 if the argument
-can be shown @emph{not} to have that property; otherwise they are
-left unevaluated. These functions are suitable for use with rewrite
-rules (@pxref{Conditional Rewrite Rules}) or programming constructs
-(@pxref{Conditionals in Macros}). They can be entered only using
-algebraic notation. @xref{Logical Operations}, for functions
-that perform other tests not related to declarations.
-
-For example, @samp{dint(17)} returns 1 because 17 is an integer, as
-do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
address@hidden, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
-Calc consults knowledge of its own built-in functions as well as your
-own declarations: @samp{dint(floor(x))} returns 1.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dint
address@hidden
address@hidden
address@hidden ignore
address@hidden dnumint
address@hidden
address@hidden
address@hidden ignore
address@hidden dnatnum
-The @code{dint} function checks if its argument is an integer.
-The @code{dnatnum} function checks if its argument is a natural
-number, i.e., a nonnegative integer. The @code{dnumint} function
-checks if its argument is numerically an integer, i.e., either an
-integer or an integer-valued float. Note that these and the other
-data type functions also accept vectors or matrices composed of
-suitable elements, and that real infinities @samp{inf} and @samp{-inf}
-are considered to be integers for the purposes of these functions.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden drat
-The @code{drat} function checks if its argument is rational, i.e.,
-an integer or fraction. Infinities count as rational, but intervals
-and error forms do not.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dreal
-The @code{dreal} function checks if its argument is real. This
-includes integers, fractions, floats, real error forms, and intervals.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dimag
-The @code{dimag} function checks if its argument is imaginary,
-i.e., is mathematically equal to a real number times @expr{i}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dpos
address@hidden
address@hidden
address@hidden ignore
address@hidden dneg
address@hidden
address@hidden
address@hidden ignore
address@hidden dnonneg
-The @code{dpos} function checks for positive (but nonzero) reals.
-The @code{dneg} function checks for negative reals. The @code{dnonneg}
-function checks for nonnegative reals, i.e., reals greater than or
-equal to zero. Note that the @kbd{a s} command can simplify an
-expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
address@hidden s} is effectively applied to all conditions in rewrite rules,
-so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
-are rarely necessary.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dnonzero
-The @code{dnonzero} function checks that its argument is nonzero.
-This includes all nonzero real or complex numbers, all intervals that
-do not include zero, all nonzero modulo forms, vectors all of whose
-elements are nonzero, and variables or formulas whose values can be
-deduced to be nonzero. It does not include error forms, since they
-represent values which could be anything including zero. (This is
-also the set of objects considered ``true'' in conditional contexts.)
-
address@hidden
address@hidden
address@hidden ignore
address@hidden deven
address@hidden
address@hidden
address@hidden ignore
address@hidden dodd
-The @code{deven} function returns 1 if its argument is known to be
-an even integer (or integer-valued float); it returns 0 if its argument
-is known not to be even (because it is known to be odd or a non-integer).
-The @kbd{a s} command uses this to simplify a test of the form
address@hidden % 2 = 0}. There is also an analogous @code{dodd} function.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden drange
-The @code{drange} function returns a set (an interval or a vector
-of intervals and/or numbers; @pxref{Set Operations}) that describes
-the set of possible values of its argument. If the argument is
-a variable or a function with a declaration, the range is copied
-from the declaration. Otherwise, the possible signs of the
-expression are determined using a method similar to @code{dpos},
-etc., and a suitable set like @samp{[0 .. inf]} is returned. If
-the expression is not provably real, the @code{drange} function
-remains unevaluated.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dscalar
-The @code{dscalar} function returns 1 if its argument is provably
-scalar, or 0 if its argument is provably non-scalar. It is left
-unevaluated if this cannot be determined. (If Matrix mode or Scalar
-mode is in effect, this function returns 1 or 0, respectively,
-if it has no other information.) When Calc interprets a condition
-(say, in a rewrite rule) it considers an unevaluated formula to be
-``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
-provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
-is provably non-scalar; both are ``false'' if there is insufficient
-information to tell.
-
address@hidden Display Modes, Language Modes, Declarations, Mode Settings
address@hidden Display Modes
-
address@hidden
-The commands in this section are two-key sequences beginning with the
address@hidden prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
-(@code{calc-line-breaking}) commands are described elsewhere;
address@hidden Basics} and @pxref{Normal Language Modes}, respectively.
-Display formats for vectors and matrices are also covered elsewhere;
address@hidden and Matrix Formats}.
-
-One thing all display modes have in common is their treatment of the
address@hidden prefix. This prefix causes any mode command that would normally
-refresh the stack to leave the stack display alone. The word ``Dirty''
-will appear in the mode line when Calc thinks the stack display may not
-reflect the latest mode settings.
-
address@hidden d @key{RET}
address@hidden calc-refresh-top
-The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
-top stack entry according to all the current modes. Positive prefix
-arguments reformat the top @var{n} entries; negative prefix arguments
-reformat the specified entry, and a prefix of zero is equivalent to
address@hidden @key{SPC}} (@code{calc-refresh}), which reformats the entire
stack.
-For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
-but reformats only the top two stack entries in the new mode.
-
-The @kbd{I} prefix has another effect on the display modes. The mode
-is set only temporarily; the top stack entry is reformatted according
-to that mode, then the original mode setting is restored. In other
-words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old
mode})}.
-
address@hidden
-* Radix Modes::
-* Grouping Digits::
-* Float Formats::
-* Complex Formats::
-* Fraction Formats::
-* HMS Formats::
-* Date Formats::
-* Truncating the Stack::
-* Justification::
-* Labels::
address@hidden menu
-
address@hidden Radix Modes, Grouping Digits, Display Modes, Display Modes
address@hidden Radix Modes
-
address@hidden
address@hidden Radix display
address@hidden Non-decimal numbers
address@hidden Decimal and non-decimal numbers
-Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
-notation. Calc can actually display in any radix from two (binary) to 36.
-When the radix is above 10, the letters @code{A} to @code{Z} are used as
-digits. When entering such a number, letter keys are interpreted as
-potential digits rather than terminating numeric entry mode.
-
address@hidden d 2
address@hidden d 8
address@hidden d 6
address@hidden d 0
address@hidden Hexadecimal integers
address@hidden Octal integers
-The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
-binary, octal, hexadecimal, and decimal as the current display radix,
-respectively. Numbers can always be entered in any radix, though the
-current radix is used as a default if you press @kbd{#} without any initial
-digits. A number entered without a @kbd{#} is @emph{always} interpreted
-as decimal.
-
address@hidden d r
address@hidden calc-radix
-To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
-an integer from 2 to 36. You can specify the radix as a numeric prefix
-argument; otherwise you will be prompted for it.
-
address@hidden d z
address@hidden calc-leading-zeros
address@hidden Leading zeros
-Integers normally are displayed with however many digits are necessary to
-represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
-command causes integers to be padded out with leading zeros according to the
-current binary word size. (@xref{Binary Functions}, for a discussion of
-word size.) If the absolute value of the word size is @expr{w}, all integers
-are displayed with at least enough digits to represent
address@hidden @math{2^w-1}
address@hidden @expr{(2^w)-1}
-in the current radix. (Larger integers will still be displayed in their
-entirety.)
-
address@hidden Grouping Digits, Float Formats, Radix Modes, Display Modes
address@hidden Grouping Digits
-
address@hidden
address@hidden d g
address@hidden calc-group-digits
address@hidden Grouping digits
address@hidden Digit grouping
-Long numbers can be hard to read if they have too many digits. For
-example, the factorial of 30 is 33 digits long! Press @kbd{d g}
-(@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
-are displayed in clumps of 3 or 4 (depending on the current radix)
-separated by commas.
-
-The @kbd{d g} command toggles grouping on and off.
-With a numeric prefix of 0, this command displays the current state of
-the grouping flag; with an argument of minus one it disables grouping;
-with a positive argument @expr{N} it enables grouping on every @expr{N}
-digits. For floating-point numbers, grouping normally occurs only
-before the decimal point. A negative prefix argument @expr{-N} enables
-grouping every @expr{N} digits both before and after the decimal point.
-
address@hidden d ,
address@hidden calc-group-char
-The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
-character as the grouping separator. The default is the comma character.
-If you find it difficult to read vectors of large integers grouped with
-commas, you may wish to use spaces or some other character instead.
-This command takes the next character you type, whatever it is, and
-uses it as the digit separator. As a special case, @kbd{d , \} selects
address@hidden,} (@TeX{}'s thin-space symbol) as the digit separator.
-
-Please note that grouped numbers will not generally be parsed correctly
-if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
-(@xref{Kill and Yank}, for details on these commands.) One exception is
-the @samp{\,} separator, which doesn't interfere with parsing because it
-is ignored by @TeX{} language mode.
-
address@hidden Float Formats, Complex Formats, Grouping Digits, Display Modes
address@hidden Float Formats
-
address@hidden
-Floating-point quantities are normally displayed in standard decimal
-form, with scientific notation used if the exponent is especially high
-or low. All significant digits are normally displayed. The commands
-in this section allow you to choose among several alternative display
-formats for floats.
-
address@hidden d n
address@hidden calc-normal-notation
-The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
-display format. All significant figures in a number are displayed.
-With a positive numeric prefix, numbers are rounded if necessary to
-that number of significant digits. With a negative numerix prefix,
-the specified number of significant digits less than the current
-precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
-current precision is 12.)
-
address@hidden d f
address@hidden calc-fix-notation
-The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
-notation. The numeric argument is the number of digits after the
-decimal point, zero or more. This format will relax into scientific
-notation if a nonzero number would otherwise have been rounded all the
-way to zero. Specifying a negative number of digits is the same as
-for a positive number, except that small nonzero numbers will be rounded
-to zero rather than switching to scientific notation.
-
address@hidden d s
address@hidden calc-sci-notation
address@hidden Scientific notation, display of
-The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
-notation. A positive argument sets the number of significant figures
-displayed, of which one will be before and the rest after the decimal
-point. A negative argument works the same as for @kbd{d n} format.
-The default is to display all significant digits.
-
address@hidden d e
address@hidden calc-eng-notation
address@hidden Engineering notation, display of
-The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
-notation. This is similar to scientific notation except that the
-exponent is rounded down to a multiple of three, with from one to three
-digits before the decimal point. An optional numeric prefix sets the
-number of significant digits to display, as for @kbd{d s}.
-
-It is important to distinguish between the current @emph{precision} and
-the current @emph{display format}. After the commands @kbd{C-u 10 p}
-and @kbd{C-u 6 d n} the Calculator computes all results to ten
-significant figures but displays only six. (In fact, intermediate
-calculations are often carried to one or two more significant figures,
-but values placed on the stack will be rounded down to ten figures.)
-Numbers are never actually rounded to the display precision for storage,
-except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
-actual displayed text in the Calculator buffer.
-
address@hidden d .
address@hidden calc-point-char
-The @kbd{d .} (@code{calc-point-char}) command selects the character used
-as a decimal point. Normally this is a period; users in some countries
-may wish to change this to a comma. Note that this is only a display
-style; on entry, periods must always be used to denote floating-point
-numbers, and commas to separate elements in a list.
-
address@hidden Complex Formats, Fraction Formats, Float Formats, Display Modes
address@hidden Complex Formats
-
address@hidden
address@hidden d c
address@hidden calc-complex-notation
-There are three supported notations for complex numbers in rectangular
-form. The default is as a pair of real numbers enclosed in parentheses
-and separated by a comma: @samp{(a,b)}. The @kbd{d c}
-(@code{calc-complex-notation}) command selects this style.
-
address@hidden d i
address@hidden calc-i-notation
address@hidden d j
address@hidden calc-j-notation
-The other notations are @kbd{d i} (@code{calc-i-notation}), in which
-numbers are displayed in @samp{a+bi} form, and @kbd{d j}
-(@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
-in some disciplines.
-
address@hidden @code{i} variable
address@hidden i
-Complex numbers are normally entered in @samp{(a,b)} format.
-If you enter @samp{2+3i} as an algebraic formula, it will be stored as
-the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
-this formula and you have not changed the variable @samp{i}, the @samp{i}
-will be interpreted as @samp{(0,1)} and the formula will be simplified
-to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
-interpret the formula @samp{2 + 3 * i} as a complex number.
address@hidden, under ``special constants.''
-
address@hidden Fraction Formats, HMS Formats, Complex Formats, Display Modes
address@hidden Fraction Formats
-
address@hidden
address@hidden d o
address@hidden calc-over-notation
-Display of fractional numbers is controlled by the @kbd{d o}
-(@code{calc-over-notation}) command. By default, a number like
-eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
-prompts for a one- or two-character format. If you give one character,
-that character is used as the fraction separator. Common separators are
address@hidden:} and @samp{/}. (During input of numbers, the @kbd{:} key must
be
-used regardless of the display format; in particular, the @kbd{/} is used
-for RPN-style division, @emph{not} for entering fractions.)
-
-If you give two characters, fractions use ``integer-plus-fractional-part''
-notation. For example, the format @samp{+/} would display eight thirds
-as @samp{2+2/3}. If two colons are present in a number being entered,
-the number is interpreted in this form (so that the entries @kbd{2:2:3}
-and @kbd{8:3} are equivalent).
-
-It is also possible to follow the one- or two-character format with
-a number. For example: @samp{:10} or @samp{+/3}. In this case,
-Calc adjusts all fractions that are displayed to have the specified
-denominator, if possible. Otherwise it adjusts the denominator to
-be a multiple of the specified value. For example, in @samp{:6} mode
-the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
-displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
-and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
-affected by this mode: 3 is displayed as @expr{18:6}. Note that the
-format @samp{:1} writes fractions the same as @samp{:}, but it writes
-integers as @expr{n:1}.
-
-The fraction format does not affect the way fractions or integers are
-stored, only the way they appear on the screen. The fraction format
-never affects floats.
-
address@hidden HMS Formats, Date Formats, Fraction Formats, Display Modes
address@hidden HMS Formats
-
address@hidden
address@hidden d h
address@hidden calc-hms-notation
-The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
-HMS (hours-minutes-seconds) forms. It prompts for a string which
-consists basically of an ``hours'' marker, optional punctuation, a
-``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
-Punctuation is zero or more spaces, commas, or semicolons. The hours
-marker is one or more non-punctuation characters. The minutes and
-seconds markers must be single non-punctuation characters.
-
-The default HMS format is @samp{@@ ' "}, producing HMS values of the form
address@hidden@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
-value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
-keys are recognized as synonyms for @kbd{@@} regardless of display format.
-The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
address@hidden"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o})
has
-already been typed; otherwise, they have their usual meanings
-(@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
address@hidden h 5 s} are some of the ways to enter the quantity ``five
seconds.''
-The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
address@hidden) has already been pressed; otherwise it means to switch to
algebraic
-entry.
-
address@hidden Date Formats, Truncating the Stack, HMS Formats, Display Modes
address@hidden Date Formats
-
address@hidden
address@hidden d d
address@hidden calc-date-notation
-The @kbd{d d} (@code{calc-date-notation}) command controls the display
-of date forms (@pxref{Date Forms}). It prompts for a string which
-contains letters that represent the various parts of a date and time.
-To show which parts should be omitted when the form represents a pure
-date with no time, parts of the string can be enclosed in @samp{< >}
-marks. If you don't include @samp{< >} markers in the format, Calc
-guesses at which parts, if any, should be omitted when formatting
-pure dates.
-
-The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
-An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
-If you enter a blank format string, this default format is
-reestablished.
-
-Calc uses @samp{< >} notation for nameless functions as well as for
-dates. @xref{Specifying Operators}. To avoid confusion with nameless
-functions, your date formats should avoid using the @samp{#} character.
-
address@hidden
-* Date Formatting Codes::
-* Free-Form Dates::
-* Standard Date Formats::
address@hidden menu
-
address@hidden Date Formatting Codes, Free-Form Dates, Date Formats, Date
Formats
address@hidden Date Formatting Codes
-
address@hidden
-When displaying a date, the current date format is used. All
-characters except for letters and @samp{<} and @samp{>} are
-copied literally when dates are formatted. The portion between
address@hidden< >} markers is omitted for pure dates, or included for
-date/time forms. Letters are interpreted according to the table
-below.
-
-When dates are read in during algebraic entry, Calc first tries to
-match the input string to the current format either with or without
-the time part. The punctuation characters (including spaces) must
-match exactly; letter fields must correspond to suitable text in
-the input. If this doesn't work, Calc checks if the input is a
-simple number; if so, the number is interpreted as a number of days
-since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and
-flexible algorithm which is described in the next section.
-
-Weekday names are ignored during reading.
-
-Two-digit year numbers are interpreted as lying in the range
-from 1941 to 2039. Years outside that range are always
-entered and displayed in full. Year numbers with a leading
address@hidden sign are always interpreted exactly, allowing the
-entry and display of the years 1 through 99 AD.
-
-Here is a complete list of the formatting codes for dates:
-
address@hidden @asis
address@hidden Y
-Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
address@hidden YY
-Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
address@hidden BY
-Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
address@hidden YYY
-Year: ``1991'' for 1991, ``23'' for 23 AD.
address@hidden YYYY
-Year: ``1991'' for 1991, ``+23'' for 23 AD.
address@hidden aa
-Year: ``ad'' or blank.
address@hidden AA
-Year: ``AD'' or blank.
address@hidden aaa
-Year: ``ad '' or blank. (Note trailing space.)
address@hidden AAA
-Year: ``AD '' or blank.
address@hidden aaaa
-Year: ``a.d.'' or blank.
address@hidden AAAA
-Year: ``A.D.'' or blank.
address@hidden bb
-Year: ``bc'' or blank.
address@hidden BB
-Year: ``BC'' or blank.
address@hidden bbb
-Year: `` bc'' or blank. (Note leading space.)
address@hidden BBB
-Year: `` BC'' or blank.
address@hidden bbbb
-Year: ``b.c.'' or blank.
address@hidden BBBB
-Year: ``B.C.'' or blank.
address@hidden M
-Month: ``8'' for August.
address@hidden MM
-Month: ``08'' for August.
address@hidden BM
-Month: `` 8'' for August.
address@hidden MMM
-Month: ``AUG'' for August.
address@hidden Mmm
-Month: ``Aug'' for August.
address@hidden mmm
-Month: ``aug'' for August.
address@hidden MMMM
-Month: ``AUGUST'' for August.
address@hidden Mmmm
-Month: ``August'' for August.
address@hidden D
-Day: ``7'' for 7th day of month.
address@hidden DD
-Day: ``07'' for 7th day of month.
address@hidden BD
-Day: `` 7'' for 7th day of month.
address@hidden W
-Weekday: ``0'' for Sunday, ``6'' for Saturday.
address@hidden WWW
-Weekday: ``SUN'' for Sunday.
address@hidden Www
-Weekday: ``Sun'' for Sunday.
address@hidden www
-Weekday: ``sun'' for Sunday.
address@hidden WWWW
-Weekday: ``SUNDAY'' for Sunday.
address@hidden Wwww
-Weekday: ``Sunday'' for Sunday.
address@hidden d
-Day of year: ``34'' for Feb. 3.
address@hidden ddd
-Day of year: ``034'' for Feb. 3.
address@hidden bdd
-Day of year: `` 34'' for Feb. 3.
address@hidden h
-Hour: ``5'' for 5 AM; ``17'' for 5 PM.
address@hidden hh
-Hour: ``05'' for 5 AM; ``17'' for 5 PM.
address@hidden bh
-Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
address@hidden H
-Hour: ``5'' for 5 AM and 5 PM.
address@hidden HH
-Hour: ``05'' for 5 AM and 5 PM.
address@hidden BH
-Hour: `` 5'' for 5 AM and 5 PM.
address@hidden p
-AM/PM: ``a'' or ``p''.
address@hidden P
-AM/PM: ``A'' or ``P''.
address@hidden pp
-AM/PM: ``am'' or ``pm''.
address@hidden PP
-AM/PM: ``AM'' or ``PM''.
address@hidden pppp
-AM/PM: ``a.m.'' or ``p.m.''.
address@hidden PPPP
-AM/PM: ``A.M.'' or ``P.M.''.
address@hidden m
-Minutes: ``7'' for 7.
address@hidden mm
-Minutes: ``07'' for 7.
address@hidden bm
-Minutes: `` 7'' for 7.
address@hidden s
-Seconds: ``7'' for 7; ``7.23'' for 7.23.
address@hidden ss
-Seconds: ``07'' for 7; ``07.23'' for 7.23.
address@hidden bs
-Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
address@hidden SS
-Optional seconds: ``07'' for 7; blank for 0.
address@hidden BS
-Optional seconds: `` 7'' for 7; blank for 0.
address@hidden N
-Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
address@hidden n
-Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
address@hidden J
-Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
address@hidden j
-Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
address@hidden U
-Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
address@hidden X
-Brackets suppression. An ``X'' at the front of the format
-causes the surrounding @address@hidden< >}} delimiters to be omitted
-when formatting dates. Note that the brackets are still
-required for algebraic entry.
address@hidden table
-
-If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
-colon is also omitted if the seconds part is zero.
-
-If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
-appear in the format, then negative year numbers are displayed
-without a minus sign. Note that ``aa'' and ``bb'' are mutually
-exclusive. Some typical usages would be @samp{YYYY AABB};
address@hidden; @samp{YYYYBBB}.
-
-The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
-``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
-reading unless several of these codes are strung together with no
-punctuation in between, in which case the input must have exactly as
-many digits as there are letters in the format.
-
-The ``j,'' ``J,'' and ``U'' formats do not make any time zone
-adjustment. They effectively use @samp{julian(x,0)} and
address@hidden(x,0)} to make the conversion; @pxref{Date Arithmetic}.
-
address@hidden Free-Form Dates, Standard Date Formats, Date Formatting Codes,
Date Formats
address@hidden Free-Form Dates
-
address@hidden
-When reading a date form during algebraic entry, Calc falls back
-on the algorithm described here if the input does not exactly
-match the current date format. This algorithm generally
-``does the right thing'' and you don't have to worry about it,
-but it is described here in full detail for the curious.
-
-Calc does not distinguish between upper- and lower-case letters
-while interpreting dates.
-
-First, the time portion, if present, is located somewhere in the
-text and then removed. The remaining text is then interpreted as
-the date.
-
-A time is of the form @samp{hh:mm:ss}, possibly with the seconds
-part omitted and possibly with an AM/PM indicator added to indicate
-12-hour time. If the AM/PM is present, the minutes may also be
-omitted. The AM/PM part may be any of the words @samp{am},
address@hidden, @samp{noon}, or @samp{midnight}; each of these may be
-abbreviated to one letter, and the alternate forms @samp{a.m.},
address@hidden, and @samp{mid} are also understood. Obviously
address@hidden and @samp{midnight} are allowed only on 12:00:00.
-The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
-recognized with no number attached.
-
-If there is no AM/PM indicator, the time is interpreted in 24-hour
-format.
-
-To read the date portion, all words and numbers are isolated
-from the string; other characters are ignored. All words must
-be either month names or day-of-week names (the latter of which
-are ignored). Names can be written in full or as three-letter
-abbreviations.
-
-Large numbers, or numbers with @samp{+} or @samp{-} signs,
-are interpreted as years. If one of the other numbers is
-greater than 12, then that must be the day and the remaining
-number in the input is therefore the month. Otherwise, Calc
-assumes the month, day and year are in the same order that they
-appear in the current date format. If the year is omitted, the
-current year is taken from the system clock.
-
-If there are too many or too few numbers, or any unrecognizable
-words, then the input is rejected.
-
-If there are any large numbers (of five digits or more) other than
-the year, they are ignored on the assumption that they are something
-like Julian dates that were included along with the traditional
-date components when the date was formatted.
-
-One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
-may optionally be used; the latter two are equivalent to a
-minus sign on the year value.
-
-If you always enter a four-digit year, and use a name instead
-of a number for the month, there is no danger of ambiguity.
-
address@hidden Standard Date Formats, , Free-Form Dates, Date Formats
address@hidden Standard Date Formats
-
address@hidden
-There are actually ten standard date formats, numbered 0 through 9.
-Entering a blank line at the @kbd{d d} command's prompt gives
-you format number 1, Calc's usual format. You can enter any digit
-to select the other formats.
-
-To create your own standard date formats, give a numeric prefix
-argument from 0 to 9 to the @address@hidden d}} command. The format you
-enter will be recorded as the new standard format of that
-number, as well as becoming the new current date format.
-You can save your formats permanently with the @address@hidden m}}
-command (@pxref{Mode Settings}).
-
address@hidden @asis
address@hidden 0
address@hidden (Numerical format)
address@hidden 1
address@hidden<H:mm:SSpp >Www Mmm D, YYYY} (American format)
address@hidden 2
address@hidden Mmm YYYY<, h:mm:SS>} (European format)
address@hidden 3
address@hidden Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
address@hidden 4
address@hidden/D/Y< H:mm:SSpp>} (American slashed format)
address@hidden 5
address@hidden< h:mm:SS>} (European dotted format)
address@hidden 6
address@hidden< H:mm:SSpp>} (American dashed format)
address@hidden 7
address@hidden< h:mm:SS>} (European dashed format)
address@hidden 8
address@hidden<, h:mm:ss>} (Julian day plus time)
address@hidden 9
address@hidden< hh:mm:ss>} (Year-day format)
address@hidden table
-
address@hidden Truncating the Stack, Justification, Date Formats, Display Modes
address@hidden Truncating the Stack
-
address@hidden
address@hidden d t
address@hidden calc-truncate-stack
address@hidden Truncating the stack
address@hidden Narrowing the stack
-The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
-line that marks the top-of-stack up or down in the Calculator buffer.
-The number right above that line is considered to the be at the top of
-the stack. Any numbers below that line are ``hidden'' from all stack
-operations (although still visible to the user). This is similar to the
-Emacs ``narrowing'' feature, except that the values below the @samp{.}
-are @emph{visible}, just temporarily frozen. This feature allows you to
-keep several independent calculations running at once in different parts
-of the stack, or to apply a certain command to an element buried deep in
-the stack.
-
-Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
-is on. Thus, this line and all those below it become hidden. To un-hide
-these lines, move down to the end of the buffer and press @address@hidden t}}.
-With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
-bottom @expr{n} values in the buffer. With a negative argument, it hides
-all but the top @expr{n} values. With an argument of zero, it hides zero
-values, i.e., moves the @samp{.} all the way down to the bottom.
-
address@hidden d [
address@hidden calc-truncate-up
address@hidden d ]
address@hidden calc-truncate-down
-The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
-(@code{calc-truncate-down}) commands move the @samp{.} up or down one
-line at a time (or several lines with a prefix argument).
-
address@hidden Justification, Labels, Truncating the Stack, Display Modes
address@hidden Justification
-
address@hidden
address@hidden d <
address@hidden calc-left-justify
address@hidden d =
address@hidden calc-center-justify
address@hidden d >
address@hidden calc-right-justify
-Values on the stack are normally left-justified in the window. You can
-control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
address@hidden >} (@code{calc-right-justify}), or @kbd{d =}
-(@code{calc-center-justify}). For example, in Right-Justification mode,
-stack entries are displayed flush-right against the right edge of the
-window.
-
-If you change the width of the Calculator window you may have to type
address@hidden @key{SPC}} (@code{calc-refresh}) to re-align right-justified or
centered
-text.
-
-Right-justification is especially useful together with fixed-point
-notation (see @code{d f}; @code{calc-fix-notation}). With these modes
-together, the decimal points on numbers will always line up.
-
-With a numeric prefix argument, the justification commands give you
-a little extra control over the display. The argument specifies the
-horizontal ``origin'' of a display line. It is also possible to
-specify a maximum line width using the @kbd{d b} command (@pxref{Normal
-Language Modes}). For reference, the precise rules for formatting and
-breaking lines are given below. Notice that the interaction between
-origin and line width is slightly different in each justification
-mode.
-
-In Left-Justified mode, the line is indented by a number of spaces
-given by the origin (default zero). If the result is longer than the
-maximum line width, if given, or too wide to fit in the Calc window
-otherwise, then it is broken into lines which will fit; each broken
-line is indented to the origin.
-
-In Right-Justified mode, lines are shifted right so that the rightmost
-character is just before the origin, or just before the current
-window width if no origin was specified. If the line is too long
-for this, then it is broken; the current line width is used, if
-specified, or else the origin is used as a width if that is
-specified, or else the line is broken to fit in the window.
-
-In Centering mode, the origin is the column number of the center of
-each stack entry. If a line width is specified, lines will not be
-allowed to go past that width; Calc will either indent less or
-break the lines if necessary. If no origin is specified, half the
-line width or Calc window width is used.
-
-Note that, in each case, if line numbering is enabled the display
-is indented an additional four spaces to make room for the line
-number. The width of the line number is taken into account when
-positioning according to the current Calc window width, but not
-when positioning by explicit origins and widths. In the latter
-case, the display is formatted as specified, and then uniformly
-shifted over four spaces to fit the line numbers.
-
address@hidden Labels, , Justification, Display Modes
address@hidden Labels
-
address@hidden
address@hidden d @{
address@hidden calc-left-label
-The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
-then displays that string to the left of every stack entry. If the
-entries are left-justified (@pxref{Justification}), then they will
-appear immediately after the label (unless you specified an origin
-greater than the length of the label). If the entries are centered
-or right-justified, the label appears on the far left and does not
-affect the horizontal position of the stack entry.
-
-Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
-
address@hidden d @}
address@hidden calc-right-label
-The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
-label on the righthand side. It does not affect positioning of
-the stack entries unless they are right-justified. Also, if both
-a line width and an origin are given in Right-Justified mode, the
-stack entry is justified to the origin and the righthand label is
-justified to the line width.
-
-One application of labels would be to add equation numbers to
-formulas you are manipulating in Calc and then copying into a
-document (possibly using Embedded mode). The equations would
-typically be centered, and the equation numbers would be on the
-left or right as you prefer.
-
address@hidden Language Modes, Modes Variable, Display Modes, Mode Settings
address@hidden Language Modes
-
address@hidden
-The commands in this section change Calc to use a different notation for
-entry and display of formulas, corresponding to the conventions of some
-other common language such as Pascal or address@hidden Objects displayed on
the
-stack or yanked from the Calculator to an editing buffer will be formatted
-in the current language; objects entered in algebraic entry or yanked from
-another buffer will be interpreted according to the current language.
-
-The current language has no effect on things written to or read from the
-trail buffer, nor does it affect numeric entry. Only algebraic entry is
-affected. You can make even algebraic entry ignore the current language
-and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
-
-For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
-program; elsewhere in the program you need the derivatives of this formula
-with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
-to switch to C notation. Now use @code{C-u C-x * g} to grab the formula
-into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
-to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
-back into your C program. Press @kbd{U} to undo the differentiation and
-repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
-
-Without being switched into C mode first, Calc would have misinterpreted
-the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
address@hidden was equivalent to Calc's built-in @code{arctan} function,
-and would have written the formula back with notations (like implicit
-multiplication) which would not have been valid for a C program.
-
-As another example, suppose you are maintaining a C program and a
address@hidden
-document, each of which needs a copy of the same formula. You can grab the
-formula from the program in C mode, switch to address@hidden mode, and yank the
-formula into the document in address@hidden math-mode format.
-
-Language modes are selected by typing the letter @kbd{d} followed by a
-shifted letter key.
-
address@hidden
-* Normal Language Modes::
-* C FORTRAN Pascal::
-* TeX and LaTeX Language Modes::
-* Eqn Language Mode::
-* Mathematica Language Mode::
-* Maple Language Mode::
-* Compositions::
-* Syntax Tables::
address@hidden menu
-
address@hidden Normal Language Modes, C FORTRAN Pascal, Language Modes,
Language Modes
address@hidden Normal Language Modes
-
address@hidden
address@hidden d N
address@hidden calc-normal-language
-The @kbd{d N} (@code{calc-normal-language}) command selects the usual
-notation for Calc formulas, as described in the rest of this manual.
-Matrices are displayed in a multi-line tabular format, but all other
-objects are written in linear form, as they would be typed from the
-keyboard.
-
address@hidden d O
address@hidden calc-flat-language
address@hidden Matrix display
-The @kbd{d O} (@code{calc-flat-language}) command selects a language
-identical with the normal one, except that matrices are written in
-one-line form along with everything else. In some applications this
-form may be more suitable for yanking data into other buffers.
-
address@hidden d b
address@hidden calc-line-breaking
address@hidden Line breaking
address@hidden Breaking up long lines
-Even in one-line mode, long formulas or vectors will still be split
-across multiple lines if they exceed the width of the Calculator window.
-The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
-feature on and off. (It works independently of the current language.)
-If you give a numeric prefix argument of five or greater to the @kbd{d b}
-command, that argument will specify the line width used when breaking
-long lines.
-
address@hidden d B
address@hidden calc-big-language
-The @kbd{d B} (@code{calc-big-language}) command selects a language
-which uses textual approximations to various mathematical notations,
-such as powers, quotients, and square roots:
-
address@hidden
- ____________
- | a + 1 2
- | ----- + c
-\| b
address@hidden example
-
address@hidden
-in place of @samp{sqrt((a+1)/b + c^2)}.
-
-Subscripts like @samp{a_i} are displayed as actual subscripts in Big
-mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
-are displayed as @samp{a} with subscripts separated by commas:
address@hidden, j}. They must still be entered in the usual underscore
-notation.
-
-One slight ambiguity of Big notation is that
-
address@hidden
- 3
-- -
- 4
address@hidden example
-
address@hidden
-can represent either the negative rational number @expr{-3:4}, or the
-actual expression @samp{-(3/4)}; but the latter formula would normally
-never be displayed because it would immediately be evaluated to
address@hidden:4} or @expr{-0.75}, so this ambiguity is not a problem in
-typical use.
-
-Non-decimal numbers are displayed with subscripts. Thus there is no
-way to tell the difference between @samp{16#C2} and @samp{C2_16},
-though generally you will know which interpretation is correct.
-Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
-in Big mode.
-
-In Big mode, stack entries often take up several lines. To aid
-readability, stack entries are separated by a blank line in this mode.
-You may find it useful to expand the Calc window's height using
address@hidden ^} (@code{enlarge-window}) or to make the Calc window the only
-one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
-
-Long lines are currently not rearranged to fit the window width in
-Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
-to scroll across a wide formula. For really big formulas, you may
-even need to use @address@hidden and @address@hidden to scroll up and down.
-
address@hidden d U
address@hidden calc-unformatted-language
-The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
-the use of operator notation in formulas. In this mode, the formula
-shown above would be displayed:
-
address@hidden
-sqrt(add(div(add(a, 1), b), pow(c, 2)))
address@hidden example
-
-These four modes differ only in display format, not in the format
-expected for algebraic entry. The standard Calc operators work in
-all four modes, and unformatted notation works in any language mode
-(except that Mathematica mode expects square brackets instead of
-parentheses).
-
address@hidden C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language
Modes, Language Modes
address@hidden C, FORTRAN, and Pascal Modes
-
address@hidden
address@hidden d C
address@hidden calc-c-language
address@hidden C language
-The @kbd{d C} (@code{calc-c-language}) command selects the conventions
-of the C language for display and entry of formulas. This differs from
-the normal language mode in a variety of (mostly minor) ways. In
-particular, C language operators and operator precedences are used in
-place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
-in C mode; a value raised to a power is written as a function call,
address@hidden(a,b)}.
-
-In C mode, vectors and matrices use curly braces instead of brackets.
-Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
-rather than using the @samp{#} symbol. Array subscripting is
-translated into @code{subscr} calls, so that @samp{a[i]} in C
-mode is the same as @samp{a_i} in Normal mode. Assignments
-turn into the @code{assign} function, which Calc normally displays
-using the @samp{:=} symbol.
-
-The variables @code{pi} and @code{e} would be displayed @samp{pi}
-and @samp{e} in Normal mode, but in C mode they are displayed as
address@hidden and @samp{M_E}, corresponding to the names of constants
-typically provided in the @file{<math.h>} header. Functions whose
-names are different in C are translated automatically for entry and
-display purposes. For example, entering @samp{asin(x)} will push the
-formula @samp{arcsin(x)} onto the stack; this formula will be displayed
-as @samp{asin(x)} as long as C mode is in effect.
-
address@hidden d P
address@hidden calc-pascal-language
address@hidden Pascal language
-The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
-conventions. Like C mode, Pascal mode interprets array brackets and uses
-a different table of operators. Hexadecimal numbers are entered and
-displayed with a preceding dollar sign. (Thus the regular meaning of
address@hidden during algebraic entry does not work in Pascal mode, though
address@hidden (and @kbd{$$}, etc.) not followed by digits works the same as
-always.) No special provisions are made for other non-decimal numbers,
-vectors, and so on, since there is no universally accepted standard way
-of handling these in Pascal.
-
address@hidden d F
address@hidden calc-fortran-language
address@hidden FORTRAN language
-The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
-conventions. Various function names are transformed into FORTRAN
-equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
-entered this way or using square brackets. Since FORTRAN uses round
-parentheses for both function calls and array subscripts, Calc displays
-both in the same way; @samp{a(i)} is interpreted as a function call
-upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
-Also, if the variable @code{a} has been declared to have type
address@hidden or @code{matrix} then @samp{a(i)} will be parsed as a
-subscript. (@xref{Declarations}.) Usually it doesn't matter, though;
-if you enter the subscript expression @samp{a(i)} and Calc interprets
-it as a function call, you'll never know the difference unless you
-switch to another language mode or replace @code{a} with an actual
-vector (or unless @code{a} happens to be the name of a built-in
-function!).
-
-Underscores are allowed in variable and function names in all of these
-language modes. The underscore here is equivalent to the @samp{#} in
-Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
-
-FORTRAN and Pascal modes normally do not adjust the case of letters in
-formulas. Most built-in Calc names use lower-case letters. If you use a
-positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
-modes will use upper-case letters exclusively for display, and will
-convert to lower-case on input. With a negative prefix, these modes
-convert to lower-case for display and input.
-
address@hidden TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN
Pascal, Language Modes
address@hidden @TeX{} and address@hidden Language Modes
-
address@hidden
address@hidden d T
address@hidden calc-tex-language
address@hidden TeX language
address@hidden d L
address@hidden calc-latex-language
address@hidden LaTeX language
-The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
-of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
-and the @kbd{d L} (@code{calc-latex-language}) command selects the
-conventions of ``math mode'' in address@hidden, a typesetting language that
-uses @TeX{} as its formatting engine. Calc's address@hidden language mode can
-read any formula that the @TeX{} language mode can, although address@hidden
-mode may display it differently.
-
-Formulas are entered and displayed in the appropriate notation;
address@hidden @math{\sin(a/b)}
address@hidden @expr{sin(a/b)}
-will appear as @samp{\sin\left( a \over b \right)} in @TeX{} mode and
address@hidden(address@hidden@address@hidden@}\right)} in address@hidden mode.
-Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
address@hidden; these should be omitted when interfacing with Calc. To Calc,
-the @samp{$} sign has the same meaning it always does in algebraic
-formulas (a reference to an existing entry on the stack).
-
-Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
-quotients are written using @code{\over} in @TeX{} mode (as in
address@hidden@{a \over address@hidden) and @code{\frac} in address@hidden mode
(as in
address@hidden@address@hidden@address@hidden); binomial coefficients are
written with
address@hidden in @TeX{} mode (as in @address@hidden \choose address@hidden) and
address@hidden in address@hidden mode (as in @address@hidden@address@hidden@}}).
-Interval forms are written with @code{\ldots}, and error forms are
-written with @code{\pm}. Absolute values are written as in
address@hidden|x + 1|}, and the floor and ceiling functions are written with
address@hidden, @code{\rfloor}, etc. The words @code{\left} and
address@hidden are ignored when reading formulas in @TeX{} and address@hidden
-modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
-when read, @code{\infty} always translates to @code{inf}.
-
-Function calls are written the usual way, with the function name followed
-by the arguments in parentheses. However, functions for which @TeX{}
-and address@hidden have special names (like @code{\sin}) will use curly braces
-instead of parentheses for very simple arguments. During input, curly
-braces and parentheses work equally well for grouping, but when the
-document is formatted the curly braces will be invisible. Thus the
-printed result is
address@hidden @math{\sin{2 x}}
address@hidden @expr{sin 2x}
-but
address@hidden @math{\sin(2 + x)}.
address@hidden @expr{sin(2 + x)}.
-
-Function and variable names not treated specially by @TeX{} and address@hidden
-are simply written out as-is, which will cause them to come out in
-italic letters in the printed document. If you invoke @kbd{d T} or
address@hidden L} with a positive numeric prefix argument, names of more than
-one character will instead be enclosed in a protective commands that
-will prevent them from being typeset in the math italics; they will be
-written @address@hidden@address@hidden in @TeX{} mode and
address@hidden@address@hidden@}} in address@hidden mode. The
address@hidden@{ @}} and @address@hidden @}} notations are ignored during
-reading. If you use a negative prefix argument, such function names are
-written @address@hidden, and function names that begin with @code{\} during
-reading have the @code{\} removed. (Note that in this mode, long
-variable names are still written with @code{\hbox} or @code{\text}.
-However, you can always make an actual variable name like @code{\bar} in
-any @TeX{} mode.)
-
-During reading, text of the form @address@hidden ...@: @}} is replaced
-by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
address@hidden In address@hidden mode this also applies to
address@hidden@address@hidden ... address@hidden@}},
address@hidden@address@hidden ... address@hidden@}},
address@hidden@address@hidden ... address@hidden@}}, as well as
address@hidden@address@hidden ... address@hidden@}}.
-The symbol @samp{&} is interpreted as a comma,
-and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
-During output, matrices are displayed in @address@hidden a & b \\ c &
address@hidden
-format in @TeX{} mode and in
address@hidden@address@hidden a & b \\ c & d address@hidden@}} format in
address@hidden mode; you may need to edit this afterwards to change to your
-preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
-argument of 2 or -2, then matrices will be displayed in two-dimensional
-form, such as
-
address@hidden
address@hidden@}
-a & b \\
-c & d
address@hidden@}
address@hidden example
-
address@hidden
-This may be convenient for isolated matrices, but could lead to
-expressions being displayed like
-
address@hidden
address@hidden@} \times x
-a & b \\
-c & d
address@hidden@}
address@hidden example
-
address@hidden
-While this wouldn't bother Calc, it is incorrect address@hidden
-(Similarly for @TeX{}.)
-
-Accents like @code{\tilde} and @code{\bar} translate into function
-calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
-sequence is treated as an accent. The @code{\vec} accent corresponds
-to the function name @code{Vec}, because @code{vec} is the name of
-a built-in Calc function. The following table shows the accents
-in Calc, @TeX{}, address@hidden and @dfn{eqn} (described in the next section):
-
address@hidden
address@hidden
address@hidden@address@hidden @c Suppress marginal notes
address@hidden@address@hidden
address@hidden iftex
address@hidden
address@hidden
address@hidden ignore
address@hidden acute
address@hidden
address@hidden
address@hidden ignore
address@hidden Acute
address@hidden
address@hidden
address@hidden ignore
address@hidden bar
address@hidden
address@hidden
address@hidden ignore
address@hidden Bar
address@hidden
address@hidden
address@hidden ignore
address@hidden breve
address@hidden
address@hidden
address@hidden ignore
address@hidden Breve
address@hidden
address@hidden
address@hidden ignore
address@hidden check
address@hidden
address@hidden
address@hidden ignore
address@hidden Check
address@hidden
address@hidden
address@hidden ignore
address@hidden dddot
address@hidden
address@hidden
address@hidden ignore
address@hidden ddddot
address@hidden
address@hidden
address@hidden ignore
address@hidden dot
address@hidden
address@hidden
address@hidden ignore
address@hidden Dot
address@hidden
address@hidden
address@hidden ignore
address@hidden dotdot
address@hidden
address@hidden
address@hidden ignore
address@hidden DotDot
address@hidden
address@hidden
address@hidden ignore
address@hidden dyad
address@hidden
address@hidden
address@hidden ignore
address@hidden grave
address@hidden
address@hidden
address@hidden ignore
address@hidden Grave
address@hidden
address@hidden
address@hidden ignore
address@hidden hat
address@hidden
address@hidden
address@hidden ignore
address@hidden Hat
address@hidden
address@hidden
address@hidden ignore
address@hidden Prime
address@hidden
address@hidden
address@hidden ignore
address@hidden tilde
address@hidden
address@hidden
address@hidden ignore
address@hidden Tilde
address@hidden
address@hidden
address@hidden ignore
address@hidden under
address@hidden
address@hidden
address@hidden ignore
address@hidden Vec
address@hidden
address@hidden
address@hidden ignore
address@hidden VEC
address@hidden
address@hidden
address@hidden iftex
address@hidden
-Calc TeX LaTeX eqn
----- --- ----- ---
-acute \acute \acute
-Acute \Acute
-bar \bar \bar bar
-Bar \Bar
-breve \breve \breve
-Breve \Breve
-check \check \check
-Check \Check
-dddot \dddot
-ddddot \ddddot
-dot \dot \dot dot
-Dot \Dot
-dotdot \ddot \ddot dotdot
-DotDot \Ddot
-dyad dyad
-grave \grave \grave
-Grave \Grave
-hat \hat \hat hat
-Hat \Hat
-Prime prime
-tilde \tilde \tilde tilde
-Tilde \Tilde
-under \underline \underline under
-Vec \vec \vec vec
-VEC \Vec
address@hidden example
-
-The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
address@hidden@address@hidden \to @address@hidden @TeX{} defines @code{\to} as
an
-alias for @code{\rightarrow}. However, if the @samp{=>} is the
-top-level expression being formatted, a slightly different notation
-is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
-word is ignored by Calc's input routines, and is undefined in @TeX{}.
-You will typically want to include one of the following definitions
-at the top of a @TeX{} file that uses @code{\evalto}:
-
address@hidden
address@hidden@}
address@hidden@}
address@hidden example
-
-The first definition formats evaluates-to operators in the usual
-way. The second causes only the @var{b} part to appear in the
-printed document; the @var{a} part and the arrow are hidden.
-Another definition you may wish to use is @samp{\let\to=\Rightarrow}
-which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
address@hidden Operator}, for a discussion of @code{evalto}.
-
-The complete set of @TeX{} control sequences that are ignored during
-reading is:
-
address@hidden
-\hbox \mbox \text \left \right
-\, \> \: \; \! \quad \qquad \hfil \hfill
-\displaystyle \textstyle \dsize \tsize
-\scriptstyle \scriptscriptstyle \ssize \ssize
-\rm \bf \it \sl \roman \bold \italic \slanted
-\cal \mit \Cal \Bbb \frak \goth
-\evalto
address@hidden example
-
-Note that, because these symbols are ignored, reading a @TeX{} or
address@hidden formula into Calc and writing it back out may lose spacing and
-font information.
-
-Also, the ``discretionary multiplication sign'' @samp{\*} is read
-the same as @samp{*}.
-
address@hidden
-The @TeX{} version of this manual includes some printed examples at the
-end of this section.
address@hidden ifnottex
address@hidden
-Here are some examples of how various Calc formulas are formatted in @TeX{}:
-
address@hidden
address@hidden
-sin(a^2 / b_i)
-\sin\left( {a^2 \over b_i} \right)
address@hidden group
address@hidden example
address@hidden
-$$ \sin\left( a^2 \over b_i \right) $$
address@hidden tex
address@hidden 1
-
address@hidden
address@hidden
-[(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
-[3 + 4i, @{3 \over address@hidden, 3 \pm 4, [3 \ldots \infty)]
address@hidden group
address@hidden example
address@hidden
-\turnoffactive
-$$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
address@hidden tex
address@hidden 1
-
address@hidden
address@hidden
-[abs(a), abs(a / b), floor(a), ceil(a / b)]
-[|a|, \left| a \over b \right|,
- \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
address@hidden group
address@hidden example
address@hidden
-$$ [|a|, \left| a \over b \right|,
- \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
address@hidden tex
address@hidden 1
-
address@hidden
address@hidden
-[sin(a), sin(2 a), sin(2 + a), sin(a / b)]
address@hidden@}, address@hidden address@hidden, \sin(2 + a),
- \sin\left( @{a \over address@hidden \right)]
address@hidden group
address@hidden example
address@hidden
-\turnoffactive
-$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
address@hidden tex
address@hidden 2
-
-First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
address@hidden - d T} (using the example definition
address@hidden@{\tilde F(#1)@}}:
-
address@hidden
address@hidden
-[f(a), foo(bar), sin(pi)]
-[f(a), foo(bar), \sin{\pi}]
-[f(a), address@hidden@}(address@hidden@}), address@hidden@}]
-[f(a), address@hidden@address@hidden@}, address@hidden@}]
address@hidden group
address@hidden example
address@hidden
-$$ [f(a), foo(bar), \sin{\pi}] $$
-$$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
-$$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
address@hidden tex
address@hidden 2
-
-First with @address@hidden@}}, then with @address@hidden@}}:
-
address@hidden
address@hidden
-2 + 3 => 5
-\evalto 2 + 3 \to 5
address@hidden group
address@hidden example
address@hidden
-\turnoffactive
-$$ 2 + 3 \to 5 $$
-$$ 5 $$
address@hidden tex
address@hidden 2
-
-First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
-
address@hidden
address@hidden
-[2 + 3 => 5, a / 2 => (b + c) / 2]
address@hidden + 3 \to address@hidden, @address@hidden \over address@hidden \to
@{b + c \over address@hidden@}]
address@hidden group
address@hidden example
address@hidden
-\turnoffactive
-$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
-{\let\to\Rightarrow
-$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
address@hidden tex
address@hidden 2
-
-Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
-
address@hidden
address@hidden
-[ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
address@hidden @{a \over address@hidden & 0 \\ 0 & address@hidden(x + 1)@} @}
address@hidden @{a \over address@hidden & 0 \\ 0 & address@hidden(x + 1)@} @}
address@hidden group
address@hidden example
address@hidden
-\turnoffactive
-$$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
-$$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
address@hidden tex
address@hidden 2
address@hidden iftex
-
address@hidden Eqn Language Mode, Mathematica Language Mode, TeX and LaTeX
Language Modes, Language Modes
address@hidden Eqn Language Mode
-
address@hidden
address@hidden d E
address@hidden calc-eqn-language
address@hidden is another popular formatter for math formulas. It is
-designed for use with the TROFF text formatter, and comes standard
-with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
-command selects @dfn{eqn} notation.
-
-The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
-a significant part in the parsing of the language. For example,
address@hidden x+1 + y} treats @samp{x+1} as the argument of the
address@hidden operator. @dfn{Eqn} also understands more conventional
-grouping using curly braces: @address@hidden@} + y}. Braces are
-required only when the argument contains spaces.
-
-In Calc's @dfn{eqn} mode, however, curly braces are required to
-delimit arguments of operators like @code{sqrt}. The first of the
-above examples would treat only the @samp{x} as the argument of
address@hidden, and in fact @samp{sin x+1} would be interpreted as
address@hidden * x + 1}, because @code{sin} is not a special operator
-in the @dfn{eqn} language. If you always surround the argument
-with curly braces, Calc will never misunderstand.
-
-Calc also understands parentheses as grouping characters. Another
-peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
-words with spaces from any surrounding characters that aren't curly
-braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
-(The spaces around @code{sin} are important to make @dfn{eqn}
-recognize that @code{sin} should be typeset in a roman font, and
-the spaces around @code{x} and @code{y} are a good idea just in
-case the @dfn{eqn} document has defined special meanings for these
-names, too.)
-
-Powers and subscripts are written with the @code{sub} and @code{sup}
-operators, respectively. Note that the caret symbol @samp{^} is
-treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
-symbol (these are used to introduce spaces of various widths into
-the typeset output of @dfn{eqn}).
-
-As in address@hidden mode, Calc's formatter omits parentheses around the
-arguments of functions like @code{ln} and @code{sin} if they are
-``simple-looking''; in this case Calc surrounds the argument with
-braces, separated by a @samp{~} from the function name: @address@hidden@}}.
-
-Font change codes (like @samp{roman @var{x}}) and positioning codes
-(like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
address@hidden reader. Also ignored are the words @code{left}, @code{right},
address@hidden, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
-are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
address@hidden @address@hidden; this is only an approximation to the true
meaning
-of quotes in @dfn{eqn}, but it is good enough for most uses.
-
-Accent codes (@address@hidden dot}) are handled by treating them as
-function calls (@samp{dot(@var{x})}) internally.
address@hidden and LaTeX Language Modes}, for a table of these accent
-functions. The @code{prime} accent is treated specially if it occurs on
-a variable or function name: @samp{f prime prime @w{( x prime )}} is
-stored internally as @samp{f'@w{'}(x')}. For example, taking the
-derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
-x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
-
-Assignments are written with the @samp{<-} (left-arrow) symbol,
-and @code{evalto} operators are written with @samp{->} or
address@hidden ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
-of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
-recognized for these operators during reading.
-
-Vectors in @dfn{eqn} mode use regular Calc square brackets, but
-matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
-The words @code{lcol} and @code{rcol} are recognized as synonyms
-for @code{ccol} during input, and are generated instead of @code{ccol}
-if the matrix justification mode so specifies.
-
address@hidden Mathematica Language Mode, Maple Language Mode, Eqn Language
Mode, Language Modes
address@hidden Mathematica Language Mode
-
address@hidden
address@hidden d M
address@hidden calc-mathematica-language
address@hidden Mathematica language
-The @kbd{d M} (@code{calc-mathematica-language}) command selects the
-conventions of Mathematica. Notable differences in Mathematica mode
-are that the names of built-in functions are capitalized, and function
-calls use square brackets instead of parentheses. Thus the Calc
-formula @samp{sin(2 x)} is entered and displayed @address@hidden x]}} in
-Mathematica mode.
-
-Vectors and matrices use curly braces in Mathematica. Complex numbers
-are written @samp{3 + 4 I}. The standard special constants in Calc are
-written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
address@hidden, @code{ComplexInfinity}, and @code{Indeterminate} in
-Mathematica mode.
-Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
-numbers in scientific notation are written @samp{1.23*10.^3}.
-Subscripts use double square brackets: @samp{a[[i]]}.
-
address@hidden Maple Language Mode, Compositions, Mathematica Language Mode,
Language Modes
address@hidden Maple Language Mode
-
address@hidden
address@hidden d W
address@hidden calc-maple-language
address@hidden Maple language
-The @kbd{d W} (@code{calc-maple-language}) command selects the
-conventions of Maple.
-
-Maple's language is much like C. Underscores are allowed in symbol
-names; square brackets are used for subscripts; explicit @samp{*}s for
-multiplications are required. Use either @samp{^} or @samp{**} to
-denote powers.
-
-Maple uses square brackets for lists and curly braces for sets. Calc
-interprets both notations as vectors, and displays vectors with square
-brackets. This means Maple sets will be converted to lists when they
-pass through Calc. As a special case, matrices are written as calls
-to the function @code{matrix}, given a list of lists as the argument,
-and can be read in this form or with all-capitals @code{MATRIX}.
-
-The Maple interval notation @samp{2 .. 3} has no surrounding brackets;
-Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and
-writes any kind of interval as @samp{2 .. 3}. This means you cannot
-see the difference between an open and a closed interval while in
-Maple display mode.
-
-Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
-are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
address@hidden, @code{uinf}, and @code{nan} display as @code{infinity}).
-Floating-point numbers are written @samp{1.23*10.^3}.
-
-Among things not currently handled by Calc's Maple mode are the
-various quote symbols, procedures and functional operators, and
-inert (@samp{&}) operators.
-
address@hidden Compositions, Syntax Tables, Maple Language Mode, Language Modes
address@hidden Compositions
-
address@hidden
address@hidden Compositions
-There are several @dfn{composition functions} which allow you to get
-displays in a variety of formats similar to those in Big language
-mode. Most of these functions do not evaluate to anything; they are
-placeholders which are left in symbolic form by Calc's evaluator but
-are recognized by Calc's display formatting routines.
-
-Two of these, @code{string} and @code{bstring}, are described elsewhere.
address@hidden For example, @samp{string("ABC")} is displayed as
address@hidden When viewed on the stack it will be indistinguishable from
-the variable @code{ABC}, but internally it will be stored as
address@hidden([65, 66, 67])} and can still be manipulated this way; for
-example, the selection and vector commands @kbd{j 1 v v j u} would
-select the vector portion of this object and reverse the elements, then
-deselect to reveal a string whose characters had been reversed.
-
-The composition functions do the same thing in all language modes
-(although their components will of course be formatted in the current
-language mode). The one exception is Unformatted mode (@kbd{d U}),
-which does not give the composition functions any special treatment.
-The functions are discussed here because of their relationship to
-the language modes.
-
address@hidden
-* Composition Basics::
-* Horizontal Compositions::
-* Vertical Compositions::
-* Other Compositions::
-* Information about Compositions::
-* User-Defined Compositions::
address@hidden menu
-
address@hidden Composition Basics, Horizontal Compositions, Compositions,
Compositions
address@hidden Composition Basics
-
address@hidden
-Compositions are generally formed by stacking formulas together
-horizontally or vertically in various ways. Those formulas are
-themselves compositions. @TeX{} users will find this analogous
-to @TeX{}'s ``boxes.'' Each multi-line composition has a
address@hidden; horizontal compositions use the baselines to
-decide how formulas should be positioned relative to one another.
-For example, in the Big mode formula
-
address@hidden
address@hidden
- 2
- a + b
-17 + ------
- c
address@hidden group
address@hidden example
-
address@hidden
-the second term of the sum is four lines tall and has line three as
-its baseline. Thus when the term is combined with 17, line three
-is placed on the same level as the baseline of 17.
-
address@hidden
-\bigskip
address@hidden tex
-
-Another important composition concept is @dfn{precedence}. This is
-an integer that represents the binding strength of various operators.
-For example, @samp{*} has higher precedence (195) than @samp{+} (180),
-which means that @samp{(a * b) + c} will be formatted without the
-parentheses, but @samp{a * (b + c)} will keep the parentheses.
-
-The operator table used by normal and Big language modes has the
-following precedences:
-
address@hidden
-_ 1200 @r{(subscripts)}
-% 1100 @r{(as in address@hidden)}
-- 1000 @r{(as in address@hidden)}
-! 1000 @r{(as in address@hidden)}
-mod 400
-+/- 300
-!! 210 @r{(as in address@hidden)}
-! 210 @r{(as in address@hidden)}
-^ 200
-* 195 @r{(or implicit multiplication)}
-/ % \ 190
-+ - 180 @r{(as in address@hidden)}
-| 170
-< = 160 @r{(and other relations)}
-&& 110
-|| 100
-? : 90
-!!! 85
-&&& 80
-||| 75
-:= 50
-:: 45
-=> 40
address@hidden example
-
-The general rule is that if an operator with precedence @expr{n}
-occurs as an argument to an operator with precedence @expr{m}, then
-the argument is enclosed in parentheses if @expr{n < m}. Top-level
-expressions and expressions which are function arguments, vector
-components, etc., are formatted with precedence zero (so that they
-normally never get additional parentheses).
-
-For binary left-associative operators like @samp{+}, the righthand
-argument is actually formatted with one-higher precedence than shown
-in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
-but the unnatural form @samp{a + (b + c)} keeps its parentheses.
-Right-associative operators like @samp{^} format the lefthand argument
-with one-higher precedence.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cprec
-The @code{cprec} function formats an expression with an arbitrary
-precedence. For example, @samp{cprec(abc, 185)} will combine into
-sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
-this @code{cprec} form has higher precedence than addition, but lower
-precedence than multiplication).
-
address@hidden
-\bigskip
address@hidden tex
-
-A final composition issue is @dfn{line breaking}. Calc uses two
-different strategies for ``flat'' and ``non-flat'' compositions.
-A non-flat composition is anything that appears on multiple lines
-(not counting line breaking). Examples would be matrices and Big
-mode powers and quotients. Non-flat compositions are displayed
-exactly as specified. If they come out wider than the current
-window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
-view them.
-
-Flat compositions, on the other hand, will be broken across several
-lines if they are too wide to fit the window. Certain points in a
-composition are noted internally as @dfn{break points}. Calc's
-general strategy is to fill each line as much as possible, then to
-move down to the next line starting at the first break point that
-didn't fit. However, the line breaker understands the hierarchical
-structure of formulas. It will not break an ``inner'' formula if
-it can use an earlier break point from an ``outer'' formula instead.
-For example, a vector of sums might be formatted as:
-
address@hidden
address@hidden
-[ a + b + c, d + e + f,
- g + h + i, j + k + l, m ]
address@hidden group
address@hidden example
-
address@hidden
-If the @samp{m} can fit, then so, it seems, could the @samp{g}.
-But Calc prefers to break at the comma since the comma is part
-of a ``more outer'' formula. Calc would break at a plus sign
-only if it had to, say, if the very first sum in the vector had
-itself been too large to fit.
-
-Of the composition functions described below, only @code{choriz}
-generates break points. The @code{bstring} function (@pxref{Strings})
-also generates breakable items: A break point is added after every
-space (or group of spaces) except for spaces at the very beginning or
-end of the string.
-
-Composition functions themselves count as levels in the formula
-hierarchy, so a @code{choriz} that is a component of a larger
address@hidden will be less likely to be broken. As a special case,
-if a @code{bstring} occurs as a component of a @code{choriz} or
address@hidden object (such as a vector or a list of arguments
-in a function call), then the break points in that @code{bstring}
-will be on the same level as the break points of the surrounding
-object.
-
address@hidden Horizontal Compositions, Vertical Compositions, Composition
Basics, Compositions
address@hidden Horizontal Compositions
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden choriz
-The @code{choriz} function takes a vector of objects and composes
-them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
-as @address@hidden b / cd}} in Normal language mode, or as
-
address@hidden
address@hidden
- a b
-17---d
- c
address@hidden group
address@hidden example
-
address@hidden
-in Big language mode. This is actually one case of the general
-function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
-either or both of @var{sep} and @var{prec} may be omitted.
address@hidden gives the @dfn{precedence} to use when formatting
-each of the components of @var{vec}. The default precedence is
-the precedence from the surrounding environment.
-
address@hidden is a string (i.e., a vector of character codes as might
-be entered with @code{" "} notation) which should separate components
-of the composition. Also, if @var{sep} is given, the line breaker
-will allow lines to be broken after each occurrence of @var{sep}.
-If @var{sep} is omitted, the composition will not be breakable
-(unless any of its component compositions are breakable).
-
-For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
-formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
-to have precedence 180 ``outwards'' as well as ``inwards,''
-enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
-formats as @samp{2 (a + b c + (d = e))}.
-
-The baseline of a horizontal composition is the same as the
-baselines of the component compositions, which are all aligned.
-
address@hidden Vertical Compositions, Other Compositions, Horizontal
Compositions, Compositions
address@hidden Vertical Compositions
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden cvert
-The @code{cvert} function makes a vertical composition. Each
-component of the vector is centered in a column. The baseline of
-the result is by default the top line of the resulting composition.
-For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
-formats in Big mode as
-
address@hidden
address@hidden
-f( a , 2 )
- bb a + 1
- ccc 2
- b
address@hidden group
address@hidden example
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cbase
-There are several special composition functions that work only as
-components of a vertical composition. The @code{cbase} function
-controls the baseline of the vertical composition; the baseline
-will be the same as the baseline of whatever component is enclosed
-in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
-cvert([a^2 + 1, cbase(b^2)]))} displays as
-
address@hidden
address@hidden
- 2
- a + 1
- a 2
-f(bb , b )
- ccc
address@hidden group
address@hidden example
-
address@hidden
address@hidden
address@hidden ignore
address@hidden ctbase
address@hidden
address@hidden
address@hidden ignore
address@hidden cbbase
-There are also @code{ctbase} and @code{cbbase} functions which
-make the baseline of the vertical composition equal to the top
-or bottom line (rather than the baseline) of that component.
-Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
-cvert([cbbase(a / b)])} gives
-
address@hidden
address@hidden
- a
-a -
-- + a + b
-b -
- b
address@hidden group
address@hidden example
-
-There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
-function in a given vertical composition. These functions can also
-be written with no arguments: @samp{ctbase()} is a zero-height object
-which means the baseline is the top line of the following item, and
address@hidden()} means the baseline is the bottom line of the preceding
-item.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden crule
-The @code{crule} function builds a ``rule,'' or horizontal line,
-across a vertical composition. By itself @samp{crule()} uses @samp{-}
-characters to build the rule. You can specify any other character,
-e.g., @samp{crule("=")}. The argument must be a character code or
-vector of exactly one character code. It is repeated to match the
-width of the widest item in the stack. For example, a quotient
-with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
-
address@hidden
address@hidden
-a + 1
-=====
- 2
- b
address@hidden group
address@hidden example
-
address@hidden
address@hidden
address@hidden ignore
address@hidden clvert
address@hidden
address@hidden
address@hidden ignore
address@hidden crvert
-Finally, the functions @code{clvert} and @code{crvert} act exactly
-like @code{cvert} except that the items are left- or right-justified
-in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
-gives:
-
address@hidden
address@hidden
-a + a
-bb bb
-ccc ccc
address@hidden group
address@hidden example
-
-Like @code{choriz}, the vertical compositions accept a second argument
-which gives the precedence to use when formatting the components.
-Vertical compositions do not support separator strings.
-
address@hidden Other Compositions, Information about Compositions, Vertical
Compositions, Compositions
address@hidden Other Compositions
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden csup
-The @code{csup} function builds a superscripted expression. For
-example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
-language mode. This is essentially a horizontal composition of
address@hidden and @samp{b}, where @samp{b} is shifted up so that its
-bottom line is one above the baseline.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden csub
-Likewise, the @code{csub} function builds a subscripted expression.
-This shifts @samp{b} down so that its top line is one below the
-bottom line of @samp{a} (note that this is not quite analogous to
address@hidden). Other arrangements can be obtained by using
address@hidden and @code{cvert} directly.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cflat
-The @code{cflat} function formats its argument in ``flat'' mode,
-as obtained by @samp{d O}, if the current language mode is normal
-or Big. It has no effect in other language modes. For example,
address@hidden(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
-to improve its readability.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cspace
-The @code{cspace} function creates horizontal space. For example,
address@hidden(4)} is effectively the same as @samp{string(" ")}.
-A second string (i.e., vector of characters) argument is repeated
-instead of the space character. For example, @samp{cspace(4, "ab")}
-looks like @samp{abababab}. If the second argument is not a string,
-it is formatted in the normal way and then several copies of that
-are composed together: @samp{cspace(4, a^2)} yields
-
address@hidden
address@hidden
- 2 2 2 2
-a a a a
address@hidden group
address@hidden example
-
address@hidden
-If the number argument is zero, this is a zero-width object.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cvspace
-The @code{cvspace} function creates vertical space, or a vertical
-stack of copies of a certain string or formatted object. The
-baseline is the center line of the resulting stack. A numerical
-argument of zero will produce an object which contributes zero
-height if used in a vertical composition.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden ctspace
address@hidden
address@hidden
address@hidden ignore
address@hidden cbspace
-There are also @code{ctspace} and @code{cbspace} functions which
-create vertical space with the baseline the same as the baseline
-of the top or bottom copy, respectively, of the second argument.
-Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
-displays as:
-
address@hidden
address@hidden
- a
- -
-a b
-- a a
-b + - + -
-a b b
-- a
-b -
- b
address@hidden group
address@hidden example
-
address@hidden Information about Compositions, User-Defined Compositions, Other
Compositions, Compositions
address@hidden Information about Compositions
-
address@hidden
-The functions in this section are actual functions; they compose their
-arguments according to the current language and other display modes,
-then return a certain measurement of the composition as an integer.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cwidth
-The @code{cwidth} function measures the width, in characters, of a
-composition. For example, @samp{cwidth(a + b)} is 5, and
address@hidden(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
address@hidden mode (for @address@hidden \over address@hidden). The argument
may involve
-the composition functions described in this section.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cheight
-The @code{cheight} function measures the height of a composition.
-This is the total number of lines in the argument's printed form.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden cascent
address@hidden
address@hidden
address@hidden ignore
address@hidden cdescent
-The functions @code{cascent} and @code{cdescent} measure the amount
-of the height that is above (and including) the baseline, or below
-the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
-always equals @samp{cheight(@var{x})}. For a one-line formula like
address@hidden + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
-For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
-returns 1. The only formula for which @code{cascent} will return zero
-is @samp{cvspace(0)} or equivalents.
-
address@hidden User-Defined Compositions, , Information about Compositions,
Compositions
address@hidden User-Defined Compositions
-
address@hidden
address@hidden Z C
address@hidden calc-user-define-composition
-The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
-define the display format for any algebraic function. You provide a
-formula containing a certain number of argument variables on the stack.
-Any time Calc formats a call to the specified function in the current
-language mode and with that number of arguments, Calc effectively
-replaces the function call with that formula with the arguments
-replaced.
-
-Calc builds the default argument list by sorting all the variable names
-that appear in the formula into alphabetical order. You can edit this
-argument list before pressing @key{RET} if you wish. Any variables in
-the formula that do not appear in the argument list will be displayed
-literally; any arguments that do not appear in the formula will not
-affect the display at all.
-
-You can define formats for built-in functions, for functions you have
-defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
-which have no definitions but are being used as purely syntactic objects.
-You can define different formats for each language mode, and for each
-number of arguments, using a succession of @kbd{Z C} commands. When
-Calc formats a function call, it first searches for a format defined
-for the current language mode (and number of arguments); if there is
-none, it uses the format defined for the Normal language mode. If
-neither format exists, Calc uses its built-in standard format for that
-function (usually just @address@hidden(@var{args})}).
-
-If you execute @kbd{Z C} with the number 0 on the stack instead of a
-formula, any defined formats for the function in the current language
-mode will be removed. The function will revert to its standard format.
-
-For example, the default format for the binomial coefficient function
address@hidden(n, m)} in the Big language mode is
-
address@hidden
address@hidden
- n
-( )
- m
address@hidden group
address@hidden example
-
address@hidden
-You might prefer the notation,
-
address@hidden
address@hidden
- C
-n m
address@hidden group
address@hidden example
-
address@hidden
-To define this notation, first make sure you are in Big mode,
-then put the formula
-
address@hidden
-choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
address@hidden smallexample
-
address@hidden
-on the stack and type @kbd{Z C}. Answer the first prompt with
address@hidden The second prompt will be the default argument list
-of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
address@hidden Now, try it out: For example, turn simplification
-off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
-as an algebraic entry.
-
address@hidden
address@hidden
- C + C
-a b 7 3
address@hidden group
address@hidden example
-
-As another example, let's define the usual notation for Stirling
-numbers of the first kind, @samp{stir1(n, m)}. This is just like
-the regular format for binomial coefficients but with square brackets
-instead of parentheses.
-
address@hidden
-choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
address@hidden smallexample
-
-Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
address@hidden(n m)}, and type @key{RET}.
-
-The formula provided to @kbd{Z C} usually will involve composition
-functions, but it doesn't have to. Putting the formula @samp{a + b + c}
-onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
-the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
-This ``sum'' will act exactly like a real sum for all formatting
-purposes (it will be parenthesized the same, and so on). However
-it will be computationally unrelated to a sum. For example, the
-formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
-Operator precedences have caused the ``sum'' to be written in
-parentheses, but the arguments have not actually been summed.
-(Generally a display format like this would be undesirable, since
-it can easily be confused with a real sum.)
-
-The special function @code{eval} can be used inside a @kbd{Z C}
-composition formula to cause all or part of the formula to be
-evaluated at display time. For example, if the formula is
address@hidden + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
-as @samp{1 + 5}. Evaluation will use the default simplifications,
-regardless of the current simplification mode. There are also
address@hidden and @code{evalextsimp} which simplify as if by
address@hidden s} and @kbd{a e} (respectively). Note that these ``functions''
-operate only in the context of composition formulas (and also in
-rewrite rules, where they serve a similar purpose; @pxref{Rewrite
-Rules}). On the stack, a call to @code{eval} will be left in
-symbolic form.
-
-It is not a good idea to use @code{eval} except as a last resort.
-It can cause the display of formulas to be extremely slow. For
-example, while @samp{eval(a + b)} might seem quite fast and simple,
-there are several situations where it could be slow. For example,
address@hidden and/or @samp{b} could be polar complex numbers, in which
-case doing the sum requires trigonometry. Or, @samp{a} could be
-the factorial @samp{fact(100)} which is unevaluated because you
-have typed @kbd{m O}; @code{eval} will evaluate it anyway to
-produce a large, unwieldy integer.
-
-You can save your display formats permanently using the @kbd{Z P}
-command (@pxref{Creating User Keys}).
-
address@hidden Syntax Tables, , Compositions, Language Modes
address@hidden Syntax Tables
-
address@hidden
address@hidden Syntax tables
address@hidden Parsing formulas, customized
-Syntax tables do for input what compositions do for output: They
-allow you to teach custom notations to Calc's formula parser.
-Calc keeps a separate syntax table for each language mode.
-
-(Note that the Calc ``syntax tables'' discussed here are completely
-unrelated to the syntax tables described in the Emacs manual.)
-
address@hidden Z S
address@hidden calc-edit-user-syntax
-The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
-syntax table for the current language mode. If you want your
-syntax to work in any language, define it in the Normal language
-mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
address@hidden k} to cancel the edit. The @kbd{m m} command saves all
-the syntax tables along with the other mode settings;
address@hidden Mode Commands}.
-
address@hidden
-* Syntax Table Basics::
-* Precedence in Syntax Tables::
-* Advanced Syntax Patterns::
-* Conditional Syntax Rules::
address@hidden menu
-
address@hidden Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables,
Syntax Tables
address@hidden Syntax Table Basics
-
address@hidden
address@hidden is the process of converting a raw string of characters,
-such as you would type in during algebraic entry, into a Calc formula.
-Calc's parser works in two stages. First, the input is broken down
-into @dfn{tokens}, such as words, numbers, and punctuation symbols
-like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
-ignored (except when it serves to separate adjacent words). Next,
-the parser matches this string of tokens against various built-in
-syntactic patterns, such as ``an expression followed by @samp{+}
-followed by another expression'' or ``a name followed by @samp{(},
-zero or more expressions separated by commas, and @samp{)}.''
-
-A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
-which allow you to specify new patterns to define your own
-favorite input notations. Calc's parser always checks the syntax
-table for the current language mode, then the table for the Normal
-language mode, before it uses its built-in rules to parse an
-algebraic formula you have entered. Each syntax rule should go on
-its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
-and a Calc formula with an optional @dfn{condition}. (Syntax rules
-resemble algebraic rewrite rules, but the notation for patterns is
-completely different.)
-
-A syntax pattern is a list of tokens, separated by spaces.
-Except for a few special symbols, tokens in syntax patterns are
-matched literally, from left to right. For example, the rule,
-
address@hidden
-foo ( ) := 2+3
address@hidden example
-
address@hidden
-would cause Calc to parse the formula @samp{4+foo()*5} as if it
-were @samp{4+(2+3)*5}. Notice that the parentheses were written
-as two separate tokens in the rule. As a result, the rule works
-for both @samp{foo()} and @address@hidden ( )}}. If we had written
-the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
-as a single, indivisible token, so that @address@hidden( )}} would
-not be recognized by the rule. (It would be parsed as a regular
-zero-argument function call instead.) In fact, this rule would
-also make trouble for the rest of Calc's parser: An unrelated
-formula like @samp{bar()} would now be tokenized into @samp{bar ()}
-instead of @samp{bar ( )}, so that the standard parser for function
-calls would no longer recognize it!
-
-While it is possible to make a token with a mixture of letters
-and punctuation symbols, this is not recommended. It is better to
-break it into several tokens, as we did with @samp{foo()} above.
-
-The symbol @samp{#} in a syntax pattern matches any Calc expression.
-On the righthand side, the things that matched the @samp{#}s can
-be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
-matches the leftmost @samp{#} in the pattern). For example, these
-rules match a user-defined function, prefix operator, infix operator,
-and postfix operator, respectively:
-
address@hidden
-foo ( # ) := myfunc(#1)
-foo # := myprefix(#1)
-# foo # := myinfix(#1,#2)
-# foo := mypostfix(#1)
address@hidden example
-
-Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
-will parse as @samp{mypostfix(2+3)}.
-
-It is important to write the first two rules in the order shown,
-because Calc tries rules in order from first to last. If the
-pattern @samp{foo #} came first, it would match anything that could
-match the @samp{foo ( # )} rule, since an expression in parentheses
-is itself a valid expression. Thus the @address@hidden ( # )}} rule would
-never get to match anything. Likewise, the last two rules must be
-written in the order shown or else @samp{3 foo 4} will be parsed as
address@hidden(3) * 4}. (Of course, the best way to avoid these
-ambiguities is not to use the same symbol in more than one way at
-the same time! In case you're not convinced, try the following
-exercise: How will the above rules parse the input @samp{foo(3,4)},
-if at all? Work it out for yourself, then try it in Calc and see.)
-
-Calc is quite flexible about what sorts of patterns are allowed.
-The only rule is that every pattern must begin with a literal
-token (like @samp{foo} in the first two patterns above), or with
-a @samp{#} followed by a literal token (as in the last two
-patterns). After that, any mixture is allowed, although putting
-two @samp{#}s in a row will not be very useful since two
-expressions with nothing between them will be parsed as one
-expression that uses implicit multiplication.
-
-As a more practical example, Maple uses the notation
address@hidden(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
-recognize at present. To handle this syntax, we simply add the
-rule,
-
address@hidden
-sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
address@hidden example
-
address@hidden
-to the Maple mode syntax table. As another example, C mode can't
-read assignment operators like @samp{++} and @samp{*=}. We can
-define these operators quite easily:
-
address@hidden
-# *= # := muleq(#1,#2)
-# ++ := postinc(#1)
-++ # := preinc(#1)
address@hidden example
-
address@hidden
-To complete the job, we would use corresponding composition functions
-and @kbd{Z C} to cause these functions to display in their respective
-Maple and C notations. (Note that the C example ignores issues of
-operator precedence, which are discussed in the next section.)
-
-You can enclose any token in quotes to prevent its usual
-interpretation in syntax patterns:
-
address@hidden
-# ":=" # := becomes(#1,#2)
address@hidden example
-
-Quotes also allow you to include spaces in a token, although once
-again it is generally better to use two tokens than one token with
-an embedded space. To include an actual quotation mark in a quoted
-token, precede it with a backslash. (This also works to include
-backslashes in tokens.)
-
address@hidden
-# "bad token" # "/\"\\" # := silly(#1,#2,#3)
address@hidden example
-
address@hidden
-This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
-
-The token @kbd{#} has a predefined meaning in Calc's formula parser;
-it is not valid to use @samp{"#"} in a syntax rule. However, longer
-tokens that include the @samp{#} character are allowed. Also, while
address@hidden"$"} and @samp{"\""} are allowed as tokens, their presence in
-the syntax table will prevent those characters from working in their
-usual ways (referring to stack entries and quoting strings,
-respectively).
-
-Finally, the notation @samp{%%} anywhere in a syntax table causes
-the rest of the line to be ignored as a comment.
-
address@hidden Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax
Table Basics, Syntax Tables
address@hidden Precedence
-
address@hidden
-Different operators are generally assigned different @dfn{precedences}.
-By default, an operator defined by a rule like
-
address@hidden
-# foo # := foo(#1,#2)
address@hidden example
-
address@hidden
-will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
-will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
-precedence of an operator, use the notation @samp{#/@var{p}} in
-place of @samp{#}, where @var{p} is an integer precedence level.
-For example, 185 lies between the precedences for @samp{+} and
address@hidden, so if we change this rule to
-
address@hidden
-#/185 foo #/186 := foo(#1,#2)
address@hidden example
-
address@hidden
-then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
-Also, because we've given the righthand expression slightly higher
-precedence, our new operator will be left-associative:
address@hidden foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
-By raising the precedence of the lefthand expression instead, we
-can create a right-associative operator.
-
address@hidden Basics}, for a table of precedences of the
-standard Calc operators. For the precedences of operators in other
-language modes, look in the Calc source file @file{calc-lang.el}.
-
address@hidden Advanced Syntax Patterns, Conditional Syntax Rules, Precedence
in Syntax Tables, Syntax Tables
address@hidden Advanced Syntax Patterns
-
address@hidden
-To match a function with a variable number of arguments, you could
-write
-
address@hidden
-foo ( # ) := myfunc(#1)
-foo ( # , # ) := myfunc(#1,#2)
-foo ( # , # , # ) := myfunc(#1,#2,#3)
address@hidden example
-
address@hidden
-but this isn't very elegant. To match variable numbers of items,
-Calc uses some notations inspired regular expressions and the
-``extended BNF'' style used by some language designers.
-
address@hidden
-foo ( @{ # @}*, ) := apply(myfunc,#1)
address@hidden example
-
-The token @address@hidden introduces a repeated or optional portion.
-One of the three tokens @address@hidden, @address@hidden, or @address@hidden
-ends the portion. These will match zero or more, one or more,
-or zero or one copies of the enclosed pattern, respectively.
-In addition, @address@hidden and @address@hidden can be followed by a
-separator token (with no space in between, as shown above).
-Thus @address@hidden # @}*,} matches nothing, or one expression, or
-several expressions separated by commas.
-
-A complete @address@hidden ... @}} item matches as a vector of the
-items that matched inside it. For example, the above rule will
-match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
-The Calc @code{apply} function takes a function name and a vector
-of arguments and builds a call to the function with those
-arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
-
-If the body of a @address@hidden ... @}} contains several @samp{#}s
-(or nested @address@hidden ... @}} constructs), then the items will be
-strung together into the resulting vector. If the body
-does not contain anything but literal tokens, the result will
-always be an empty vector.
-
address@hidden
-foo ( @{ # , # @}+, ) := bar(#1)
-foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
address@hidden example
-
address@hidden
-will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
address@hidden(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
-some thought it's easy to see how this pair of rules will parse
address@hidden(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
-rule will only match an even number of arguments. The rule
-
address@hidden
-foo ( # @{ , # , # @}? ) := bar(#1,#2)
address@hidden example
-
address@hidden
-will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
address@hidden(2)} as @samp{bar(2,[])}.
-
-The notation @address@hidden ... @}?.} (note the trailing period) works
-just the same as regular @address@hidden ... @}?}, except that it does not
-count as an argument; the following two rules are equivalent:
-
address@hidden
-foo ( # , @{ also @}? # ) := bar(#1,#3)
-foo ( # , @{ also @}?. # ) := bar(#1,#2)
address@hidden example
-
address@hidden
-Note that in the first case the optional text counts as @samp{#2},
-which will always be an empty vector, but in the second case no
-empty vector is produced.
-
-Another variant is @address@hidden ... @}?$}, which means the body is
-optional only at the end of the input formula. All built-in syntax
-rules in Calc use this for closing delimiters, so that during
-algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
-the closing parenthesis and bracket. Calc does this automatically
-for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
-rules, but you can use @address@hidden ... @}?$} explicitly to get
-this effect with any token (such as @samp{"@}"} or @samp{end}).
-Like @address@hidden ... @}?.}, this notation does not count as an
-argument. Conversely, you can use quotes, as in @samp{")"}, to
-prevent a closing-delimiter token from being automatically treated
-as optional.
-
-Calc's parser does not have full backtracking, which means some
-patterns will not work as you might expect:
-
address@hidden
-foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
address@hidden example
-
address@hidden
-Here we are trying to make the first argument optional, so that
address@hidden(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
-first tries to match @samp{2,} against the optional part of the
-pattern, finds a match, and so goes ahead to match the rest of the
-pattern. Later on it will fail to match the second comma, but it
-doesn't know how to go back and try the other alternative at that
-point. One way to get around this would be to use two rules:
-
address@hidden
-foo ( # , # , # ) := bar([#1],#2,#3)
-foo ( # , # ) := bar([],#1,#2)
address@hidden example
-
-More precisely, when Calc wants to match an optional or repeated
-part of a pattern, it scans forward attempting to match that part.
-If it reaches the end of the optional part without failing, it
-``finalizes'' its choice and proceeds. If it fails, though, it
-backs up and tries the other alternative. Thus Calc has ``partial''
-backtracking. A fully backtracking parser would go on to make sure
-the rest of the pattern matched before finalizing the choice.
-
address@hidden Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax
Tables
address@hidden Conditional Syntax Rules
-
address@hidden
-It is possible to attach a @dfn{condition} to a syntax rule. For
-example, the rules
-
address@hidden
-foo ( # ) := ifoo(#1) :: integer(#1)
-foo ( # ) := gfoo(#1)
address@hidden example
-
address@hidden
-will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
address@hidden(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
-number of conditions may be attached; all must be true for the
-rule to succeed. A condition is ``true'' if it evaluates to a
-nonzero number. @xref{Logical Operations}, for a list of Calc
-functions like @code{integer} that perform logical tests.
-
-The exact sequence of events is as follows: When Calc tries a
-rule, it first matches the pattern as usual. It then substitutes
address@hidden, @samp{#2}, etc., in the conditions, if any. Next, the
-conditions are simplified and evaluated in order from left to right,
-as if by the @address@hidden s}} algebra command (@pxref{Simplifying
Formulas}).
-Each result is true if it is a nonzero number, or an expression
-that can be proven to be nonzero (@pxref{Declarations}). If the
-results of all conditions are true, the expression (such as
address@hidden(#1)}) has its @samp{#}s substituted, and that is the
-result of the parse. If the result of any condition is false, Calc
-goes on to try the next rule in the syntax table.
-
-Syntax rules also support @code{let} conditions, which operate in
-exactly the same way as they do in algebraic rewrite rules.
address@hidden Features of Rewrite Rules}, for details. A @code{let}
-condition is always true, but as a side effect it defines a
-variable which can be used in later conditions, and also in the
-expression after the @samp{:=} sign:
-
address@hidden
-foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
address@hidden example
-
address@hidden
-The @code{dnumint} function tests if a value is numerically an
-integer, i.e., either a true integer or an integer-valued float.
-This rule will parse @code{foo} with a half-integer argument,
-like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
-
-The lefthand side of a syntax rule @code{let} must be a simple
-variable, not the arbitrary pattern that is allowed in rewrite
-rules.
-
-The @code{matches} function is also treated specially in syntax
-rule conditions (again, in the same way as in rewrite rules).
address@hidden Commands}. If the matching pattern contains
-meta-variables, then those meta-variables may be used in later
-conditions and in the result expression. The arguments to
address@hidden are not evaluated in this situation.
-
address@hidden
-sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
address@hidden example
-
address@hidden
-This is another way to implement the Maple mode @code{sum} notation.
-In this approach, we allow @samp{#2} to equal the whole expression
address@hidden Then, we use @code{matches} to break it apart into
-its components. If the expression turns out not to match the pattern,
-the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
-Normal language mode for editing expressions in syntax rules, so we
-must use regular Calc notation for the interval @samp{[b..c]} that
-will correspond to the Maple mode interval @samp{1..10}.
-
address@hidden Modes Variable, Calc Mode Line, Language Modes, Mode Settings
address@hidden The @code{Modes} Variable
-
address@hidden
address@hidden m g
address@hidden calc-get-modes
-The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
-a vector of numbers that describes the various mode settings that
-are in effect. With a numeric prefix argument, it pushes only the
address@hidden mode, i.e., the @var{n}th element of this vector. Keyboard
-macros can use the @kbd{m g} command to modify their behavior based
-on the current mode settings.
-
address@hidden @code{Modes} variable
address@hidden Modes
-The modes vector is also available in the special variable
address@hidden In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
-It will not work to store into this variable; in fact, if you do,
address@hidden will cease to track the current modes. (The @kbd{m g}
-command will continue to work, however.)
-
-In general, each number in this vector is suitable as a numeric
-prefix argument to the associated mode-setting command. (Recall
-that the @kbd{~} key takes a number from the stack and gives it as
-a numeric prefix to the next command.)
-
-The elements of the modes vector are as follows:
-
address@hidden
address@hidden
-Current precision. Default is 12; associated command is @kbd{p}.
-
address@hidden
-Binary word size. Default is 32; associated command is @kbd{b w}.
-
address@hidden
-Stack size (not counting the value about to be pushed by @kbd{m g}).
-This is zero if @kbd{m g} is executed with an empty stack.
-
address@hidden
-Number radix. Default is 10; command is @kbd{d r}.
-
address@hidden
-Floating-point format. This is the number of digits, plus the
-constant 0 for normal notation, 10000 for scientific notation,
-20000 for engineering notation, or 30000 for fixed-point notation.
-These codes are acceptable as prefix arguments to the @kbd{d n}
-command, but note that this may lose information: For example,
address@hidden s} and @kbd{C-u 12 d s} have similar (but not quite
-identical) effects if the current precision is 12, but they both
-produce a code of 10012, which will be treated by @kbd{d n} as
address@hidden 12 d s}. If the precision then changes, the float format
-will still be frozen at 12 significant figures.
-
address@hidden
-Angular mode. Default is 1 (degrees). Other values are 2 (radians)
-and 3 (HMS). The @kbd{m d} command accepts these prefixes.
-
address@hidden
-Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
-
address@hidden
-Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
-
address@hidden
-Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
-Command is @kbd{m p}.
-
address@hidden
-Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
-mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
-or @var{N} for
address@hidden @math{N\times N}
address@hidden @address@hidden
-Matrix mode. Command is @kbd{m v}.
-
address@hidden
-Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
-0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
-or 5 for @address@hidden U}}. The @kbd{m D} command accepts these prefixes.
-
address@hidden
-Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
-or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
address@hidden enumerate
-
-For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
-precision by two, leaving a copy of the old precision on the stack.
-Later, @kbd{~ p} will restore the original precision using that
-stack value. (This sequence might be especially useful inside a
-keyboard macro.)
-
-As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
-oldest (bottommost) stack entry.
-
-Yet another example: The HP-48 ``round'' command rounds a number
-to the current displayed precision. You could roughly emulate this
-in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
-would not work for fixed-point mode, but it wouldn't be hard to
-do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
-programming commands. @xref{Conditionals in Macros}.)
-
address@hidden Calc Mode Line, , Modes Variable, Mode Settings
address@hidden The Calc Mode Line
-
address@hidden
address@hidden Mode line indicators
-This section is a summary of all symbols that can appear on the
-Calc mode line, the highlighted bar that appears under the Calc
-stack window (or under an editing window in Embedded mode).
-
-The basic mode line format is:
-
address@hidden
---%%-Calc: 12 Deg @var{other modes} (Calculator)
address@hidden example
-
-The @samp{%%} is the Emacs symbol for ``read-only''; it shows that
-regular Emacs commands are not allowed to edit the stack buffer
-as if it were text.
-
-The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
-is enabled. The words after this describe the various Calc modes
-that are in effect.
-
-The first mode is always the current precision, an integer.
-The second mode is always the angular mode, either @code{Deg},
address@hidden, or @code{Hms}.
-
-Here is a complete list of the remaining symbols that can appear
-on the mode line:
-
address@hidden @code
address@hidden Alg
-Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
-
address@hidden Alg[(
-Incomplete algebraic mode (@kbd{C-u m a}).
-
address@hidden Alg*
-Total algebraic mode (@kbd{m t}).
-
address@hidden Symb
-Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
-
address@hidden Matrix
-Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
-
address@hidden address@hidden
-Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).
-
address@hidden SqMatrix
-Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).
-
address@hidden Scalar
-Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
-
address@hidden Polar
-Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
-
address@hidden Frac
-Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
-
address@hidden Inf
-Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
-
address@hidden +Inf
-Positive Infinite mode (@kbd{C-u 0 m i}).
-
address@hidden NoSimp
-Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
-
address@hidden NumSimp
-Default simplifications for numeric arguments only (@kbd{m N}).
-
address@hidden address@hidden
-Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
-
address@hidden AlgSimp
-Algebraic simplification mode (@kbd{m A}).
-
address@hidden ExtSimp
-Extended algebraic simplification mode (@kbd{m E}).
-
address@hidden UnitSimp
-Units simplification mode (@kbd{m U}).
-
address@hidden Bin
-Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
-
address@hidden Oct
-Current radix is 8 (@kbd{d 8}).
-
address@hidden Hex
-Current radix is 16 (@kbd{d 6}).
-
address@hidden address@hidden
-Current radix is @var{n} (@kbd{d r}).
-
address@hidden Zero
-Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
-
address@hidden Big
-Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
-
address@hidden Flat
-One-line normal language mode (@kbd{d O}).
-
address@hidden Unform
-Unformatted language mode (@kbd{d U}).
-
address@hidden C
-C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
-
address@hidden Pascal
-Pascal language mode (@kbd{d P}).
-
address@hidden Fortran
-FORTRAN language mode (@kbd{d F}).
-
address@hidden TeX
address@hidden language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
-
address@hidden LaTeX
address@hidden language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
-
address@hidden Eqn
address@hidden language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
-
address@hidden Math
-Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
-
address@hidden Maple
-Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
-
address@hidden address@hidden
-Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
-
address@hidden address@hidden
-Fixed point mode with @var{n} digits after the point (@kbd{d f}).
-
address@hidden Sci
-Scientific notation mode (@kbd{d s}).
-
address@hidden address@hidden
-Scientific notation with @var{n} digits (@kbd{d s}).
-
address@hidden Eng
-Engineering notation mode (@kbd{d e}).
-
address@hidden address@hidden
-Engineering notation with @var{n} digits (@kbd{d e}).
-
address@hidden address@hidden
-Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
-
address@hidden Right
-Right-justified display (@kbd{d >}).
-
address@hidden address@hidden
-Right-justified display with width @var{n} (@kbd{d >}).
-
address@hidden Center
-Centered display (@kbd{d =}).
-
address@hidden address@hidden
-Centered display with center column @var{n} (@kbd{d =}).
-
address@hidden address@hidden
-Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
-
address@hidden Wide
-No line breaking (@kbd{d b}).
-
address@hidden Break
-Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
-
address@hidden Save
-Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
-
address@hidden Local
-Record modes in Embedded buffer (@kbd{m R}).
-
address@hidden LocEdit
-Record modes as editing-only in Embedded buffer (@kbd{m R}).
-
address@hidden LocPerm
-Record modes as permanent-only in Embedded buffer (@kbd{m R}).
-
address@hidden Global
-Record modes as global in Embedded buffer (@kbd{m R}).
-
address@hidden Manual
-Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
-Recomputation}).
-
address@hidden Graph
-GNUPLOT process is alive in background (@pxref{Graphics}).
-
address@hidden Sel
-Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
-
address@hidden Dirty
-The stack display may not be up-to-date (@pxref{Display Modes}).
-
address@hidden Inv
-``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
-
address@hidden Hyp
-``Hyperbolic'' prefix was pressed (@kbd{H}).
-
address@hidden Keep
-``Keep-arguments'' prefix was pressed (@kbd{K}).
-
address@hidden Narrow
-Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
address@hidden table
-
-In addition, the symbols @code{Active} and @code{~Active} can appear
-as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
-
address@hidden Arithmetic, Scientific Functions, Mode Settings, Top
address@hidden Arithmetic Functions
-
address@hidden
-This chapter describes the Calc commands for doing simple calculations
-on numbers, such as addition, absolute value, and square roots. These
-commands work by removing the top one or two values from the stack,
-performing the desired operation, and pushing the result back onto the
-stack. If the operation cannot be performed, the result pushed is a
-formula instead of a number, such as @samp{2/0} (because division by zero
-is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
-
-Most of the commands described here can be invoked by a single keystroke.
-Some of the more obscure ones are two-letter sequences beginning with
-the @kbd{f} (``functions'') prefix key.
-
address@hidden Arguments}, for a discussion of the effect of numeric
-prefix arguments on commands in this chapter which do not otherwise
-interpret a prefix argument.
-
address@hidden
-* Basic Arithmetic::
-* Integer Truncation::
-* Complex Number Functions::
-* Conversions::
-* Date Arithmetic::
-* Financial Functions::
-* Binary Functions::
address@hidden menu
-
address@hidden Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
address@hidden Basic Arithmetic
-
address@hidden
address@hidden +
address@hidden calc-plus
address@hidden
address@hidden @null
address@hidden ignore
address@hidden +
-The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
-be any of the standard Calc data types. The resulting sum is pushed back
-onto the stack.
-
-If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
-the result is a vector or matrix sum. If one argument is a vector and the
-other a scalar (i.e., a non-vector), the scalar is added to each of the
-elements of the vector to form a new vector. If the scalar is not a
-number, the operation is left in symbolic form: Suppose you added @samp{x}
-to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
-you may plan to substitute a 2-vector for @samp{x} in the future. Since
-the Calculator can't tell which interpretation you want, it makes the
-safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
-to every element of a vector.
-
-If either argument of @kbd{+} is a complex number, the result will in general
-be complex. If one argument is in rectangular form and the other polar,
-the current Polar mode determines the form of the result. If Symbolic
-mode is enabled, the sum may be left as a formula if the necessary
-conversions for polar addition are non-trivial.
-
-If both arguments of @kbd{+} are HMS forms, the forms are added according to
-the usual conventions of hours-minutes-seconds notation. If one argument
-is an HMS form and the other is a number, that number is converted from
-degrees or radians (depending on the current Angular mode) to HMS format
-and then the two HMS forms are added.
-
-If one argument of @kbd{+} is a date form, the other can be either a
-real number, which advances the date by a certain number of days, or
-an HMS form, which advances the date by a certain amount of time.
-Subtracting two date forms yields the number of days between them.
-Adding two date forms is meaningless, but Calc interprets it as the
-subtraction of one date form and the negative of the other. (The
-negative of a date form can be understood by remembering that dates
-are stored as the number of days before or after Jan 1, 1 AD.)
-
-If both arguments of @kbd{+} are error forms, the result is an error form
-with an appropriately computed standard deviation. If one argument is an
-error form and the other is a number, the number is taken to have zero error.
-Error forms may have symbolic formulas as their mean and/or error parts;
-adding these will produce a symbolic error form result. However, adding an
-error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
-work, for the same reasons just mentioned for vectors. Instead you must
-write @samp{(a +/- b) + (c +/- 0)}.
-
-If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
-or if one argument is a modulo form and the other a plain number, the
-result is a modulo form which represents the sum, modulo @expr{M}, of
-the two values.
-
-If both arguments of @kbd{+} are intervals, the result is an interval
-which describes all possible sums of the possible input values. If
-one argument is a plain number, it is treated as the interval
address@hidden@samp{[x ..@: x]}}.
-
-If one argument of @kbd{+} is an infinity and the other is not, the
-result is that same infinity. If both arguments are infinite and in
-the same direction, the result is the same infinity, but if they are
-infinite in different directions the result is @code{nan}.
-
address@hidden -
address@hidden calc-minus
address@hidden
address@hidden @null
address@hidden ignore
address@hidden -
-The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
-number on the stack is subtracted from the one behind it, so that the
-computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
-available for @kbd{+} are available for @kbd{-} as well.
-
address@hidden *
address@hidden calc-times
address@hidden
address@hidden @null
address@hidden ignore
address@hidden *
-The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
-argument is a vector and the other a scalar, the scalar is multiplied by
-the elements of the vector to produce a new vector. If both arguments
-are vectors, the interpretation depends on the dimensions of the
-vectors: If both arguments are matrices, a matrix multiplication is
-done. If one argument is a matrix and the other a plain vector, the
-vector is interpreted as a row vector or column vector, whichever is
-dimensionally correct. If both arguments are plain vectors, the result
-is a single scalar number which is the dot product of the two vectors.
-
-If one argument of @kbd{*} is an HMS form and the other a number, the
-HMS form is multiplied by that amount. It is an error to multiply two
-HMS forms together, or to attempt any multiplication involving date
-forms. Error forms, modulo forms, and intervals can be multiplied;
-see the comments for addition of those forms. When two error forms
-or intervals are multiplied they are considered to be statistically
-independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
-whereas @address@hidden ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
-
address@hidden /
address@hidden calc-divide
address@hidden
address@hidden @null
address@hidden ignore
address@hidden /
-The @kbd{/} (@code{calc-divide}) command divides two numbers.
-
-When combining multiplication and division in an algebraic formula, it
-is good style to use parentheses to distinguish between possible
-interpretations; the expression @samp{a/b*c} should be written
address@hidden(a/b)*c} or @samp{a/(b*c)}, as appropriate. Without the
-parentheses, Calc will interpret @samp{a/b*c} as @samp{a/(b*c)}, since
-in algebraic entry Calc gives division a lower precedence than
-multiplication. (This is not standard across all computer languages, and
-Calc may change the precedence depending on the language mode being used.
address@hidden Modes}.) This default ordering can be changed by setting
-the customizable variable @code{calc-multiplication-has-precedence} to
address@hidden (@pxref{Customizing Calc}); this will give multiplication and
-division equal precedences. Note that Calc's default choice of
-precedence allows @samp{a b / c d} to be used as a shortcut for
address@hidden
address@hidden
-a b
----.
-c d
address@hidden group
address@hidden smallexample
-
-When dividing a scalar @expr{B} by a square matrix @expr{A}, the
-computation performed is @expr{B} times the inverse of @expr{A}. This
-also occurs if @expr{B} is itself a vector or matrix, in which case the
-effect is to solve the set of linear equations represented by @expr{B}.
-If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
-plain vector (which is interpreted here as a column vector), then the
-equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
-Otherwise, if @expr{B} is a non-square matrix with the same number of
address@hidden as @expr{A}, the equation @expr{X A = B} is solved. If
-you wish a vector @expr{B} to be interpreted as a row vector to be
-solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
-v p} first. To force a left-handed solution with a square matrix
address@hidden, transpose @expr{A} and @expr{B} before dividing, then
-transpose the result.
-
-HMS forms can be divided by real numbers or by other HMS forms. Error
-forms can be divided in any combination of ways. Modulo forms where both
-values and the modulo are integers can be divided to get an integer modulo
-form result. Intervals can be divided; dividing by an interval that
-encompasses zero or has zero as a limit will result in an infinite
-interval.
-
address@hidden ^
address@hidden calc-power
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ^
-The @kbd{^} (@code{calc-power}) command raises a number to a power. If
-the power is an integer, an exact result is computed using repeated
-multiplications. For non-integer powers, Calc uses Newton's method or
-logarithms and exponentials. Square matrices can be raised to integer
-powers. If either argument is an error (or interval or modulo) form,
-the result is also an error (or interval or modulo) form.
-
address@hidden I ^
address@hidden nroot
-If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
-computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
-(This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
-
address@hidden \
address@hidden calc-idiv
address@hidden idiv
address@hidden
address@hidden @null
address@hidden ignore
address@hidden \
-The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
-to produce an integer result. It is equivalent to dividing with
address@hidden/}, then rounding down with @kbd{F} (@code{calc-floor}), only a
bit
-more convenient and efficient. Also, since it is an all-integer
-operation when the arguments are integers, it avoids problems that
address@hidden/ F} would have with floating-point roundoff.
-
address@hidden %
address@hidden calc-mod
address@hidden
address@hidden @null
address@hidden ignore
address@hidden %
-The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
-operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
-for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
-positive @expr{b}, the result will always be between 0 (inclusive) and
address@hidden (exclusive). Modulo does not work for HMS forms and error forms.
-If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
-must be positive real number.
-
address@hidden :
address@hidden calc-fdiv
address@hidden fdiv
-The @kbd{:} (@code{calc-fdiv}) address@hidden command
-divides the two integers on the top of the stack to produce a fractional
-result. This is a convenient shorthand for enabling Fraction mode (with
address@hidden f}) temporarily and using @samp{/}. Note that during numeric
entry
-the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
-you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
-this case, it would be much easier simply to enter the fraction directly
-as @kbd{8:6 @key{RET}}!)
-
address@hidden n
address@hidden calc-change-sign
-The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
-of the stack. It works on numbers, vectors and matrices, HMS forms, date
-forms, error forms, intervals, and modulo forms.
-
address@hidden A
address@hidden calc-abs
address@hidden abs
-The @kbd{A} (@code{calc-abs}) address@hidden command computes the absolute
-value of a number. The result of @code{abs} is always a nonnegative
-real number: With a complex argument, it computes the complex magnitude.
-With a vector or matrix argument, it computes the Frobenius norm, i.e.,
-the square root of the sum of the squares of the absolute values of the
-elements. The absolute value of an error form is defined by replacing
-the mean part with its absolute value and leaving the error part the same.
-The absolute value of a modulo form is undefined. The absolute value of
-an interval is defined in the obvious way.
-
address@hidden f A
address@hidden calc-abssqr
address@hidden abssqr
-The @kbd{f A} (@code{calc-abssqr}) address@hidden command computes the
-absolute value squared of a number, vector or matrix, or error form.
-
address@hidden f s
address@hidden calc-sign
address@hidden sign
-The @kbd{f s} (@code{calc-sign}) address@hidden command returns 1 if its
-argument is positive, @mathit{-1} if its argument is negative, or 0 if its
-argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
-which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
-zero depending on the sign of @samp{a}.
-
address@hidden &
address@hidden calc-inv
address@hidden inv
address@hidden Reciprocal
-The @kbd{&} (@code{calc-inv}) address@hidden command computes the
-reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
-matrix, it computes the inverse of that matrix.
-
address@hidden Q
address@hidden calc-sqrt
address@hidden sqrt
-The @kbd{Q} (@code{calc-sqrt}) address@hidden command computes the square
-root of a number. For a negative real argument, the result will be a
-complex number whose form is determined by the current Polar mode.
-
address@hidden f h
address@hidden calc-hypot
address@hidden hypot
-The @kbd{f h} (@code{calc-hypot}) address@hidden command computes the square
-root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
-is the length of the hypotenuse of a right triangle with sides @expr{a}
-and @expr{b}. If the arguments are complex numbers, their squared
-magnitudes are used.
-
address@hidden f Q
address@hidden calc-isqrt
address@hidden isqrt
-The @kbd{f Q} (@code{calc-isqrt}) address@hidden command computes the
-integer square root of an integer. This is the true square root of the
-number, rounded down to an integer. For example, @samp{isqrt(10)}
-produces 3. Note that, like @kbd{\} address@hidden, this uses exact
-integer arithmetic throughout to avoid roundoff problems. If the input
-is a floating-point number or other non-integer value, this is exactly
-the same as @samp{floor(sqrt(x))}.
-
address@hidden f n
address@hidden f x
address@hidden calc-min
address@hidden min
address@hidden calc-max
address@hidden max
-The @kbd{f n} (@code{calc-min}) address@hidden and @kbd{f x} (@code{calc-max})
address@hidden commands take the minimum or maximum of two real numbers,
-respectively. These commands also work on HMS forms, date forms,
-intervals, and infinities. (In algebraic expressions, these functions
-take any number of arguments and return the maximum or minimum among
-all the arguments.)
-
address@hidden f M
address@hidden f X
address@hidden calc-mant-part
address@hidden mant
address@hidden calc-xpon-part
address@hidden xpon
-The @kbd{f M} (@code{calc-mant-part}) address@hidden function extracts
-the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
-(@code{calc-xpon-part}) address@hidden extracts the ``exponent'' part
address@hidden The original number is equal to
address@hidden @math{m \times 10^e},
address@hidden @expr{m * 10^e},
-where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
address@hidden if the original number is zero. For integers
-and fractions, @code{mant} returns the number unchanged and @code{xpon}
-returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
-used to ``unpack'' a floating-point number; this produces an integer
-mantissa and exponent, with the constraint that the mantissa is not
-a multiple of ten (again except for the @expr{m=e=0} case).
-
address@hidden f S
address@hidden calc-scale-float
address@hidden scf
-The @kbd{f S} (@code{calc-scale-float}) address@hidden function scales a number
-by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
-real @samp{x}. The second argument must be an integer, but the first
-may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
-or @samp{1:20} depending on the current Fraction mode.
-
address@hidden f [
address@hidden f ]
address@hidden calc-decrement
address@hidden calc-increment
address@hidden decr
address@hidden incr
-The @kbd{f [} (@code{calc-decrement}) address@hidden and @kbd{f ]}
-(@code{calc-increment}) address@hidden functions decrease or increase
-a number by one unit. For integers, the effect is obvious. For
-floating-point numbers, the change is by one unit in the last place.
-For example, incrementing @samp{12.3456} when the current precision
-is 6 digits yields @samp{12.3457}. If the current precision had been
-8 digits, the result would have been @samp{12.345601}. Incrementing
address@hidden produces
address@hidden @math{10^{-p}},
address@hidden @expr{10^-p},
-where @expr{p} is the current
-precision. These operations are defined only on integers and floats.
-With numeric prefix arguments, they change the number by @expr{n} units.
-
-Note that incrementing followed by decrementing, or vice-versa, will
-almost but not quite always cancel out. Suppose the precision is
-6 digits and the number @samp{9.99999} is on the stack. Incrementing
-will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
-One digit has been dropped. This is an unavoidable consequence of the
-way floating-point numbers work.
-
-Incrementing a date/time form adjusts it by a certain number of seconds.
-Incrementing a pure date form adjusts it by a certain number of days.
-
address@hidden Integer Truncation, Complex Number Functions, Basic Arithmetic,
Arithmetic
address@hidden Integer Truncation
-
address@hidden
-There are four commands for truncating a real number to an integer,
-differing mainly in their treatment of negative numbers. All of these
-commands have the property that if the argument is an integer, the result
-is the same integer. An integer-valued floating-point argument is converted
-to integer form.
-
-If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
-expressed as an integer-valued floating-point number.
-
address@hidden Integer part of a number
address@hidden F
address@hidden calc-floor
address@hidden floor
address@hidden ffloor
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H F
-The @kbd{F} (@code{calc-floor}) address@hidden or @code{ffloor}] command
-truncates a real number to the next lower integer, i.e., toward minus
-infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
address@hidden
-
address@hidden I F
address@hidden calc-ceiling
address@hidden ceil
address@hidden fceil
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I F
-The @kbd{I F} (@code{calc-ceiling}) address@hidden or @code{fceil}]
-command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
-4, and @kbd{_3.6 I F} produces @mathit{-3}.
-
address@hidden R
address@hidden calc-round
address@hidden round
address@hidden fround
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H R
-The @kbd{R} (@code{calc-round}) address@hidden or @code{fround}] command
-rounds to the nearest integer. When the fractional part is .5 exactly,
-this command rounds away from zero. (All other rounding in the
-Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
-but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
-
address@hidden I R
address@hidden calc-trunc
address@hidden trunc
address@hidden ftrunc
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I R
-The @kbd{I R} (@code{calc-trunc}) address@hidden or @code{ftrunc}]
-command truncates toward zero. In other words, it ``chops off''
-everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
address@hidden I R} produces @mathit{-3}.
-
-These functions may not be applied meaningfully to error forms, but they
-do work for intervals. As a convenience, applying @code{floor} to a
-modulo form floors the value part of the form. Applied to a vector,
-these functions operate on all elements of the vector one by one.
-Applied to a date form, they operate on the internal numerical
-representation of dates, converting a date/time form into a pure date.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden rounde
address@hidden
address@hidden
address@hidden ignore
address@hidden roundu
address@hidden
address@hidden
address@hidden ignore
address@hidden frounde
address@hidden
address@hidden
address@hidden ignore
address@hidden froundu
-There are two more rounding functions which can only be entered in
-algebraic notation. The @code{roundu} function is like @code{round}
-except that it rounds up, toward plus infinity, when the fractional
-part is .5. This distinction matters only for negative arguments.
-Also, @code{rounde} rounds to an even number in the case of a tie,
-rounding up or down as necessary. For example, @samp{rounde(3.5)} and
address@hidden(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
-The advantage of round-to-even is that the net error due to rounding
-after a long calculation tends to cancel out to zero. An important
-subtle point here is that the number being fed to @code{rounde} will
-already have been rounded to the current precision before @code{rounde}
-begins. For example, @samp{rounde(2.500001)} with a current precision
-of 6 will incorrectly, or at least surprisingly, yield 2 because the
-argument will first have been rounded down to @expr{2.5} (which
address@hidden sees as an exact tie between 2 and 3).
-
-Each of these functions, when written in algebraic formulas, allows
-a second argument which specifies the number of digits after the
-decimal point to keep. For example, @samp{round(123.4567, 2)} will
-produce the answer 123.46, and @samp{round(123.4567, -1)} will
-produce 120 (i.e., the cutoff is one digit to the @emph{left} of
-the decimal point). A second argument of zero is equivalent to
-no second argument at all.
-
address@hidden Fractional part of a number
-To compute the fractional part of a number (i.e., the amount which, when
-added to address@hidden(address@hidden@tfn{)}', will produce @var{n}) just
take @var{n}
-modulo 1 using the @code{%} command.
-
-Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
-and @kbd{f Q} (integer square root) commands, which are analogous to
address@hidden/}, @kbd{B}, and @kbd{Q}, respectively, except that they take
integer
-arguments and return the result rounded down to an integer.
-
address@hidden Complex Number Functions, Conversions, Integer Truncation,
Arithmetic
address@hidden Complex Number Functions
-
address@hidden
address@hidden J
address@hidden calc-conj
address@hidden conj
-The @kbd{J} (@code{calc-conj}) address@hidden command computes the
-complex conjugate of a number. For complex number @expr{a+bi}, the
-complex conjugate is @expr{a-bi}. If the argument is a real number,
-this command leaves it the same. If the argument is a vector or matrix,
-this command replaces each element by its complex conjugate.
-
address@hidden G
address@hidden calc-argument
address@hidden arg
-The @kbd{G} (@code{calc-argument}) address@hidden command computes the
-``argument'' or polar angle of a complex number. For a number in polar
-notation, this is simply the second component of the pair
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'.
address@hidden address@hidden(address@hidden@tfn{;address@hidden@tfn{)}'.
-The result is expressed according to the current angular mode and will
-be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
-(inclusive), or the equivalent range in radians.
-
address@hidden calc-imaginary
-The @code{calc-imaginary} command multiplies the number on the
-top of the stack by the imaginary number @expr{i = (0,1)}. This
-command is not normally bound to a key in Calc, but it is available
-on the @key{IMAG} button in Keypad mode.
-
address@hidden f r
address@hidden calc-re
address@hidden re
-The @kbd{f r} (@code{calc-re}) address@hidden command replaces a complex number
-by its real part. This command has no effect on real numbers. (As an
-added convenience, @code{re} applied to a modulo form extracts
-the value part.)
-
address@hidden f i
address@hidden calc-im
address@hidden im
-The @kbd{f i} (@code{calc-im}) address@hidden command replaces a complex number
-by its imaginary part; real numbers are converted to zero. With a vector
-or matrix argument, these functions operate element-wise.
-
address@hidden
address@hidden v p
address@hidden ignore
address@hidden v p (complex)
address@hidden calc-pack
-The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
-the stack into a composite object such as a complex number. With
-a prefix argument of @mathit{-1}, it produces a rectangular complex number;
-with an argument of @mathit{-2}, it produces a polar complex number.
-(Also, @pxref{Building Vectors}.)
-
address@hidden
address@hidden v u
address@hidden ignore
address@hidden v u (complex)
address@hidden calc-unpack
-The @kbd{v u} (@code{calc-unpack}) command takes the complex number
-(or other composite object) on the top of the stack and unpacks it
-into its separate components.
-
address@hidden Conversions, Date Arithmetic, Complex Number Functions,
Arithmetic
address@hidden Conversions
-
address@hidden
-The commands described in this section convert numbers from one form
-to another; they are two-key sequences beginning with the letter @kbd{c}.
-
address@hidden c f
address@hidden calc-float
address@hidden pfloat
-The @kbd{c f} (@code{calc-float}) address@hidden command converts the
-number on the top of the stack to floating-point form. For example,
address@hidden is converted to @expr{23.0}, @expr{3:2} is converted to
address@hidden, and @expr{2.3} is left the same. If the value is a composite
-object such as a complex number or vector, each of the components is
-converted to floating-point. If the value is a formula, all numbers
-in the formula are converted to floating-point. Note that depending
-on the current floating-point precision, conversion to floating-point
-format may lose information.
-
-As a special exception, integers which appear as powers or subscripts
-are not floated by @kbd{c f}. If you really want to float a power,
-you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
-Because @kbd{c f} cannot examine the formula outside of the selection,
-it does not notice that the thing being floated is a power.
address@hidden Subformulas}.
-
-The normal @kbd{c f} command is ``pervasive'' in the sense that it
-applies to all numbers throughout the formula. The @code{pfloat}
-algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
-changes to @samp{a + 1.0} as soon as it is evaluated.
-
address@hidden H c f
address@hidden float
-With the Hyperbolic flag, @kbd{H c f} address@hidden operates
-only on the number or vector of numbers at the top level of its
-argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
-is left unevaluated because its argument is not a number.
-
-You should use @kbd{H c f} if you wish to guarantee that the final
-value, once all the variables have been assigned, is a float; you
-would use @kbd{c f} if you wish to do the conversion on the numbers
-that appear right now.
-
address@hidden c F
address@hidden calc-fraction
address@hidden pfrac
-The @kbd{c F} (@code{calc-fraction}) address@hidden command converts a
-floating-point number into a fractional approximation. By default, it
-produces a fraction whose decimal representation is the same as the
-input number, to within the current precision. You can also give a
-numeric prefix argument to specify a tolerance, either directly, or,
-if the prefix argument is zero, by using the number on top of the stack
-as the tolerance. If the tolerance is a positive integer, the fraction
-is correct to within that many significant figures. If the tolerance is
-a non-positive integer, it specifies how many digits fewer than the current
-precision to use. If the tolerance is a floating-point number, the
-fraction is correct to within that absolute amount.
-
address@hidden H c F
address@hidden frac
-The @code{pfrac} function is pervasive, like @code{pfloat}.
-There is also a non-pervasive version, @kbd{H c F} address@hidden,
-which is analogous to @kbd{H c f} discussed above.
-
address@hidden c d
address@hidden calc-to-degrees
address@hidden deg
-The @kbd{c d} (@code{calc-to-degrees}) address@hidden command converts a
-number into degrees form. The value on the top of the stack may be an
-HMS form (interpreted as degrees-minutes-seconds), or a real number which
-will be interpreted in radians regardless of the current angular mode.
-
address@hidden c r
address@hidden calc-to-radians
address@hidden rad
-The @kbd{c r} (@code{calc-to-radians}) address@hidden command converts an
-HMS form or angle in degrees into an angle in radians.
-
address@hidden c h
address@hidden calc-to-hms
address@hidden hms
-The @kbd{c h} (@code{calc-to-hms}) address@hidden command converts a real
-number, interpreted according to the current angular mode, to an HMS
-form describing the same angle. In algebraic notation, the @code{hms}
-function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
-(The three-argument version is independent of the current angular mode.)
-
address@hidden calc-from-hms
-The @code{calc-from-hms} command converts the HMS form on the top of the
-stack into a real number according to the current angular mode.
-
address@hidden c p
address@hidden I c p
address@hidden calc-polar
address@hidden polar
address@hidden rect
-The @kbd{c p} (@code{calc-polar}) command converts the complex number on
-the top of the stack from polar to rectangular form, or from rectangular
-to polar form, whichever is appropriate. Real numbers are left the same.
-This command is equivalent to the @code{rect} or @code{polar}
-functions in algebraic formulas, depending on the direction of
-conversion. (It uses @code{polar}, except that if the argument is
-already a polar complex number, it uses @code{rect} instead. The
address@hidden c p} command always uses @code{rect}.)
-
address@hidden c c
address@hidden calc-clean
address@hidden pclean
-The @kbd{c c} (@code{calc-clean}) address@hidden command ``cleans'' the
-number on the top of the stack. Floating point numbers are re-rounded
-according to the current precision. Polar numbers whose angular
-components have strayed from the @mathit{-180} to @mathit{+180} degree range
-are normalized. (Note that results will be undesirable if the current
-angular mode is different from the one under which the number was
-produced!) Integers and fractions are generally unaffected by this
-operation. Vectors and formulas are cleaned by cleaning each component
-number (i.e., pervasively).
-
-If the simplification mode is set below the default level, it is raised
-to the default level for the purposes of this command. Thus, @kbd{c c}
-applies the default simplifications even if their automatic application
-is disabled. @xref{Simplification Modes}.
-
address@hidden Roundoff errors, correcting
-A numeric prefix argument to @kbd{c c} sets the floating-point precision
-to that value for the duration of the command. A positive prefix (of at
-least 3) sets the precision to the specified value; a negative or zero
-prefix decreases the precision by the specified amount.
-
address@hidden c 0-9
address@hidden calc-clean-num
-The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
-to @kbd{c c} with the corresponding negative prefix argument. If roundoff
-errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
-decimal place often conveniently does the trick.
-
-The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
-through @kbd{c 9} commands, also ``clip'' very small floating-point
-numbers to zero. If the exponent is less than or equal to the negative
-of the specified precision, the number is changed to 0.0. For example,
-if the current precision is 12, then @kbd{c 2} changes the vector
address@hidden, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
-Numbers this small generally arise from roundoff noise.
-
-If the numbers you are using really are legitimately this small,
-you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
-(The plain @kbd{c c} command rounds to the current precision but
-does not clip small numbers.)
-
-One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
-a prefix argument, is that integer-valued floats are converted to
-plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
-produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
-numbers (@samp{1e100} is technically an integer-valued float, but
-you wouldn't want it automatically converted to a 100-digit integer).
-
address@hidden H c 0-9
address@hidden H c c
address@hidden clean
-With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
-operate non-pervasively address@hidden
-
address@hidden Date Arithmetic, Financial Functions, Conversions, Arithmetic
address@hidden Date Arithmetic
-
address@hidden
address@hidden Date arithmetic, additional functions
-The commands described in this section perform various conversions
-and calculations involving date forms (@pxref{Date Forms}). They
-use the @kbd{t} (for time/date) prefix key followed by shifted
-letters.
-
-The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
-commands. In particular, adding a number to a date form advances the
-date form by a certain number of days; adding an HMS form to a date
-form advances the date by a certain amount of time; and subtracting two
-date forms produces a difference measured in days. The commands
-described here provide additional, more specialized operations on dates.
-
-Many of these commands accept a numeric prefix argument; if you give
-plain @kbd{C-u} as the prefix, these commands will instead take the
-additional argument from the top of the stack.
-
address@hidden
-* Date Conversions::
-* Date Functions::
-* Time Zones::
-* Business Days::
address@hidden menu
-
address@hidden Date Conversions, Date Functions, Date Arithmetic, Date
Arithmetic
address@hidden Date Conversions
-
address@hidden
address@hidden t D
address@hidden calc-date
address@hidden date
-The @kbd{t D} (@code{calc-date}) address@hidden command converts a
-date form into a number, measured in days since Jan 1, 1 AD. The
-result will be an integer if @var{date} is a pure date form, or a
-fraction or float if @var{date} is a date/time form. Or, if its
-argument is a number, it converts this number into a date form.
-
-With a numeric prefix argument, @kbd{t D} takes that many objects
-(up to six) from the top of the stack and interprets them in one
-of the following ways:
-
-The @samp{date(@var{year}, @var{month}, @var{day})} function
-builds a pure date form out of the specified year, month, and
-day, which must all be integers. @var{Year} is a year number,
-such as 1991 (@emph{not} the same as 91!). @var{Month} must be
-an integer in the range 1 to 12; @var{day} must be in the range
-1 to 31. If the specified month has fewer than 31 days and
address@hidden is too large, the equivalent day in the following
-month will be used.
-
-The @samp{date(@var{month}, @var{day})} function builds a
-pure date form using the current year, as determined by the
-real-time clock.
-
-The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
-function builds a date/time form using an @var{hms} form.
-
-The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
address@hidden, @var{second})} function builds a date/time form.
address@hidden should be an integer in the range 0 to 23;
address@hidden should be an integer in the range 0 to 59;
address@hidden should be any real number in the range @samp{[0 .. 60)}.
-The last two arguments default to zero if omitted.
-
address@hidden t J
address@hidden calc-julian
address@hidden julian
address@hidden Julian day counts, conversions
-The @kbd{t J} (@code{calc-julian}) address@hidden command converts
-a date form into a Julian day count, which is the number of days
-since noon (GMT) on Jan 1, 4713 BC. A pure date is converted to an
-integer Julian count representing noon of that day. A date/time form
-is converted to an exact floating-point Julian count, adjusted to
-interpret the date form in the current time zone but the Julian
-day count in Greenwich Mean Time. A numeric prefix argument allows
-you to specify the time zone; @pxref{Time Zones}. Use a prefix of
-zero to suppress the time zone adjustment. Note that pure date forms
-are never time-zone adjusted.
-
-This command can also do the opposite conversion, from a Julian day
-count (either an integer day, or a floating-point day and time in
-the GMT zone), into a pure date form or a date/time form in the
-current or specified time zone.
-
address@hidden t U
address@hidden calc-unix-time
address@hidden unixtime
address@hidden Unix time format, conversions
-The @kbd{t U} (@code{calc-unix-time}) address@hidden command
-converts a date form into a Unix time value, which is the number of
-seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
-will be an integer if the current precision is 12 or less; for higher
-precisions, the result may be a float with (@address@hidden)
-digits after the decimal. Just as for @kbd{t J}, the numeric time
-is interpreted in the GMT time zone and the date form is interpreted
-in the current or specified zone. Some systems use Unix-like
-numbering but with the local time zone; give a prefix of zero to
-suppress the adjustment if so.
-
address@hidden t C
address@hidden calc-convert-time-zones
address@hidden tzconv
address@hidden Time Zones, converting between
-The @kbd{t C} (@code{calc-convert-time-zones}) address@hidden
-command converts a date form from one time zone to another. You
-are prompted for each time zone name in turn; you can answer with
-any suitable Calc time zone expression (@pxref{Time Zones}).
-If you answer either prompt with a blank line, the local time
-zone is used for that prompt. You can also answer the first
-prompt with @kbd{$} to take the two time zone names from the
-stack (and the date to be converted from the third stack level).
-
address@hidden Date Functions, Business Days, Date Conversions, Date Arithmetic
address@hidden Date Functions
-
address@hidden
address@hidden t N
address@hidden calc-now
address@hidden now
-The @kbd{t N} (@code{calc-now}) address@hidden command pushes the
-current date and time on the stack as a date form. The time is
-reported in terms of the specified time zone; with no numeric prefix
-argument, @kbd{t N} reports for the current time zone.
-
address@hidden t P
address@hidden calc-date-part
-The @kbd{t P} (@code{calc-date-part}) command extracts one part
-of a date form. The prefix argument specifies the part; with no
-argument, this command prompts for a part code from 1 to 9.
-The various part codes are described in the following paragraphs.
-
address@hidden year
-The @kbd{M-1 t P} address@hidden function extracts the year number
-from a date form as an integer, e.g., 1991. This and the
-following functions will also accept a real number for an
-argument, which is interpreted as a standard Calc day number.
-Note that this function will never return zero, since the year
-1 BC immediately precedes the year 1 AD.
-
address@hidden month
-The @kbd{M-2 t P} address@hidden function extracts the month number
-from a date form as an integer in the range 1 to 12.
-
address@hidden day
-The @kbd{M-3 t P} address@hidden function extracts the day number
-from a date form as an integer in the range 1 to 31.
-
address@hidden hour
-The @kbd{M-4 t P} address@hidden function extracts the hour from
-a date form as an integer in the range 0 (midnight) to 23. Note
-that 24-hour time is always used. This returns zero for a pure
-date form. This function (and the following two) also accept
-HMS forms as input.
-
address@hidden minute
-The @kbd{M-5 t P} address@hidden function extracts the minute
-from a date form as an integer in the range 0 to 59.
-
address@hidden second
-The @kbd{M-6 t P} address@hidden function extracts the second
-from a date form. If the current precision is 12 or less,
-the result is an integer in the range 0 to 59. For higher
-precisions, the result may instead be a floating-point number.
-
address@hidden weekday
-The @kbd{M-7 t P} address@hidden function extracts the weekday
-number from a date form as an integer in the range 0 (Sunday)
-to 6 (Saturday).
-
address@hidden yearday
-The @kbd{M-8 t P} address@hidden function extracts the day-of-year
-number from a date form as an integer in the range 1 (January 1)
-to 366 (December 31 of a leap year).
-
address@hidden time
-The @kbd{M-9 t P} address@hidden function extracts the time portion
-of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
-for a pure date form.
-
address@hidden t M
address@hidden calc-new-month
address@hidden newmonth
-The @kbd{t M} (@code{calc-new-month}) address@hidden command
-computes a new date form that represents the first day of the month
-specified by the input date. The result is always a pure date
-form; only the year and month numbers of the input are retained.
-With a numeric prefix argument @var{n} in the range from 1 to 31,
address@hidden M} computes the @var{n}th day of the month. (If @var{n}
-is greater than the actual number of days in the month, or if
address@hidden is zero, the last day of the month is used.)
-
address@hidden t Y
address@hidden calc-new-year
address@hidden newyear
-The @kbd{t Y} (@code{calc-new-year}) address@hidden command
-computes a new pure date form that represents the first day of
-the year specified by the input. The month, day, and time
-of the input date form are lost. With a numeric prefix argument
address@hidden in the range from 1 to 366, @kbd{t Y} computes the
address@hidden day of the year (366 is treated as 365 in non-leap
-years). A prefix argument of 0 computes the last day of the
-year (December 31). A negative prefix argument from @mathit{-1} to
address@hidden computes the first day of the @var{n}th month of the year.
-
address@hidden t W
address@hidden calc-new-week
address@hidden newweek
-The @kbd{t W} (@code{calc-new-week}) address@hidden command
-computes a new pure date form that represents the Sunday on or before
-the input date. With a numeric prefix argument, it can be made to
-use any day of the week as the starting day; the argument must be in
-the range from 0 (Sunday) to 6 (Saturday). This function always
-subtracts between 0 and 6 days from the input date.
-
-Here's an example use of @code{newweek}: Find the date of the next
-Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
-will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
-will give you the following Wednesday. A further look at the definition
-of @code{newweek} shows that if the input date is itself a Wednesday,
-this formula will return the Wednesday one week in the future. An
-exercise for the reader is to modify this formula to yield the same day
-if the input is already a Wednesday. Another interesting exercise is
-to preserve the time-of-day portion of the input (@code{newweek} resets
-the time to midnight; hint:@: how can @code{newweek} be defined in terms
-of the @code{weekday} function?).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden pwday
-The @samp{pwday(@var{date})} function (not on any key) computes the
-day-of-month number of the Sunday on or before @var{date}. With
-two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
-number of the Sunday on or before day number @var{day} of the month
-specified by @var{date}. The @var{day} must be in the range from
-7 to 31; if the day number is greater than the actual number of days
-in the month, the true number of days is used instead. Thus
address@hidden(@var{date}, 7)} finds the first Sunday of the month, and
address@hidden(@var{date}, 31)} finds the last Sunday of the month.
-With a third @var{weekday} argument, @code{pwday} can be made to look
-for any day of the week instead of Sunday.
-
address@hidden t I
address@hidden calc-inc-month
address@hidden incmonth
-The @kbd{t I} (@code{calc-inc-month}) address@hidden command
-increases a date form by one month, or by an arbitrary number of
-months specified by a numeric prefix argument. The time portion,
-if any, of the date form stays the same. The day also stays the
-same, except that if the new month has fewer days the day
-number may be reduced to lie in the valid range. For example,
address@hidden(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
-Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
-the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
-in this case).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden incyear
-The @samp{incyear(@var{date}, @var{step})} function increases
-a date form by the specified number of years, which may be
-any positive or negative integer. Note that @samp{incyear(d, n)}
-is equivalent to @address@hidden(d, 12*n)}}, but these do not have
-simple equivalents in terms of day arithmetic because
-months and years have varying lengths. If the @var{step}
-argument is omitted, 1 year is assumed. There is no keyboard
-command for this function; use @kbd{C-u 12 t I} instead.
-
-There is no @code{newday} function at all because @kbd{F} address@hidden
-serves this purpose. Similarly, instead of @code{incday} and
address@hidden simply use @expr{d + n} or @expr{d + 7 n}.
-
address@hidden Arithmetic}, for the @kbd{f ]} address@hidden command
-which can adjust a date/time form by a certain number of seconds.
-
address@hidden Business Days, Time Zones, Date Functions, Date Arithmetic
address@hidden Business Days
-
address@hidden
-Often time is measured in ``business days'' or ``working days,''
-where weekends and holidays are skipped. Calc's normal date
-arithmetic functions use calendar days, so that subtracting two
-consecutive Mondays will yield a difference of 7 days. By contrast,
-subtracting two consecutive Mondays would yield 5 business days
-(assuming two-day weekends and the absence of holidays).
-
address@hidden t +
address@hidden t -
address@hidden badd
address@hidden bsub
address@hidden calc-business-days-plus
address@hidden calc-business-days-minus
-The @kbd{t +} (@code{calc-business-days-plus}) address@hidden
-and @kbd{t -} (@code{calc-business-days-minus}) address@hidden
-commands perform arithmetic using business days. For @kbd{t +},
-one argument must be a date form and the other must be a real
-number (positive or negative). If the number is not an integer,
-then a certain amount of time is added as well as a number of
-days; for example, adding 0.5 business days to a time in Friday
-evening will produce a time in Monday morning. It is also
-possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
-half a business day. For @kbd{t -}, the arguments are either a
-date form and a number or HMS form, or two date forms, in which
-case the result is the number of business days between the two
-dates.
-
address@hidden @code{Holidays} variable
address@hidden Holidays
-By default, Calc considers any day that is not a Saturday or
-Sunday to be a business day. You can define any number of
-additional holidays by editing the variable @code{Holidays}.
-(There is an @address@hidden H}} convenience command for editing this
-variable.) Initially, @code{Holidays} contains the vector
address@hidden, sun]}. Entries in the @code{Holidays} vector may
-be any of the following kinds of objects:
-
address@hidden @bullet
address@hidden
-Date forms (pure dates, not date/time forms). These specify
-particular days which are to be treated as holidays.
-
address@hidden
-Intervals of date forms. These specify a range of days, all of
-which are holidays (e.g., Christmas week). @xref{Interval Forms}.
-
address@hidden
-Nested vectors of date forms. Each date form in the vector is
-considered to be a holiday.
-
address@hidden
-Any Calc formula which evaluates to one of the above three things.
-If the formula involves the variable @expr{y}, it stands for a
-yearly repeating holiday; @expr{y} will take on various year
-numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
-Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
-Thanksgiving (which is held on the fourth Thursday of November).
-If the formula involves the variable @expr{m}, that variable
-takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
-a holiday that takes place on the 15th of every month.
-
address@hidden
-A weekday name, such as @code{sat} or @code{sun}. This is really
-a variable whose name is a three-letter, lower-case day name.
-
address@hidden
-An interval of year numbers (integers). This specifies the span of
-years over which this holiday list is to be considered valid. Any
-business-day arithmetic that goes outside this range will result
-in an error message. Use this if you are including an explicit
-list of holidays, rather than a formula to generate them, and you
-want to make sure you don't accidentally go beyond the last point
-where the holidays you entered are complete. If there is no
-limiting interval in the @code{Holidays} vector, the default
address@hidden .. 2737]} is used. (This is the absolute range of years
-for which Calc's business-day algorithms will operate.)
-
address@hidden
-An interval of HMS forms. This specifies the span of hours that
-are to be considered one business day. For example, if this
-range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
-the business day is only eight hours long, so that @kbd{1.5 t +}
-on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
-four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
-Likewise, @kbd{t -} will now express differences in time as
-fractions of an eight-hour day. Times before 9am will be treated
-as 9am by business date arithmetic, and times at or after 5pm will
-be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
-the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
-(Regardless of the type of bounds you specify, the interval is
-treated as inclusive on the low end and exclusive on the high end,
-so that the work day goes from 9am up to, but not including, 5pm.)
address@hidden itemize
-
-If the @code{Holidays} vector is empty, then @kbd{t +} and
address@hidden -} will act just like @kbd{+} and @kbd{-} because there will
-then be no difference between business days and calendar days.
-
-Calc expands the intervals and formulas you give into a complete
-list of holidays for internal use. This is done mainly to make
-sure it can detect multiple holidays. (For example,
address@hidden<Jan 1, 1989>} is both New Year's Day and a Sunday, but
-Calc's algorithms take care to count it only once when figuring
-the number of holidays between two dates.)
-
-Since the complete list of holidays for all the years from 1 to
-2737 would be huge, Calc actually computes only the part of the
-list between the smallest and largest years that have been involved
-in business-day calculations so far. Normally, you won't have to
-worry about this. Keep in mind, however, that if you do one
-calculation for 1992, and another for 1792, even if both involve
-only a small range of years, Calc will still work out all the
-holidays that fall in that 200-year span.
-
-If you add a (positive) number of days to a date form that falls on a
-weekend or holiday, the date form is treated as if it were the most
-recent business day. (Thus adding one business day to a Friday,
-Saturday, or Sunday will all yield the following Monday.) If you
-subtract a number of days from a weekend or holiday, the date is
-effectively on the following business day. (So subtracting one business
-day from Saturday, Sunday, or Monday yields the preceding Friday.) The
-difference between two dates one or both of which fall on holidays
-equals the number of actual business days between them. These
-conventions are consistent in the sense that, if you add @var{n}
-business days to any date, the difference between the result and the
-original date will come out to @var{n} business days. (It can't be
-completely consistent though; a subtraction followed by an addition
-might come out a bit differently, since @kbd{t +} is incapable of
-producing a date that falls on a weekend or holiday.)
-
address@hidden
address@hidden
address@hidden ignore
address@hidden holiday
-There is a @code{holiday} function, not on any keys, that takes
-any date form and returns 1 if that date falls on a weekend or
-holiday, as defined in @code{Holidays}, or 0 if the date is a
-business day.
-
address@hidden Time Zones, , Business Days, Date Arithmetic
address@hidden Time Zones
-
address@hidden
address@hidden Time zones
address@hidden Daylight saving time
-Time zones and daylight saving time are a complicated business.
-The conversions to and from Julian and Unix-style dates automatically
-compute the correct time zone and daylight saving adjustment to use,
-provided they can figure out this information. This section describes
-Calc's time zone adjustment algorithm in detail, in case you want to
-do conversions in different time zones or in case Calc's algorithms
-can't determine the right correction to use.
-
-Adjustments for time zones and daylight saving time are done by
address@hidden U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
-commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
-to exactly 30 days even though there is a daylight-saving
-transition in between. This is also true for Julian pure dates:
address@hidden(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
-and Unix date/times will adjust for daylight saving time: using Calc's
-default daylight saving time rule (see the explanation below),
address@hidden(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
-evaluates to @samp{29.95833} (that's 29 days and 23 hours)
-because one hour was lost when daylight saving commenced on
-April 7, 1991.
-
-In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
-computes the actual number of 24-hour periods between two dates, whereas
address@hidden@var{date1} - @var{date2}} computes the number of calendar
-days between two dates without taking daylight saving into account.
-
address@hidden calc-time-zone
address@hidden
address@hidden
address@hidden ignore
address@hidden tzone
-The @code{calc-time-zone} address@hidden command converts the time
-zone specified by its numeric prefix argument into a number of
-seconds difference from Greenwich mean time (GMT). If the argument
-is a number, the result is simply that value multiplied by 3600.
-Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
-Daylight Saving time is in effect, one hour should be subtracted from
-the normal difference.
-
-If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
-date arithmetic commands that include a time zone argument) takes the
-zone argument from the top of the stack. (In the case of @kbd{t J}
-and @kbd{t U}, the normal argument is then taken from the second-to-top
-stack position.) This allows you to give a non-integer time zone
-adjustment. The time-zone argument can also be an HMS form, or
-it can be a variable which is a time zone name in upper- or lower-case.
-For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
-(for Pacific standard and daylight saving times, respectively).
-
-North American and European time zone names are defined as follows;
-note that for each time zone there is one name for standard time,
-another for daylight saving time, and a third for ``generalized'' time
-in which the daylight saving adjustment is computed from context.
-
address@hidden
address@hidden
-YST PST MST CST EST AST NST GMT WET MET MEZ
- 9 8 7 6 5 4 3.5 0 -1 -2 -2
-
-YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
- 8 7 6 5 4 3 2.5 -1 -2 -3 -3
-
-YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
-9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
address@hidden group
address@hidden smallexample
-
address@hidden math-tzone-names
-To define time zone names that do not appear in the above table,
-you must modify the Lisp variable @code{math-tzone-names}. This
-is a list of lists describing the different time zone names; its
-structure is best explained by an example. The three entries for
-Pacific Time look like this:
-
address@hidden
address@hidden
-( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
- ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment.
- ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
address@hidden group
address@hidden smallexample
-
address@hidden @code{TimeZone} variable
address@hidden TimeZone
-With no arguments, @code{calc-time-zone} or @samp{tzone()} will by
-default get the time zone and daylight saving information from the
-calendar (@pxref{Daylight Saving,Calendar/Diary,The Calendar and the Diary,
-emacs,The GNU Emacs Manual}). To use a different time zone, or if the
-calendar does not give the desired result, you can set the Calc variable
address@hidden (which is by default @code{nil}) to an appropriate
-time zone name. (The easiest way to do this is to edit the
address@hidden variable using Calc's @kbd{s T} command, then use the
address@hidden p} (@code{calc-permanent-variable}) command to save the value of
address@hidden permanently.)
-If the time zone given by @code{TimeZone} is a generalized time zone,
-e.g., @code{EGT}, Calc examines the date being converted to tell whether
-to use standard or daylight saving time. But if the current time zone
-is explicit, e.g., @code{EST} or @code{EDT}, then that adjustment is
-used exactly and Calc's daylight saving algorithm is not consulted.
-The special time zone name @code{local}
-is equivalent to no argument; i.e., it uses the information obtained
-from the calendar.
-
-The @kbd{t J} and @code{t U} commands with no numeric prefix
-arguments do the same thing as @samp{tzone()}; namely, use the
-information from the calendar if @code{TimeZone} is @code{nil},
-otherwise use the time zone given by @code{TimeZone}.
-
address@hidden math-daylight-savings-hook
address@hidden math-std-daylight-savings
-When Calc computes the daylight saving information itself (i.e., when
-the @code{TimeZone} variable is set), it will by default consider
-daylight saving time to begin at 2 a.m.@: on the second Sunday of March
-(for years from 2007 on) or on the last Sunday in April (for years
-before 2007), and to end at 2 a.m.@: on the first Sunday of
-November. (for years from 2007 on) or the last Sunday in October (for
-years before 2007). These are the rules that have been in effect in
-much of North America since 1966 and take into account the rule change
-that began in 2007. If you are in a country that uses different rules
-for computing daylight saving time, you have two choices: Write your own
-daylight saving hook, or control time zones explicitly by setting the
address@hidden variable and/or always giving a time-zone argument for
-the conversion functions.
-
-The Lisp variable @code{math-daylight-savings-hook} holds the
-name of a function that is used to compute the daylight saving
-adjustment for a given date. The default is
address@hidden, which computes an adjustment
-(either 0 or @mathit{-1}) using the North American rules given above.
-
-The daylight saving hook function is called with four arguments:
-The date, as a floating-point number in standard Calc format;
-a six-element list of the date decomposed into year, month, day,
-hour, minute, and second, respectively; a string which contains
-the generalized time zone name in upper-case, e.g., @code{"WEGT"};
-and a special adjustment to be applied to the hour value when
-converting into a generalized time zone (see below).
-
address@hidden math-prev-weekday-in-month
-The Lisp function @code{math-prev-weekday-in-month} is useful for
-daylight saving computations. This is an internal version of
-the user-level @code{pwday} function described in the previous
-section. It takes four arguments: The floating-point date value,
-the corresponding six-element date list, the day-of-month number,
-and the weekday number (0-6).
-
-The default daylight saving hook ignores the time zone name, but a
-more sophisticated hook could use different algorithms for different
-time zones. It would also be possible to use different algorithms
-depending on the year number, but the default hook always uses the
-algorithm for 1987 and later. Here is a listing of the default
-daylight saving hook:
-
address@hidden
-(defun math-std-daylight-savings (date dt zone bump)
- (cond ((< (nth 1 dt) 4) 0)
- ((= (nth 1 dt) 4)
- (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
- (cond ((< (nth 2 dt) sunday) 0)
- ((= (nth 2 dt) sunday)
- (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
- (t -1))))
- ((< (nth 1 dt) 10) -1)
- ((= (nth 1 dt) 10)
- (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
- (cond ((< (nth 2 dt) sunday) -1)
- ((= (nth 2 dt) sunday)
- (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
- (t 0))))
- (t 0))
-)
address@hidden smallexample
-
address@hidden
-The @code{bump} parameter is equal to zero when Calc is converting
-from a date form in a generalized time zone into a GMT date value.
-It is @mathit{-1} when Calc is converting in the other direction. The
-adjustments shown above ensure that the conversion behaves correctly
-and reasonably around the 2 a.m.@: transition in each direction.
-
-There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
-beginning of daylight saving time; converting a date/time form that
-falls in this hour results in a time value for the following hour,
-from 3 a.m.@: to 4 a.m. At the end of daylight saving time, the
-hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
-form that falls in this hour results in a time value for the first
-manifestation of that time (@emph{not} the one that occurs one hour
-later).
-
-If @code{math-daylight-savings-hook} is @code{nil}, then the
-daylight saving adjustment is always taken to be zero.
-
-In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
-computes the time zone adjustment for a given zone name at a
-given date. The @var{date} is ignored unless @var{zone} is a
-generalized time zone. If @var{date} is a date form, the
-daylight saving computation is applied to it as it appears.
-If @var{date} is a numeric date value, it is adjusted for the
-daylight-saving version of @var{zone} before being given to
-the daylight saving hook. This odd-sounding rule ensures
-that the daylight-saving computation is always done in
-local time, not in the GMT time that a numeric @var{date}
-is typically represented in.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden dsadj
-The @samp{dsadj(@var{date}, @var{zone})} function computes the
-daylight saving adjustment that is appropriate for @var{date} in
-time zone @var{zone}. If @var{zone} is explicitly in or not in
-daylight saving time (e.g., @code{PDT} or @code{PST}) the
address@hidden is ignored. If @var{zone} is a generalized time zone,
-the algorithms described above are used. If @var{zone} is omitted,
-the computation is done for the current time zone.
-
address@hidden Bugs}, for the address of Calc's author, if you
-should wish to contribute your improved versions of
address@hidden and @code{math-daylight-savings-hook}
-to the Calc distribution.
-
address@hidden Financial Functions, Binary Functions, Date Arithmetic,
Arithmetic
address@hidden Financial Functions
-
address@hidden
-Calc's financial or business functions use the @kbd{b} prefix
-key followed by a shifted letter. (The @kbd{b} prefix followed by
-a lower-case letter is used for operations on binary numbers.)
-
-Note that the rate and the number of intervals given to these
-functions must be on the same time scale, e.g., both months or
-both years. Mixing an annual interest rate with a time expressed
-in months will give you very wrong answers!
-
-It is wise to compute these functions to a higher precision than
-you really need, just to make sure your answer is correct to the
-last penny; also, you may wish to check the definitions at the end
-of this section to make sure the functions have the meaning you expect.
-
address@hidden
-* Percentages::
-* Future Value::
-* Present Value::
-* Related Financial Functions::
-* Depreciation Functions::
-* Definitions of Financial Functions::
address@hidden menu
-
address@hidden Percentages, Future Value, Financial Functions, Financial
Functions
address@hidden Percentages
-
address@hidden M-%
address@hidden calc-percent
address@hidden %
address@hidden percent
-The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
-say 5.4, and converts it to an equivalent actual number. For example,
address@hidden M-%} enters 0.054 on the stack. (That's the @key{META} or
address@hidden key combined with @kbd{%}.)
-
-Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
-You can enter @samp{5.4%} yourself during algebraic entry. The
address@hidden operator simply means, ``the preceding value divided by
-100.'' The @samp{%} operator has very high precedence, so that
address@hidden is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
-(The @samp{%} operator is just a postfix notation for the
address@hidden function, just like @samp{20!} is the notation for
address@hidden(20)}, or twenty-factorial.)
-
-The formula @samp{5.4%} would normally evaluate immediately to
-0.054, but the @kbd{M-%} command suppresses evaluation as it puts
-the formula onto the stack. However, the next Calc command that
-uses the formula @samp{5.4%} will evaluate it as its first step.
-The net effect is that you get to look at @samp{5.4%} on the stack,
-but Calc commands see it as @samp{0.054}, which is what they expect.
-
-In particular, @samp{5.4%} and @samp{0.054} are suitable values
-for the @var{rate} arguments of the various financial functions,
-but the number @samp{5.4} is probably @emph{not} suitable---it
-represents a rate of 540 percent!
-
-The key sequence @kbd{M-% *} effectively means ``percent-of.''
-For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
-68 (and also 68% of 25, which comes out to the same thing).
-
address@hidden c %
address@hidden calc-convert-percent
-The @kbd{c %} (@code{calc-convert-percent}) command converts the
-value on the top of the stack from numeric to percentage form.
-For example, if 0.08 is on the stack, @kbd{c %} converts it to
address@hidden The quantity is the same, it's just represented
-differently. (Contrast this with @kbd{M-%}, which would convert
-this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
-to convert a formula like @samp{8%} back to numeric form, 0.08.
-
-To compute what percentage one quantity is of another quantity,
-use @kbd{/ c %}. For example, @address@hidden @key{RET} 68 / c %}} displays
address@hidden
-
address@hidden b %
address@hidden calc-percent-change
address@hidden relch
-The @kbd{b %} (@code{calc-percent-change}) address@hidden command
-calculates the percentage change from one number to another.
-For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
-since 50 is 25% larger than 40. A negative result represents a
-decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
-20% smaller than 50. (The answers are different in magnitude
-because, in the first case, we're increasing by 25% of 40, but
-in the second case, we're decreasing by 20% of 50.) The effect
-of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
-the answer to percentage form as if by @kbd{c %}.
-
address@hidden Future Value, Present Value, Percentages, Financial Functions
address@hidden Future Value
-
address@hidden
address@hidden b F
address@hidden calc-fin-fv
address@hidden fv
-The @kbd{b F} (@code{calc-fin-fv}) address@hidden command computes
-the future value of an investment. It takes three arguments
-from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
-If you give payments of @var{payment} every year for @var{n}
-years, and the money you have paid earns interest at @var{rate} per
-year, then this function tells you what your investment would be
-worth at the end of the period. (The actual interval doesn't
-have to be years, as long as @var{n} and @var{rate} are expressed
-in terms of the same intervals.) This function assumes payments
-occur at the @emph{end} of each interval.
-
address@hidden I b F
address@hidden fvb
-The @kbd{I b F} address@hidden command does the same computation,
-but assuming your payments are at the beginning of each interval.
-Suppose you plan to deposit $1000 per year in a savings account
-earning 5.4% interest, starting right now. How much will be
-in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
-Thus you will have earned $870 worth of interest over the years.
-Using the stack, this calculation would have been
address@hidden M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
-as a number between 0 and 1, @emph{not} as a percentage.
-
address@hidden H b F
address@hidden fvl
-The @kbd{H b F} address@hidden command computes the future value
-of an initial lump sum investment. Suppose you could deposit
-those five thousand dollars in the bank right now; how much would
-they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
-
-The algebraic functions @code{fv} and @code{fvb} accept an optional
-fourth argument, which is used as an initial lump sum in the sense
-of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
address@hidden, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
-+ fvl(@var{rate}, @var{n}, @var{initial})}.
-
-To illustrate the relationships between these functions, we could
-do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
-final balance will be the sum of the contributions of our five
-deposits at various times. The first deposit earns interest for
-five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
-deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
-1234.13}. And so on down to the last deposit, which earns one
-year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
-these five values is, sure enough, $5870.73, just as was computed
-by @code{fvb} directly.
-
-What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
-are now at the ends of the periods. The end of one year is the same
-as the beginning of the next, so what this really means is that we've
-lost the payment at year zero (which contributed $1300.78), but we're
-now counting the payment at year five (which, since it didn't have
-a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
-5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
-
address@hidden Present Value, Related Financial Functions, Future Value,
Financial Functions
address@hidden Present Value
-
address@hidden
address@hidden b P
address@hidden calc-fin-pv
address@hidden pv
-The @kbd{b P} (@code{calc-fin-pv}) address@hidden command computes
-the present value of an investment. Like @code{fv}, it takes
-three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
-It computes the present value of a series of regular payments.
-Suppose you have the chance to make an investment that will
-pay $2000 per year over the next four years; as you receive
-these payments you can put them in the bank at 9% interest.
-You want to know whether it is better to make the investment, or
-to keep the money in the bank where it earns 9% interest right
-from the start. The calculation @code{pv(9%, 4, 2000)} gives the
-result 6479.44. If your initial investment must be less than this,
-say, $6000, then the investment is worthwhile. But if you had to
-put up $7000, then it would be better just to leave it in the bank.
-
-Here is the interpretation of the result of @code{pv}: You are
-trying to compare the return from the investment you are
-considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
-the return from leaving the money in the bank, which is
address@hidden(9%, 4, @var{x})} where @var{x} is the amount of money
-you would have to put up in advance. The @code{pv} function
-finds the break-even point, @expr{x = 6479.44}, at which
address@hidden(9%, 4, 6479.44)} is also equal to 9146.26. This is
-the largest amount you should be willing to invest.
-
address@hidden I b P
address@hidden pvb
-The @kbd{I b P} address@hidden command solves the same problem,
-but with payments occurring at the beginning of each interval.
-It has the same relationship to @code{fvb} as @code{pv} has
-to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
-a larger number than @code{pv} produced because we get to start
-earning interest on the return from our investment sooner.
-
address@hidden H b P
address@hidden pvl
-The @kbd{H b P} address@hidden command computes the present value of
-an investment that will pay off in one lump sum at the end of the
-period. For example, if we get our $8000 all at the end of the
-four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
-less than @code{pv} reported, because we don't earn any interest
-on the return from this investment. Note that @code{pvl} and
address@hidden are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
-
-You can give an optional fourth lump-sum argument to @code{pv}
-and @code{pvb}; this is handled in exactly the same way as the
-fourth argument for @code{fv} and @code{fvb}.
-
address@hidden b N
address@hidden calc-fin-npv
address@hidden npv
-The @kbd{b N} (@code{calc-fin-npv}) address@hidden command computes
-the net present value of a series of irregular investments.
-The first argument is the interest rate. The second argument is
-a vector which represents the expected return from the investment
-at the end of each interval. For example, if the rate represents
-a yearly interest rate, then the vector elements are the return
-from the first year, second year, and so on.
-
-Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
-Obviously this function is more interesting when the payments are
-not all the same!
-
-The @code{npv} function can actually have two or more arguments.
-Multiple arguments are interpreted in the same way as for the
-vector statistical functions like @code{vsum}.
address@hidden Statistics}. Basically, if there are several
-payment arguments, each either a vector or a plain number, all these
-values are collected left-to-right into the complete list of payments.
-A numeric prefix argument on the @kbd{b N} command says how many
-payment values or vectors to take from the stack.
-
address@hidden I b N
address@hidden npvb
-The @kbd{I b N} address@hidden command computes the net present
-value where payments occur at the beginning of each interval
-rather than at the end.
-
address@hidden Related Financial Functions, Depreciation Functions, Present
Value, Financial Functions
address@hidden Related Financial Functions
-
address@hidden
-The functions in this section are basically inverses of the
-present value functions with respect to the various arguments.
-
address@hidden b M
address@hidden calc-fin-pmt
address@hidden pmt
-The @kbd{b M} (@code{calc-fin-pmt}) address@hidden command computes
-the amount of periodic payment necessary to amortize a loan.
-Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
-value of @var{payment} such that @code{pv(@var{rate}, @var{n},
address@hidden) = @var{amount}}.
-
address@hidden I b M
address@hidden pmtb
-The @kbd{I b M} address@hidden command does the same computation
-but using @code{pvb} instead of @code{pv}. Like @code{pv} and
address@hidden, these functions can also take a fourth argument which
-represents an initial lump-sum investment.
-
address@hidden H b M
-The @kbd{H b M} key just invokes the @code{fvl} function, which is
-the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
-
address@hidden b #
address@hidden calc-fin-nper
address@hidden nper
-The @kbd{b #} (@code{calc-fin-nper}) address@hidden command computes
-the number of regular payments necessary to amortize a loan.
-Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
-the value of @var{n} such that @code{pv(@var{rate}, @var{n},
address@hidden) = @var{amount}}. If @var{payment} is too small
-ever to amortize a loan for @var{amount} at interest rate @var{rate},
-the @code{nper} function is left in symbolic form.
-
address@hidden I b #
address@hidden nperb
-The @kbd{I b #} address@hidden command does the same computation
-but using @code{pvb} instead of @code{pv}. You can give a fourth
-lump-sum argument to these functions, but the computation will be
-rather slow in the four-argument case.
-
address@hidden H b #
address@hidden nperl
-The @kbd{H b #} address@hidden command does the same computation
-using @code{pvl}. By exchanging @var{payment} and @var{amount} you
-can also get the solution for @code{fvl}. For example,
address@hidden(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
-bank account earning 8%, it will take nine years to grow to $2000.
-
address@hidden b T
address@hidden calc-fin-rate
address@hidden rate
-The @kbd{b T} (@code{calc-fin-rate}) address@hidden command computes
-the rate of return on an investment. This is also an inverse of @code{pv}:
address@hidden(@var{n}, @var{payment}, @var{amount})} computes the value of
address@hidden such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
address@hidden The result is expressed as a formula like @samp{6.3%}.
-
address@hidden I b T
address@hidden H b T
address@hidden rateb
address@hidden ratel
-The @kbd{I b T} address@hidden and @kbd{H b T} address@hidden
-commands solve the analogous equations with @code{pvb} or @code{pvl}
-in place of @code{pv}. Also, @code{rate} and @code{rateb} can
-accept an optional fourth argument just like @code{pv} and @code{pvb}.
-To redo the above example from a different perspective,
address@hidden(9, 2000, 1000) = 8.00597%}, which says you will need an
-interest rate of 8% in order to double your account in nine years.
-
address@hidden b I
address@hidden calc-fin-irr
address@hidden irr
-The @kbd{b I} (@code{calc-fin-irr}) address@hidden command is the
-analogous function to @code{rate} but for net present value.
-Its argument is a vector of payments. Thus @code{irr(@var{payments})}
-computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
-this rate is known as the @dfn{internal rate of return}.
-
address@hidden I b I
address@hidden irrb
-The @kbd{I b I} address@hidden command computes the internal rate of
-return assuming payments occur at the beginning of each period.
-
address@hidden Depreciation Functions, Definitions of Financial Functions,
Related Financial Functions, Financial Functions
address@hidden Depreciation Functions
-
address@hidden
-The functions in this section calculate @dfn{depreciation}, which is
-the amount of value that a possession loses over time. These functions
-are characterized by three parameters: @var{cost}, the original cost
-of the asset; @var{salvage}, the value the asset will have at the end
-of its expected ``useful life''; and @var{life}, the number of years
-(or other periods) of the expected useful life.
-
-There are several methods for calculating depreciation that differ in
-the way they spread the depreciation over the lifetime of the asset.
-
address@hidden b S
address@hidden calc-fin-sln
address@hidden sln
-The @kbd{b S} (@code{calc-fin-sln}) address@hidden command computes the
-``straight-line'' depreciation. In this method, the asset depreciates
-by the same amount every year (or period). For example,
address@hidden(12000, 2000, 5)} returns 2000. The asset costs $12000
-initially and will be worth $2000 after five years; it loses $2000
-per year.
-
address@hidden b Y
address@hidden calc-fin-syd
address@hidden syd
-The @kbd{b Y} (@code{calc-fin-syd}) address@hidden command computes the
-accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
-is higher during the early years of the asset's life. Since the
-depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
-parameter which specifies which year is requested, from 1 to @var{life}.
-If @var{period} is outside this range, the @code{syd} function will
-return zero.
-
address@hidden b D
address@hidden calc-fin-ddb
address@hidden ddb
-The @kbd{b D} (@code{calc-fin-ddb}) address@hidden command computes an
-accelerated depreciation using the double-declining balance method.
-It also takes a fourth @var{period} parameter.
-
-For symmetry, the @code{sln} function will accept a @var{period}
-parameter as well, although it will ignore its value except that the
-return value will as usual be zero if @var{period} is out of range.
-
-For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
-and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
-ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
-the three depreciation methods:
-
address@hidden
address@hidden
-[ [ 2000, 3333, 4800 ]
- [ 2000, 2667, 2880 ]
- [ 2000, 2000, 1728 ]
- [ 2000, 1333, 592 ]
- [ 2000, 667, 0 ] ]
address@hidden group
address@hidden example
-
address@hidden
-(Values have been rounded to nearest integers in this figure.)
-We see that @code{sln} depreciates by the same amount each year,
address@hidden depreciates more at the beginning and less at the end,
-and @kbd{ddb} weights the depreciation even more toward the beginning.
-
-Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
-the total depreciation in any method is (by definition) the
-difference between the cost and the salvage value.
-
address@hidden Definitions of Financial Functions, , Depreciation Functions,
Financial Functions
address@hidden Definitions
-
address@hidden
-For your reference, here are the actual formulas used to compute
-Calc's financial functions.
-
-Calc will not evaluate a financial function unless the @var{rate} or
address@hidden argument is known. However, @var{payment} or @var{amount} can
-be a variable. Calc expands these functions according to the
-formulas below for symbolic arguments only when you use the @kbd{a "}
-(@code{calc-expand-formula}) command, or when taking derivatives or
-integrals or solving equations involving the functions.
-
address@hidden
-These formulas are shown using the conventions of Big display
-mode (@kbd{d B}); for example, the formula for @code{fv} written
-linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
-
address@hidden
- n
- (1 + rate) - 1
-fv(rate, n, pmt) = pmt * ---------------
- rate
-
- n
- ((1 + rate) - 1) (1 + rate)
-fvb(rate, n, pmt) = pmt * ----------------------------
- rate
-
- n
-fvl(rate, n, pmt) = pmt * (1 + rate)
-
- -n
- 1 - (1 + rate)
-pv(rate, n, pmt) = pmt * ----------------
- rate
-
- -n
- (1 - (1 + rate) ) (1 + rate)
-pvb(rate, n, pmt) = pmt * -----------------------------
- rate
-
- -n
-pvl(rate, n, pmt) = pmt * (1 + rate)
-
- -1 -2 -3
-npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
-
- -1 -2
-npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
-
- -n
- (amt - x * (1 + rate) ) * rate
-pmt(rate, n, amt, x) = -------------------------------
- -n
- 1 - (1 + rate)
-
- -n
- (amt - x * (1 + rate) ) * rate
-pmtb(rate, n, amt, x) = -------------------------------
- -n
- (1 - (1 + rate) ) (1 + rate)
-
- amt * rate
-nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
- pmt
-
- amt * rate
-nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
- pmt * (1 + rate)
-
- amt
-nperl(rate, pmt, amt) = - log(---, 1 + rate)
- pmt
-
- 1/n
- pmt
-ratel(n, pmt, amt) = ------ - 1
- 1/n
- amt
-
- cost - salv
-sln(cost, salv, life) = -----------
- life
-
- (cost - salv) * (life - per + 1)
-syd(cost, salv, life, per) = --------------------------------
- life * (life + 1) / 2
-
- book * 2
-ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
- life
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
-$$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
-$$ \code{fvl}(r, n, p) = p (1 + r)^n $$
-$$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
-$$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
-$$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
-$$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
-$$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
-$$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
-$$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
- (1 - (1 + r)^{-n}) (1 + r) } $$
-$$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
-$$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
-$$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
-$$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
-$$ \code{sln}(c, s, l) = { c - s \over l } $$
-$$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
-$$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
address@hidden tex
-
address@hidden
-In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
-
-These functions accept any numeric objects, including error forms,
-intervals, and even (though not very usefully) complex numbers. The
-above formulas specify exactly the behavior of these functions with
-all sorts of inputs.
-
-Note that if the first argument to the @code{log} in @code{nper} is
-negative, @code{nper} leaves itself in symbolic form rather than
-returning a (financially meaningless) complex number.
-
address@hidden(num, pmt, amt)} solves the equation
address@hidden(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
-(@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
-for an initial guess. The @code{rateb} function is the same except
-that it uses @code{pvb}. Note that @code{ratel} can be solved
-directly; its formula is shown in the above list.
-
-Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
-for @samp{rate}.
-
-If you give a fourth argument to @code{nper} or @code{nperb}, Calc
-will also use @kbd{H a R} to solve the equation using an initial
-guess interval of @samp{[0 .. 100]}.
-
-A fourth argument to @code{fv} simply sums the two components
-calculated from the above formulas for @code{fv} and @code{fvl}.
-The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
-
-The @kbd{ddb} function is computed iteratively; the ``book'' value
-starts out equal to @var{cost}, and decreases according to the above
-formula for the specified number of periods. If the book value
-would decrease below @var{salvage}, it only decreases to @var{salvage}
-and the depreciation is zero for all subsequent periods. The @code{ddb}
-function returns the amount the book value decreased in the specified
-period.
-
address@hidden Binary Functions, , Financial Functions, Arithmetic
address@hidden Binary Number Functions
-
address@hidden
-The commands in this chapter all use two-letter sequences beginning with
-the @kbd{b} prefix.
-
address@hidden Binary numbers
-The ``binary'' operations actually work regardless of the currently
-displayed radix, although their results make the most sense in a radix
-like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @address@hidden
6}}
-commands, respectively). You may also wish to enable display of leading
-zeros with @kbd{d z}. @xref{Radix Modes}.
-
address@hidden Word size for binary operations
-The Calculator maintains a current @dfn{word size} @expr{w}, an
-arbitrary positive or negative integer. For a positive word size, all
-of the binary operations described here operate modulo @expr{2^w}. In
-particular, negative arguments are converted to positive integers modulo
address@hidden by all binary functions.
-
-If the word size is negative, binary operations produce 2's complement
-integers from
address@hidden @math{-2^{-w-1}}
address@hidden @expr{-(2^(-w-1))}
-to
address@hidden @math{2^{-w-1}-1}
address@hidden @expr{2^(-w-1)-1}
-inclusive. Either mode accepts inputs in any range; the sign of
address@hidden affects only the results produced.
-
address@hidden b c
address@hidden calc-clip
address@hidden clip
-The @kbd{b c} (@code{calc-clip})
address@hidden command can be used to clip a number by reducing it modulo
address@hidden The commands described in this chapter automatically clip
-their results to the current word size. Note that other operations like
-addition do not use the current word size, since integer addition
-generally is not ``binary.'' (However, @pxref{Simplification Modes},
address@hidden) For example, with a word size of 8
-bits @kbd{b c} converts a number to the range 0 to 255; with a word
-size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
-
address@hidden b w
address@hidden calc-word-size
-The default word size is 32 bits. All operations except the shifts and
-rotates allow you to specify a different word size for that one
-operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
-top of stack to the range 0 to 255 regardless of the current word size.
-To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
-This command displays a prompt with the current word size; press @key{RET}
-immediately to keep this word size, or type a new word size at the prompt.
-
-When the binary operations are written in symbolic form, they take an
-optional second (or third) word-size parameter. When a formula like
address@hidden(a,b)} is finally evaluated, the word size current at that time
-will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
address@hidden will always be used. A symbolic binary function will be left
-in symbolic form unless the all of its argument(s) are integers or
-integer-valued floats.
-
-If either or both arguments are modulo forms for which @expr{M} is a
-power of two, that power of two is taken as the word size unless a
-numeric prefix argument overrides it. The current word size is never
-consulted when modulo-power-of-two forms are involved.
-
address@hidden b a
address@hidden calc-and
address@hidden and
-The @kbd{b a} (@code{calc-and}) address@hidden command computes the bitwise
-AND of the two numbers on the top of the stack. In other words, for each
-of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
-bit of the result is 1 if and only if both input bits are 1:
address@hidden(2#1100, 2#1010) = 2#1000}.
-
address@hidden b o
address@hidden calc-or
address@hidden or
-The @kbd{b o} (@code{calc-or}) address@hidden command computes the bitwise
-inclusive OR of two numbers. A bit is 1 if either of the input bits, or
-both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
-
address@hidden b x
address@hidden calc-xor
address@hidden xor
-The @kbd{b x} (@code{calc-xor}) address@hidden command computes the bitwise
-exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
-is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
-
address@hidden b d
address@hidden calc-diff
address@hidden diff
-The @kbd{b d} (@code{calc-diff}) address@hidden command computes the bitwise
-difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
-so that @samp{diff(2#1100, 2#1010) = 2#0100}.
-
address@hidden b n
address@hidden calc-not
address@hidden not
-The @kbd{b n} (@code{calc-not}) address@hidden command computes the bitwise
-NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
-
address@hidden b l
address@hidden calc-lshift-binary
address@hidden lsh
-The @kbd{b l} (@code{calc-lshift-binary}) address@hidden command shifts a
-number left by one bit, or by the number of bits specified in the numeric
-prefix argument. A negative prefix argument performs a logical right shift,
-in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
-is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
-Bits shifted ``off the end,'' according to the current word size, are lost.
-
address@hidden H b l
address@hidden H b r
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden H b L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H b R
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H b t
-The @kbd{H b l} command also does a left shift, but it takes two arguments
-from the stack (the value to shift, and, at top-of-stack, the number of
-bits to shift). This version interprets the prefix argument just like
-the regular binary operations, i.e., as a word size. The Hyperbolic flag
-has a similar effect on the rest of the binary shift and rotate commands.
-
address@hidden b r
address@hidden calc-rshift-binary
address@hidden rsh
-The @kbd{b r} (@code{calc-rshift-binary}) address@hidden command shifts a
-number right by one bit, or by the number of bits specified in the numeric
-prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
-
address@hidden b L
address@hidden calc-lshift-arith
address@hidden ash
-The @kbd{b L} (@code{calc-lshift-arith}) address@hidden command shifts a
-number left. It is analogous to @code{lsh}, except that if the shift
-is rightward (the prefix argument is negative), an arithmetic shift
-is performed as described below.
-
address@hidden b R
address@hidden calc-rshift-arith
address@hidden rash
-The @kbd{b R} (@code{calc-rshift-arith}) address@hidden command performs
-an ``arithmetic'' shift to the right, in which the leftmost bit (according
-to the current word size) is duplicated rather than shifting in zeros.
-This corresponds to dividing by a power of two where the input is interpreted
-as a signed, twos-complement number. (The distinction between the @samp{rsh}
-and @samp{rash} operations is totally independent from whether the word
-size is positive or negative.) With a negative prefix argument, this
-performs a standard left shift.
-
address@hidden b t
address@hidden calc-rotate-binary
address@hidden rot
-The @kbd{b t} (@code{calc-rotate-binary}) address@hidden command rotates a
-number one bit to the left. The leftmost bit (according to the current
-word size) is dropped off the left and shifted in on the right. With a
-numeric prefix argument, the number is rotated that many bits to the left
-or right.
-
address@hidden Operations}, for the @kbd{b p} and @kbd{b u} commands that
-pack and unpack binary integers into sets. (For example, @kbd{b u}
-unpacks the number @samp{2#11001} to the set of bit-numbers
address@hidden, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
-bits in a binary integer.
-
-Another interesting use of the set representation of binary integers
-is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
-unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
-with 31 minus that bit-number; type @kbd{b p} to pack the set back
-into a binary integer.
-
address@hidden Scientific Functions, Matrix Functions, Arithmetic, Top
address@hidden Scientific Functions
-
address@hidden
-The functions described here perform trigonometric and other transcendental
-calculations. They generally produce floating-point answers correct to the
-full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
-flag keys must be used to get some of these functions from the keyboard.
-
address@hidden P
address@hidden calc-pi
address@hidden @code{pi} variable
address@hidden pi
address@hidden H P
address@hidden @code{e} variable
address@hidden e
address@hidden I P
address@hidden @code{gamma} variable
address@hidden gamma
address@hidden Gamma constant, Euler's
address@hidden Euler's gamma constant
address@hidden H I P
address@hidden @code{phi} variable
address@hidden Phi, golden ratio
address@hidden Golden ratio
-One miscellaneous command is address@hidden (@code{calc-pi}), which pushes
-the value of @cpi{} (at the current precision) onto the stack. With the
-Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
-With the Inverse flag, it pushes Euler's constant
address@hidden @math{\gamma}
address@hidden @expr{gamma}
-(about 0.5772). With both Inverse and Hyperbolic, it
-pushes the ``golden ratio''
address@hidden @math{\phi}
address@hidden @expr{phi}
-(about 1.618). (At present, Euler's constant is not available
-to unlimited precision; Calc knows only the first 100 digits.)
-In Symbolic mode, these commands push the
-actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
-respectively, instead of their values; @pxref{Symbolic Mode}.
-
address@hidden
address@hidden Q
address@hidden ignore
address@hidden
address@hidden I Q
address@hidden ignore
address@hidden I Q
address@hidden sqr
-The @kbd{Q} (@code{calc-sqrt}) address@hidden function is described elsewhere;
address@hidden Arithmetic}. With the Inverse flag address@hidden, this command
-computes the square of the argument.
-
address@hidden Arguments}, for a discussion of the effect of numeric
-prefix arguments on commands in this chapter which do not otherwise
-interpret a prefix argument.
-
address@hidden
-* Logarithmic Functions::
-* Trigonometric and Hyperbolic Functions::
-* Advanced Math Functions::
-* Branch Cuts::
-* Random Numbers::
-* Combinatorial Functions::
-* Probability Distribution Functions::
address@hidden menu
-
address@hidden Logarithmic Functions, Trigonometric and Hyperbolic Functions,
Scientific Functions, Scientific Functions
address@hidden Logarithmic Functions
-
address@hidden
address@hidden L
address@hidden calc-ln
address@hidden ln
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I E
-The address@hidden (@code{calc-ln}) address@hidden command computes the natural
-logarithm of the real or complex number on the top of the stack. With
-the Inverse flag it computes the exponential function instead, although
-this is redundant with the @kbd{E} command.
-
address@hidden E
address@hidden calc-exp
address@hidden exp
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I L
-The address@hidden (@code{calc-exp}) address@hidden command computes the
-exponential, i.e., @expr{e} raised to the power of the number on the stack.
-The meanings of the Inverse and Hyperbolic flags follow from those for
-the @code{calc-ln} command.
-
address@hidden H L
address@hidden H E
address@hidden calc-log10
address@hidden log10
address@hidden exp10
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I E
-The @kbd{H L} (@code{calc-log10}) address@hidden command computes the common
-(base-10) logarithm of a number. (With the Inverse flag address@hidden,
-it raises ten to a given power.) Note that the common logarithm of a
-complex number is computed by taking the natural logarithm and dividing
-by
address@hidden @math{\ln10}.
address@hidden @expr{ln(10)}.
-
address@hidden B
address@hidden I B
address@hidden calc-log
address@hidden log
address@hidden alog
-The @kbd{B} (@code{calc-log}) address@hidden command computes a logarithm
-to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
address@hidden @math{2^{10} = 1024}.
address@hidden @expr{2^10 = 1024}.
-In certain cases like @samp{log(3,9)}, the result
-will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
-mode setting. With the Inverse flag address@hidden, this command is
-similar to @kbd{^} except that the order of the arguments is reversed.
-
address@hidden f I
address@hidden calc-ilog
address@hidden ilog
-The @kbd{f I} (@code{calc-ilog}) address@hidden command computes the
-integer logarithm of a number to any base. The number and the base must
-themselves be positive integers. This is the true logarithm, rounded
-down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
-range from 1000 to 9999. If both arguments are positive integers, exact
-integer arithmetic is used; otherwise, this is equivalent to
address@hidden(log(x,b))}.
-
address@hidden f E
address@hidden calc-expm1
address@hidden expm1
-The @kbd{f E} (@code{calc-expm1}) address@hidden command computes
address@hidden @math{e^x - 1},
address@hidden @expr{exp(x)-1},
-but using an algorithm that produces a more accurate
-answer when the result is close to zero, i.e., when
address@hidden @math{e^x}
address@hidden @expr{exp(x)}
-is close to one.
-
address@hidden f L
address@hidden calc-lnp1
address@hidden lnp1
-The @kbd{f L} (@code{calc-lnp1}) address@hidden command computes
address@hidden @math{\ln(x+1)},
address@hidden @expr{ln(x+1)},
-producing a more accurate answer when @expr{x} is close to zero.
-
address@hidden Trigonometric and Hyperbolic Functions, Advanced Math Functions,
Logarithmic Functions, Scientific Functions
address@hidden Trigonometric/Hyperbolic Functions
-
address@hidden
address@hidden S
address@hidden calc-sin
address@hidden sin
-The address@hidden (@code{calc-sin}) address@hidden command computes the sine
-of an angle or complex number. If the input is an HMS form, it is interpreted
-as degrees-minutes-seconds; otherwise, the input is interpreted according
-to the current angular mode. It is best to use Radians mode when operating
-on complex numbers.
-
-Calc's ``units'' mechanism includes angular units like @code{deg},
address@hidden, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
-all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
-simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
-of the current angular mode. @xref{Basic Operations on Units}.
-
-Also, the symbolic variable @code{pi} is not ordinarily recognized in
-arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
-the @kbd{a s} (@code{calc-simplify}) command recognizes many such
-formulas when the current angular mode is Radians @emph{and} Symbolic
-mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
address@hidden Mode}. Beware, this simplification occurs even if you
-have stored a different value in the variable @samp{pi}; this is one
-reason why changing built-in variables is a bad idea. Arguments of
-the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
-Calc includes similar formulas for @code{cos} and @code{tan}.
-
-The @kbd{a s} command knows all angles which are integer multiples of
address@hidden, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
-analogous simplifications occur for integer multiples of 15 or 18
-degrees, and for arguments plus multiples of 90 degrees.
-
address@hidden I S
address@hidden calc-arcsin
address@hidden arcsin
-With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
-available as the @code{calc-arcsin} command or @code{arcsin} algebraic
-function. The returned argument is converted to degrees, radians, or HMS
-notation depending on the current angular mode.
-
address@hidden H S
address@hidden calc-sinh
address@hidden sinh
address@hidden H I S
address@hidden calc-arcsinh
address@hidden arcsinh
-With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
-sine, also available as @code{calc-sinh} address@hidden With the
-Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
-(@code{calc-arcsinh}) address@hidden
-
address@hidden C
address@hidden calc-cos
address@hidden cos
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I C
address@hidden calc-arccos
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arccos
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H C
address@hidden calc-cosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden cosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I C
address@hidden calc-arccosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arccosh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden T
address@hidden calc-tan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden tan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden I T
address@hidden calc-arctan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arctan
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H T
address@hidden calc-tanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden tanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I T
address@hidden calc-arctanh
address@hidden
address@hidden @null
address@hidden ignore
address@hidden arctanh
-The address@hidden (@code{calc-cos}) address@hidden command computes the cosine
-of an angle or complex number, and address@hidden (@code{calc-tan})
address@hidden
-computes the tangent, along with all the various inverse and hyperbolic
-variants of these functions.
-
address@hidden f T
address@hidden calc-arctan2
address@hidden arctan2
-The @kbd{f T} (@code{calc-arctan2}) address@hidden command takes two
-numbers from the stack and computes the arc tangent of their ratio. The
-result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
-(inclusive) degrees, or the analogous range in radians. A similar
-result would be obtained with @kbd{/} followed by @kbd{I T}, but the
-value would only be in the range from @mathit{-90} to @mathit{+90} degrees
-since the division loses information about the signs of the two
-components, and an error might result from an explicit division by zero
-which @code{arctan2} would avoid. By (arbitrary) definition,
address@hidden(0,0)=0}.
-
address@hidden calc-sincos
address@hidden
address@hidden
address@hidden ignore
address@hidden sincos
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden arcsincos
-The @code{calc-sincos} address@hidden command computes the sine and
-cosine of a number, returning them as a vector of the form
address@hidden@var{cos}, @var{sin}]}.
-With the Inverse flag address@hidden, this command takes a two-element
-vector as an argument and computes @code{arctan2} of the elements.
-(This command does not accept the Hyperbolic flag.)
-
address@hidden calc-sec
address@hidden sec
address@hidden calc-csc
address@hidden csc
address@hidden calc-cot
address@hidden cot
address@hidden calc-sech
address@hidden sech
address@hidden calc-csch
address@hidden csch
address@hidden calc-coth
address@hidden coth
-The remaining trigonometric functions, @code{calc-sec} address@hidden,
address@hidden address@hidden and @code{calc-sec} address@hidden, are also
-available. With the Hyperbolic flag, these compute their hyperbolic
-counterparts, which are also available separately as @code{calc-sech}
address@hidden, @code{calc-csch} address@hidden and @code{calc-sech}
address@hidden (These commmands do not accept the Inverse flag.)
-
address@hidden Advanced Math Functions, Branch Cuts, Trigonometric and
Hyperbolic Functions, Scientific Functions
address@hidden Advanced Mathematical Functions
-
address@hidden
-Calc can compute a variety of less common functions that arise in
-various branches of mathematics. All of the functions described in
-this section allow arbitrary complex arguments and, except as noted,
-will work to arbitrarily large precisions. They can not at present
-handle error forms or intervals as arguments.
-
-NOTE: These functions are still experimental. In particular, their
-accuracy is not guaranteed in all domains. It is advisable to set the
-current precision comfortably higher than you actually need when
-using these functions. Also, these functions may be impractically
-slow for some values of the arguments.
-
address@hidden f g
address@hidden calc-gamma
address@hidden gamma
-The @kbd{f g} (@code{calc-gamma}) address@hidden command computes the Euler
-gamma function. For positive integer arguments, this is related to the
-factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
-arguments the gamma function can be defined by the following definite
-integral:
address@hidden @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
address@hidden @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
-(The actual implementation uses far more efficient computational methods.)
-
address@hidden f G
address@hidden gammaP
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I f G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H f G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H I f G
address@hidden calc-inc-gamma
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammaQ
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammag
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gammaG
-The @kbd{f G} (@code{calc-inc-gamma}) address@hidden command computes
-the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
-the integral,
address@hidden @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) /
\Gamma(a)}.
address@hidden @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
-This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
-definition of the normal gamma function).
-
-Several other varieties of incomplete gamma function are defined.
-The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
-some authors, is computed by the @kbd{I f G} address@hidden command.
-You can think of this as taking the other half of the integral, from
address@hidden to infinity.
-
address@hidden
-The functions corresponding to the integrals that define @expr{P(a,x)}
-and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
-factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
-(where @expr{g} and @expr{G} represent the lower- and upper-case Greek
-letter gamma). You can obtain these using the @kbd{H f G} address@hidden
-and @kbd{H I f G} address@hidden commands.
address@hidden ifnottex
address@hidden
-\turnoffactive
-The functions corresponding to the integrals that define $P(a,x)$
-and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
-factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
-You can obtain these using the \kbd{H f G} [\code{gammag}] and
-\kbd{I H f G} [\code{gammaG}] commands.
address@hidden tex
-
address@hidden f b
address@hidden calc-beta
address@hidden beta
-The @kbd{f b} (@code{calc-beta}) address@hidden command computes the
-Euler beta function, which is defined in terms of the gamma function as
address@hidden @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
address@hidden @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
-or by
address@hidden @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
address@hidden @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
-
address@hidden f B
address@hidden H f B
address@hidden calc-inc-beta
address@hidden betaI
address@hidden betaB
-The @kbd{f B} (@code{calc-inc-beta}) address@hidden command computes
-the incomplete beta function @expr{I(x,a,b)}. It is defined by
address@hidden @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right)
/ B(a,b)}.
address@hidden @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) /
beta(a,b)}.
-Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
-un-normalized version address@hidden
-
address@hidden f e
address@hidden I f e
address@hidden calc-erf
address@hidden erf
address@hidden erfc
-The @kbd{f e} (@code{calc-erf}) address@hidden command computes the
-error function
address@hidden @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
address@hidden @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
-The complementary error function @kbd{I f e} (@code{calc-erfc}) address@hidden
-is the corresponding integral from @samp{x} to infinity; the sum
address@hidden @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
address@hidden @expr{erf(x) + erfc(x) = 1}.
-
address@hidden f j
address@hidden f y
address@hidden calc-bessel-J
address@hidden calc-bessel-Y
address@hidden besJ
address@hidden besY
-The @kbd{f j} (@code{calc-bessel-J}) address@hidden and @kbd{f y}
-(@code{calc-bessel-Y}) address@hidden commands compute the Bessel
-functions of the first and second kinds, respectively.
-In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
address@hidden is often an integer, but is not required to be one.
-Calc's implementation of the Bessel functions currently limits the
-precision to 8 digits, and may not be exact even to that precision.
-Use with care!
-
address@hidden Branch Cuts, Random Numbers, Advanced Math Functions, Scientific
Functions
address@hidden Branch Cuts and Principal Values
-
address@hidden
address@hidden Branch cuts
address@hidden Principal values
-All of the logarithmic, trigonometric, and other scientific functions are
-defined for complex numbers as well as for reals.
-This section describes the values
-returned in cases where the general result is a family of possible values.
-Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
-second edition, in these matters. This section will describe each
-function briefly; for a more detailed discussion (including some nifty
-diagrams), consult Steele's book.
-
-Note that the branch cuts for @code{arctan} and @code{arctanh} were
-changed between the first and second editions of Steele. Versions of
-Calc starting with 2.00 follow the second edition.
-
-The new branch cuts exactly match those of the HP-28/48 calculators.
-They also match those of Mathematica 1.2, except that Mathematica's
address@hidden cut is always in the right half of the complex plane,
-and its @code{arctanh} cut is always in the top half of the plane.
-Calc's cuts are continuous with quadrants I and III for @code{arctan},
-or II and IV for @code{arctanh}.
-
-Note: The current implementations of these functions with complex arguments
-are designed with proper behavior around the branch cuts in mind, @emph{not}
-efficiency or accuracy. You may need to increase the floating precision
-and wait a while to get suitable answers from them.
-
-For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
-or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
-negative, the result is close to the @expr{-i} axis. The result always lies
-in the right half of the complex plane.
-
-For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
-The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
-Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
-negative real axis.
-
-The following table describes these branch cuts in another way.
-If the real and imaginary parts of @expr{z} are as shown, then
-the real and imaginary parts of @expr{f(z)} will be as shown.
-Here @code{eps} stands for a small positive value; each
-occurrence of @code{eps} may stand for a different small value.
-
address@hidden
- z sqrt(z) ln(z)
-----------------------------------------
- +, 0 +, 0 any, 0
- -, 0 0, + any, pi
- -, +eps +eps, + +eps, +
- -, -eps +eps, - +eps, -
address@hidden smallexample
-
-For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
-One interesting consequence of this is that @samp{(-8)^1:3} does
-not evaluate to @mathit{-2} as you might expect, but to the complex
-number @expr{(1., 1.732)}. Both of these are valid cube roots
-of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
-less-obvious root for the sake of mathematical consistency.
-
-For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
-The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
-
-For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
-or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
-the real axis, less than @mathit{-1} and greater than 1.
-
-For @samp{arctan(z)}: This is defined by
address@hidden(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
-imaginary axis, below @expr{-i} and above @expr{i}.
-
-For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
-The branch cuts are on the imaginary axis, below @expr{-i} and
-above @expr{i}.
-
-For @samp{arccosh(z)}: This is defined by
address@hidden(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
-real axis less than 1.
-
-For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
-The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
-
-The following tables for @code{arcsin}, @code{arccos}, and
address@hidden assume the current angular mode is Radians. The
-hyperbolic functions operate independently of the angular mode.
-
address@hidden
- z arcsin(z) arccos(z)
--------------------------------------------------------
- (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
- (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
- (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
- <-1, 0 -pi/2, + pi, -
- <-1, +eps -pi/2 + eps, + pi - eps, -
- <-1, -eps -pi/2 + eps, - pi - eps, +
- >1, 0 pi/2, - 0, +
- >1, +eps pi/2 - eps, + +eps, -
- >1, -eps pi/2 - eps, - +eps, +
address@hidden smallexample
-
address@hidden
- z arccosh(z) arctanh(z)
------------------------------------------------------
- (-1..1), 0 0, (0..pi) any, 0
- (-1..1), +eps +eps, (0..pi) any, +eps
- (-1..1), -eps +eps, (-pi..0) any, -eps
- <-1, 0 +, pi -, pi/2
- <-1, +eps +, pi - eps -, pi/2 - eps
- <-1, -eps +, -pi + eps -, -pi/2 + eps
- >1, 0 +, 0 +, -pi/2
- >1, +eps +, +eps +, pi/2 - eps
- >1, -eps +, -eps +, -pi/2 + eps
address@hidden smallexample
-
address@hidden
- z arcsinh(z) arctan(z)
------------------------------------------------------
- 0, (-1..1) 0, (-pi/2..pi/2) 0, any
- 0, <-1 -, -pi/2 -pi/2, -
- +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
- -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
- 0, >1 +, pi/2 pi/2, +
- +eps, >1 +, pi/2 - eps pi/2 - eps, +
- -eps, >1 -, pi/2 - eps -pi/2 + eps, +
address@hidden smallexample
-
-Finally, the following identities help to illustrate the relationship
-between the complex trigonometric and hyperbolic functions. They
-are valid everywhere, including on the branch cuts.
-
address@hidden
-sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
-cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
-tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
-sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
address@hidden smallexample
-
-The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
-for general complex arguments, but their branch cuts and principal values
-are not rigorously specified at present.
-
address@hidden Random Numbers, Combinatorial Functions, Branch Cuts, Scientific
Functions
address@hidden Random Numbers
-
address@hidden
address@hidden k r
address@hidden calc-random
address@hidden random
-The @kbd{k r} (@code{calc-random}) address@hidden command produces
-random numbers of various sorts.
-
-Given a positive numeric prefix argument @expr{M}, it produces a random
-integer @expr{N} in the range
address@hidden @math{0 \le N < M}.
address@hidden @expr{0 <= N < M}.
-Each of the @expr{M} values appears with equal probability.
-
-With no numeric prefix argument, the @kbd{k r} command takes its argument
-from the stack instead. Once again, if this is a positive integer @expr{M}
-the result is a random integer less than @expr{M}. However, note that
-while numeric prefix arguments are limited to six digits or so, an @expr{M}
-taken from the stack can be arbitrarily large. If @expr{M} is negative,
-the result is a random integer in the range
address@hidden @math{M < N \le 0}.
address@hidden @expr{M < N <= 0}.
-
-If the value on the stack is a floating-point number @expr{M}, the result
-is a random floating-point number @expr{N} in the range
address@hidden @math{0 \le N < M}
address@hidden @expr{0 <= N < M}
-or
address@hidden @math{M < N \le 0},
address@hidden @expr{M < N <= 0},
-according to the sign of @expr{M}.
-
-If @expr{M} is zero, the result is a Gaussian-distributed random real
-number; the distribution has a mean of zero and a standard deviation
-of one. The algorithm used generates random numbers in pairs; thus,
-every other call to this function will be especially fast.
-
-If @expr{M} is an error form
address@hidden @math{m} @code{+/-} @math{\sigma}
address@hidden @samp{m +/- s}
-where @var{m} and
address@hidden @math{\sigma}
address@hidden @var{s}
-are both real numbers, the result uses a Gaussian distribution with mean
address@hidden and standard deviation
address@hidden @math{\sigma}.
address@hidden @var{s}.
-
-If @expr{M} is an interval form, the lower and upper bounds specify the
-acceptable limits of the random numbers. If both bounds are integers,
-the result is a random integer in the specified range. If either bound
-is floating-point, the result is a random real number in the specified
-range. If the interval is open at either end, the result will be sure
-not to equal that end value. (This makes a big difference for integer
-intervals, but for floating-point intervals it's relatively minor:
-with a precision of 6, @samp{random([1.0..2.0))} will return any of one
-million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
-additionally return 2.00000, but the probability of this happening is
-extremely small.)
-
-If @expr{M} is a vector, the result is one element taken at random from
-the vector. All elements of the vector are given equal probabilities.
-
address@hidden RandSeed
-The sequence of numbers produced by @kbd{k r} is completely random by
-default, i.e., the sequence is seeded each time you start Calc using
-the current time and other information. You can get a reproducible
-sequence by storing a particular ``seed value'' in the Calc variable
address@hidden Any integer will do for a seed; integers of from 1
-to 12 digits are good. If you later store a different integer into
address@hidden, Calc will switch to a different pseudo-random
-sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
-from the current time. If you store the same integer that you used
-before back into @code{RandSeed}, you will get the exact same sequence
-of random numbers as before.
-
address@hidden calc-rrandom
-The @code{calc-rrandom} command (not on any key) produces a random real
-number between zero and one. It is equivalent to @samp{random(1.0)}.
-
address@hidden k a
address@hidden calc-random-again
-The @kbd{k a} (@code{calc-random-again}) command produces another random
-number, re-using the most recent value of @expr{M}. With a numeric
-prefix argument @var{n}, it produces @var{n} more random numbers using
-that value of @expr{M}.
-
address@hidden k h
address@hidden calc-shuffle
address@hidden shuffle
-The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
-random values with no duplicates. The value on the top of the stack
-specifies the set from which the random values are drawn, and may be any
-of the @expr{M} formats described above. The numeric prefix argument
-gives the length of the desired list. (If you do not provide a numeric
-prefix argument, the length of the list is taken from the top of the
-stack, and @expr{M} from second-to-top.)
-
-If @expr{M} is a floating-point number, zero, or an error form (so
-that the random values are being drawn from the set of real numbers)
-there is little practical difference between using @kbd{k h} and using
address@hidden r} several times. But if the set of possible values consists
-of just a few integers, or the elements of a vector, then there is
-a very real chance that multiple @kbd{k r}'s will produce the same
-number more than once. The @kbd{k h} command produces a vector whose
-elements are always distinct. (Actually, there is a slight exception:
-If @expr{M} is a vector, no given vector element will be drawn more
-than once, but if several elements of @expr{M} are equal, they may
-each make it into the result vector.)
-
-One use of @kbd{k h} is to rearrange a list at random. This happens
-if the prefix argument is equal to the number of values in the list:
address@hidden, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
address@hidden, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
address@hidden is negative it is replaced by the size of the set represented
-by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
-a small discrete set of possibilities.
-
-To do the equivalent of @kbd{k h} but with duplications allowed,
-given @expr{M} on the stack and with @var{n} just entered as a numeric
-prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
address@hidden M k r} to ``map'' the normal @kbd{k r} function over the
-elements of this vector. @xref{Matrix Functions}.
-
address@hidden
-* Random Number Generator:: (Complete description of Calc's algorithm)
address@hidden menu
-
address@hidden Random Number Generator, , Random Numbers, Random Numbers
address@hidden Random Number Generator
-
-Calc's random number generator uses several methods to ensure that
-the numbers it produces are highly random. Knuth's @emph{Art of
-Computer Programming}, Volume II, contains a thorough description
-of the theory of random number generators and their measurement and
-characterization.
-
-If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
address@hidden function to get a stream of random numbers, which it
-then treats in various ways to avoid problems inherent in the simple
-random number generators that many systems use to implement @code{random}.
-
-When Calc's random number generator is first invoked, it ``seeds''
-the low-level random sequence using the time of day, so that the
-random number sequence will be different every time you use Calc.
-
-Since Emacs Lisp doesn't specify the range of values that will be
-returned by its @code{random} function, Calc exercises the function
-several times to estimate the range. When Calc subsequently uses
-the @code{random} function, it takes only 10 bits of the result
-near the most-significant end. (It avoids at least the bottom
-four bits, preferably more, and also tries to avoid the top two
-bits.) This strategy works well with the linear congruential
-generators that are typically used to implement @code{random}.
-
-If @code{RandSeed} contains an integer, Calc uses this integer to
-seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
-computing
address@hidden @math{X_{n-55} - X_{n-24}}.
address@hidden @expr{X_n-55 - X_n-24}).
-This method expands the seed
-value into a large table which is maintained internally; the variable
address@hidden is changed from, e.g., 42 to the vector @expr{[42]}
-to indicate that the seed has been absorbed into this table. When
address@hidden contains a vector, @kbd{k r} and related commands
-continue to use the same internal table as last time. There is no
-way to extract the complete state of the random number generator
-so that you can restart it from any point; you can only restart it
-from the same initial seed value. A simple way to restart from the
-same seed is to type @kbd{s r RandSeed} to get the seed vector,
address@hidden u} to unpack it back into a number, then @kbd{s t RandSeed}
-to reseed the generator with that number.
-
-Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
-of Knuth. It fills a table with 13 random 10-bit numbers. Then,
-to generate a new random number, it uses the previous number to
-index into the table, picks the value it finds there as the new
-random number, then replaces that table entry with a new value
-obtained from a call to the base random number generator (either
-the additive congruential generator or the @code{random} function
-supplied by the system). If there are any flaws in the base
-generator, shuffling will tend to even them out. But if the system
-provides an excellent @code{random} function, shuffling will not
-damage its randomness.
-
-To create a random integer of a certain number of digits, Calc
-builds the integer three decimal digits at a time. For each group
-of three digits, Calc calls its 10-bit shuffling random number generator
-(which returns a value from 0 to 1023); if the random value is 1000
-or more, Calc throws it out and tries again until it gets a suitable
-value.
-
-To create a random floating-point number with precision @var{p}, Calc
-simply creates a random @var{p}-digit integer and multiplies by
address@hidden @math{10^{-p}}.
address@hidden @expr{10^-p}.
-The resulting random numbers should be very clean, but note
-that relatively small numbers will have few significant random digits.
-In other words, with a precision of 12, you will occasionally get
-numbers on the order of
address@hidden @math{10^{-9}}
address@hidden @expr{10^-9}
-or
address@hidden @math{10^{-10}},
address@hidden @expr{10^-10},
-but those numbers will only have two or three random digits since they
-correspond to small integers times
address@hidden @math{10^{-12}}.
address@hidden @expr{10^-12}.
-
-To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
-counts the digits in @var{m}, creates a random integer with three
-additional digits, then reduces modulo @var{m}. Unless @var{m} is a
-power of ten the resulting values will be very slightly biased toward
-the lower numbers, but this bias will be less than 0.1%. (For example,
-if @var{m} is 42, Calc will reduce a random integer less than 100000
-modulo 42 to get a result less than 42. It is easy to show that the
-numbers 40 and 41 will be only 2380/2381 as likely to result from this
-modulo operation as numbers 39 and below.) If @var{m} is a power of
-ten, however, the numbers should be completely unbiased.
-
-The Gaussian random numbers generated by @samp{random(0.0)} use the
-``polar'' method described in Knuth section 3.4.1C. This method
-generates a pair of Gaussian random numbers at a time, so only every
-other call to @samp{random(0.0)} will require significant calculations.
-
address@hidden Combinatorial Functions, Probability Distribution Functions,
Random Numbers, Scientific Functions
address@hidden Combinatorial Functions
-
address@hidden
-Commands relating to combinatorics and number theory begin with the
address@hidden key prefix.
-
address@hidden k g
address@hidden calc-gcd
address@hidden gcd
-The @kbd{k g} (@code{calc-gcd}) address@hidden command computes the
-Greatest Common Divisor of two integers. It also accepts fractions;
-the GCD of two fractions is defined by taking the GCD of the
-numerators, and the LCM of the denominators. This definition is
-consistent with the idea that @samp{a / gcd(a,x)} should yield an
-integer for any @samp{a} and @samp{x}. For other types of arguments,
-the operation is left in symbolic form.
-
address@hidden k l
address@hidden calc-lcm
address@hidden lcm
-The @kbd{k l} (@code{calc-lcm}) address@hidden command computes the
-Least Common Multiple of two integers or fractions. The product of
-the LCM and GCD of two numbers is equal to the product of the
-numbers.
-
address@hidden k E
address@hidden calc-extended-gcd
address@hidden egcd
-The @kbd{k E} (@code{calc-extended-gcd}) address@hidden command computes
-the GCD of two integers @expr{x} and @expr{y} and returns a vector
address@hidden, a, b]} where
address@hidden @math{g = \gcd(x,y) = a x + b y}.
address@hidden @expr{g = gcd(x,y) = a x + b y}.
-
address@hidden !
address@hidden calc-factorial
address@hidden fact
address@hidden
address@hidden @null
address@hidden ignore
address@hidden !
-The @kbd{!} (@code{calc-factorial}) address@hidden command computes the
-factorial of the number at the top of the stack. If the number is an
-integer, the result is an exact integer. If the number is an
-integer-valued float, the result is a floating-point approximation. If
-the number is a non-integral real number, the generalized factorial is used,
-as defined by the Euler Gamma function. Please note that computation of
-large factorials can be slow; using floating-point format will help
-since fewer digits must be maintained. The same is true of many of
-the commands in this section.
-
address@hidden k d
address@hidden calc-double-factorial
address@hidden dfact
address@hidden
address@hidden @null
address@hidden ignore
address@hidden !!
-The @kbd{k d} (@code{calc-double-factorial}) address@hidden command
-computes the ``double factorial'' of an integer. For an even integer,
-this is the product of even integers from 2 to @expr{N}. For an odd
-integer, this is the product of odd integers from 3 to @expr{N}. If
-the argument is an integer-valued float, the result is a floating-point
-approximation. This function is undefined for negative even integers.
-The notation @expr{N!!} is also recognized for double factorials.
-
address@hidden k c
address@hidden calc-choose
address@hidden choose
-The @kbd{k c} (@code{calc-choose}) address@hidden command computes the
-binomial coefficient @address@hidden, where @expr{M} is the number
-on the top of the stack and @expr{N} is second-to-top. If both arguments
-are integers, the result is an exact integer. Otherwise, the result is a
-floating-point approximation. The binomial coefficient is defined for all
-real numbers by
address@hidden @math{N! \over M! (N-M)!\,}.
address@hidden @expr{N! / M! (N-M)!}.
-
address@hidden H k c
address@hidden calc-perm
address@hidden perm
address@hidden
-The @kbd{H k c} (@code{calc-perm}) address@hidden command computes the
-number-of-permutations function @expr{N! / (N-M)!}.
address@hidden ifnottex
address@hidden
-The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
-number-of-perm\-utations function $N! \over (N-M)!\,$.
address@hidden tex
-
address@hidden k b
address@hidden H k b
address@hidden calc-bernoulli-number
address@hidden bern
-The @kbd{k b} (@code{calc-bernoulli-number}) address@hidden command
-computes a given Bernoulli number. The value at the top of the stack
-is a nonnegative integer @expr{n} that specifies which Bernoulli number
-is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
-taking @expr{n} from the second-to-top position and @expr{x} from the
-top of the stack. If @expr{x} is a variable or formula the result is
-a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
-
address@hidden k e
address@hidden H k e
address@hidden calc-euler-number
address@hidden euler
-The @kbd{k e} (@code{calc-euler-number}) address@hidden command similarly
-computes an Euler number, and @address@hidden k e}} computes an Euler
polynomial.
-Bernoulli and Euler numbers occur in the Taylor expansions of several
-functions.
-
address@hidden k s
address@hidden H k s
address@hidden calc-stirling-number
address@hidden stir1
address@hidden stir2
-The @kbd{k s} (@code{calc-stirling-number}) address@hidden command
-computes a Stirling number of the first
address@hidden address@hidden@math{n \brack m},
address@hidden kind,
-given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
address@hidden command computes a Stirling number of the second
address@hidden address@hidden@math{n \brace m}.
address@hidden kind.
-These are the number of @expr{m}-cycle permutations of @expr{n} objects,
-and the number of ways to partition @expr{n} objects into @expr{m}
-non-empty sets, respectively.
-
address@hidden k p
address@hidden calc-prime-test
address@hidden Primes
-The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
-the top of the stack is prime. For integers less than eight million, the
-answer is always exact and reasonably fast. For larger integers, a
-probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
-The number is first checked against small prime factors (up to 13). Then,
-any number of iterations of the algorithm are performed. Each step either
-discovers that the number is non-prime, or substantially increases the
-certainty that the number is prime. After a few steps, the chance that
-a number was mistakenly described as prime will be less than one percent.
-(Indeed, this is a worst-case estimate of the probability; in practice
-even a single iteration is quite reliable.) After the @kbd{k p} command,
-the number will be reported as definitely prime or non-prime if possible,
-or otherwise ``probably'' prime with a certain probability of error.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden prime
-The normal @kbd{k p} command performs one iteration of the primality
-test. Pressing @kbd{k p} repeatedly for the same integer will perform
-additional iterations. Also, @kbd{k p} with a numeric prefix performs
-the specified number of iterations. There is also an algebraic function
address@hidden(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
-is (probably) prime and 0 if not.
-
address@hidden k f
address@hidden calc-prime-factors
address@hidden prfac
-The @kbd{k f} (@code{calc-prime-factors}) address@hidden command
-attempts to decompose an integer into its prime factors. For numbers up
-to 25 million, the answer is exact although it may take some time. The
-result is a vector of the prime factors in increasing order. For larger
-inputs, prime factors above 5000 may not be found, in which case the
-last number in the vector will be an unfactored integer greater than 25
-million (with a warning message). For negative integers, the first
-element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0},
and
address@hidden, the result is a list of the same number.
-
address@hidden k n
address@hidden calc-next-prime
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden nextprime
-The @kbd{k n} (@code{calc-next-prime}) address@hidden command finds
-the next prime above a given number. Essentially, it searches by calling
address@hidden on successive integers until it finds one that
-passes the test. This is quite fast for integers less than eight million,
-but once the probabilistic test comes into play the search may be rather
-slow. Ordinarily this command stops for any prime that passes one iteration
-of the primality test. With a numeric prefix argument, a number must pass
-the specified number of iterations before the search stops. (This only
-matters when searching above eight million.) You can always use additional
address@hidden p} commands to increase your certainty that the number is indeed
-prime.
-
address@hidden I k n
address@hidden calc-prev-prime
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden prevprime
-The @kbd{I k n} (@code{calc-prev-prime}) address@hidden command
-analogously finds the next prime less than a given number.
-
address@hidden k t
address@hidden calc-totient
address@hidden totient
-The @kbd{k t} (@code{calc-totient}) address@hidden command computes the
-Euler ``totient''
address@hidden address@hidden@math{\phi(n)},
address@hidden function,
-the number of integers less than @expr{n} which
-are relatively prime to @expr{n}.
-
address@hidden k m
address@hidden calc-moebius
address@hidden moebius
-The @kbd{k m} (@code{calc-moebius}) address@hidden command computes the
address@hidden M@"obius @math{\mu}
address@hidden Moebius ``mu''
-function. If the input number is a product of @expr{k}
-distinct factors, this is @expr{(-1)^k}. If the input number has any
-duplicate factors (i.e., can be divided by the same prime more than once),
-the result is zero.
-
address@hidden Probability Distribution Functions, , Combinatorial Functions,
Scientific Functions
address@hidden Probability Distribution Functions
-
address@hidden
-The functions in this section compute various probability distributions.
-For continuous distributions, this is the integral of the probability
-density function from @expr{x} to infinity. (These are the ``upper
-tail'' distribution functions; there are also corresponding ``lower
-tail'' functions which integrate from minus infinity to @expr{x}.)
-For discrete distributions, the upper tail function gives the sum
-from @expr{x} to infinity; the lower tail function gives the sum
-from minus infinity up to, but not including,@w{ address@hidden
-
-To integrate from @expr{x} to @expr{y}, just use the distribution
-function twice and subtract. For example, the probability that a
-Gaussian random variable with mean 2 and standard deviation 1 will
-lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
-(``the probability that it is greater than 2.5, but not greater than 2.8''),
-or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
-
address@hidden k B
address@hidden I k B
address@hidden calc-utpb
address@hidden utpb
address@hidden ltpb
-The @kbd{k B} (@code{calc-utpb}) address@hidden function uses the
-binomial distribution. Push the parameters @var{n}, @var{p}, and
-then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
-probability that an event will occur @var{x} or more times out
-of @var{n} trials, if its probability of occurring in any given
-trial is @var{p}. The @kbd{I k B} address@hidden function is
-the probability that the event will occur fewer than @var{x} times.
-
-The other probability distribution functions similarly take the
-form @kbd{k @var{X}} (@address@hidden) address@hidden@var{x}}]
-and @kbd{I k @var{X}} address@hidden@var{x}}], for various letters
address@hidden The arguments to the algebraic functions are the value of
-the random variable first, then whatever other parameters define the
-distribution. Note these are among the few Calc functions where the
-order of the arguments in algebraic form differs from the order of
-arguments as found on the stack. (The random variable comes last on
-the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
-k N address@hidden @key{DEL} 2.8 k N -}, using @address@hidden @key{DEL}} to
-recover the original arguments but substitute a new value for @expr{x}.)
-
address@hidden k C
address@hidden calc-utpc
address@hidden utpc
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k C
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpc
-The @samp{utpc(x,v)} function uses the chi-square distribution with
address@hidden @math{\nu}
address@hidden @expr{v}
-degrees of freedom. It is the probability that a model is
-correct if its chi-square statistic is @expr{x}.
-
address@hidden k F
address@hidden calc-utpf
address@hidden utpf
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k F
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpf
-The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
-various statistical tests. The parameters
address@hidden @math{\nu_1}
address@hidden @expr{v1}
-and
address@hidden @math{\nu_2}
address@hidden @expr{v2}
-are the degrees of freedom in the numerator and denominator,
-respectively, used in computing the statistic @expr{F}.
-
address@hidden k N
address@hidden calc-utpn
address@hidden utpn
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k N
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpn
-The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
-with mean @expr{m} and standard deviation
address@hidden @math{\sigma}.
address@hidden @expr{s}.
-It is the probability that such a normal-distributed random variable
-would exceed @expr{x}.
-
address@hidden k P
address@hidden calc-utpp
address@hidden utpp
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k P
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpp
-The @samp{utpp(n,x)} function uses a Poisson distribution with
-mean @expr{x}. It is the probability that @expr{n} or more such
-Poisson random events will occur.
-
address@hidden k T
address@hidden calc-ltpt
address@hidden utpt
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden I k T
address@hidden
address@hidden @null
address@hidden ignore
address@hidden ltpt
-The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
-with
address@hidden @math{\nu}
address@hidden @expr{v}
-degrees of freedom. It is the probability that a
-t-distributed random variable will be greater than @expr{t}.
-(Note: This computes the distribution function
address@hidden @math{A(t|\nu)}
address@hidden @expr{A(t|v)}
-where
address@hidden @math{A(0|\nu) = 1}
address@hidden @expr{A(0|v) = 1}
-and
address@hidden @math{A(\infty|\nu) \to 0}.
address@hidden @expr{A(inf|v) -> 0}.
-The @code{UTPT} operation on the HP-48 uses a different definition which
-returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
-
-While Calc does not provide inverses of the probability distribution
-functions, the @kbd{a R} command can be used to solve for the inverse.
-Since the distribution functions are monotonic, @kbd{a R} is guaranteed
-to be able to find a solution given any initial guess.
address@hidden Solutions}.
-
address@hidden Matrix Functions, Algebra, Scientific Functions, Top
address@hidden Vector/Matrix Functions
-
address@hidden
-Many of the commands described here begin with the @kbd{v} prefix.
-(For convenience, the address@hidden prefix is equivalent to @kbd{v}.)
-The commands usually apply to both plain vectors and matrices; some
-apply only to matrices or only to square matrices. If the argument
-has the wrong dimensions the operation is left in symbolic form.
-
-Vectors are entered and displayed using @samp{[a,b,c]} notation.
-Matrices are vectors of which all elements are vectors of equal length.
-(Though none of the standard Calc commands use this concept, a
-three-dimensional matrix or rank-3 tensor could be defined as a
-vector of matrices, and so on.)
-
address@hidden
-* Packing and Unpacking::
-* Building Vectors::
-* Extracting Elements::
-* Manipulating Vectors::
-* Vector and Matrix Arithmetic::
-* Set Operations::
-* Statistical Operations::
-* Reducing and Mapping::
-* Vector and Matrix Formats::
address@hidden menu
-
address@hidden Packing and Unpacking, Building Vectors, Matrix Functions,
Matrix Functions
address@hidden Packing and Unpacking
-
address@hidden
-Calc's ``pack'' and ``unpack'' commands collect stack entries to build
-composite objects such as vectors and complex numbers. They are
-described in this chapter because they are most often used to build
-vectors.
-
address@hidden v p
address@hidden calc-pack
-The @kbd{v p} (@code{calc-pack}) address@hidden command collects several
-elements from the stack into a matrix, complex number, HMS form, error
-form, etc. It uses a numeric prefix argument to specify the kind of
-object to be built; this argument is referred to as the ``packing mode.''
-If the packing mode is a nonnegative integer, a vector of that
-length is created. For example, @kbd{C-u 5 v p} will pop the top
-five stack elements and push back a single vector of those five
-elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
-
-The same effect can be had by pressing @kbd{[} to push an incomplete
-vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
-the incomplete object up past a certain number of elements, and
-then pressing @kbd{]} to complete the vector.
-
-Negative packing modes create other kinds of composite objects:
-
address@hidden @cite
address@hidden -1
-Two values are collected to build a complex number. For example,
address@hidden @key{RET} 7 C-u -1 v p} creates the complex number
address@hidden(5, 7)}. The result is always a rectangular complex
-number. The two input values must both be real numbers,
-i.e., integers, fractions, or floats. If they are not, Calc
-will instead build a formula like @samp{a + (0, 1) b}. (The
-other packing modes also create a symbolic answer if the
-components are not suitable.)
-
address@hidden -2
-Two values are collected to build a polar complex number.
-The first is the magnitude; the second is the phase expressed
-in either degrees or radians according to the current angular
-mode.
-
address@hidden -3
-Three values are collected into an HMS form. The first
-two values (hours and minutes) must be integers or
-integer-valued floats. The third value may be any real
-number.
-
address@hidden -4
-Two values are collected into an error form. The inputs
-may be real numbers or formulas.
-
address@hidden -5
-Two values are collected into a modulo form. The inputs
-must be real numbers.
-
address@hidden -6
-Two values are collected into the interval @samp{[a .. b]}.
-The inputs may be real numbers, HMS or date forms, or formulas.
-
address@hidden -7
-Two values are collected into the interval @samp{[a .. b)}.
-
address@hidden -8
-Two values are collected into the interval @samp{(a .. b]}.
-
address@hidden -9
-Two values are collected into the interval @samp{(a .. b)}.
-
address@hidden -10
-Two integer values are collected into a fraction.
-
address@hidden -11
-Two values are collected into a floating-point number.
-The first is the mantissa; the second, which must be an
-integer, is the exponent. The result is the mantissa
-times ten to the power of the exponent.
-
address@hidden -12
-This is treated the same as @mathit{-11} by the @kbd{v p} command.
-When unpacking, @mathit{-12} specifies that a floating-point mantissa
-is desired.
-
address@hidden -13
-A real number is converted into a date form.
-
address@hidden -14
-Three numbers (year, month, day) are packed into a pure date form.
-
address@hidden -15
-Six numbers are packed into a date/time form.
address@hidden table
-
-With any of the two-input negative packing modes, either or both
-of the inputs may be vectors. If both are vectors of the same
-length, the result is another vector made by packing corresponding
-elements of the input vectors. If one input is a vector and the
-other is a plain number, the number is packed along with each vector
-element to produce a new vector. For example, @kbd{C-u -4 v p}
-could be used to convert a vector of numbers and a vector of errors
-into a single vector of error forms; @kbd{C-u -5 v p} could convert
-a vector of numbers and a single number @var{M} into a vector of
-numbers modulo @var{M}.
-
-If you don't give a prefix argument to @kbd{v p}, it takes
-the packing mode from the top of the stack. The elements to
-be packed then begin at stack level 2. Thus
address@hidden @key{RET} 2 @key{RET} 4 n v p} is another way to
-enter the error form @samp{1 +/- 2}.
-
-If the packing mode taken from the stack is a vector, the result is a
-matrix with the dimensions specified by the elements of the vector,
-which must each be integers. For example, if the packing mode is
address@hidden, 3]}, then six numbers will be taken from the stack and
-returned in the form @address@hidden, b, c]}, [d, e, f]]}.
-
-If any elements of the vector are negative, other kinds of
-packing are done at that level as described above. For
-example, @samp{[2, 3, -4]} takes 12 objects and creates a
address@hidden @math{2\times3}
address@hidden 2x3
-matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
-Also, @samp{[-4, -10]} will convert four integers into an
-error form consisting of two fractions: @samp{a:b +/- c:d}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden pack
-There is an equivalent algebraic function,
address@hidden(@var{mode}, @var{items})} where @var{mode} is a
-packing mode (an integer or a vector of integers) and @var{items}
-is a vector of objects to be packed (re-packed, really) according
-to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
-yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
-left in symbolic form if the packing mode is invalid, or if the
-number of data items does not match the number of items required
-by the mode.
-
address@hidden v u
address@hidden calc-unpack
-The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
-number, HMS form, or other composite object on the top of the stack and
-``unpacks'' it, pushing each of its elements onto the stack as separate
-objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
-at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
-each of the arguments of the top-level operator onto the stack.
-
-You can optionally give a numeric prefix argument to @kbd{v u}
-to specify an explicit (un)packing mode. If the packing mode is
-negative and the input is actually a vector or matrix, the result
-will be two or more similar vectors or matrices of the elements.
-For example, given the vector @address@hidden +/- b}, c^2, d +/- 7]},
-the result of @kbd{C-u -4 v u} will be the two vectors
address@hidden, c^2, d]} and @address@hidden, 0, 7]}}.
-
-Note that the prefix argument can have an effect even when the input is
-not a vector. For example, if the input is the number @mathit{-5}, then
address@hidden -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
-when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
-and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
-and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
-number). Plain @kbd{v u} with this input would complain that the input
-is not a composite object.
-
-Unpacking mode @mathit{-11} converts a float into an integer mantissa and
-an integer exponent, where the mantissa is not divisible by 10
-(except that 0.0 is represented by a mantissa and exponent of 0).
-Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
-and integer exponent, where the mantissa (for non-zero numbers)
-is guaranteed to lie in the range [1 .. 10). In both cases,
-the mantissa is shifted left or right (and the exponent adjusted
-to compensate) in order to satisfy these constraints.
-
-Positive unpacking modes are treated differently than for @kbd{v p}.
-A mode of 1 is much like plain @kbd{v u} with no prefix argument,
-except that in addition to the components of the input object,
-a suitable packing mode to re-pack the object is also pushed.
-Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
-original object.
-
-A mode of 2 unpacks two levels of the object; the resulting
-re-packing mode will be a vector of length 2. This might be used
-to unpack a matrix, say, or a vector of error forms. Higher
-unpacking modes unpack the input even more deeply.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden unpack
-There are two algebraic functions analogous to @kbd{v u}.
-The @samp{unpack(@var{mode}, @var{item})} function unpacks the
address@hidden using the given @var{mode}, returning the result as
-a vector of components. Here the @var{mode} must be an
-integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
-returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden unpackt
-The @code{unpackt} function is like @code{unpack} but instead
-of returning a simple vector of items, it returns a vector of
-two things: The mode, and the vector of items. For example,
address@hidden(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
-and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
-The identity for re-building the original object is
address@hidden(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
address@hidden function builds a function call given the function
-name and a vector of arguments.)
-
address@hidden Numerator of a fraction, extracting
-Subscript notation is a useful way to extract a particular part
-of an object. For example, to get the numerator of a rational
-number, you can use @samp{unpack(-10, @var{x})_1}.
-
address@hidden Building Vectors, Extracting Elements, Packing and Unpacking,
Matrix Functions
address@hidden Building Vectors
-
address@hidden
-Vectors and matrices can be added,
-subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
-
address@hidden |
address@hidden calc-concat
address@hidden
address@hidden @null
address@hidden ignore
address@hidden |
-The @kbd{|} (@code{calc-concat}) address@hidden command ``concatenates'' two
vectors
-into one. For example, after @address@hidden 1 , 2 ]} [ 3 , 4 ] |}, the stack
-will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
-are matrices, the rows of the first matrix are concatenated with the
-rows of the second. (In other words, two matrices are just two vectors
-of row-vectors as far as @kbd{|} is concerned.)
-
-If either argument to @kbd{|} is a scalar (a non-vector), it is treated
-like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
-produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
-matrix and the other is a plain vector, the vector is treated as a
-one-row matrix.
-
address@hidden H |
address@hidden append
-The @kbd{H |} (@code{calc-append}) address@hidden command concatenates
-two vectors without any special cases. Both inputs must be vectors.
-Whether or not they are matrices is not taken into account. If either
-argument is a scalar, the @code{append} function is left in symbolic form.
-See also @code{cons} and @code{rcons} below.
-
address@hidden I |
address@hidden H I |
-The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
-two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
-to @address@hidden |}, but possibly more convenient and also a bit faster.
-
address@hidden v d
address@hidden calc-diag
address@hidden diag
-The @kbd{v d} (@code{calc-diag}) address@hidden function builds a diagonal
-square matrix. The optional numeric prefix gives the number of rows
-and columns in the matrix. If the value at the top of the stack is a
-vector, the elements of the vector are used as the diagonal elements; the
-prefix, if specified, must match the size of the vector. If the value on
-the stack is a scalar, it is used for each element on the diagonal, and
-the prefix argument is required.
-
-To build a constant square matrix, e.g., a
address@hidden @math{3\times3}
address@hidden 3x3
-matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
-matrix first and then add a constant value to that matrix. (Another
-alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
-
address@hidden v i
address@hidden calc-ident
address@hidden idn
-The @kbd{v i} (@code{calc-ident}) address@hidden function builds an identity
-matrix of the specified size. It is a convenient form of @kbd{v d}
-where the diagonal element is always one. If no prefix argument is given,
-this command prompts for one.
-
-In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
-except that @expr{a} is required to be a scalar (non-vector) quantity.
-If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
-identity matrix of unknown size. Calc can operate algebraically on
-such generic identity matrices, and if one is combined with a matrix
-whose size is known, it is converted automatically to an identity
-matrix of a suitable matching size. The @kbd{v i} command with an
-argument of zero creates a generic identity matrix, @samp{idn(1)}.
-Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
-identity matrices are immediately expanded to the current default
-dimensions.
-
address@hidden v x
address@hidden calc-index
address@hidden index
-The @kbd{v x} (@code{calc-index}) address@hidden function builds a vector
-of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
-prefix argument. If you do not provide a prefix argument, you will be
-prompted to enter a suitable number. If @var{n} is negative, the result
-is a vector of negative integers from @var{n} to @mathit{-1}.
-
-With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
-three values from the stack: @var{n}, @var{start}, and @var{incr} (with
address@hidden at top-of-stack). Counting starts at @var{start} and increases
-by @var{incr} for successive vector elements. If @var{start} or @var{n}
-is in floating-point format, the resulting vector elements will also be
-floats. Note that @var{start} and @var{incr} may in fact be any kind
-of numbers or formulas.
-
-When @var{start} and @var{incr} are specified, a negative @var{n} has a
-different interpretation: It causes a geometric instead of arithmetic
-sequence to be generated. For example, @samp{index(-3, a, b)} produces
address@hidden, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
address@hidden(@var{n}, @var{start})}, the default value for @var{incr}
-is one for positive @var{n} or two for negative @var{n}.
-
address@hidden v b
address@hidden calc-build-vector
address@hidden cvec
-The @kbd{v b} (@code{calc-build-vector}) address@hidden function builds a
-vector of @var{n} copies of the value on the top of the stack, where @var{n}
-is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
-can also be used to build an @address@hidden matrix of copies of @var{x}.
-(Interactively, just use @kbd{v b} twice: once to build a row, then again
-to build a matrix of copies of that row.)
-
address@hidden v h
address@hidden I v h
address@hidden calc-head
address@hidden calc-tail
address@hidden head
address@hidden tail
-The @kbd{v h} (@code{calc-head}) address@hidden function returns the first
-element of a vector. The @kbd{I v h} (@code{calc-tail}) address@hidden
-function returns the vector with its first element removed. In both
-cases, the argument must be a non-empty vector.
-
address@hidden v k
address@hidden calc-cons
address@hidden cons
-The @kbd{v k} (@code{calc-cons}) address@hidden function takes a value @var{h}
-and a vector @var{t} from the stack, and produces the vector whose head is
address@hidden and whose tail is @var{t}. This is similar to @kbd{|}, except
-if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
-whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
-
address@hidden H v h
address@hidden rhead
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden H I v h
address@hidden
address@hidden @null
address@hidden ignore
address@hidden H v k
address@hidden
address@hidden @null
address@hidden ignore
address@hidden rtail
address@hidden
address@hidden @null
address@hidden ignore
address@hidden rcons
-Each of these three functions also accepts the Hyperbolic flag address@hidden,
address@hidden, @code{rcons}] in which case @var{t} instead represents
-the @emph{last} single element of the vector, with @var{h}
-representing the remainder of the vector. Thus the vector
address@hidden, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
-Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
address@hidden([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
-
address@hidden Extracting Elements, Manipulating Vectors, Building Vectors,
Matrix Functions
address@hidden Extracting Vector Elements
-
address@hidden
address@hidden v r
address@hidden calc-mrow
address@hidden mrow
-The @kbd{v r} (@code{calc-mrow}) address@hidden command extracts one row of
-the matrix on the top of the stack, or one element of the plain vector on
-the top of the stack. The row or element is specified by the numeric
-prefix argument; the default is to prompt for the row or element number.
-The matrix or vector is replaced by the specified row or element in the
-form of a vector or scalar, respectively.
-
address@hidden Permutations, applying
-With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
-the element or row from the top of the stack, and the vector or matrix
-from the second-to-top position. If the index is itself a vector of
-integers, the result is a vector of the corresponding elements of the
-input vector, or a matrix of the corresponding rows of the input matrix.
-This command can be used to obtain any permutation of a vector.
-
-With @kbd{C-u}, if the index is an interval form with integer components,
-it is interpreted as a range of indices and the corresponding subvector or
-submatrix is returned.
-
address@hidden Subscript notation
address@hidden a _
address@hidden calc-subscript
address@hidden subscr
address@hidden _
-Subscript notation in algebraic formulas (@samp{a_b}) stands for the
-Calc function @code{subscr}, which is synonymous with @code{mrow}.
-Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
address@hidden is one, two, or three, respectively. A double subscript
-(@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
-access the element at row @expr{i}, column @expr{j} of a matrix.
-The @kbd{a _} (@code{calc-subscript}) command creates a subscript
-formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
-``algebra'' prefix because subscripted variables are often used
-purely as an algebraic notation.)
-
address@hidden mrrow
-Given a negative prefix argument, @kbd{v r} instead deletes one row or
-element from the matrix or vector on the top of the stack. Thus
address@hidden 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v
r}
-replaces the matrix with the same matrix with its second row removed.
-In algebraic form this function is called @code{mrrow}.
-
address@hidden getdiag
-Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
-of a square matrix in the form of a vector. In algebraic form this
-function is called @code{getdiag}.
-
address@hidden v c
address@hidden calc-mcol
address@hidden mcol
address@hidden mrcol
-The @kbd{v c} (@code{calc-mcol}) address@hidden or @code{mrcol}] command is
-the analogous operation on columns of a matrix. Given a plain vector
-it extracts (or removes) one element, just like @kbd{v r}. If the
-index in @kbd{C-u v c} is an interval or vector and the argument is a
-matrix, the result is a submatrix with only the specified columns
-retained (and possibly permuted in the case of a vector index).
-
-To extract a matrix element at a given row and column, use @kbd{v r} to
-extract the row as a vector, then @kbd{v c} to extract the column element
-from that vector. In algebraic formulas, it is often more convenient to
-use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
-of matrix @expr{m}.
-
address@hidden v s
address@hidden calc-subvector
address@hidden subvec
-The @kbd{v s} (@code{calc-subvector}) address@hidden command extracts
-a subvector of a vector. The arguments are the vector, the starting
-index, and the ending index, with the ending index in the top-of-stack
-position. The starting index indicates the first element of the vector
-to take. The ending index indicates the first element @emph{past} the
-range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
-the subvector @samp{[b, c]}. You could get the same result using
address@hidden([a, b, c, d, e], @w{[2 .. 4)})}.
-
-If either the start or the end index is zero or negative, it is
-interpreted as relative to the end of the vector. Thus
address@hidden([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
-the algebraic form, the end index can be omitted in which case it
-is taken as zero, i.e., elements from the starting element to the
-end of the vector are used. The infinity symbol, @code{inf}, also
-has this effect when used as the ending index.
-
address@hidden I v s
address@hidden rsubvec
-With the Inverse flag, @kbd{I v s} address@hidden removes a subvector
-from a vector. The arguments are interpreted the same as for the
-normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
-produces @samp{[a, d, e]}. It is always true that @code{subvec} and
address@hidden return complementary parts of the input vector.
-
address@hidden Subformulas}, for an alternative way to operate on
-vectors one element at a time.
-
address@hidden Manipulating Vectors, Vector and Matrix Arithmetic, Extracting
Elements, Matrix Functions
address@hidden Manipulating Vectors
-
address@hidden
address@hidden v l
address@hidden calc-vlength
address@hidden vlen
-The @kbd{v l} (@code{calc-vlength}) address@hidden command computes the
-length of a vector. The length of a non-vector is considered to be zero.
-Note that matrices are just vectors of vectors for the purposes of this
-command.
-
address@hidden H v l
address@hidden mdims
-With the Hyperbolic flag, @kbd{H v l} address@hidden computes a vector
-of the dimensions of a vector, matrix, or higher-order object. For
-example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
-its argument is a
address@hidden @math{2\times3}
address@hidden 2x3
-matrix.
-
address@hidden v f
address@hidden calc-vector-find
address@hidden find
-The @kbd{v f} (@code{calc-vector-find}) address@hidden command searches
-along a vector for the first element equal to a given target. The target
-is on the top of the stack; the vector is in the second-to-top position.
-If a match is found, the result is the index of the matching element.
-Otherwise, the result is zero. The numeric prefix argument, if given,
-allows you to select any starting index for the search.
-
address@hidden v a
address@hidden calc-arrange-vector
address@hidden arrange
address@hidden Arranging a matrix
address@hidden Reshaping a matrix
address@hidden Flattening a matrix
-The @kbd{v a} (@code{calc-arrange-vector}) address@hidden command
-rearranges a vector to have a certain number of columns and rows. The
-numeric prefix argument specifies the number of columns; if you do not
-provide an argument, you will be prompted for the number of columns.
-The vector or matrix on the top of the stack is @dfn{flattened} into a
-plain vector. If the number of columns is nonzero, this vector is
-then formed into a matrix by taking successive groups of @var{n} elements.
-If the number of columns does not evenly divide the number of elements
-in the vector, the last row will be short and the result will not be
-suitable for use as a matrix. For example, with the matrix
address@hidden, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
address@hidden, 2, 3, 4]]} (a
address@hidden @math{1\times4}
address@hidden 1x4
-matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
address@hidden @math{4\times1}
address@hidden 4x1
-matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
address@hidden @math{2\times2}
address@hidden 2x2
-matrix), @address@hidden a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
-matrix), and @kbd{v a 0} produces the flattened list
address@hidden, 2, @w{3, 4}]}.
-
address@hidden Sorting data
address@hidden V S
address@hidden I V S
address@hidden calc-sort
address@hidden sort
address@hidden rsort
-The @kbd{V S} (@code{calc-sort}) address@hidden command sorts the elements of
-a vector into increasing order. Real numbers, real infinities, and
-constant interval forms come first in this ordering; next come other
-kinds of numbers, then variables (in alphabetical order), then finally
-come formulas and other kinds of objects; these are sorted according
-to a kind of lexicographic ordering with the useful property that
-one vector is less or greater than another if the first corresponding
-unequal elements are less or greater, respectively. Since quoted strings
-are stored by Calc internally as vectors of ASCII character codes
-(@pxref{Strings}), this means vectors of strings are also sorted into
-alphabetical order by this command.
-
-The @kbd{I V S} address@hidden command sorts a vector into decreasing order.
-
address@hidden Permutation, inverse of
address@hidden Inverse of permutation
address@hidden Index tables
address@hidden Rank tables
address@hidden V G
address@hidden I V G
address@hidden calc-grade
address@hidden grade
address@hidden rgrade
-The @kbd{V G} (@code{calc-grade}) address@hidden, @code{rgrade}] command
-produces an index table or permutation vector which, if applied to the
-input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
-A permutation vector is just a vector of integers from 1 to @var{n}, where
-each integer occurs exactly once. One application of this is to sort a
-matrix of data rows using one column as the sort key; extract that column,
-grade it with @kbd{V G}, then use the result to reorder the original matrix
-with @kbd{C-u v r}. Another interesting property of the @code{V G} command
-is that, if the input is itself a permutation vector, the result will
-be the inverse of the permutation. The inverse of an index table is
-a rank table, whose @var{k}th element says where the @var{k}th original
-vector element will rest when the vector is sorted. To get a rank
-table, just use @kbd{V G V G}.
-
-With the Inverse flag, @kbd{I V G} produces an index table that would
-sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
-use a ``stable'' sorting algorithm, i.e., any two elements which are equal
-will not be moved out of their original order. Generally there is no way
-to tell with @kbd{V S}, since two elements which are equal look the same,
-but with @kbd{V G} this can be an important issue. In the matrix-of-rows
-example, suppose you have names and telephone numbers as two columns and
-you wish to sort by phone number primarily, and by name when the numbers
-are equal. You can sort the data matrix by names first, and then again
-by phone numbers. Because the sort is stable, any two rows with equal
-phone numbers will remain sorted by name even after the second sort.
-
address@hidden Histograms
address@hidden V H
address@hidden calc-histogram
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden histogram
-The @kbd{V H} (@code{calc-histogram}) address@hidden command builds a
-histogram of a vector of numbers. Vector elements are assumed to be
-integers or real numbers in the range address@hidden) for some ``number of
-bins'' @var{n}, which is the numeric prefix argument given to the
-command. The result is a vector of @var{n} counts of how many times
-each value appeared in the original vector. Non-integers in the input
-are rounded down to integers. Any vector elements outside the specified
-range are ignored. (You can tell if elements have been ignored by noting
-that the counts in the result vector don't add up to the length of the
-input vector.)
-
address@hidden H V H
-With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
-The second-to-top vector is the list of numbers as before. The top
-vector is an equal-sized list of ``weights'' to attach to the elements
-of the data vector. For example, if the first data element is 4.2 and
-the first weight is 10, then 10 will be added to bin 4 of the result
-vector. Without the hyperbolic flag, every element has a weight of one.
-
address@hidden v t
address@hidden calc-transpose
address@hidden trn
-The @kbd{v t} (@code{calc-transpose}) address@hidden command computes
-the transpose of the matrix at the top of the stack. If the argument
-is a plain vector, it is treated as a row vector and transposed into
-a one-column matrix.
-
address@hidden v v
address@hidden calc-reverse-vector
address@hidden rev
-The @kbd{v v} (@code{calc-reverse-vector}) address@hidden command reverses
-a vector end-for-end. Given a matrix, it reverses the order of the rows.
-(To reverse the columns instead, just use @kbd{v t v v v t}. The same
-principle can be used to apply other vector commands to the columns of
-a matrix.)
-
address@hidden v m
address@hidden calc-mask-vector
address@hidden vmask
-The @kbd{v m} (@code{calc-mask-vector}) address@hidden command uses
-one vector as a mask to extract elements of another vector. The mask
-is in the second-to-top position; the target vector is on the top of
-the stack. These vectors must have the same length. The result is
-the same as the target vector, but with all elements which correspond
-to zeros in the mask vector deleted. Thus, for example,
address@hidden([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
address@hidden Operations}.
-
address@hidden v e
address@hidden calc-expand-vector
address@hidden vexp
-The @kbd{v e} (@code{calc-expand-vector}) address@hidden command
-expands a vector according to another mask vector. The result is a
-vector the same length as the mask, but with nonzero elements replaced
-by successive elements from the target vector. The length of the target
-vector is normally the number of nonzero elements in the mask. If the
-target vector is longer, its last few elements are lost. If the target
-vector is shorter, the last few nonzero mask elements are left
-unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
-produces @samp{[a, 0, b, 0, 7]}.
-
address@hidden H v e
-With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
-top of the stack; the mask and target vectors come from the third and
-second elements of the stack. This filler is used where the mask is
-zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
address@hidden, z, c, z, 7]}. If the filler value is itself a vector,
-then successive values are taken from it, so that the effect is to
-interleave two vectors according to the mask:
address@hidden([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
address@hidden, x, b, 7, y, 0]}.
-
-Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
-with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
-You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
-operation across the two vectors. @xref{Logical Operations}. Note that
-the @code{? :} operation also discussed there allows other types of
-masking using vectors.
-
address@hidden Vector and Matrix Arithmetic, Set Operations, Manipulating
Vectors, Matrix Functions
address@hidden Vector and Matrix Arithmetic
-
address@hidden
-Basic arithmetic operations like addition and multiplication are defined
-for vectors and matrices as well as for numbers. Division of matrices, in
-the sense of multiplying by the inverse, is supported. (Division by a
-matrix actually uses LU-decomposition for greater accuracy and speed.)
address@hidden Arithmetic}.
-
-The following functions are applied element-wise if their arguments are
-vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
address@hidden, @code{im}, @code{polar}, @code{rect}, @code{clean},
address@hidden, @code{frac}. @xref{Function Index}.
-
address@hidden V J
address@hidden calc-conj-transpose
address@hidden ctrn
-The @kbd{V J} (@code{calc-conj-transpose}) address@hidden command computes
-the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
-
address@hidden
address@hidden A
address@hidden ignore
address@hidden A (vectors)
address@hidden calc-abs (vectors)
address@hidden
address@hidden abs
address@hidden ignore
address@hidden abs (vectors)
-The @kbd{A} (@code{calc-abs}) address@hidden command computes the
-Frobenius norm of a vector or matrix argument. This is the square
-root of the sum of the squares of the absolute values of the
-elements of the vector or matrix. If the vector is interpreted as
-a point in two- or three-dimensional space, this is the distance
-from that point to the origin.
-
address@hidden v n
address@hidden calc-rnorm
address@hidden rnorm
-The @kbd{v n} (@code{calc-rnorm}) address@hidden command computes
-the row norm, or infinity-norm, of a vector or matrix. For a plain
-vector, this is the maximum of the absolute values of the elements.
-For a matrix, this is the maximum of the row-absolute-value-sums,
-i.e., of the sums of the absolute values of the elements along the
-various rows.
-
address@hidden V N
address@hidden calc-cnorm
address@hidden cnorm
-The @kbd{V N} (@code{calc-cnorm}) address@hidden command computes
-the column norm, or one-norm, of a vector or matrix. For a plain
-vector, this is the sum of the absolute values of the elements.
-For a matrix, this is the maximum of the column-absolute-value-sums.
-General @expr{k}-norms for @expr{k} other than one or infinity are
-not provided.
-
address@hidden V C
address@hidden calc-cross
address@hidden cross
-The @kbd{V C} (@code{calc-cross}) address@hidden command computes the
-right-handed cross product of two vectors, each of which must have
-exactly three elements.
-
address@hidden
address@hidden &
address@hidden ignore
address@hidden & (matrices)
address@hidden calc-inv (matrices)
address@hidden
address@hidden inv
address@hidden ignore
address@hidden inv (matrices)
-The @kbd{&} (@code{calc-inv}) address@hidden command computes the
-inverse of a square matrix. If the matrix is singular, the inverse
-operation is left in symbolic form. Matrix inverses are recorded so
-that once an inverse (or determinant) of a particular matrix has been
-computed, the inverse and determinant of the matrix can be recomputed
-quickly in the future.
-
-If the argument to @kbd{&} is a plain number @expr{x}, this
-command simply computes @expr{1/x}. This is okay, because the
address@hidden/} operator also does a matrix inversion when dividing one
-by a matrix.
-
address@hidden V D
address@hidden calc-mdet
address@hidden det
-The @kbd{V D} (@code{calc-mdet}) address@hidden command computes the
-determinant of a square matrix.
-
address@hidden V L
address@hidden calc-mlud
address@hidden lud
-The @kbd{V L} (@code{calc-mlud}) address@hidden command computes the
-LU decomposition of a matrix. The result is a list of three matrices
-which, when multiplied together left-to-right, form the original matrix.
-The first is a permutation matrix that arises from pivoting in the
-algorithm, the second is lower-triangular with ones on the diagonal,
-and the third is upper-triangular.
-
address@hidden V T
address@hidden calc-mtrace
address@hidden tr
-The @kbd{V T} (@code{calc-mtrace}) address@hidden command computes the
-trace of a square matrix. This is defined as the sum of the diagonal
-elements of the matrix.
-
address@hidden Set Operations, Statistical Operations, Vector and Matrix
Arithmetic, Matrix Functions
address@hidden Set Operations using Vectors
-
address@hidden
address@hidden Sets, as vectors
-Calc includes several commands which interpret vectors as @dfn{sets} of
-objects. A set is a collection of objects; any given object can appear
-only once in the set. Calc stores sets as vectors of objects in
-sorted order. Objects in a Calc set can be any of the usual things,
-such as numbers, variables, or formulas. Two set elements are considered
-equal if they are identical, except that numerically equal numbers like
-the integer 4 and the float 4.0 are considered equal even though they
-are not ``identical.'' Variables are treated like plain symbols without
-attached values by the set operations; subtracting the set @samp{[b]}
-from @samp{[a, b]} always yields the set @samp{[a]} even though if
-the variables @samp{a} and @samp{b} both equaled 17, you might
-expect the answer @samp{[]}.
-
-If a set contains interval forms, then it is assumed to be a set of
-real numbers. In this case, all set operations require the elements
-of the set to be only things that are allowed in intervals: Real
-numbers, plus and minus infinity, HMS forms, and date forms. If
-there are variables or other non-real objects present in a real set,
-all set operations on it will be left in unevaluated form.
-
-If the input to a set operation is a plain number or interval form
address@hidden, it is treated like the one-element vector @address@hidden
-The result is always a vector, except that if the set consists of a
-single interval, the interval itself is returned instead.
-
address@hidden Operations}, for the @code{in} function which tests if
-a certain value is a member of a given set. To test if the set @expr{A}
-is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
-
address@hidden V +
address@hidden calc-remove-duplicates
address@hidden rdup
-The @kbd{V +} (@code{calc-remove-duplicates}) address@hidden command
-converts an arbitrary vector into set notation. It works by sorting
-the vector as if by @kbd{V S}, then removing duplicates. (For example,
address@hidden, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
-reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
-necessary. You rarely need to use @kbd{V +} explicitly, since all the
-other set-based commands apply @kbd{V +} to their inputs before using
-them.
-
address@hidden V V
address@hidden calc-set-union
address@hidden vunion
-The @kbd{V V} (@code{calc-set-union}) address@hidden command computes
-the union of two sets. An object is in the union of two sets if and
-only if it is in either (or both) of the input sets. (You could
-accomplish the same thing by concatenating the sets with @kbd{|},
-then using @kbd{V +}.)
-
address@hidden V ^
address@hidden calc-set-intersect
address@hidden vint
-The @kbd{V ^} (@code{calc-set-intersect}) address@hidden command computes
-the intersection of two sets. An object is in the intersection if
-and only if it is in both of the input sets. Thus if the input
-sets are disjoint, i.e., if they share no common elements, the result
-will be the empty vector @samp{[]}. Note that the characters @kbd{V}
-and @kbd{^} were chosen to be close to the conventional mathematical
-notation for set
address@hidden address@hidden(@math{A \cup B})
address@hidden union
-and
address@hidden address@hidden(@math{A \cap B}).
address@hidden intersection.
-
address@hidden V -
address@hidden calc-set-difference
address@hidden vdiff
-The @kbd{V -} (@code{calc-set-difference}) address@hidden command computes
-the difference between two sets. An object is in the difference
address@hidden - B} if and only if it is in @expr{A} but not in @expr{B}.
-Thus subtracting @samp{[y,z]} from a set will remove the elements
address@hidden and @samp{z} if they are present. You can also think of this
-as a general @dfn{set complement} operator; if @expr{A} is the set of
-all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
-Obviously this is only practical if the set of all possible values in
-your problem is small enough to list in a Calc vector (or simple
-enough to express in a few intervals).
-
address@hidden V X
address@hidden calc-set-xor
address@hidden vxor
-The @kbd{V X} (@code{calc-set-xor}) address@hidden command computes
-the ``exclusive-or,'' or ``symmetric difference'' of two sets.
-An object is in the symmetric difference of two sets if and only
-if it is in one, but @emph{not} both, of the sets. Objects that
-occur in both sets ``cancel out.''
-
address@hidden V ~
address@hidden calc-set-complement
address@hidden vcompl
-The @kbd{V ~} (@code{calc-set-complement}) address@hidden command
-computes the complement of a set with respect to the real numbers.
-Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
-For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
address@hidden .. 2), (2 .. 3], (4 .. inf]]}.
-
address@hidden V F
address@hidden calc-set-floor
address@hidden vfloor
-The @kbd{V F} (@code{calc-set-floor}) address@hidden command
-reinterprets a set as a set of integers. Any non-integer values,
-and intervals that do not enclose any integers, are removed. Open
-intervals are converted to equivalent closed intervals. Successive
-integers are converted into intervals of integers. For example, the
-complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
-the complement with respect to the set of integers you could type
address@hidden ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
-
address@hidden V E
address@hidden calc-set-enumerate
address@hidden venum
-The @kbd{V E} (@code{calc-set-enumerate}) address@hidden command
-converts a set of integers into an explicit vector. Intervals in
-the set are expanded out to lists of all integers encompassed by
-the intervals. This only works for finite sets (i.e., sets which
-do not involve @samp{-inf} or @samp{inf}).
-
address@hidden V :
address@hidden calc-set-span
address@hidden vspan
-The @kbd{V :} (@code{calc-set-span}) address@hidden command converts any
-set of reals into an interval form that encompasses all its elements.
-The lower limit will be the smallest element in the set; the upper
-limit will be the largest element. For an empty set, @samp{vspan([])}
-returns the empty interval @address@hidden .. 0)}}.
-
address@hidden V #
address@hidden calc-set-cardinality
address@hidden vcard
-The @kbd{V #} (@code{calc-set-cardinality}) address@hidden command counts
-the number of integers in a set. The result is the length of the vector
-that would be produced by @kbd{V E}, although the computation is much
-more efficient than actually producing that vector.
-
address@hidden Sets, as binary numbers
-Another representation for sets that may be more appropriate in some
-cases is binary numbers. If you are dealing with sets of integers
-in the range 0 to 49, you can use a 50-bit binary number where a
-particular bit is 1 if the corresponding element is in the set.
address@hidden Functions}, for a list of commands that operate on
-binary numbers. Note that many of the above set operations have
-direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
address@hidden a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
address@hidden x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
-respectively. You can use whatever representation for sets is most
-convenient to you.
-
address@hidden b p
address@hidden b u
address@hidden calc-pack-bits
address@hidden calc-unpack-bits
address@hidden vpack
address@hidden vunpack
-The @kbd{b u} (@code{calc-unpack-bits}) address@hidden command
-converts an integer that represents a set in binary into a set
-in vector/interval notation. For example, @samp{vunpack(67)}
-returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
-it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
-Use @kbd{V E} afterwards to expand intervals to individual
-values if you wish. Note that this command uses the @kbd{b}
-(binary) prefix key.
-
-The @kbd{b p} (@code{calc-pack-bits}) address@hidden command
-converts the other way, from a vector or interval representing
-a set of nonnegative integers into a binary integer describing
-the same set. The set may include positive infinity, but must
-not include any negative numbers. The input is interpreted as a
-set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
-that a simple input like @samp{[100]} can result in a huge integer
-representation
address@hidden (@math{2^{100}}, a 31-digit integer, in this case).
address@hidden (@expr{2^100}, a 31-digit integer, in this case).
-
address@hidden Statistical Operations, Reducing and Mapping, Set Operations,
Matrix Functions
address@hidden Statistical Operations on Vectors
-
address@hidden
address@hidden Statistical functions
-The commands in this section take vectors as arguments and compute
-various statistical measures on the data stored in the vectors. The
-references used in the definitions of these functions are Bevington's
address@hidden Reduction and Error Analysis for the Physical Sciences},
-and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
-Vetterling.
-
-The statistical commands use the @kbd{u} prefix key followed by
-a shifted letter or other character.
-
address@hidden Vectors}, for a description of @kbd{V H}
-(@code{calc-histogram}).
-
address@hidden Fitting}, for the @kbd{a F} command for doing
-least-squares fits to statistical data.
-
address@hidden Distribution Functions}, for several common
-probability distribution functions.
-
address@hidden
-* Single-Variable Statistics::
-* Paired-Sample Statistics::
address@hidden menu
-
address@hidden Single-Variable Statistics, Paired-Sample Statistics,
Statistical Operations, Statistical Operations
address@hidden Single-Variable Statistics
-
address@hidden
-These functions do various statistical computations on single
-vectors. Given a numeric prefix argument, they actually pop
address@hidden objects from the stack and combine them into a data
-vector. Each object may be either a number or a vector; if a
-vector, any sub-vectors inside it are ``flattened'' as if by
address@hidden a 0}; @pxref{Manipulating Vectors}. By default one object
-is popped, which (in order to be useful) is usually a vector.
-
-If an argument is a variable name, and the value stored in that
-variable is a vector, then the stored vector is used. This method
-has the advantage that if your data vector is large, you can avoid
-the slow process of manipulating it directly on the stack.
-
-These functions are left in symbolic form if any of their arguments
-are not numbers or vectors, e.g., if an argument is a formula, or
-a non-vector variable. However, formulas embedded within vector
-arguments are accepted; the result is a symbolic representation
-of the computation, based on the assumption that the formula does
-not itself represent a vector. All varieties of numbers such as
-error forms and interval forms are acceptable.
-
-Some of the functions in this section also accept a single error form
-or interval as an argument. They then describe a property of the
-normal or uniform (respectively) statistical distribution described
-by the argument. The arguments are interpreted in the same way as
-the @var{M} argument of the random number function @kbd{k r}. In
-particular, an interval with integer limits is considered an integer
-distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
-An interval with at least one floating-point limit is a continuous
-distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
address@hidden .. 5.0]}!
-
address@hidden u #
address@hidden calc-vector-count
address@hidden vcount
-The @kbd{u #} (@code{calc-vector-count}) address@hidden command
-computes the number of data values represented by the inputs.
-For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
-If the argument is a single vector with no sub-vectors, this
-simply computes the length of the vector.
-
address@hidden u +
address@hidden u *
address@hidden calc-vector-sum
address@hidden calc-vector-prod
address@hidden vsum
address@hidden vprod
address@hidden Summations (statistical)
-The @kbd{u +} (@code{calc-vector-sum}) address@hidden command
-computes the sum of the data values. The @kbd{u *}
-(@code{calc-vector-prod}) address@hidden command computes the
-product of the data values. If the input is a single flat vector,
-these are the same as @kbd{V R +} and @kbd{V R *}
-(@pxref{Reducing and Mapping}).
-
address@hidden u X
address@hidden u N
address@hidden calc-vector-max
address@hidden calc-vector-min
address@hidden vmax
address@hidden vmin
-The @kbd{u X} (@code{calc-vector-max}) address@hidden command
-computes the maximum of the data values, and the @kbd{u N}
-(@code{calc-vector-min}) address@hidden command computes the minimum.
-If the argument is an interval, this finds the minimum or maximum
-value in the interval. (Note that @samp{vmax([2..6)) = 5} as
-described above.) If the argument is an error form, this returns
-plus or minus infinity.
-
address@hidden u M
address@hidden calc-vector-mean
address@hidden vmean
address@hidden Mean of data values
-The @kbd{u M} (@code{calc-vector-mean}) address@hidden command
-computes the average (arithmetic mean) of the data values.
-If the inputs are error forms
address@hidden @math{x \pm \sigma},
address@hidden @samp{x +/- s},
-this is the weighted mean of the @expr{x} values with weights
address@hidden @math{1 /\sigma^2}.
address@hidden @expr{1 / s^2}.
address@hidden
-\turnoffactive
-$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
- \displaystyle \sum { 1 \over \sigma_i^2 } } $$
address@hidden tex
-If the inputs are not error forms, this is simply the sum of the
-values divided by the count of the values.
-
-Note that a plain number can be considered an error form with
-error
address@hidden @math{\sigma = 0}.
address@hidden @expr{s = 0}.
-If the input to @kbd{u M} is a mixture of
-plain numbers and error forms, the result is the mean of the
-plain numbers, ignoring all values with non-zero errors. (By the
-above definitions it's clear that a plain number effectively
-has an infinite weight, next to which an error form with a finite
-weight is completely negligible.)
-
-This function also works for distributions (error forms or
-intervals). The mean of an error form address@hidden @tfn{+/-} @var{b}' is
simply
address@hidden The mean of an interval is the mean of the minimum
-and maximum values of the interval.
-
address@hidden I u M
address@hidden calc-vector-mean-error
address@hidden vmeane
-The @kbd{I u M} (@code{calc-vector-mean-error}) address@hidden
-command computes the mean of the data points expressed as an
-error form. This includes the estimated error associated with
-the mean. If the inputs are error forms, the error is the square
-root of the reciprocal of the sum of the reciprocals of the squares
-of the input errors. (I.e., the variance is the reciprocal of the
-sum of the reciprocals of the variances.)
address@hidden
-\turnoffactive
-$$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
address@hidden tex
-If the inputs are plain
-numbers, the error is equal to the standard deviation of the values
-divided by the square root of the number of values. (This works
-out to be equivalent to calculating the standard deviation and
-then assuming each value's error is equal to this standard
-deviation.)
address@hidden
-\turnoffactive
-$$ \sigma_\mu^2 = {\sigma^2 \over N} $$
address@hidden tex
-
address@hidden H u M
address@hidden calc-vector-median
address@hidden vmedian
address@hidden Median of data values
-The @kbd{H u M} (@code{calc-vector-median}) address@hidden
-command computes the median of the data values. The values are
-first sorted into numerical order; the median is the middle
-value after sorting. (If the number of data values is even,
-the median is taken to be the average of the two middle values.)
-The median function is different from the other functions in
-this section in that the arguments must all be real numbers;
-variables are not accepted even when nested inside vectors.
-(Otherwise it is not possible to sort the data values.) If
-any of the input values are error forms, their error parts are
-ignored.
-
-The median function also accepts distributions. For both normal
-(error form) and uniform (interval) distributions, the median is
-the same as the mean.
-
address@hidden H I u M
address@hidden calc-vector-harmonic-mean
address@hidden vhmean
address@hidden Harmonic mean
-The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) address@hidden
-command computes the harmonic mean of the data values. This is
-defined as the reciprocal of the arithmetic mean of the reciprocals
-of the values.
address@hidden
-\turnoffactive
-$$ { N \over \displaystyle \sum {1 \over x_i} } $$
address@hidden tex
-
address@hidden u G
address@hidden calc-vector-geometric-mean
address@hidden vgmean
address@hidden Geometric mean
-The @kbd{u G} (@code{calc-vector-geometric-mean}) address@hidden
-command computes the geometric mean of the data values. This
-is the @var{n}th root of the product of the values. This is also
-equal to the @code{exp} of the arithmetic mean of the logarithms
-of the data values.
address@hidden
-\turnoffactive
-$$ \exp \left ( \sum { \ln x_i } \right ) =
- \left ( \prod { x_i } \right)^{1 / N} $$
address@hidden tex
-
address@hidden H u G
address@hidden agmean
-The @kbd{H u G} address@hidden command computes the ``arithmetic-geometric
-mean'' of two numbers taken from the stack. This is computed by
-replacing the two numbers with their arithmetic mean and geometric
-mean, then repeating until the two values converge.
address@hidden
-\turnoffactive
-$$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
address@hidden tex
-
address@hidden Root-mean-square
-Another commonly used mean, the RMS (root-mean-square), can be computed
-for a vector of numbers simply by using the @kbd{A} command.
-
address@hidden u S
address@hidden calc-vector-sdev
address@hidden vsdev
address@hidden Standard deviation
address@hidden Sample statistics
-The @kbd{u S} (@code{calc-vector-sdev}) address@hidden command
-computes the standard
address@hidden address@hidden@math{\sigma}
address@hidden deviation
-of the data values. If the values are error forms, the errors are used
-as weights just as for @kbd{u M}. This is the @emph{sample} standard
-deviation, whose value is the square root of the sum of the squares of
-the differences between the values and the mean of the @expr{N} values,
-divided by @expr{N-1}.
address@hidden
-\turnoffactive
-$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
address@hidden tex
-
-This function also applies to distributions. The standard deviation
-of a single error form is simply the error part. The standard deviation
-of a continuous interval happens to equal the difference between the
-limits, divided by
address@hidden @math{\sqrt{12}}.
address@hidden @expr{sqrt(12)}.
-The standard deviation of an integer interval is the same as the
-standard deviation of a vector of those integers.
-
address@hidden I u S
address@hidden calc-vector-pop-sdev
address@hidden vpsdev
address@hidden Population statistics
-The @kbd{I u S} (@code{calc-vector-pop-sdev}) address@hidden
-command computes the @emph{population} standard deviation.
-It is defined by the same formula as above but dividing
-by @expr{N} instead of by @expr{N-1}. The population standard
-deviation is used when the input represents the entire set of
-data values in the distribution; the sample standard deviation
-is used when the input represents a sample of the set of all
-data values, so that the mean computed from the input is itself
-only an estimate of the true mean.
address@hidden
-\turnoffactive
-$$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
address@hidden tex
-
-For error forms and continuous intervals, @code{vpsdev} works
-exactly like @code{vsdev}. For integer intervals, it computes the
-population standard deviation of the equivalent vector of integers.
-
address@hidden H u S
address@hidden H I u S
address@hidden calc-vector-variance
address@hidden calc-vector-pop-variance
address@hidden vvar
address@hidden vpvar
address@hidden Variance of data values
-The @kbd{H u S} (@code{calc-vector-variance}) address@hidden and
address@hidden I u S} (@code{calc-vector-pop-variance}) address@hidden
-commands compute the variance of the data values. The variance
-is the
address@hidden address@hidden@math{\sigma^2}
address@hidden square
-of the standard deviation, i.e., the sum of the
-squares of the deviations of the data values from the mean.
-(This definition also applies when the argument is a distribution.)
-
address@hidden
address@hidden
address@hidden ignore
address@hidden vflat
-The @code{vflat} algebraic function returns a vector of its
-arguments, interpreted in the same way as the other functions
-in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
-returns @samp{[1, 2, 3, 4, 5]}.
-
address@hidden Paired-Sample Statistics, , Single-Variable Statistics,
Statistical Operations
address@hidden Paired-Sample Statistics
-
address@hidden
-The functions in this section take two arguments, which must be
-vectors of equal size. The vectors are each flattened in the same
-way as by the single-variable statistical functions. Given a numeric
-prefix argument of 1, these functions instead take one object from
-the stack, which must be an
address@hidden @math{N\times2}
address@hidden Nx2
-matrix of data values. Once again, variable names can be used in place
-of actual vectors and matrices.
-
address@hidden u C
address@hidden calc-vector-covariance
address@hidden vcov
address@hidden Covariance
-The @kbd{u C} (@code{calc-vector-covariance}) address@hidden command
-computes the sample covariance of two vectors. The covariance
-of vectors @var{x} and @var{y} is the sum of the products of the
-differences between the elements of @var{x} and the mean of @var{x}
-times the differences between the corresponding elements of @var{y}
-and the mean of @var{y}, all divided by @expr{N-1}. Note that
-the variance of a vector is just the covariance of the vector
-with itself. Once again, if the inputs are error forms the
-errors are used as weight factors. If both @var{x} and @var{y}
-are composed of error forms, the error for a given data point
-is taken as the square root of the sum of the squares of the two
-input errors.
address@hidden
-\turnoffactive
-$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
-$$ \sigma_{x\!y}^2 =
- {\displaystyle {1 \over N-1}
- \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
- \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
-$$
address@hidden tex
-
address@hidden I u C
address@hidden calc-vector-pop-covariance
address@hidden vpcov
-The @kbd{I u C} (@code{calc-vector-pop-covariance}) address@hidden
-command computes the population covariance, which is the same as the
-sample covariance computed by @kbd{u C} except dividing by @expr{N}
-instead of @expr{N-1}.
-
address@hidden H u C
address@hidden calc-vector-correlation
address@hidden vcorr
address@hidden Correlation coefficient
address@hidden Linear correlation
-The @kbd{H u C} (@code{calc-vector-correlation}) address@hidden
-command computes the linear correlation coefficient of two vectors.
-This is defined by the covariance of the vectors divided by the
-product of their standard deviations. (There is no difference
-between sample or population statistics here.)
address@hidden
-\turnoffactive
-$$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
address@hidden tex
-
address@hidden Reducing and Mapping, Vector and Matrix Formats, Statistical
Operations, Matrix Functions
address@hidden Reducing and Mapping Vectors
-
address@hidden
-The commands in this section allow for more general operations on the
-elements of vectors.
-
address@hidden V A
address@hidden calc-apply
address@hidden apply
-The simplest of these operations is @kbd{V A} (@code{calc-apply})
address@hidden, which applies a given operator to the elements of a vector.
-For example, applying the hypothetical function @code{f} to the vector
address@hidden@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2,
3)}.
-Applying the @code{+} function to the vector @samp{[a, b]} gives
address@hidden + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
-error, since the @code{+} function expects exactly two arguments.
-
-While @kbd{V A} is useful in some cases, you will usually find that either
address@hidden R} or @kbd{V M}, described below, is closer to what you want.
-
address@hidden
-* Specifying Operators::
-* Mapping::
-* Reducing::
-* Nesting and Fixed Points::
-* Generalized Products::
address@hidden menu
-
address@hidden Specifying Operators, Mapping, Reducing and Mapping, Reducing
and Mapping
address@hidden Specifying Operators
-
address@hidden
-Commands in this section (like @kbd{V A}) prompt you to press the key
-corresponding to the desired operator. Press @kbd{?} for a partial
-list of the available operators. Generally, an operator is any key or
-sequence of keys that would normally take one or more arguments from
-the stack and replace them with a result. For example, @kbd{V A H C}
-uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
-expects one argument, @kbd{V A H C} requires a vector with a single
-element as its argument.)
-
-You can press @kbd{x} at the operator prompt to select any algebraic
-function by name to use as the operator. This includes functions you
-have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
-Definitions}.) If you give a name for which no function has been
-defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
-Calc will prompt for the number of arguments the function takes if it
-can't figure it out on its own (say, because you named a function that
-is currently undefined). It is also possible to type a digit key before
-the function name to specify the number of arguments, e.g.,
address@hidden M 3 x f @key{RET}} calls @code{f} with three arguments even if it
-looks like it ought to have only two. This technique may be necessary
-if the function allows a variable number of arguments. For example,
-the @kbd{v e} address@hidden function accepts two or three arguments;
-if you want to map with the three-argument version, you will have to
-type @kbd{V M 3 v e}.
-
-It is also possible to apply any formula to a vector by treating that
-formula as a function. When prompted for the operator to use, press
address@hidden'} (the apostrophe) and type your formula as an algebraic entry.
-You will then be prompted for the argument list, which defaults to a
-list of all variables that appear in the formula, sorted into alphabetic
-order. For example, suppose you enter the formula @address@hidden + 2y^x}}.
-The default argument list would be @samp{(x y)}, which means that if
-this function is applied to the arguments @samp{[3, 10]} the result will
-be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
-way often, you might consider defining it as a function with @kbd{Z F}.)
-
-Another way to specify the arguments to the formula you enter is with
address@hidden, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
-has the same effect as the previous example. The argument list is
-automatically taken to be @samp{($$ $)}. (The order of the arguments
-may seem backwards, but it is analogous to the way normal algebraic
-entry interacts with the stack.)
-
-If you press @kbd{$} at the operator prompt, the effect is similar to
-the apostrophe except that the relevant formula is taken from top-of-stack
-instead. The actual vector arguments of the @kbd{V A $} or related command
-then start at the second-to-top stack position. You will still be
-prompted for an argument list.
-
address@hidden Nameless functions
address@hidden Generic functions
-A function can be written without a name using the notation @samp{<#1 - #2>},
-which means ``a function of two arguments that computes the first
-argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
-are placeholders for the arguments. You can use any names for these
-placeholders if you wish, by including an argument list followed by a
-colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
-Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
-to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET}
@key{RET}},
-Calc builds the nameless function @address@hidden<x, y : x + 2 y^x>}}. In both
-cases, Calc also writes the nameless function to the Trail so that you
-can get it back later if you wish.
-
-If there is only one argument, you can write @samp{#} in place of @samp{#1}.
-(Note that @samp{< >} notation is also used for date forms. Calc tells
-that @samp{<@var{stuff}>} is a nameless function by the presence of
address@hidden signs inside @var{stuff}, or by the fact that @var{stuff}
-begins with a list of variables followed by a colon.)
-
-You can type a nameless function directly to @kbd{V A '}, or put one on
-the stack and use it with @address@hidden A $}}. Calc will not prompt for an
-argument list in this case, since the nameless function specifies the
-argument list as well as the function itself. In @kbd{V A '}, you can
-omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
-so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2>
@key{RET}},
-which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
-
address@hidden Lambda expressions
address@hidden
address@hidden
address@hidden ignore
address@hidden lambda
-The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
-(The word @code{lambda} derives from Lisp notation and the theory of
-functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
-ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
address@hidden; the whole point is that the @code{lambda} expression is
-used in its symbolic form, not evaluated for an answer until it is applied
-to specific arguments by a command like @kbd{V A} or @kbd{V M}.
-
-(Actually, @code{lambda} does have one special property: Its arguments
-are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
-will not simplify the @samp{2/3} until the nameless function is actually
-called.)
-
address@hidden add
address@hidden sub
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden mul
address@hidden
address@hidden @null
address@hidden ignore
address@hidden div
address@hidden
address@hidden @null
address@hidden ignore
address@hidden pow
address@hidden
address@hidden @null
address@hidden ignore
address@hidden neg
address@hidden
address@hidden @null
address@hidden ignore
address@hidden mod
address@hidden
address@hidden @null
address@hidden ignore
address@hidden vconcat
-As usual, commands like @kbd{V A} have algebraic function name equivalents.
-For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
address@hidden(gcd, v)}. The first argument specifies the operator name,
-and is either a variable whose name is the same as the function name,
-or a nameless function like @samp{<#^3+1>}. Operators that are normally
-written as algebraic symbols have the names @code{add}, @code{sub},
address@hidden, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
address@hidden
-
address@hidden
address@hidden
address@hidden ignore
address@hidden call
-The @code{call} function builds a function call out of several arguments:
address@hidden(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
-in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
-like the other functions described here, may be either a variable naming a
-function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
-as @samp{x + 2y}).
-
-(Experts will notice that it's not quite proper to use a variable to name
-a function, since the name @code{gcd} corresponds to the Lisp variable
address@hidden but to the Lisp function @code{calcFunc-gcd}. Calc
-automatically makes this translation, so you don't have to worry
-about it.)
-
address@hidden Mapping, Reducing, Specifying Operators, Reducing and Mapping
address@hidden Mapping
-
address@hidden
address@hidden V M
address@hidden calc-map
address@hidden map
-The @kbd{V M} (@code{calc-map}) address@hidden command applies a given
-operator elementwise to one or more vectors. For example, mapping
address@hidden address@hidden produces a vector of the absolute values of the
-elements in the input vector. Mapping @code{+} pops two vectors from
-the stack, which must be of equal length, and produces a vector of the
-pairwise sums of the elements. If either argument is a non-vector, it
-is duplicated for each element of the other vector. For example,
address@hidden,2,3] 2 V M ^} squares the elements of the specified vector.
-With the 2 listed first, it would have computed a vector of powers of
-two. Mapping a user-defined function pops as many arguments from the
-stack as the function requires. If you give an undefined name, you will
-be prompted for the number of arguments to use.
-
-If any argument to @kbd{V M} is a matrix, the operator is normally mapped
-across all elements of the matrix. For example, given the matrix
address@hidden, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
-produce another
address@hidden @math{3\times2}
address@hidden 3x2
-matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
-
address@hidden mapr
-The command @kbd{V M _} address@hidden (i.e., type an underscore at the
-operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
-the above matrix as a vector of two 3-element row vectors. It produces
-a new vector which contains the absolute values of those row vectors,
-namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
-defined as the square root of the sum of the squares of the elements.)
-Some operators accept vectors and return new vectors; for example,
address@hidden v} reverses a vector, so @kbd{V M _ v v} would reverse each row
-of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
-
-Sometimes a vector of vectors (representing, say, strings, sets, or lists)
-happens to look like a matrix. If so, remember to use @kbd{V M _} if you
-want to map a function across the whole strings or sets rather than across
-their individual elements.
-
address@hidden mapc
-The command @kbd{V M :} address@hidden maps by columns. Basically, it
-transposes the input matrix, maps by rows, and then, if the result is a
-matrix, transposes again. For example, @kbd{V M : A} takes the absolute
-values of the three columns of the matrix, treating each as a 2-vector,
-and @kbd{V M : v v} reverses the columns to get the matrix
address@hidden, 5, -6], [1, -2, 3]]}.
-
-(The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
-and column-like appearances, and were not already taken by useful
-operators. Also, they appear shifted on most keyboards so they are easy
-to type after @kbd{V M}.)
-
-The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
-not matrices (so if none of the arguments are matrices, they have no
-effect at all). If some of the arguments are matrices and others are
-plain numbers, the plain numbers are held constant for all rows of the
-matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
-a vector takes a dot product of the vector with itself).
-
-If some of the arguments are vectors with the same lengths as the
-rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
-arguments, those vectors are also held constant for every row or
-column.
-
-Sometimes it is useful to specify another mapping command as the operator
-to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
-to each row of the input matrix, which in turn adds the two values on that
-row. If you give another vector-operator command as the operator for
address@hidden M}, it automatically uses map-by-rows mode if you don't specify
-otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
-you really want to map-by-elements another mapping command, you can use
-a triple-nested mapping command: @kbd{V M V M V A +} means to map
address@hidden M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
-mapped over the elements of each row.)
-
address@hidden mapa
address@hidden mapd
-Previous versions of Calc had ``map across'' and ``map down'' modes
-that are now considered obsolete; the old ``map across'' is now simply
address@hidden M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
-functions @code{mapa} and @code{mapd} are still supported, though.
-Note also that, while the old mapping modes were persistent (once you
-set the mode, it would apply to later mapping commands until you reset
-it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
-mapping command. The default @kbd{V M} always means map-by-elements.
-
address@hidden Manipulation}, for the @kbd{a M} command, which is like
address@hidden M} but for equations and inequalities instead of vectors.
address@hidden Variables}, for the @kbd{s m} command which modifies a
-variable's stored value using a @kbd{V M}-like operator.
-
address@hidden Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
address@hidden Reducing
-
address@hidden
address@hidden V R
address@hidden calc-reduce
address@hidden reduce
-The @kbd{V R} (@code{calc-reduce}) address@hidden command applies a given
-binary operator across all the elements of a vector. A binary operator is
-a function such as @code{+} or @code{max} which takes two arguments. For
-example, reducing @code{+} over a vector computes the sum of the elements
-of the vector. Reducing @code{-} computes the first element minus each of
-the remaining elements. Reducing @code{max} computes the maximum element
-and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
-produces @samp{f(f(f(a, b), c), d)}.
-
address@hidden I V R
address@hidden rreduce
-The @kbd{I V R} address@hidden command is similar to @kbd{V R} except
-that works from right to left through the vector. For example, plain
address@hidden R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c
- d}
-but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
-or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
-in power series expansions.
-
address@hidden V U
address@hidden accum
-The @kbd{V U} (@code{calc-accumulate}) address@hidden command does an
-accumulation operation. Here Calc does the corresponding reduction
-operation, but instead of producing only the final result, it produces
-a vector of all the intermediate results. Accumulating @code{+} over
-the vector @samp{[a, b, c, d]} produces the vector
address@hidden, a + b, a + b + c, a + b + c + d]}.
-
address@hidden I V U
address@hidden raccum
-The @kbd{I V U} address@hidden command does a right-to-left accumulation.
-For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
-vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
-
address@hidden reducea
address@hidden rreducea
address@hidden reduced
address@hidden rreduced
-As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
-example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
-compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
address@hidden R :} to modify this behavior. The @kbd{V R _} address@hidden
-command reduces ``across'' the matrix; it reduces each row of the matrix
-as a vector, then collects the results. Thus @kbd{V R _ +} of this
-matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
address@hidden reduces down; @kbd{V R : +} would produce @expr{[a + d,
-b + e, c + f]}.
-
address@hidden reducer
address@hidden rreducer
-There is a third ``by rows'' mode for reduction that is occasionally
-useful; @kbd{V R =} address@hidden simply reduces the operator over
-the rows of the matrix themselves. Thus @kbd{V R = +} on the above
-matrix would get the same result as @kbd{V R : +}, since adding two
-row vectors is equivalent to adding their elements. But @kbd{V R = *}
-would multiply the two rows (to get a single number, their dot product),
-while @kbd{V R : *} would produce a vector of the products of the columns.
-
-These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
-but they are not currently supported with @kbd{V U} or @kbd{I V U}.
-
address@hidden reducec
address@hidden rreducec
-The obsolete reduce-by-columns function, @code{reducec}, is still
-supported but there is no way to get it through the @kbd{V R} command.
-
-The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
address@hidden * r} to grab a rectangle of data into Calc, and then typing
address@hidden R : +} or @kbd{V R _ +}, respectively, to sum the columns or
-rows of the matrix. @xref{Grabbing From Buffers}.
-
address@hidden Nesting and Fixed Points, Generalized Products, Reducing,
Reducing and Mapping
address@hidden Nesting and Fixed Points
-
address@hidden
address@hidden H V R
address@hidden nest
-The @kbd{H V R} address@hidden command applies a function to a given
-argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
-the stack, where @samp{n} must be an integer. It then applies the
-function nested @samp{n} times; if the function is @samp{f} and @samp{n}
-is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
-negative if Calc knows an inverse for the function @samp{f}; for
-example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
-
address@hidden H V U
address@hidden anest
-The @kbd{H V U} address@hidden command is an accumulating version of
address@hidden: It returns a vector of @samp{n+1} values, e.g.,
address@hidden, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
address@hidden is the inverse of @samp{f}, then the result is of the
-form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
-
address@hidden H I V R
address@hidden fixp
address@hidden Fixed points
-The @kbd{H I V R} address@hidden command is like @kbd{H V R}, except
-that it takes only an @samp{a} value from the stack; the function is
-applied until it reaches a ``fixed point,'' i.e., until the result
-no longer changes.
-
address@hidden H I V U
address@hidden afixp
-The @kbd{H I V U} address@hidden command is an accumulating @code{fixp}.
-The first element of the return vector will be the initial value @samp{a};
-the last element will be the final result that would have been returned
-by @code{fixp}.
-
-For example, 0.739085 is a fixed point of the cosine function (in radians):
address@hidden(0.739085) = 0.739085}. You can find this value by putting, say,
-1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
-version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
-0.65329, ...]}. With a precision of six, this command will take 36 steps
-to converge to 0.739085.)
-
-Newton's method for finding roots is a classic example of iteration
-to a fixed point. To find the square root of five starting with an
-initial guess, Newton's method would look for a fixed point of the
-function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
-and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
-2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
-command to find a root of the equation @samp{x^2 = 5}.
-
-These examples used numbers for @samp{a} values. Calc keeps applying
-the function until two successive results are equal to within the
-current precision. For complex numbers, both the real parts and the
-imaginary parts must be equal to within the current precision. If
address@hidden is a formula (say, a variable name), then the function is
-applied until two successive results are exactly the same formula.
-It is up to you to ensure that the function will eventually converge;
-if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
-
-The algebraic @code{fixp} function takes two optional arguments, @samp{n}
-and @samp{tol}. The first is the maximum number of steps to be allowed,
-and must be either an integer or the symbol @samp{inf} (infinity, the
-default). The second is a convergence tolerance. If a tolerance is
-specified, all results during the calculation must be numbers, not
-formulas, and the iteration stops when the magnitude of the difference
-between two successive results is less than or equal to the tolerance.
-(This implies that a tolerance of zero iterates until the results are
-exactly equal.)
-
-Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
-computes the square root of @samp{A} given the initial guess @samp{B},
-stopping when the result is correct within the specified tolerance, or
-when 20 steps have been taken, whichever is sooner.
-
address@hidden Generalized Products, , Nesting and Fixed Points, Reducing and
Mapping
address@hidden Generalized Products
-
address@hidden V O
address@hidden calc-outer-product
address@hidden outer
-The @kbd{V O} (@code{calc-outer-product}) address@hidden command applies
-a given binary operator to all possible pairs of elements from two
-vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
-and @samp{[x, y, z]} on the stack produces a multiplication table:
address@hidden x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
-the result matrix is obtained by applying the operator to element @var{r}
-of the lefthand vector and element @var{c} of the righthand vector.
-
address@hidden V I
address@hidden calc-inner-product
address@hidden inner
-The @kbd{V I} (@code{calc-inner-product}) address@hidden command computes
-the generalized inner product of two vectors or matrices, given a
-``multiplicative'' operator and an ``additive'' operator. These can each
-actually be any binary operators; if they are @samp{*} and @samp{+},
-respectively, the result is a standard matrix multiplication. Element
address@hidden,@var{c} of the result matrix is obtained by mapping the
-multiplicative operator across row @var{r} of the lefthand matrix and
-column @var{c} of the righthand matrix, and then reducing with the additive
-operator. Just as for the standard @kbd{*} command, this can also do a
-vector-matrix or matrix-vector inner product, or a vector-vector
-generalized dot product.
-
-Since @kbd{V I} requires two operators, it prompts twice. In each case,
-you can use any of the usual methods for entering the operator. If you
-use @kbd{$} twice to take both operator formulas from the stack, the
-first (multiplicative) operator is taken from the top of the stack
-and the second (additive) operator is taken from second-to-top.
-
address@hidden Vector and Matrix Formats, , Reducing and Mapping, Matrix
Functions
address@hidden Vector and Matrix Display Formats
-
address@hidden
-Commands for controlling vector and matrix display use the @kbd{v} prefix
-instead of the usual @kbd{d} prefix. But they are display modes; in
-particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
-in the same way (@pxref{Display Modes}). Matrix display is also
-influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
address@hidden Language Modes}.
-
address@hidden V <
address@hidden calc-matrix-left-justify
address@hidden V =
address@hidden calc-matrix-center-justify
address@hidden V >
address@hidden calc-matrix-right-justify
-The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
-(@code{calc-matrix-right-justify}), and @address@hidden =}}
-(@code{calc-matrix-center-justify}) control whether matrix elements
-are justified to the left, right, or center of their columns.
-
address@hidden V [
address@hidden calc-vector-brackets
address@hidden V @{
address@hidden calc-vector-braces
address@hidden V (
address@hidden calc-vector-parens
-The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
-brackets that surround vectors and matrices displayed in the stack on
-and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
-(@code{calc-vector-parens}) commands use curly braces or parentheses,
-respectively, instead of square brackets. For example, @kbd{v @{} might
-be used in preparation for yanking a matrix into a buffer running
-Mathematica. (In fact, the Mathematica language mode uses this mode;
address@hidden Language Mode}.) Note that, regardless of the
-display mode, either brackets or braces may be used to enter vectors,
-and parentheses may never be used for this purpose.
-
address@hidden V ]
address@hidden calc-matrix-brackets
-The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
-``big'' style display of matrices. It prompts for a string of code
-letters; currently implemented letters are @code{R}, which enables
-brackets on each row of the matrix; @code{O}, which enables outer
-brackets in opposite corners of the matrix; and @code{C}, which
-enables commas or semicolons at the ends of all rows but the last.
-The default format is @samp{RO}. (Before Calc 2.00, the format
-was fixed at @samp{ROC}.) Here are some example matrices:
-
address@hidden
address@hidden
-[ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
- [ 0, 123, 0 ] [ 0, 123, 0 ],
- [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
-
- RO ROC
-
address@hidden group
address@hidden example
address@hidden
address@hidden
address@hidden
- [ 123, 0, 0 [ 123, 0, 0 ;
- 0, 123, 0 0, 123, 0 ;
- 0, 0, 123 ] 0, 0, 123 ]
-
- O OC
-
address@hidden group
address@hidden example
address@hidden
address@hidden
address@hidden
- [ 123, 0, 0 ] 123, 0, 0
- [ 0, 123, 0 ] 0, 123, 0
- [ 0, 0, 123 ] 0, 0, 123
-
- R @r{blank}
address@hidden group
address@hidden example
-
address@hidden
-Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
address@hidden are all recognized as matrices during reading, while
-the others are useful for display only.
-
address@hidden V ,
address@hidden calc-vector-commas
-The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
-off in vector and matrix display.
-
-In vectors of length one, and in all vectors when commas have been
-turned off, Calc adds extra parentheses around formulas that might
-otherwise be ambiguous. For example, @samp{[a b]} could be a vector
-of the one formula @samp{a b}, or it could be a vector of two
-variables with commas turned off. Calc will display the former
-case as @samp{[(a b)]}. You can disable these extra parentheses
-(to make the output less cluttered at the expense of allowing some
-ambiguity) by adding the letter @code{P} to the control string you
-give to @kbd{v ]} (as described above).
-
address@hidden V .
address@hidden calc-full-vectors
-The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
-display of long vectors on and off. In this mode, vectors of six
-or more elements, or matrices of six or more rows or columns, will
-be displayed in an abbreviated form that displays only the first
-three elements and the last element: @samp{[a, b, c, ..., z]}.
-When very large vectors are involved this will substantially
-improve Calc's display speed.
-
address@hidden t .
address@hidden calc-full-trail-vectors
-The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
-similar mode for recording vectors in the Trail. If you turn on
-this mode, vectors of six or more elements and matrices of six or
-more rows or columns will be abbreviated when they are put in the
-Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
-unable to recover those vectors. If you are working with very
-large vectors, this mode will improve the speed of all operations
-that involve the trail.
-
address@hidden V /
address@hidden calc-break-vectors
-The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
-vector display on and off. Normally, matrices are displayed with one
-row per line but all other types of vectors are displayed in a single
-line. This mode causes all vectors, whether matrices or not, to be
-displayed with a single element per line. Sub-vectors within the
-vectors will still use the normal linear form.
-
address@hidden Algebra, Units, Matrix Functions, Top
address@hidden Algebra
-
address@hidden
-This section covers the Calc features that help you work with
-algebraic formulas. First, the general sub-formula selection
-mechanism is described; this works in conjunction with any Calc
-commands. Then, commands for specific algebraic operations are
-described. Finally, the flexible @dfn{rewrite rule} mechanism
-is discussed.
-
-The algebraic commands use the @kbd{a} key prefix; selection
-commands use the @kbd{j} (for ``just a letter that wasn't used
-for anything else'') prefix.
-
address@hidden Stack Entries}, to see how to manipulate formulas
-using regular Emacs editing commands.
-
-When doing algebraic work, you may find several of the Calculator's
-modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
-or No-Simplification mode (@kbd{m O}),
-Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
-Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
-of these modes. You may also wish to select Big display mode (@kbd{d B}).
address@hidden Language Modes}.
-
address@hidden
-* Selecting Subformulas::
-* Algebraic Manipulation::
-* Simplifying Formulas::
-* Polynomials::
-* Calculus::
-* Solving Equations::
-* Numerical Solutions::
-* Curve Fitting::
-* Summations::
-* Logical Operations::
-* Rewrite Rules::
address@hidden menu
-
address@hidden Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
address@hidden Selecting Sub-Formulas
-
address@hidden
address@hidden Selections
address@hidden Sub-formulas
address@hidden Parts of formulas
-When working with an algebraic formula it is often necessary to
-manipulate a portion of the formula rather than the formula as a
-whole. Calc allows you to ``select'' a portion of any formula on
-the stack. Commands which would normally operate on that stack
-entry will now operate only on the sub-formula, leaving the
-surrounding part of the stack entry alone.
-
-One common non-algebraic use for selection involves vectors. To work
-on one element of a vector in-place, simply select that element as a
-``sub-formula'' of the vector.
-
address@hidden
-* Making Selections::
-* Changing Selections::
-* Displaying Selections::
-* Operating on Selections::
-* Rearranging with Selections::
address@hidden menu
-
address@hidden Making Selections, Changing Selections, Selecting Subformulas,
Selecting Subformulas
address@hidden Making Selections
-
address@hidden
address@hidden j s
address@hidden calc-select-here
-To select a sub-formula, move the Emacs cursor to any character in that
-sub-formula, and press @address@hidden s}} (@code{calc-select-here}). Calc
will
-highlight the smallest portion of the formula that contains that
-character. By default the sub-formula is highlighted by blanking out
-all of the rest of the formula with dots. Selection works in any
-display mode but is perhaps easiest in Big mode (@kbd{d B}).
-Suppose you enter the following formula:
-
address@hidden
address@hidden
- 3 ___
- (a + b) + V c
-1: ---------------
- 2 x + 1
address@hidden group
address@hidden smallexample
-
address@hidden
-(by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
-cursor to the letter @samp{b} and press @address@hidden s}}, the display
changes
-to
-
address@hidden
address@hidden
- . ...
- .. . b. . . .
-1* ...............
- . . . .
address@hidden group
address@hidden smallexample
-
address@hidden
-Every character not part of the sub-formula @samp{b} has been changed
-to a dot. The @samp{*} next to the line number is to remind you that
-the formula has a portion of it selected. (In this case, it's very
-obvious, but it might not always be. If Embedded mode is enabled,
-the word @samp{Sel} also appears in the mode line because the stack
-may not be visible. @pxref{Embedded Mode}.)
-
-If you had instead placed the cursor on the parenthesis immediately to
-the right of the @samp{b}, the selection would have been:
-
address@hidden
address@hidden
- . ...
- (a + b) . . .
-1* ...............
- . . . .
address@hidden group
address@hidden smallexample
-
address@hidden
-The portion selected is always large enough to be considered a complete
-formula all by itself, so selecting the parenthesis selects the whole
-formula that it encloses. Putting the cursor on the @samp{+} sign
-would have had the same effect.
-
-(Strictly speaking, the Emacs cursor is really the manifestation of
-the Emacs ``point,'' which is a position @emph{between} two characters
-in the buffer. So purists would say that Calc selects the smallest
-sub-formula which contains the character to the right of ``point.'')
-
-If you supply a numeric prefix argument @var{n}, the selection is
-expanded to the @var{n}th enclosing sub-formula. Thus, positioning
-the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
address@hidden + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
-and so on.
-
-If the cursor is not on any part of the formula, or if you give a
-numeric prefix that is too large, the entire formula is selected.
-
-If the cursor is on the @samp{.} line that marks the top of the stack
-(i.e., its normal ``rest position''), this command selects the entire
-formula at stack level 1. Most selection commands similarly operate
-on the formula at the top of the stack if you haven't positioned the
-cursor on any stack entry.
-
address@hidden j a
address@hidden calc-select-additional
-The @kbd{j a} (@code{calc-select-additional}) command enlarges the
-current selection to encompass the cursor. To select the smallest
-sub-formula defined by two different points, move to the first and
-press @kbd{j s}, then move to the other and press @kbd{j a}. This
-is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
-select the two ends of a region of text during normal Emacs editing.
-
address@hidden j o
address@hidden calc-select-once
-The @kbd{j o} (@code{calc-select-once}) command selects a formula in
-exactly the same way as @kbd{j s}, except that the selection will
-last only as long as the next command that uses it. For example,
address@hidden o 1 +} is a handy way to add one to the sub-formula indicated
-by the cursor.
-
-(A somewhat more precise definition: The @kbd{j o} command sets a flag
-such that the next command involving selected stack entries will clear
-the selections on those stack entries afterwards. All other selection
-commands except @kbd{j a} and @kbd{j O} clear this flag.)
-
address@hidden j S
address@hidden j O
address@hidden calc-select-here-maybe
address@hidden calc-select-once-maybe
-The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
-(@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
-and @kbd{j o}, respectively, except that if the formula already
-has a selection they have no effect. This is analogous to the
-behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
address@hidden with Rewrite Rules}) and is mainly intended to be
-used in keyboard macros that implement your own selection-oriented
-commands.
-
-Selection of sub-formulas normally treats associative terms like
address@hidden + b - c + d} and @samp{x * y * z} as single levels of the
formula.
-If you place the cursor anywhere inside @samp{a + b - c + d} except
-on one of the variable names and use @kbd{j s}, you will select the
-entire four-term sum.
-
address@hidden j b
address@hidden calc-break-selections
-The @kbd{j b} (@code{calc-break-selections}) command controls a mode
-in which the ``deep structure'' of these associative formulas shows
-through. Calc actually stores the above formulas as @samp{((a + b) - c) + d}
-and @samp{x * (y * z)}. (Note that for certain obscure reasons, Calc
-treats multiplication as right-associative.) Once you have enabled
address@hidden b} mode, selecting with the cursor on the @samp{-} sign would
-only select the @samp{a + b - c} portion, which makes sense when the
-deep structure of the sum is considered. There is no way to select
-the @samp{b - c + d} portion; although this might initially look
-like just as legitimate a sub-formula as @samp{a + b - c}, the deep
-structure shows that it isn't. The @kbd{d U} command can be used
-to view the deep structure of any formula (@pxref{Normal Language Modes}).
-
-When @kbd{j b} mode has not been enabled, the deep structure is
-generally hidden by the selection commands---what you see is what
-you get.
-
address@hidden j u
address@hidden calc-unselect
-The @kbd{j u} (@code{calc-unselect}) command unselects the formula
-that the cursor is on. If there was no selection in the formula,
-this command has no effect. With a numeric prefix argument, it
-unselects the @var{n}th stack element rather than using the cursor
-position.
-
address@hidden j c
address@hidden calc-clear-selections
-The @kbd{j c} (@code{calc-clear-selections}) command unselects all
-stack elements.
-
address@hidden Changing Selections, Displaying Selections, Making Selections,
Selecting Subformulas
address@hidden Changing Selections
-
address@hidden
address@hidden j m
address@hidden calc-select-more
-Once you have selected a sub-formula, you can expand it using the
address@hidden@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
-selected, pressing @address@hidden m}} repeatedly works as follows:
-
address@hidden
address@hidden
- 3 ... 3 ___ 3 ___
- (a + b) . . . (a + b) + V c (a + b) + V c
-1* ............... 1* ............... 1* ---------------
- . . . . . . . . 2 x + 1
address@hidden group
address@hidden smallexample
-
address@hidden
-In the last example, the entire formula is selected. This is roughly
-the same as having no selection at all, but because there are subtle
-differences the @samp{*} character is still there on the line number.
-
-With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
-times (or until the entire formula is selected). Note that @kbd{j s}
-with argument @var{n} is equivalent to plain @kbd{j s} followed by
address@hidden m} with argument @var{n}. If @address@hidden m}} is used when
there
-is no current selection, it is equivalent to @address@hidden s}}.
-
-Even though @kbd{j m} does not explicitly use the location of the
-cursor within the formula, it nevertheless uses the cursor to determine
-which stack element to operate on. As usual, @kbd{j m} when the cursor
-is not on any stack element operates on the top stack element.
-
address@hidden j l
address@hidden calc-select-less
-The @kbd{j l} (@code{calc-select-less}) command reduces the current
-selection around the cursor position. That is, it selects the
-immediate sub-formula of the current selection which contains the
-cursor, the opposite of @kbd{j m}. If the cursor is not inside the
-current selection, the command de-selects the formula.
-
address@hidden j 1-9
address@hidden calc-select-part
-The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
-select the @var{n}th sub-formula of the current selection. They are
-like @kbd{j l} (@code{calc-select-less}) except they use counting
-rather than the cursor position to decide which sub-formula to select.
-For example, if the current selection is @kbd{a + b + c} or
address@hidden(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
address@hidden 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
-these cases, @kbd{j 4} through @kbd{j 9} would be errors.
-
-If there is no current selection, @kbd{j 1} through @kbd{j 9} select
-the @var{n}th top-level sub-formula. (In other words, they act as if
-the entire stack entry were selected first.) To select the @var{n}th
-sub-formula where @var{n} is greater than nine, you must instead invoke
address@hidden@kbd{j 1}} with @var{n} as a numeric prefix argument.
-
address@hidden j n
address@hidden j p
address@hidden calc-select-next
address@hidden calc-select-previous
-The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
-(@code{calc-select-previous}) commands change the current selection
-to the next or previous sub-formula at the same level. For example,
-if @samp{b} is selected in @address@hidden + a*b*c + x}}, then @kbd{j n}
-selects @samp{c}. Further @kbd{j n} commands would be in error because,
-even though there is something to the right of @samp{c} (namely, @samp{x}),
-it is not at the same level; in this case, it is not a term of the
-same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
-the whole product @samp{a*b*c} as a term of the sum) followed by
address@hidden@kbd{j n}} would successfully select the @samp{x}.
-
-Similarly, @kbd{j p} moves the selection from the @samp{b} in this
-sample formula to the @samp{a}. Both commands accept numeric prefix
-arguments to move several steps at a time.
-
-It is interesting to compare Calc's selection commands with the
-Emacs Info system's commands for navigating through hierarchically
-organized documentation. Calc's @kbd{j n} command is completely
-analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
address@hidden, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
-(Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
-The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
address@hidden l}; in each case, you can jump directly to a sub-component
-of the hierarchy simply by pointing to it with the cursor.
-
address@hidden Displaying Selections, Operating on Selections, Changing
Selections, Selecting Subformulas
address@hidden Displaying Selections
-
address@hidden
address@hidden j d
address@hidden calc-show-selections
-The @kbd{j d} (@code{calc-show-selections}) command controls how
-selected sub-formulas are displayed. One of the alternatives is
-illustrated in the above examples; if we press @kbd{j d} we switch
-to the other style in which the selected portion itself is obscured
-by @samp{#} signs:
-
address@hidden
address@hidden
- 3 ... # ___
- (a + b) . . . ## # ## + V c
-1* ............... 1* ---------------
- . . . . 2 x + 1
address@hidden group
address@hidden smallexample
-
address@hidden Operating on Selections, Rearranging with Selections, Displaying
Selections, Selecting Subformulas
address@hidden Operating on Selections
-
address@hidden
-Once a selection is made, all Calc commands that manipulate items
-on the stack will operate on the selected portions of the items
-instead. (Note that several stack elements may have selections
-at once, though there can be only one selection at a time in any
-given stack element.)
-
address@hidden j e
address@hidden calc-enable-selections
-The @kbd{j e} (@code{calc-enable-selections}) command disables the
-effect that selections have on Calc commands. The current selections
-still exist, but Calc commands operate on whole stack elements anyway.
-This mode can be identified by the fact that the @samp{*} markers on
-the line numbers are gone, even though selections are visible. To
-reactivate the selections, press @kbd{j e} again.
-
-To extract a sub-formula as a new formula, simply select the
-sub-formula and press @key{RET}. This normally duplicates the top
-stack element; here it duplicates only the selected portion of that
-element.
-
-To replace a sub-formula with something different, you can enter the
-new value onto the stack and press @key{TAB}. This normally exchanges
-the top two stack elements; here it swaps the value you entered into
-the selected portion of the formula, returning the old selected
-portion to the top of the stack.
-
address@hidden
address@hidden
- 3 ... ... ___
- (a + b) . . . 17 x y . . . 17 x y + V c
-2* ............... 2* ............. 2: -------------
- . . . . . . . . 2 x + 1
-
- 3 3
-1: 17 x y 1: (a + b) 1: (a + b)
address@hidden group
address@hidden smallexample
-
-In this example we select a sub-formula of our original example,
-enter a new formula, @key{TAB} it into place, then deselect to see
-the complete, edited formula.
-
-If you want to swap whole formulas around even though they contain
-selections, just use @kbd{j e} before and after.
-
address@hidden j '
address@hidden calc-enter-selection
-The @kbd{j '} (@code{calc-enter-selection}) command is another way
-to replace a selected sub-formula. This command does an algebraic
-entry just like the regular @kbd{'} key. When you press @key{RET},
-the formula you type replaces the original selection. You can use
-the @samp{$} symbol in the formula to refer to the original
-selection. If there is no selection in the formula under the cursor,
-the cursor is used to make a temporary selection for the purposes of
-the command. Thus, to change a term of a formula, all you have to
-do is move the Emacs cursor to that term and press @kbd{j '}.
-
address@hidden j `
address@hidden calc-edit-selection
-The @kbd{j `} (@code{calc-edit-selection}) command is a similar
-analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
-selected sub-formula in a separate buffer. If there is no
-selection, it edits the sub-formula indicated by the cursor.
-
-To delete a sub-formula, press @key{DEL}. This generally replaces
-the sub-formula with the constant zero, but in a few suitable contexts
-it uses the constant one instead. The @key{DEL} key automatically
-deselects and re-simplifies the entire formula afterwards. Thus:
-
address@hidden
address@hidden
- ###
- 17 x y + # # 17 x y 17 # y 17 y
-1* ------------- 1: ------- 1* ------- 1: -------
- 2 x + 1 2 x + 1 2 x + 1 2 x + 1
address@hidden group
address@hidden smallexample
-
-In this example, we first delete the @samp{sqrt(c)} term; Calc
-accomplishes this by replacing @samp{sqrt(c)} with zero and
-resimplifying. We then delete the @kbd{x} in the numerator;
-since this is part of a product, Calc replaces it with @samp{1}
-and resimplifies.
-
-If you select an element of a vector and press @key{DEL}, that
-element is deleted from the vector. If you delete one side of
-an equation or inequality, only the opposite side remains.
-
address@hidden j @key{DEL}
address@hidden calc-del-selection
-The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
address@hidden but with the auto-selecting behavior of @kbd{j '} and
address@hidden `}. It deletes the selected portion of the formula
-indicated by the cursor, or, in the absence of a selection, it
-deletes the sub-formula indicated by the cursor position.
-
address@hidden j @key{RET}
address@hidden calc-grab-selection
-(There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
-command.)
-
-Normal arithmetic operations also apply to sub-formulas. Here we
-select the denominator, press @kbd{5 -} to subtract five from the
-denominator, press @kbd{n} to negate the denominator, then
-press @kbd{Q} to take the square root.
-
address@hidden
address@hidden
- .. . .. . .. . .. .
-1* ....... 1* ....... 1* ....... 1* ..........
- 2 x + 1 2 x - 4 4 - 2 x _________
- V 4 - 2 x
address@hidden group
address@hidden smallexample
-
-Certain types of operations on selections are not allowed. For
-example, for an arithmetic function like @kbd{-} no more than one of
-the arguments may be a selected sub-formula. (As the above example
-shows, the result of the subtraction is spliced back into the argument
-which had the selection; if there were more than one selection involved,
-this would not be well-defined.) If you try to subtract two selections,
-the command will abort with an error message.
-
-Operations on sub-formulas sometimes leave the formula as a whole
-in an ``un-natural'' state. Consider negating the @samp{2 x} term
-of our sample formula by selecting it and pressing @kbd{n}
-(@code{calc-change-sign}).
-
address@hidden
address@hidden
- .. . .. .
-1* .......... 1* ...........
- ......... ..........
- . . . 2 x . . . -2 x
address@hidden group
address@hidden smallexample
-
-Unselecting the sub-formula reveals that the minus sign, which would
-normally have cancelled out with the subtraction automatically, has
-not been able to do so because the subtraction was not part of the
-selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
-any other mathematical operation on the whole formula will cause it
-to be simplified.
-
address@hidden
address@hidden
- 17 y 17 y
-1: ----------- 1: ----------
- __________ _________
- V 4 - -2 x V 4 + 2 x
address@hidden group
address@hidden smallexample
-
address@hidden Rearranging with Selections, , Operating on Selections,
Selecting Subformulas
address@hidden Rearranging Formulas using Selections
-
address@hidden
address@hidden j R
address@hidden calc-commute-right
-The @kbd{j R} (@code{calc-commute-right}) command moves the selected
-sub-formula to the right in its surrounding formula. Generally the
-selection is one term of a sum or product; the sum or product is
-rearranged according to the commutative laws of algebra.
-
-As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
-if there is no selection in the current formula. All commands described
-in this section share this property. In this example, we place the
-cursor on the @samp{a} and type @kbd{j R}, then repeat.
-
address@hidden
-1: a + b - c 1: b + a - c 1: b - c + a
address@hidden smallexample
-
address@hidden
-Note that in the final step above, the @samp{a} is switched with
-the @samp{c} but the signs are adjusted accordingly. When moving
-terms of sums and products, @kbd{j R} will never change the
-mathematical meaning of the formula.
-
-The selected term may also be an element of a vector or an argument
-of a function. The term is exchanged with the one to its right.
-In this case, the ``meaning'' of the vector or function may of
-course be drastically changed.
-
address@hidden
-1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
-
-1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
address@hidden smallexample
-
address@hidden j L
address@hidden calc-commute-left
-The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
-except that it swaps the selected term with the one to its left.
-
-With numeric prefix arguments, these commands move the selected
-term several steps at a time. It is an error to try to move a
-term left or right past the end of its enclosing formula.
-With numeric prefix arguments of zero, these commands move the
-selected term as far as possible in the given direction.
-
address@hidden j D
address@hidden calc-sel-distribute
-The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
-sum or product into the surrounding formula using the distributive
-law. For example, in @samp{a * (b - c)} with the @samp{b - c}
-selected, the result is @samp{a b - a c}. This also distributes
-products or quotients into surrounding powers, and can also do
-transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
-where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
-to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
-
-For multiple-term sums or products, @kbd{j D} takes off one term
-at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
-with the @samp{c - d} selected so that you can type @kbd{j D}
-repeatedly to expand completely. The @kbd{j D} command allows a
-numeric prefix argument which specifies the maximum number of
-times to expand at once; the default is one time only.
-
address@hidden DistribRules
-The @kbd{j D} command is implemented using rewrite rules.
address@hidden with Rewrite Rules}. The rules are stored in
-the Calc variable @code{DistribRules}. A convenient way to view
-these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
-displays and edits the stored value of a variable. Press @kbd{C-c C-c}
-to return from editing mode; be careful not to make any actual changes
-or else you will affect the behavior of future @kbd{j D} commands!
-
-To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
-as described above. You can then use the @kbd{s p} command to save
-this variable's value permanently for future Calc sessions.
address@hidden on Variables}.
-
address@hidden j M
address@hidden calc-sel-merge
address@hidden MergeRules
-The @kbd{j M} (@code{calc-sel-merge}) command is the complement
-of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
address@hidden c} selected, the result is @samp{a * (b - c)}. Once
-again, @kbd{j M} can also merge calls to functions like @code{exp}
-and @code{ln}; examine the variable @code{MergeRules} to see all
-the relevant rules.
-
address@hidden j C
address@hidden calc-sel-commute
address@hidden CommuteRules
-The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
-of the selected sum, product, or equation. It always behaves as
-if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
-treated as the nested sums @samp{(a + b) + c} by this command.
-If you put the cursor on the first @samp{+}, the result is
address@hidden(b + a) + c}; if you put the cursor on the second @samp{+}, the
-result is @samp{c + (a + b)} (which the default simplifications
-will rearrange to @samp{(c + a) + b}). The relevant rules are stored
-in the variable @code{CommuteRules}.
-
-You may need to turn default simplifications off (with the @kbd{m O}
-command) in order to get the full benefit of @kbd{j C}. For example,
-commuting @samp{a - b} produces @samp{-b + a}, but the default
-simplifications will ``simplify'' this right back to @samp{a - b} if
-you don't turn them off. The same is true of some of the other
-manipulations described in this section.
-
address@hidden j N
address@hidden calc-sel-negate
address@hidden NegateRules
-The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
-term with the negative of that term, then adjusts the surrounding
-formula in order to preserve the meaning. For example, given
address@hidden(a - b)} where @samp{a - b} is selected, the result is
address@hidden / exp(b - a)}. By contrast, selecting a term and using the
-regular @kbd{n} (@code{calc-change-sign}) command negates the
-term without adjusting the surroundings, thus changing the meaning
-of the formula as a whole. The rules variable is @code{NegateRules}.
-
address@hidden j &
address@hidden calc-sel-invert
address@hidden InvertRules
-The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
-except it takes the reciprocal of the selected term. For example,
-given @samp{a - ln(b)} with @samp{b} selected, the result is
address@hidden + ln(1/b)}. The rules variable is @code{InvertRules}.
-
address@hidden j E
address@hidden calc-sel-jump-equals
address@hidden JumpRules
-The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
-selected term from one side of an equation to the other. Given
address@hidden + b = c + d} with @samp{c} selected, the result is
address@hidden + b - c = d}. This command also works if the selected
-term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
-relevant rules variable is @code{JumpRules}.
-
address@hidden j I
address@hidden H j I
address@hidden calc-sel-isolate
-The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
-selected term on its side of an equation. It uses the @kbd{a S}
-(@code{calc-solve-for}) command to solve the equation, and the
-Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
-When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
-It understands more rules of algebra, and works for inequalities
-as well as equations.
-
address@hidden j *
address@hidden j /
address@hidden calc-sel-mult-both-sides
address@hidden calc-sel-div-both-sides
-The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
-formula using algebraic entry, then multiplies both sides of the
-selected quotient or equation by that formula. It simplifies each
-side with @kbd{a s} (@code{calc-simplify}) before re-forming the
-quotient or equation. You can suppress this simplification by
-providing any numeric prefix argument. There is also a @kbd{j /}
-(@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
-dividing instead of multiplying by the factor you enter.
-
-As a special feature, if the numerator of the quotient is 1, then
-the denominator is expanded at the top level using the distributive
-law (i.e., using the @kbd{C-u -1 a x} command). Suppose the
-formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
-to eliminate the square root in the denominator by multiplying both
-sides by @samp{sqrt(a) - 1}. Calc's default simplifications would
-change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
-right back to the original form by cancellation; Calc expands the
-denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
-this. (You would now want to use an @kbd{a x} command to expand
-the rest of the way, whereupon the denominator would cancel out to
-the desired form, @samp{a - 1}.) When the numerator is not 1, this
-initial expansion is not necessary because Calc's default
-simplifications will not notice the potential cancellation.
-
-If the selection is an inequality, @kbd{j *} and @kbd{j /} will
-accept any factor, but will warn unless they can prove the factor
-is either positive or negative. (In the latter case the direction
-of the inequality will be switched appropriately.) @xref{Declarations},
-for ways to inform Calc that a given variable is positive or
-negative. If Calc can't tell for sure what the sign of the factor
-will be, it will assume it is positive and display a warning
-message.
-
-For selections that are not quotients, equations, or inequalities,
-these commands pull out a multiplicative factor: They divide (or
-multiply) by the entered formula, simplify, then multiply (or divide)
-back by the formula.
-
address@hidden j +
address@hidden j -
address@hidden calc-sel-add-both-sides
address@hidden calc-sel-sub-both-sides
-The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
-(@code{calc-sel-sub-both-sides}) commands analogously add to or
-subtract from both sides of an equation or inequality. For other
-types of selections, they extract an additive factor. A numeric
-prefix argument suppresses simplification of the intermediate
-results.
-
address@hidden j U
address@hidden calc-sel-unpack
-The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
-selected function call with its argument. For example, given
address@hidden + sin(x^2)} with @samp{sin(x^2)} selected, the result
-is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
-wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
-now to take the cosine of the selected part.)
-
address@hidden j v
address@hidden calc-sel-evaluate
-The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
-normal default simplifications on the selected sub-formula.
-These are the simplifications that are normally done automatically
-on all results, but which may have been partially inhibited by
-previous selection-related operations, or turned off altogether
-by the @kbd{m O} command. This command is just an auto-selecting
-version of the @address@hidden v}} command (@pxref{Algebraic Manipulation}).
-
-With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
-the @kbd{a s} (@code{calc-simplify}) command to the selected
-sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
-applies the @kbd{a e} (@code{calc-simplify-extended}) command.
address@hidden Formulas}. With a negative prefix argument
-it simplifies at the top level only, just as with @kbd{a v}.
-Here the ``top'' level refers to the top level of the selected
-sub-formula.
-
address@hidden j "
address@hidden calc-sel-expand-formula
-The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
-(@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
-
-You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
-to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
-
address@hidden Algebraic Manipulation, Simplifying Formulas, Selecting
Subformulas, Algebra
address@hidden Algebraic Manipulation
-
address@hidden
-The commands in this section perform general-purpose algebraic
-manipulations. They work on the whole formula at the top of the
-stack (unless, of course, you have made a selection in that
-formula).
-
-Many algebra commands prompt for a variable name or formula. If you
-answer the prompt with a blank line, the variable or formula is taken
-from top-of-stack, and the normal argument for the command is taken
-from the second-to-top stack level.
-
address@hidden a v
address@hidden calc-alg-evaluate
-The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
-default simplifications on a formula; for example, @samp{a - -b} is
-changed to @samp{a + b}. These simplifications are normally done
-automatically on all Calc results, so this command is useful only if
-you have turned default simplifications off with an @kbd{m O}
-command. @xref{Simplification Modes}.
-
-It is often more convenient to type @kbd{=}, which is like @kbd{a v}
-but which also substitutes stored values for variables in the formula.
-Use @kbd{a v} if you want the variables to ignore their stored values.
-
-If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
-as if in Algebraic Simplification mode. This is equivalent to typing
address@hidden s}; @pxref{Simplifying Formulas}. If you give a numeric prefix
-of 3 or more, it uses Extended Simplification mode (@kbd{a e}).
-
-If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or
@mathit{-3},
-it simplifies in the corresponding mode but only works on the top-level
-function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
-simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
address@hidden + 3}. As another example, typing @kbd{V R +} to sum the vector
address@hidden, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
-in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
-10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
-(@xref{Reducing and Mapping}.)
-
address@hidden evalv
address@hidden evalvn
-The @kbd{=} command corresponds to the @code{evalv} function, and
-the related @kbd{N} command, which is like @kbd{=} but temporarily
-disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
-to the @code{evalvn} function. (These commands interpret their prefix
-arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
-the number of stack elements to evaluate at once, and @kbd{N} treats
-it as a temporary different working precision.)
-
-The @code{evalvn} function can take an alternate working precision
-as an optional second argument. This argument can be either an
-integer, to set the precision absolutely, or a vector containing
-a single integer, to adjust the precision relative to the current
-precision. Note that @code{evalvn} with a larger than current
-precision will do the calculation at this higher precision, but the
-result will as usual be rounded back down to the current precision
-afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
-of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
-will return @samp{9.26535897932e-5} (computing a 25-digit result which
-is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
-will return @samp{9.2654e-5}.
-
address@hidden a "
address@hidden calc-expand-formula
-The @kbd{a "} (@code{calc-expand-formula}) command expands functions
-into their defining formulas wherever possible. For example,
address@hidden(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
-like @code{sin} and @code{gcd}, are not defined by simple formulas
-and so are unaffected by this command. One important class of
-functions which @emph{can} be expanded is the user-defined functions
-created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
-Other functions which @kbd{a "} can expand include the probability
-distribution functions, most of the financial functions, and the
-hyperbolic and inverse hyperbolic functions. A numeric prefix argument
-affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
-argument expands all functions in the formula and then simplifies in
-various ways; a negative argument expands and simplifies only the
-top-level function call.
-
address@hidden a M
address@hidden calc-map-equation
address@hidden mapeq
-The @kbd{a M} (@code{calc-map-equation}) address@hidden command applies
-a given function or operator to one or more equations. It is analogous
-to @kbd{V M}, which operates on vectors instead of equations.
address@hidden and Mapping}. For example, @kbd{a M S} changes
address@hidden = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
address@hidden = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
-With two equations on the stack, @kbd{a M +} would add the lefthand
-sides together and the righthand sides together to get the two
-respective sides of a new equation.
-
-Mapping also works on inequalities. Mapping two similar inequalities
-produces another inequality of the same type. Mapping an inequality
-with an equation produces an inequality of the same type. Mapping a
address@hidden<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
-If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
-are mapped, the direction of the second inequality is reversed to
-match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
-reverses the latter to get @samp{2 < a}, which then allows the
-combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
-then simplify to get @samp{2 < b}.
-
-Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
-or invert an inequality will reverse the direction of the inequality.
-Other adjustments to inequalities are @emph{not} done automatically;
address@hidden M S} will change @address@hidden < y}} to @samp{sin(x) < sin(y)}
even
-though this is not true for all values of the variables.
-
address@hidden H a M
address@hidden mapeqp
-With the Hyperbolic flag, @kbd{H a M} address@hidden does a plain
-mapping operation without reversing the direction of any inequalities.
-Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
-(This change is mathematically incorrect, but perhaps you were
-fixing an inequality which was already incorrect.)
-
address@hidden I a M
address@hidden mapeqr
-With the Inverse flag, @kbd{I a M} address@hidden always reverses
-the direction of the inequality. You might use @kbd{I a M C} to
-change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
-working with small positive angles.
-
address@hidden a b
address@hidden calc-substitute
address@hidden subst
-The @kbd{a b} (@code{calc-substitute}) address@hidden command substitutes
-all occurrences
-of some variable or sub-expression of an expression with a new
-sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
-in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
address@hidden cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
-Note that this is a purely structural substitution; the lone @samp{x} and
-the @samp{sin(2 x)} stayed the same because they did not look like
address@hidden(x)}. @xref{Rewrite Rules}, for a more general method for
-doing substitutions.
-
-The @kbd{a b} command normally prompts for two formulas, the old
-one and the new one. If you enter a blank line for the first
-prompt, all three arguments are taken from the stack (new, then old,
-then target expression). If you type an old formula but then enter a
-blank line for the new one, the new formula is taken from top-of-stack
-and the target from second-to-top. If you answer both prompts, the
-target is taken from top-of-stack as usual.
-
-Note that @kbd{a b} has no understanding of commutativity or
-associativity. The pattern @samp{x+y} will not match the formula
address@hidden Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
-because the @samp{+} operator is left-associative, so the ``deep
-structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
-(@code{calc-unformatted-language}) mode to see the true structure of
-a formula. The rewrite rule mechanism, discussed later, does not have
-these limitations.
-
-As an algebraic function, @code{subst} takes three arguments:
-Target expression, old, new. Note that @code{subst} is always
-evaluated immediately, even if its arguments are variables, so if
-you wish to put a call to @code{subst} onto the stack you must
-turn the default simplifications off first (with @kbd{m O}).
-
address@hidden Simplifying Formulas, Polynomials, Algebraic Manipulation,
Algebra
address@hidden Simplifying Formulas
-
address@hidden
address@hidden a s
address@hidden calc-simplify
address@hidden simplify
-The @kbd{a s} (@code{calc-simplify}) address@hidden command applies
-various algebraic rules to simplify a formula. This includes rules which
-are not part of the default simplifications because they may be too slow
-to apply all the time, or may not be desirable all of the time. For
-example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
-to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
-simplified to @samp{x}.
-
-The sections below describe all the various kinds of algebraic
-simplifications Calc provides in full detail. None of Calc's
-simplification commands are designed to pull rabbits out of hats;
-they simply apply certain specific rules to put formulas into
-less redundant or more pleasing forms. Serious algebra in Calc
-must be done manually, usually with a combination of selections
-and rewrite rules. @xref{Rearranging with Selections}.
address@hidden Rules}.
-
address@hidden Modes}, for commands to control what level of
-simplification occurs automatically. Normally only the ``default
-simplifications'' occur.
-
address@hidden
-* Default Simplifications::
-* Algebraic Simplifications::
-* Unsafe Simplifications::
-* Simplification of Units::
address@hidden menu
-
address@hidden Default Simplifications, Algebraic Simplifications, Simplifying
Formulas, Simplifying Formulas
address@hidden Default Simplifications
-
address@hidden
address@hidden Default simplifications
-This section describes the ``default simplifications,'' those which are
-normally applied to all results. For example, if you enter the variable
address@hidden on the stack twice and push @kbd{+}, Calc's default
-simplifications automatically change @expr{x + x} to @expr{2 x}.
-
-The @kbd{m O} command turns off the default simplifications, so that
address@hidden + x} will remain in this form unless you give an explicit
-``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic
-Manipulation}. The @kbd{m D} command turns the default simplifications
-back on.
-
-The most basic default simplification is the evaluation of functions.
-For example, @expr{2 + 3} is evaluated to @expr{5}, and @address@hidden(9)}
-is evaluated to @expr{3}. Evaluation does not occur if the arguments
-to a function are somehow of the wrong type @address@hidden([2,3,4])}),
-range (@address@hidden(90)}), or number (@address@hidden(3,5)}),
-or if the function name is not recognized (@address@hidden(5)}), or if
-Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
-(@address@hidden(2)}).
-
-Calc simplifies (evaluates) the arguments to a function before it
-simplifies the function itself. Thus @address@hidden(5+4)} is
-simplified to @address@hidden(9)} before the @code{sqrt} function
-itself is applied. There are very few exceptions to this rule:
address@hidden, @code{lambda}, and @code{condition} (the @code{::}
-operator) do not evaluate their arguments, @code{if} (the @code{? :}
-operator) does not evaluate all of its arguments, and @code{evalto}
-does not evaluate its lefthand argument.
-
-Most commands apply the default simplifications to all arguments they
-take from the stack, perform a particular operation, then simplify
-the result before pushing it back on the stack. In the common special
-case of regular arithmetic commands like @kbd{+} and @kbd{Q} address@hidden,
-the arguments are simply popped from the stack and collected into a
-suitable function call, which is then simplified (the arguments being
-simplified first as part of the process, as described above).
-
-The default simplifications are too numerous to describe completely
-here, but this section will describe the ones that apply to the
-major arithmetic operators. This list will be rather technical in
-nature, and will probably be interesting to you only if you are
-a serious user of Calc's algebra facilities.
-
address@hidden
-\bigskip
address@hidden tex
-
-As well as the simplifications described here, if you have stored
-any rewrite rules in the variable @code{EvalRules} then these rules
-will also be applied before any built-in default simplifications.
address@hidden Rewrites}, for details.
-
address@hidden
-\bigskip
address@hidden tex
-
-And now, on with the default simplifications:
-
-Arithmetic operators like @kbd{+} and @kbd{*} always take two
-arguments in Calc's internal form. Sums and products of three or
-more terms are arranged by the associative law of algebra into
-a left-associative form for sums, @expr{((a + b) + c) + d}, and
-a right-associative form for products, @expr{a * (b * (c * d))}.
-Formulas like @expr{(a + b) + (c + d)} are rearranged to
-left-associative form, though this rarely matters since Calc's
-algebra commands are designed to hide the inner structure of
-sums and products as much as possible. Sums and products in
-their proper associative form will be written without parentheses
-in the examples below.
-
-Sums and products are @emph{not} rearranged according to the
-commutative law (@expr{a + b} to @expr{b + a}) except in a few
-special cases described below. Some algebra programs always
-rearrange terms into a canonical order, which enables them to
-see that @expr{a b + b a} can be simplified to @expr{2 a b}.
-Calc assumes you have put the terms into the order you want
-and generally leaves that order alone, with the consequence
-that formulas like the above will only be simplified if you
-explicitly give the @kbd{a s} command. @xref{Algebraic
-Simplifications}.
-
-Differences @expr{a - b} are treated like sums @expr{a + (-b)}
-for purposes of simplification; one of the default simplifications
-is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
-represents a ``negative-looking'' term, into @expr{a - b} form.
-``Negative-looking'' means negative numbers, negated formulas like
address@hidden, and products or quotients in which either term is
-negative-looking.
-
-Other simplifications involving negation are @expr{-(-x)} to @expr{x};
address@hidden(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
-negative-looking, simplified by negating that term, or else where
address@hidden or @expr{b} is any number, by negating that number;
address@hidden(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
-(This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
-cases where the order of terms in a sum is changed by the default
-simplifications.)
-
-The distributive law is used to simplify sums in some cases:
address@hidden x + b x} to @expr{(a + b) x}, where @expr{a} represents
-a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
-and similarly for @expr{b}. Use the @kbd{a c}, @address@hidden f}}, or
address@hidden M} commands to merge sums with non-numeric coefficients
-using the distributive law.
-
-The distributive law is only used for sums of two terms, or
-for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
-is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
-is not simplified. The reason is that comparing all terms of a
-sum with one another would require time proportional to the
-square of the number of terms; Calc relegates potentially slow
-operations like this to commands that have to be invoked
-explicitly, like @kbd{a s}.
-
-Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
-A consequence of the above rules is that @expr{0 - a} is simplified
-to @expr{-a}.
-
address@hidden
-\bigskip
address@hidden tex
-
-The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
address@hidden(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
address@hidden a} and @expr{a 0} are simplified to @expr{0}, except that
-in Matrix mode where @expr{a} is not provably scalar the result
-is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
-infinite the result is @samp{nan}.
-
-Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
-where this occurs for negated formulas but not for regular negative
-numbers.
-
-Products are commuted only to move numbers to the front:
address@hidden b 2} is commuted to @expr{2 a b}.
-
-The product @expr{a (b + c)} is distributed over the sum only if
address@hidden and at least one of @expr{b} and @expr{c} are numbers:
address@hidden (x + 3)} goes to @expr{2 x + 6}. The formula
address@hidden(-a) (b - c)}, where @expr{-a} is a negative number, is
-rewritten to @expr{a (c - b)}.
-
-The distributive law of products and powers is used for adjacent
-terms of the product: @expr{x^a x^b} goes to
address@hidden @math{x^{a+b}}
address@hidden @expr{x^(a+b)}
-where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
-or the implicit one-half of @address@hidden(x)}, and similarly for
address@hidden The result is written using @samp{sqrt} or @samp{1/sqrt}
-if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
-If the sum of the powers is zero, the product is simplified to
address@hidden or to @samp{idn(1)} if Matrix mode is enabled.
-
-The product of a negative power times anything but another negative
-power is changed to use division:
address@hidden @math{x^{-2} y}
address@hidden @expr{x^(-2) y}
-goes to @expr{y / x^2} unless Matrix mode is
-in effect and neither @expr{x} nor @expr{y} are scalar (in which
-case it is considered unsafe to rearrange the order of the terms).
-
-Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
address@hidden(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
-
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-
-Simplifications for quotients are analogous to those for products.
-The quotient @expr{0 / x} is simplified to @expr{0}, with the same
-exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
-and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
-respectively.
-
-The quotient @expr{x / 0} is left unsimplified or changed to an
-infinite quantity, as directed by the current infinite mode.
address@hidden Mode}.
-
-The expression
address@hidden @math{a / b^{-c}}
address@hidden @expr{a / b^(-c)}
-is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
-power. Also, @expr{1 / b^c} is changed to
address@hidden @math{b^{-c}}
address@hidden @expr{b^(-c)}
-for any power @expr{c}.
-
-Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
address@hidden(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
-goes to @expr{(a c) / b} unless Matrix mode prevents this
-rearrangement. Similarly, @expr{a / (b:c)} is simplified to
address@hidden(c:b) a} for any fraction @expr{b:c}.
-
-The distributive law is applied to @expr{(a + b) / c} only if
address@hidden and at least one of @expr{a} and @expr{b} are numbers.
-Quotients of powers and square roots are distributed just as
-described for multiplication.
-
-Quotients of products cancel only in the leading terms of the
-numerator and denominator. In other words, @expr{a x b / a y b}
-is cancelled to @expr{x b / y b} but not to @expr{x / y}. Once
-again this is because full cancellation can be slow; use @kbd{a s}
-to cancel all terms of the quotient.
-
-Quotients of negative-looking values are simplified according
-to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
-to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
-
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-
-The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
-in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
-unless @expr{x} is a negative number, complex number or zero.
-If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
-infinity or an unsimplified formula according to the current infinite
-mode. The expression @expr{0^0} is simplified to @expr{1}.
-
-Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
-are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
-is an integer, or if either @expr{a} or @expr{b} are nonnegative
-real numbers. Powers of powers @expr{(a^b)^c} are simplified to
address@hidden @math{a^{b c}}
address@hidden @expr{a^(b c)}
-only when @expr{c} is an integer and @expr{b c} also
-evaluates to an integer. Without these restrictions these simplifications
-would not be safe because of problems with principal values.
-(In other words,
address@hidden @math{((-3)^{1/2})^2}
address@hidden @expr{((-3)^1:2)^2}
-is safe to simplify, but
address@hidden @math{((-3)^2)^{1/2}}
address@hidden @expr{((-3)^2)^1:2}
-is not.) @xref{Declarations}, for ways to inform Calc that your
-variables satisfy these requirements.
-
-As a special case of this rule, @address@hidden(x)^n} is simplified to
address@hidden @math{x^{n/2}}
address@hidden @expr{x^(n/2)}
-only for even integers @expr{n}.
-
-If @expr{a} is known to be real, @expr{b} is an even integer, and
address@hidden is a half- or quarter-integer, then @expr{(a^b)^c} is
-simplified to @address@hidden(a^(b c))}.
-
-Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
-even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
-for any negative-looking expression @expr{-a}.
-
-Square roots @address@hidden(x)} generally act like one-half powers
address@hidden @math{x^{1:2}}
address@hidden @expr{x^1:2}
-for the purposes of the above-listed simplifications.
-
-Also, note that
address@hidden @math{1 / x^{1:2}}
address@hidden @expr{1 / x^1:2}
-is changed to
address@hidden @math{x^{-1:2}},
address@hidden @expr{x^(-1:2)},
-but @expr{1 / @tfn{sqrt}(x)} is left alone.
-
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-\bigskip
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-
-Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
-following rules: @address@hidden(a) + b} to @expr{a + b} if @expr{b}
-is provably scalar, or expanded out if @expr{b} is a matrix;
address@hidden@tfn{idn}(a) + @tfn{idn}(b)} to @address@hidden(a + b)};
address@hidden@tfn{idn}(a)} to @address@hidden(-a)}; @expr{a @tfn{idn}(b)} to
address@hidden@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
-if @expr{a} is provably non-scalar; @address@hidden(a) @tfn{idn}(b)} to
address@hidden@tfn{idn}(a b)}; analogous simplifications for quotients involving
address@hidden; and @address@hidden(a)^n} to @address@hidden(a^n)} where
address@hidden is an integer.
-
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-\bigskip
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-
-The @code{floor} function and other integer truncation functions
-vanish if the argument is provably integer-valued, so that
address@hidden@tfn{floor}(@tfn{round}(x))} simplifies to @address@hidden(x)}.
-Also, combinations of @code{float}, @code{floor} and its friends,
-and @code{ffloor} and its friends, are simplified in appropriate
-ways. @xref{Integer Truncation}.
-
-The expression @address@hidden(-x)} changes to @address@hidden(x)}.
-The expression @address@hidden(@tfn{abs}(x))} changes to
address@hidden@tfn{abs}(x)}; in fact, @address@hidden(x)} changes to @expr{x}
or
address@hidden if @expr{x} is provably nonnegative or nonpositive
-(@pxref{Declarations}).
-
-While most functions do not recognize the variable @code{i} as an
-imaginary number, the @code{arg} function does handle the two cases
address@hidden@tfn{arg}(@tfn{i})} and @address@hidden(address@hidden)} just for
convenience.
-
-The expression @address@hidden(@tfn{conj}(x))} simplifies to @expr{x}.
-Various other expressions involving @code{conj}, @code{re}, and
address@hidden are simplified, especially if some of the arguments are
-provably real or involve the constant @code{i}. For example,
address@hidden@tfn{conj}(a + b i)} is changed to
address@hidden@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if
@expr{a}
-and @expr{b} are known to be real.
-
-Functions like @code{sin} and @code{arctan} generally don't have
-any default simplifications beyond simply evaluating the functions
-for suitable numeric arguments and infinity. The @kbd{a s} command
-described in the next section does provide some simplifications for
-these functions, though.
-
-One important simplification that does occur is that
address@hidden@tfn{ln}(@tfn{e})} is simplified to 1, and
@address@hidden(@tfn{e}^x)} is
-simplified to @expr{x} for any @expr{x}. This occurs even if you have
-stored a different value in the Calc variable @samp{e}; but this would
-be a bad idea in any case if you were also using natural logarithms!
-
-Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
address@hidden@var{a} > @var{b}} and so on. Equations and inequalities where
both sides
-are either negative-looking or zero are simplified by negating both sides
-and reversing the inequality. While it might seem reasonable to simplify
address@hidden to @expr{x}, this would not be valid in general because
address@hidden is 1, not 2.
-
-Most other Calc functions have few if any default simplifications
-defined, aside of course from evaluation when the arguments are
-suitable numbers.
-
address@hidden Algebraic Simplifications, Unsafe Simplifications, Default
Simplifications, Simplifying Formulas
address@hidden Algebraic Simplifications
-
address@hidden
address@hidden Algebraic simplifications
-The @kbd{a s} command makes simplifications that may be too slow to
-do all the time, or that may not be desirable all of the time.
-If you find these simplifications are worthwhile, you can type
address@hidden A} to have Calc apply them automatically.
-
-This section describes all simplifications that are performed by
-the @kbd{a s} command. Note that these occur in addition to the
-default simplifications; even if the default simplifications have
-been turned off by an @kbd{m O} command, @kbd{a s} will turn them
-back on temporarily while it simplifies the formula.
-
-There is a variable, @code{AlgSimpRules}, in which you can put rewrites
-to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules},
-but without the special restrictions. Basically, the simplifier does
address@hidden@w{a r} AlgSimpRules} with an infinite repeat count on the whole
-expression being simplified, then it traverses the expression applying
-the built-in rules described below. If the result is different from
-the original expression, the process repeats with the default
-simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
-then the built-in simplifications, and so on.
-
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-
-Sums are simplified in two ways. Constant terms are commuted to the
-end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
-The only exception is that a constant will not be commuted away
-from the first position of a difference, i.e., @expr{2 - x} is not
-commuted to @expr{-x + 2}.
-
-Also, terms of sums are combined by the distributive law, as in
address@hidden + y + 2 x} to @expr{y + 3 x}. This always occurs for
-adjacent terms, but @kbd{a s} compares all pairs of terms including
-non-adjacent ones.
-
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-
-Products are sorted into a canonical order using the commutative
-law. For example, @expr{b c a} is commuted to @expr{a b c}.
-This allows easier comparison of products; for example, the default
-simplifications will not change @expr{x y + y x} to @expr{2 x y},
-but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
-and then the default simplifications are able to recognize a sum
-of identical terms.
-
-The canonical ordering used to sort terms of products has the
-property that real-valued numbers, interval forms and infinities
-come first, and are sorted into increasing order. The @kbd{V S}
-command uses the same ordering when sorting a vector.
-
-Sorting of terms of products is inhibited when Matrix mode is
-turned on; in this case, Calc will never exchange the order of
-two terms unless it knows at least one of the terms is a scalar.
-
-Products of powers are distributed by comparing all pairs of
-terms, using the same method that the default simplifications
-use for adjacent terms of products.
-
-Even though sums are not sorted, the commutative law is still
-taken into account when terms of a product are being compared.
-Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
-A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
-be simplified to @expr{-(x - y)^2}; Calc does not notice that
-one term can be written as a constant times the other, even if
-that constant is @mathit{-1}.
-
-A fraction times any expression, @expr{(a:b) x}, is changed to
-a quotient involving integers: @expr{a x / b}. This is not
-done for floating-point numbers like @expr{0.5}, however. This
-is one reason why you may find it convenient to turn Fraction mode
-on while doing algebra; @pxref{Fraction Mode}.
-
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-
-Quotients are simplified by comparing all terms in the numerator
-with all terms in the denominator for possible cancellation using
-the distributive law. For example, @expr{a x^2 b / c x^3 d} will
-cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
-(The terms in the denominator will then be rearranged to @expr{c d x}
-as described above.) If there is any common integer or fractional
-factor in the numerator and denominator, it is cancelled out;
-for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
-
-Non-constant common factors are not found even by @kbd{a s}. To
-cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
-use @kbd{j M} on the product @expr{a x} to Merge the numerator to
address@hidden (1+x)}, which can then be simplified successfully.
-
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-\bigskip
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-
-Integer powers of the variable @code{i} are simplified according
-to the identity @expr{i^2 = -1}. If you store a new value other
-than the complex number @expr{(0,1)} in @code{i}, this simplification
-will no longer occur. This is done by @kbd{a s} instead of by default
-in case someone (unwisely) uses the name @code{i} for a variable
-unrelated to complex numbers; it would be unfortunate if Calc
-quietly and automatically changed this formula for reasons the
-user might not have been thinking of.
-
-Square roots of integer or rational arguments are simplified in
-several ways. (Note that these will be left unevaluated only in
-Symbolic mode.) First, square integer or rational factors are
-pulled out so that @address@hidden(8)} is rewritten as
address@hidden @math{2\,@tfn{sqrt}(2)}.
address@hidden @expr{2 sqrt(2)}.
-Conceptually speaking this implies factoring the argument into primes
-and moving pairs of primes out of the square root, but for reasons of
-efficiency Calc only looks for primes up to 29.
-
-Square roots in the denominator of a quotient are moved to the
-numerator: @expr{1 / @tfn{sqrt}(3)} changes to @address@hidden(3) / 3}.
-The same effect occurs for the square root of a fraction:
address@hidden@tfn{sqrt}(2:3)} changes to @address@hidden(6) / 3}.
-
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-
-The @code{%} (modulo) operator is simplified in several ways
-when the modulus @expr{M} is a positive real number. First, if
-the argument is of the form @expr{x + n} for some real number
address@hidden, then @expr{n} is itself reduced modulo @expr{M}. For
-example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
-
-If the argument is multiplied by a constant, and this constant
-has a common integer divisor with the modulus, then this factor is
-cancelled out. For example, @samp{12 x % 15} is changed to
address@hidden (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
-is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
-not seem ``simpler,'' they allow Calc to discover useful information
-about modulo forms in the presence of declarations.
-
-If the modulus is 1, then Calc can use @code{int} declarations to
-evaluate the expression. For example, the idiom @samp{x % 2} is
-often used to check whether a number is odd or even. As described
-above, @address@hidden n % 2}} and @samp{(2 n + 1) % 2} are simplified to
address@hidden (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
-can simplify these to 0 and 1 (respectively) if @code{n} has been
-declared to be an integer.
-
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-\bigskip
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-
-Trigonometric functions are simplified in several ways. Whenever a
-products of two trigonometric functions can be replaced by a single
-function, the replacement is made; for example,
address@hidden@tfn{tan}(x) @tfn{cos}(x)} is simplified to @address@hidden(x)}.
-Reciprocals of trigonometric functions are replaced by their reciprocal
-function; for example, @expr{1/@tfn{sec}(x)} is simplified to
address@hidden@tfn{cos}(x)}. The corresponding simplifications for the
-hyperbolic functions are also handled.
-
-Trigonometric functions of their inverse functions are
-simplified. The expression @address@hidden(@tfn{arcsin}(x))} is
-simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
-Trigonometric functions of inverses of different trigonometric
-functions can also be simplified, as in @address@hidden(@tfn{arccos}(x))}
-to @address@hidden(1 - x^2)}.
-
-If the argument to @code{sin} is negative-looking, it is simplified to
address@hidden@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
-Finally, certain special values of the argument are recognized;
address@hidden and Hyperbolic Functions}.
-
-Hyperbolic functions of their inverses and of negative-looking
-arguments are also handled, as are exponentials of inverse
-hyperbolic functions.
-
-No simplifications for inverse trigonometric and hyperbolic
-functions are known, except for negative arguments of @code{arcsin},
address@hidden, @code{arcsinh}, and @code{arctanh}. Note that
address@hidden@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
address@hidden, since this only correct within an integer multiple of
address@hidden @math{2 \pi}
address@hidden @expr{2 pi}
-radians or 360 degrees. However, @address@hidden(@tfn{sinh}(x))} is
-simplified to @expr{x} if @expr{x} is known to be real.
-
-Several simplifications that apply to logarithms and exponentials
-are that @address@hidden(@tfn{ln}(x))},
address@hidden @address@hidden(x)}},
address@hidden @address@hidden(x)},
-and
address@hidden @math{10^{{\rm log10}(x)}}
address@hidden @address@hidden(x)}
-all reduce to @expr{x}. Also, @address@hidden(@tfn{exp}(x))}, etc., can
-reduce to @expr{x} if @expr{x} is provably real. The form
address@hidden@tfn{exp}(x)^y} is simplified to @address@hidden(x y)}. If
@expr{x}
-is a suitable multiple of
address@hidden @math{\pi i}
address@hidden @expr{pi i}
-(as described above for the trigonometric functions), then
address@hidden@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
address@hidden@tfn{ln}(x)} is simplified to a form involving @code{pi} and
address@hidden where @expr{x} is provably negative, positive imaginary, or
-negative imaginary.
-
-The error functions @code{erf} and @code{erfc} are simplified when
-their arguments are negative-looking or are calls to the @code{conj}
-function.
-
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-
-Equations and inequalities are simplified by cancelling factors
-of products, quotients, or sums on both sides. Inequalities
-change sign if a negative multiplicative factor is cancelled.
-Non-constant multiplicative factors as in @expr{a b = a c} are
-cancelled from equations only if they are provably nonzero (generally
-because they were declared so; @pxref{Declarations}). Factors
-are cancelled from inequalities only if they are nonzero and their
-sign is known.
-
-Simplification also replaces an equation or inequality with
-1 or 0 (``true'' or ``false'') if it can through the use of
-declarations. If @expr{x} is declared to be an integer greater
-than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
-all simplified to 0, but @expr{x > 3} is simplified to 1.
-By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
-as is @expr{x^2 >= 0} if @expr{x} is known to be real.
-
address@hidden Unsafe Simplifications, Simplification of Units, Algebraic
Simplifications, Simplifying Formulas
address@hidden ``Unsafe'' Simplifications
-
address@hidden
address@hidden Unsafe simplifications
address@hidden Extended simplification
address@hidden a e
address@hidden calc-simplify-extended
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden esimplify
-The @kbd{a e} (@code{calc-simplify-extended}) address@hidden command
-is like @kbd{a s}
-except that it applies some additional simplifications which are not
-``safe'' in all cases. Use this only if you know the values in your
-formula lie in the restricted ranges for which these simplifications
-are valid. The symbolic integrator uses @kbd{a e};
-one effect of this is that the integrator's results must be used with
-caution. Where an integral table will often attach conditions like
-``for positive @expr{a} only,'' Calc (like most other symbolic
-integration programs) will simply produce an unqualified result.
-
-Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
-to type @kbd{C-u -3 a v}, which does extended simplification only
-on the top level of the formula without affecting the sub-formulas.
-In fact, @kbd{C-u -3 j v} allows you to target extended simplification
-to any specific part of a formula.
-
-The variable @code{ExtSimpRules} contains rewrites to be applied by
-the @kbd{a e} command. These are applied in addition to
address@hidden and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
-step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
-
-Following is a complete list of ``unsafe'' simplifications performed
-by @kbd{a e}.
-
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-\bigskip
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-
-Inverse trigonometric or hyperbolic functions, called with their
-corresponding non-inverse functions as arguments, are simplified
-by @kbd{a e}. For example, @address@hidden(@tfn{sin}(x))} changes
-to @expr{x}. Also, @address@hidden(@tfn{cos}(x))} and
address@hidden@tfn{arccos}(@tfn{sin}(x))} both change to @address@hidden/2 - x}.
-These simplifications are unsafe because they are valid only for
-values of @expr{x} in a certain range; outside that range, values
-are folded down to the 360-degree range that the inverse trigonometric
-functions always produce.
-
-Powers of powers @expr{(x^a)^b} are simplified to
address@hidden @math{x^{a b}}
address@hidden @expr{x^(a b)}
-for all @expr{a} and @expr{b}. These results will be valid only
-in a restricted range of @expr{x}; for example, in
address@hidden @math{(x^2)^{1:2}}
address@hidden @expr{(x^2)^1:2}
-the powers cancel to get @expr{x}, which is valid for positive values
-of @expr{x} but not for negative or complex values.
-
-Similarly, @address@hidden(x^a)} and @address@hidden(x)^a} are both
-simplified (possibly unsafely) to
address@hidden @math{x^{a/2}}.
address@hidden @expr{x^(a/2)}.
-
-Forms like @address@hidden(1 - sin(x)^2)} are simplified to, e.g.,
address@hidden@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
address@hidden, @code{tan}, @code{sinh}, and @code{cosh}.
-
-Arguments of square roots are partially factored to look for
-squared terms that can be extracted. For example,
address@hidden@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
address@hidden b @tfn{sqrt}(a+b)}.
-
-The simplifications of @address@hidden(@tfn{exp}(x))},
address@hidden@tfn{ln}(@tfn{e}^x)}, and @address@hidden(10^x)} to @expr{x} are
also
-unsafe because of problems with principal values (although these
-simplifications are safe if @expr{x} is known to be real).
-
-Common factors are cancelled from products on both sides of an
-equation, even if those factors may be zero: @expr{a x / b x}
-to @expr{a / b}. Such factors are never cancelled from
-inequalities: Even @kbd{a e} is not bold enough to reduce
address@hidden x < b x} to @expr{a < b} (or @expr{a > b}, depending
-on whether you believe @expr{x} is positive or negative).
-The @kbd{a M /} command can be used to divide a factor out of
-both sides of an inequality.
-
address@hidden Simplification of Units, , Unsafe Simplifications, Simplifying
Formulas
address@hidden Simplification of Units
-
address@hidden
-The simplifications described in this section are applied by the
address@hidden s} (@code{calc-simplify-units}) command. These are in addition
-to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
-earlier. @xref{Basic Operations on Units}.
-
-The variable @code{UnitSimpRules} contains rewrites to be applied by
-the @kbd{u s} command. These are applied in addition to @code{EvalRules}
-and @code{AlgSimpRules}.
-
-Scalar mode is automatically put into effect when simplifying units.
address@hidden Mode}.
-
-Sums @expr{a + b} involving units are simplified by extracting the
-units of @expr{a} as if by the @kbd{u x} command (call the result
address@hidden), then simplifying the expression @expr{b / u_a}
-using @kbd{u b} and @kbd{u s}. If the result has units then the sum
-is inconsistent and is left alone. Otherwise, it is rewritten
-in terms of the units @expr{u_a}.
-
-If units auto-ranging mode is enabled, products or quotients in
-which the first argument is a number which is out of range for the
-leading unit are modified accordingly.
-
-When cancelling and combining units in products and quotients,
-Calc accounts for unit names that differ only in the prefix letter.
-For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
-However, compatible but different units like @code{ft} and @code{in}
-are not combined in this way.
-
-Quotients @expr{a / b} are simplified in three additional ways. First,
-if @expr{b} is a number or a product beginning with a number, Calc
-computes the reciprocal of this number and moves it to the numerator.
-
-Second, for each pair of unit names from the numerator and denominator
-of a quotient, if the units are compatible (e.g., they are both
-units of area) then they are replaced by the ratio between those
-units. For example, in @samp{3 s in N / kg cm} the units
address@hidden / cm} will be replaced by @expr{2.54}.
-
-Third, if the units in the quotient exactly cancel out, so that
-a @kbd{u b} command on the quotient would produce a dimensionless
-number for an answer, then the quotient simplifies to that number.
-
-For powers and square roots, the ``unsafe'' simplifications
address@hidden(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
-and @expr{(a^b)^c} to
address@hidden @math{a^{b c}}
address@hidden @expr{a^(b c)}
-are done if the powers are real numbers. (These are safe in the context
-of units because all numbers involved can reasonably be assumed to be
-real.)
-
-Also, if a unit name is raised to a fractional power, and the
-base units in that unit name all occur to powers which are a
-multiple of the denominator of the power, then the unit name
-is expanded out into its base units, which can then be simplified
-according to the previous paragraph. For example, @samp{acre^1.5}
-is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
-is defined in terms of @samp{m^2}, and that the 2 in the power of
address@hidden is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
-replaced by approximately
address@hidden @math{(4046 m^2)^{1.5}}
address@hidden @expr{(4046 m^2)^1.5},
-which is then changed to
address@hidden @math{4046^{1.5} \, (m^2)^{1.5}},
address@hidden @expr{4046^1.5 (m^2)^1.5},
-then to @expr{257440 m^3}.
-
-The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
-as well as @code{floor} and the other integer truncation functions,
-applied to unit names or products or quotients involving units, are
-simplified. For example, @samp{round(1.6 in)} is changed to
address@hidden(1.6) round(in)}; the lefthand term evaluates to 2,
-and the righthand term simplifies to @code{in}.
-
-The functions @code{sin}, @code{cos}, and @code{tan} with arguments
-that have angular units like @code{rad} or @code{arcmin} are
-simplified by converting to base units (radians), then evaluating
-with the angular mode temporarily set to radians.
-
address@hidden Polynomials, Calculus, Simplifying Formulas, Algebra
address@hidden Polynomials
-
-A @dfn{polynomial} is a sum of terms which are coefficients times
-various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
-is a polynomial in @expr{x}. Some formulas can be considered
-polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
-is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
-are often numbers, but they may in general be any formulas not
-involving the base variable.
-
address@hidden a f
address@hidden calc-factor
address@hidden factor
-The @kbd{a f} (@code{calc-factor}) address@hidden command factors a
-polynomial into a product of terms. For example, the polynomial
address@hidden + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
-example, @expr{a c + b d + b c + a d} is factored into the product
address@hidden(a + b) (c + d)}.
-
-Calc currently has three algorithms for factoring. Formulas which are
-linear in several variables, such as the second example above, are
-merged according to the distributive law. Formulas which are
-polynomials in a single variable, with constant integer or fractional
-coefficients, are factored into irreducible linear and/or quadratic
-terms. The first example above factors into three linear terms
-(@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
-which do not fit the above criteria are handled by the algebraic
-rewrite mechanism.
-
-Calc's polynomial factorization algorithm works by using the general
-root-finding command (@address@hidden P}}) to solve for the roots of the
-polynomial. It then looks for roots which are rational numbers
-or complex-conjugate pairs, and converts these into linear and
-quadratic terms, respectively. Because it uses floating-point
-arithmetic, it may be unable to find terms that involve large
-integers (whose number of digits approaches the current precision).
-Also, irreducible factors of degree higher than quadratic are not
-found, and polynomials in more than one variable are not treated.
-(A more robust factorization algorithm may be included in a future
-version of Calc.)
-
address@hidden FactorRules
address@hidden
address@hidden
address@hidden ignore
address@hidden thecoefs
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden thefactors
-The rewrite-based factorization method uses rules stored in the variable
address@hidden @xref{Rewrite Rules}, for a discussion of the
-operation of rewrite rules. The default @code{FactorRules} are able
-to factor quadratic forms symbolically into two linear terms,
address@hidden(a x + b) (c x + d)}. You can edit these rules to include other
-cases if you wish. To use the rules, Calc builds the formula
address@hidden(x, [a, b, c, ...])} where @code{x} is the polynomial
-base variable and @code{a}, @code{b}, etc., are polynomial coefficients
-(which may be numbers or formulas). The constant term is written first,
-i.e., in the @code{a} position. When the rules complete, they should have
-changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
-where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
-Calc then multiplies these terms together to get the complete
-factored form of the polynomial. If the rules do not change the
address@hidden call to a @code{thefactors} call, @kbd{a f} leaves the
-polynomial alone on the assumption that it is unfactorable. (Note that
-the function names @code{thecoefs} and @code{thefactors} are used only
-as placeholders; there are no actual Calc functions by those names.)
-
address@hidden H a f
address@hidden factors
-The @kbd{H a f} address@hidden command also factors a polynomial,
-but it returns a list of factors instead of an expression which is the
-product of the factors. Each factor is represented by a sub-vector
-of the factor, and the power with which it appears. For example,
address@hidden + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
-in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
-If there is an overall numeric factor, it always comes first in the list.
-The functions @code{factor} and @code{factors} allow a second argument
-when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
-respect to the specific variable @expr{v}. The default is to factor with
-respect to all the variables that appear in @expr{x}.
-
address@hidden a c
address@hidden calc-collect
address@hidden collect
-The @kbd{a c} (@code{calc-collect}) address@hidden command rearranges a
-formula as a
-polynomial in a given variable, ordered in decreasing powers of that
-variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
-the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
-and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
-The polynomial will be expanded out using the distributive law as
-necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
address@hidden - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
-not be expanded.
-
-The ``variable'' you specify at the prompt can actually be any
-expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
-by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
-in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
-treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
-
address@hidden a x
address@hidden calc-expand
address@hidden expand
-The @kbd{a x} (@code{calc-expand}) address@hidden command expands an
-expression by applying the distributive law everywhere. It applies to
-products, quotients, and powers involving sums. By default, it fully
-distributes all parts of the expression. With a numeric prefix argument,
-the distributive law is applied only the specified number of times, then
-the partially expanded expression is left on the stack.
-
-The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
address@hidden x} if you want to expand all products of sums in your formula.
-Use @kbd{j D} if you want to expand a particular specified term of
-the formula. There is an exactly analogous correspondence between
address@hidden f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
-also know many other kinds of expansions, such as
address@hidden(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
-do not do.)
-
-Calc's automatic simplifications will sometimes reverse a partial
-expansion. For example, the first step in expanding @expr{(x+1)^3} is
-to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
-to put this formula onto the stack, though, Calc will automatically
-simplify it back to @expr{(x+1)^3} form. The solution is to turn
-simplification off first (@pxref{Simplification Modes}), or to run
address@hidden x} without a numeric prefix argument so that it expands all
-the way in one step.
-
address@hidden a a
address@hidden calc-apart
address@hidden apart
-The @kbd{a a} (@code{calc-apart}) address@hidden command expands a
-rational function by partial fractions. A rational function is the
-quotient of two polynomials; @code{apart} pulls this apart into a
-sum of rational functions with simple denominators. In algebraic
-notation, the @code{apart} function allows a second argument that
-specifies which variable to use as the ``base''; by default, Calc
-chooses the base variable automatically.
-
address@hidden a n
address@hidden calc-normalize-rat
address@hidden nrat
-The @kbd{a n} (@code{calc-normalize-rat}) address@hidden command
-attempts to arrange a formula into a quotient of two polynomials.
-For example, given @expr{1 + (a + b/c) / d}, the result would be
address@hidden(b + a c + c d) / c d}. The quotient is reduced, so that
address@hidden n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
-out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
-
address@hidden a \
address@hidden calc-poly-div
address@hidden pdiv
-The @kbd{a \} (@code{calc-poly-div}) address@hidden command divides
-two polynomials @expr{u} and @expr{v}, yielding a new polynomial
address@hidden If several variables occur in the inputs, the inputs are
-considered multivariate polynomials. (Calc divides by the variable
-with the largest power in @expr{u} first, or, in the case of equal
-powers, chooses the variables in alphabetical order.) For example,
-dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
-The remainder from the division, if any, is reported at the bottom
-of the screen and is also placed in the Trail along with the quotient.
-
-Using @code{pdiv} in algebraic notation, you can specify the particular
-variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
-If @code{pdiv} is given only two arguments (as is always the case with
-the @kbd{a \} command), then it does a multivariate division as outlined
-above.
-
address@hidden a %
address@hidden calc-poly-rem
address@hidden prem
-The @kbd{a %} (@code{calc-poly-rem}) address@hidden command divides
-two polynomials and keeps the remainder @expr{r}. The quotient
address@hidden is discarded. For any formulas @expr{a} and @expr{b}, the
-results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
-(This is analogous to plain @kbd{\} and @kbd{%}, which compute the
-integer quotient and remainder from dividing two numbers.)
-
address@hidden a /
address@hidden H a /
address@hidden calc-poly-div-rem
address@hidden pdivrem
address@hidden pdivide
-The @kbd{a /} (@code{calc-poly-div-rem}) address@hidden command
-divides two polynomials and reports both the quotient and the
-remainder as a vector @expr{[q, r]}. The @kbd{H a /} address@hidden
-command divides two polynomials and constructs the formula
address@hidden + r/b} on the stack. (Naturally if the remainder is zero,
-this will immediately simplify to @expr{q}.)
-
address@hidden a g
address@hidden calc-poly-gcd
address@hidden pgcd
-The @kbd{a g} (@code{calc-poly-gcd}) address@hidden command computes
-the greatest common divisor of two polynomials. (The GCD actually
-is unique only to within a constant multiplier; Calc attempts to
-choose a GCD which will be unsurprising.) For example, the @kbd{a n}
-command uses @kbd{a g} to take the GCD of the numerator and denominator
-of a quotient, then divides each by the result using @kbd{a \}. (The
-definition of GCD ensures that this division can take place without
-leaving a remainder.)
-
-While the polynomials used in operations like @kbd{a /} and @kbd{a g}
-often have integer coefficients, this is not required. Calc can also
-deal with polynomials over the rationals or floating-point reals.
-Polynomials with modulo-form coefficients are also useful in many
-applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
-automatically transforms this into a polynomial over the field of
-integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
-
-Congratulations and thanks go to Ove Ewerlid
-(@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
-polynomial routines used in the above commands.
-
address@hidden Polynomials}, for several useful functions for
-extracting the individual coefficients of a polynomial.
-
address@hidden Calculus, Solving Equations, Polynomials, Algebra
address@hidden Calculus
-
address@hidden
-The following calculus commands do not automatically simplify their
-inputs or outputs using @code{calc-simplify}. You may find it helps
-to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
-to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
-readable way.
-
address@hidden
-* Differentiation::
-* Integration::
-* Customizing the Integrator::
-* Numerical Integration::
-* Taylor Series::
address@hidden menu
-
address@hidden Differentiation, Integration, Calculus, Calculus
address@hidden Differentiation
-
address@hidden
address@hidden a d
address@hidden H a d
address@hidden calc-derivative
address@hidden deriv
address@hidden tderiv
-The @kbd{a d} (@code{calc-derivative}) address@hidden command computes
-the derivative of the expression on the top of the stack with respect to
-some variable, which it will prompt you to enter. Normally, variables
-in the formula other than the specified differentiation variable are
-considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
-the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
-instead, in which derivatives of variables are not reduced to zero
-unless those variables are known to be ``constant,'' i.e., independent
-of any other variables. (The built-in special variables like @code{pi}
-are considered constant, as are variables that have been declared
address@hidden; @pxref{Declarations}.)
-
-With a numeric prefix argument @var{n}, this command computes the
address@hidden derivative.
-
-When working with trigonometric functions, it is best to switch to
-Radians mode first (with @address@hidden r}}). The derivative of @samp{sin(x)}
-in degrees is @samp{(pi/180) cos(x)}, probably not the expected
-answer!
-
-If you use the @code{deriv} function directly in an algebraic formula,
-you can write @samp{deriv(f,x,x0)} which represents the derivative
-of @expr{f} with respect to @expr{x}, evaluated at the point
address@hidden @math{x=x_0}.
address@hidden @expr{x=x0}.
-
-If the formula being differentiated contains functions which Calc does
-not know, the derivatives of those functions are produced by adding
-primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
-produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
-derivative of @code{f}.
-
-For functions you have defined with the @kbd{Z F} command, Calc expands
-the functions according to their defining formulas unless you have
-also defined @code{f'} suitably. For example, suppose we define
address@hidden(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
-the formula @samp{sinc(2 x)}, the formula will be expanded to
address@hidden(2 x) / (2 x)} and differentiated. However, if we also
-define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
-result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
-
-For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
-to the first argument is written @samp{f'(x,y,z)}; derivatives with
-respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
-Various higher-order derivatives can be formed in the obvious way, e.g.,
address@hidden'@var{}'(x)} (the second derivative of @code{f}) or
address@hidden'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
-argument once).
-
address@hidden Integration, Customizing the Integrator, Differentiation,
Calculus
address@hidden Integration
-
address@hidden
address@hidden a i
address@hidden calc-integral
address@hidden integ
-The @kbd{a i} (@code{calc-integral}) address@hidden command computes the
-indefinite integral of the expression on the top of the stack with
-respect to a prompted-for variable. The integrator is not guaranteed to
-work for all integrable functions, but it is able to integrate several
-large classes of formulas. In particular, any polynomial or rational
-function (a polynomial divided by a polynomial) is acceptable.
-(Rational functions don't have to be in explicit quotient form, however;
address@hidden @math{x/(1+x^{-2})}
address@hidden @expr{x/(1+x^-2)}
-is not strictly a quotient of polynomials, but it is equivalent to
address@hidden/(x^2+1)}, which is.) Also, square roots of terms involving
address@hidden and @expr{x^2} may appear in rational functions being
-integrated. Finally, rational functions involving trigonometric or
-hyperbolic functions can be integrated.
-
-With an argument (@kbd{C-u a i}), this command will compute the definite
-integral of the expression on top of the stack. In this case, the
-command will again prompt for an integration variable, then prompt for a
-lower limit and an upper limit.
-
address@hidden
-If you use the @code{integ} function directly in an algebraic formula,
-you can also write @samp{integ(f,x,v)} which expresses the resulting
-indefinite integral in terms of variable @code{v} instead of @code{x}.
-With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
-integral from @code{a} to @code{b}.
address@hidden ifnottex
address@hidden
-If you use the @code{integ} function directly in an algebraic formula,
-you can also write @samp{integ(f,x,v)} which expresses the resulting
-indefinite integral in terms of variable @code{v} instead of @code{x}.
-With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
-integral $\int_a^b f(x) \, dx$.
address@hidden tex
-
-Please note that the current implementation of Calc's integrator sometimes
-produces results that are significantly more complex than they need to
-be. For example, the integral Calc finds for
address@hidden @math{1/(x+\sqrt{x^2+1})}
address@hidden @expr{1/(x+sqrt(x^2+1))}
-is several times more complicated than the answer Mathematica
-returns for the same input, although the two forms are numerically
-equivalent. Also, any indefinite integral should be considered to have
-an arbitrary constant of integration added to it, although Calc does not
-write an explicit constant of integration in its result. For example,
-Calc's solution for
address@hidden @math{1/(1+\tan x)}
address@hidden @expr{1/(1+tan(x))}
-differs from the solution given in the @emph{CRC Math Tables} by a
-constant factor of
address@hidden @math{\pi i / 2}
address@hidden @expr{pi i / 2},
-due to a different choice of constant of integration.
-
-The Calculator remembers all the integrals it has done. If conditions
-change in a way that would invalidate the old integrals, say, a switch
-from Degrees to Radians mode, then they will be thrown out. If you
-suspect this is not happening when it should, use the
address@hidden command; @pxref{Caches}.
-
address@hidden IntegLimit
-Calc normally will pursue integration by substitution or integration by
-parts up to 3 nested times before abandoning an approach as fruitless.
-If the integrator is taking too long, you can lower this limit by storing
-a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
-command is a convenient way to edit @code{IntegLimit}.) If this variable
-has no stored value or does not contain a nonnegative integer, a limit
-of 3 is used. The lower this limit is, the greater the chance that Calc
-will be unable to integrate a function it could otherwise handle. Raising
-this limit allows the Calculator to solve more integrals, though the time
-it takes may grow exponentially. You can monitor the integrator's actions
-by creating an Emacs buffer called @code{*Trace*}. If such a buffer
-exists, the @kbd{a i} command will write a log of its actions there.
-
-If you want to manipulate integrals in a purely symbolic way, you can
-set the integration nesting limit to 0 to prevent all but fast
-table-lookup solutions of integrals. You might then wish to define
-rewrite rules for integration by parts, various kinds of substitutions,
-and so on. @xref{Rewrite Rules}.
-
address@hidden Customizing the Integrator, Numerical Integration, Integration,
Calculus
address@hidden Customizing the Integrator
-
address@hidden
address@hidden IntegRules
-Calc has two built-in rewrite rules called @code{IntegRules} and
address@hidden which you can edit to define new integration
-methods. @xref{Rewrite Rules}. At each step of the integration process,
-Calc wraps the current integrand in a call to the fictitious function
address@hidden(@var{expr},@var{var})}, where @var{expr} is the
-integrand and @var{var} is the integration variable. If your rules
-rewrite this to be a plain formula (not a call to @code{integtry}), then
-Calc will use this formula as the integral of @var{expr}. For example,
-the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
-integrate a function @code{mysin} that acts like the sine function.
-Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
-will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
-automatically made various transformations on the integral to allow it
-to use your rule; integral tables generally give rules for
address@hidden(a x + b)}, but you don't need to use this much generality
-in your @code{IntegRules}.
-
address@hidden Exponential integral Ei(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden Ei
-As a more serious example, the expression @samp{exp(x)/x} cannot be
-integrated in terms of the standard functions, so the ``exponential
-integral'' function
address@hidden @math{{\rm Ei}(x)}
address@hidden @expr{Ei(x)}
-was invented to describe it.
-We can get Calc to do this integral in terms of a made-up @code{Ei}
-function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
-to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
-and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
-work with Calc's various built-in integration methods (such as
-integration by substitution) to solve a variety of other problems
-involving @code{Ei}: For example, now Calc will also be able to
-integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
-and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
-
-Your rule may do further integration by calling @code{integ}. For
-example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
-to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
-Note that @code{integ} was called with only one argument. This notation
-is allowed only within @code{IntegRules}; it means ``integrate this
-with respect to the same integration variable.'' If Calc is unable
-to integrate @code{u}, the integration that invoked @code{IntegRules}
-also fails. Thus integrating @samp{twice(f(x))} fails, returning the
-unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
-to call @code{integ} with two or more arguments, however; in this case,
-if @code{u} is not integrable, @code{twice} itself will still be
-integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
-then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
-
-If a rule instead produces the formula @samp{integsubst(@var{sexpr},
address@hidden)}, either replacing the top-level @code{integtry} call or
-nested anywhere inside the expression, then Calc will apply the
-substitution @address@hidden = @var{sexpr}(@var{svar})} to try to
-integrate the original @var{expr}. For example, the rule
address@hidden(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
-a square root in the integrand, it should attempt the substitution
address@hidden = sqrt(x)}. (This particular rule is unnecessary because
-Calc always tries ``obvious'' substitutions where @var{sexpr} actually
-appears in the integrand.) The variable @var{svar} may be the same
-as the @var{var} that appeared in the call to @code{integtry}, but
-it need not be.
-
-When integrating according to an @code{integsubst}, Calc uses the
-equation solver to find the inverse of @var{sexpr} (if the integrand
-refers to @var{var} anywhere except in subexpressions that exactly
-match @var{sexpr}). It uses the differentiator to find the derivative
-of @var{sexpr} and/or its inverse (it has two methods that use one
-derivative or the other). You can also specify these items by adding
-extra arguments to the @code{integsubst} your rules construct; the
-general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
address@hidden)}, where @var{sinv} is the inverse of @var{sexpr} (still
-written as a function of @var{svar}), and @var{sprime} is the
-derivative of @var{sexpr} with respect to @var{svar}. If you don't
-specify these things, and Calc is not able to work them out on its
-own with the information it knows, then your substitution rule will
-work only in very specific, simple cases.
-
-Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
-in other words, Calc stops rewriting as soon as any rule in your rule
-set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
-example above would keep on adding layers of @code{integsubst} calls
-forever!)
-
address@hidden IntegSimpRules
-Another set of rules, stored in @code{IntegSimpRules}, are applied
-every time the integrator uses @kbd{a s} to simplify an intermediate
-result. For example, putting the rule @samp{twice(x) := 2 x} into
address@hidden would tell Calc to convert the @code{twice}
-function into a form it knows whenever integration is attempted.
-
-One more way to influence the integrator is to define a function with
-the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
-integrator automatically expands such functions according to their
-defining formulas, even if you originally asked for the function to
-be left unevaluated for symbolic arguments. (Certain other Calc
-systems, such as the differentiator and the equation solver, also
-do this.)
-
address@hidden IntegAfterRules
-Sometimes Calc is able to find a solution to your integral, but it
-expresses the result in a way that is unnecessarily complicated. If
-this happens, you can either use @code{integsubst} as described
-above to try to hint at a more direct path to the desired result, or
-you can use @code{IntegAfterRules}. This is an extra rule set that
-runs after the main integrator returns its result; basically, Calc does
-an @kbd{a r IntegAfterRules} on the result before showing it to you.
-(It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
-to further simplify the result.) For example, Calc's integrator
-sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
-the default @code{IntegAfterRules} rewrite this into the more readable
-form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
address@hidden and @code{IntegAfterRules} are applied any number
-of times until no further changes are possible. Rewriting by
address@hidden occurs only after the main integrator has
-finished, not at every step as for @code{IntegRules} and
address@hidden
-
address@hidden Numerical Integration, Taylor Series, Customizing the
Integrator, Calculus
address@hidden Numerical Integration
-
address@hidden
address@hidden a I
address@hidden calc-num-integral
address@hidden ninteg
-If you want a purely numerical answer to an integration problem, you can
-use the @kbd{a I} (@code{calc-num-integral}) address@hidden command. This
-command prompts for an integration variable, a lower limit, and an
-upper limit. Except for the integration variable, all other variables
-that appear in the integrand formula must have stored values. (A stored
-value, if any, for the integration variable itself is ignored.)
-
-Numerical integration works by evaluating your formula at many points in
-the specified interval. Calc uses an ``open Romberg'' method; this means
-that it does not evaluate the formula actually at the endpoints (so that
-it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
-the Romberg method works especially well when the function being
-integrated is fairly smooth. If the function is not smooth, Calc will
-have to evaluate it at quite a few points before it can accurately
-determine the value of the integral.
-
-Integration is much faster when the current precision is small. It is
-best to set the precision to the smallest acceptable number of digits
-before you use @kbd{a I}. If Calc appears to be taking too long, press
address@hidden to halt it and try a lower precision. If Calc still appears
-to need hundreds of evaluations, check to make sure your function is
-well-behaved in the specified interval.
-
-It is possible for the lower integration limit to be @samp{-inf} (minus
-infinity). Likewise, the upper limit may be plus infinity. Calc
-internally transforms the integral into an equivalent one with finite
-limits. However, integration to or across singularities is not supported:
-The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
-by Calc's symbolic integrator, for example), but @kbd{a I} will fail
-because the integrand goes to infinity at one of the endpoints.
-
address@hidden Taylor Series, , Numerical Integration, Calculus
address@hidden Taylor Series
-
address@hidden
address@hidden a t
address@hidden calc-taylor
address@hidden taylor
-The @kbd{a t} (@code{calc-taylor}) address@hidden command computes a
-power series expansion or Taylor series of a function. You specify the
-variable and the desired number of terms. You may give an expression of
-the form @address@hidden = @var{a}} or @address@hidden - @var{a}} instead
-of just a variable to produce a Taylor expansion about the point @var{a}.
-You may specify the number of terms with a numeric prefix argument;
-otherwise the command will prompt you for the number of terms. Note that
-many series expansions have coefficients of zero for some terms, so you
-may appear to get fewer terms than you asked for.
-
-If the @kbd{a i} command is unable to find a symbolic integral for a
-function, you can get an approximation by integrating the function's
-Taylor series.
-
address@hidden Solving Equations, Numerical Solutions, Calculus, Algebra
address@hidden Solving Equations
-
address@hidden
address@hidden a S
address@hidden calc-solve-for
address@hidden solve
address@hidden Equations, solving
address@hidden Solving equations
-The @kbd{a S} (@code{calc-solve-for}) address@hidden command rearranges
-an equation to solve for a specific variable. An equation is an
-expression of the form @expr{L = R}. For example, the command @kbd{a S x}
-will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
-input is not an equation, it is treated like an equation of the
-form @expr{X = 0}.
-
-This command also works for inequalities, as in @expr{y < 3x + 6}.
-Some inequalities cannot be solved where the analogous equation could
-be; for example, solving
address@hidden @math{a < b \, c}
address@hidden @expr{a < b c}
-for @expr{b} is impossible
-without knowing the sign of @expr{c}. In this case, @kbd{a S} will
-produce the result
address@hidden @math{b \mathbin{\hbox{\code{!=}}} a/c}
address@hidden @expr{b != a/c}
-(using the not-equal-to operator) to signify that the direction of the
-inequality is now unknown. The inequality
address@hidden @math{a \le b \, c}
address@hidden @expr{a <= b c}
-is not even partially solved. @xref{Declarations}, for a way to tell
-Calc that the signs of the variables in a formula are in fact known.
-
-Two useful commands for working with the result of @kbd{a S} are
address@hidden .} (@pxref{Logical Operations}), which converts @expr{x = y/3 -
2}
-to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
-another formula with @expr{x} set equal to @expr{y/3 - 2}.
-
address@hidden
-* Multiple Solutions::
-* Solving Systems of Equations::
-* Decomposing Polynomials::
address@hidden menu
-
address@hidden Multiple Solutions, Solving Systems of Equations, Solving
Equations, Solving Equations
address@hidden Multiple Solutions
-
address@hidden
address@hidden H a S
address@hidden fsolve
-Some equations have more than one solution. The Hyperbolic flag
-(@code{H a S}) address@hidden tells the solver to report the fully
-general family of solutions. It will invent variables @code{n1},
address@hidden, @dots{}, which represent independent arbitrary integers, and
address@hidden, @code{s2}, @dots{}, which represent independent arbitrary
-signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
-flag, Calc will use zero in place of all arbitrary integers, and plus
-one in place of all arbitrary signs. Note that variables like @code{n1}
-and @code{s1} are not given any special interpretation in Calc except by
-the equation solver itself. As usual, you can use the @address@hidden l}}
-(@code{calc-let}) command to obtain solutions for various actual values
-of these variables.
-
-For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
-get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
-equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
-think about it is that the square-root operation is really a
-two-valued function; since every Calc function must return a
-single result, @code{sqrt} chooses to return the positive result.
-Then @kbd{H a S} doctors this result using @code{s1} to indicate
-the full set of possible values of the mathematical square-root.
-
-There is a similar phenomenon going the other direction: Suppose
-we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
-to get @samp{y = x^2}. This is correct, except that it introduces
-some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
-Calc will report @expr{y = 9} as a valid solution, which is true
-in the mathematical sense of square-root, but false (there is no
-solution) for the actual Calc positive-valued @code{sqrt}. This
-happens for both @kbd{a S} and @kbd{H a S}.
-
address@hidden @code{GenCount} variable
address@hidden GenCount
address@hidden
address@hidden
address@hidden ignore
address@hidden an
address@hidden
address@hidden
address@hidden ignore
address@hidden as
-If you store a positive integer in the Calc variable @code{GenCount},
-then Calc will generate formulas of the form @samp{as(@var{n})} for
-arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
-where @var{n} represents successive values taken by incrementing
address@hidden by one. While the normal arbitrary sign and
-integer symbols start over at @code{s1} and @code{n1} with each
-new Calc command, the @code{GenCount} approach will give each
-arbitrary value a name that is unique throughout the entire Calc
-session. Also, the arbitrary values are function calls instead
-of variables, which is advantageous in some cases. For example,
-you can make a rewrite rule that recognizes all arbitrary signs
-using a pattern like @samp{as(n)}. The @kbd{s l} command only works
-on variables, but you can use the @kbd{a b} (@code{calc-substitute})
-command to substitute actual values for function calls like @samp{as(3)}.
-
-The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
-way to create or edit this variable. Press @kbd{C-c C-c} to finish.
-
-If you have not stored a value in @code{GenCount}, or if the value
-in that variable is not a positive integer, the regular
address@hidden/@code{n1} notation is used.
-
address@hidden I a S
address@hidden H I a S
address@hidden finv
address@hidden ffinv
-With the Inverse flag, @kbd{I a S} address@hidden treats the expression
-on top of the stack as a function of the specified variable and solves
-to find the inverse function, written in terms of the same variable.
-For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
-You can use both Inverse and Hyperbolic address@hidden to obtain a
-fully general inverse, as described above.
-
address@hidden a P
address@hidden calc-poly-roots
address@hidden roots
-Some equations, specifically polynomials, have a known, finite number
-of solutions. The @kbd{a P} (@code{calc-poly-roots}) address@hidden
-command uses @kbd{H a S} to solve an equation in general form, then, for
-all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
-variables like @code{n1} for which @code{n1} only usefully varies over
-a finite range, it expands these variables out to all their possible
-values. The results are collected into a vector, which is returned.
-For example, @samp{roots(x^4 = 1, x)} returns the four solutions
address@hidden, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
-polynomial will always have @var{n} roots on the complex plane.
-(If you have given a @code{real} declaration for the solution
-variable, then only the real-valued solutions, if any, will be
-reported; @pxref{Declarations}.)
-
-Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
-symbolic solutions if the polynomial has symbolic coefficients. Also
-note that Calc's solver is not able to get exact symbolic solutions
-to all polynomials. Polynomials containing powers up to @expr{x^4}
-can always be solved exactly; polynomials of higher degree sometimes
-can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
-which can be solved for @expr{x^3} using the quadratic equation, and then
-for @expr{x} by taking cube roots. But in many cases, like
address@hidden + x + 1}, Calc does not know how to rewrite the polynomial
-into a form it can solve. The @kbd{a P} command can still deliver a
-list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
-is not turned on. (If you work with Symbolic mode on, recall that the
address@hidden (@code{calc-eval-num}) key is a handy way to reevaluate the
-formula on the stack with Symbolic mode temporarily off.) Naturally,
address@hidden P} can only provide numerical roots if the polynomial
coefficients
-are all numbers (real or complex).
-
address@hidden Solving Systems of Equations, Decomposing Polynomials, Multiple
Solutions, Solving Equations
address@hidden Solving Systems of Equations
-
address@hidden
address@hidden Systems of equations, symbolic
-You can also use the commands described above to solve systems of
-simultaneous equations. Just create a vector of equations, then
-specify a vector of variables for which to solve. (You can omit
-the surrounding brackets when entering the vector of variables
-at the prompt.)
-
-For example, putting @samp{[x + y = a, x - y = b]} on the stack
-and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
address@hidden = a - (a-b)/2, y = (a-b)/2]}. The result vector will
-have the same length as the variables vector, and the variables
-will be listed in the same order there. Note that the solutions
-are not always simplified as far as possible; the solution for
address@hidden here could be improved by an application of the @kbd{a n}
-command.
-
-Calc's algorithm works by trying to eliminate one variable at a
-time by solving one of the equations for that variable and then
-substituting into the other equations. Calc will try all the
-possibilities, but you can speed things up by noting that Calc
-first tries to eliminate the first variable with the first
-equation, then the second variable with the second equation,
-and so on. It also helps to put the simpler (e.g., more linear)
-equations toward the front of the list. Calc's algorithm will
-solve any system of linear equations, and also many kinds of
-nonlinear systems.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden elim
-Normally there will be as many variables as equations. If you
-give fewer variables than equations (an ``over-determined'' system
-of equations), Calc will find a partial solution. For example,
-typing @kbd{a S y @key{RET}} with the above system of equations
-would produce @samp{[y = a - x]}. There are now several ways to
-express this solution in terms of the original variables; Calc uses
-the first one that it finds. You can control the choice by adding
-variable specifiers of the form @samp{elim(@var{v})} to the
-variables list. This says that @var{v} should be eliminated from
-the equations; the variable will not appear at all in the solution.
-For example, typing @kbd{a S y,elim(x)} would yield
address@hidden = a - (b+a)/2]}.
-
-If the variables list contains only @code{elim} specifiers,
-Calc simply eliminates those variables from the equations
-and then returns the resulting set of equations. For example,
address@hidden S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
-eliminated will reduce the number of equations in the system
-by one.
-
-Again, @kbd{a S} gives you one solution to the system of
-equations. If there are several solutions, you can use @kbd{H a S}
-to get a general family of solutions, or, if there is a finite
-number of solutions, you can use @kbd{a P} to get a list. (In
-the latter case, the result will take the form of a matrix where
-the rows are different solutions and the columns correspond to the
-variables you requested.)
-
-Another way to deal with certain kinds of overdetermined systems of
-equations is the @kbd{a F} command, which does least-squares fitting
-to satisfy the equations. @xref{Curve Fitting}.
-
address@hidden Decomposing Polynomials, , Solving Systems of Equations,
Solving Equations
address@hidden Decomposing Polynomials
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden poly
-The @code{poly} function takes a polynomial and a variable as
-arguments, and returns a vector of polynomial coefficients (constant
-coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
address@hidden, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
-the call to @code{poly} is left in symbolic form. If the input does
-not involve the variable @expr{x}, the input is returned in a list
-of length one, representing a polynomial with only a constant
-coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
-The last element of the returned vector is guaranteed to be nonzero;
-note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
-Note also that @expr{x} may actually be any formula; for example,
address@hidden(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
-
address@hidden Coefficients of polynomial
address@hidden Degree of polynomial
-To get the @expr{x^k} coefficient of polynomial @expr{p}, use
address@hidden(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
-use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
-returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
-gives the @expr{x^2} coefficient of this polynomial, 6.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden gpoly
-One important feature of the solver is its ability to recognize
-formulas which are ``essentially'' polynomials. This ability is
-made available to the user through the @code{gpoly} function, which
-is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
-If @var{expr} is a polynomial in some term which includes @var{var}, then
-this function will return a vector @address@hidden, @var{c}, @var{a}]}
-where @var{x} is the term that depends on @var{var}, @var{c} is a
-vector of polynomial coefficients (like the one returned by @code{poly}),
-and @var{a} is a multiplier which is usually 1. Basically,
address@hidden@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
address@hidden @var{x}^2 + ...)}. The last element of @var{c} is
-guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
-(i.e., the trivial decomposition @var{expr} = @var{x} is not
-considered a polynomial). One side effect is that @samp{gpoly(x, x)}
-and @samp{gpoly(6, x)}, both of which might be expected to recognize
-their arguments as polynomials, will not because the decomposition
-is considered trivial.
-
-For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
-since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
-
-The term @var{x} may itself be a polynomial in @var{var}. This is
-done to reduce the size of the @var{c} vector. For example,
address@hidden(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
-since a quadratic polynomial in @expr{x^2} is easier to solve than
-a quartic polynomial in @expr{x}.
-
-A few more examples of the kinds of polynomials @code{gpoly} can
-discover:
-
address@hidden
-sin(x) - 1 [sin(x), [-1, 1], 1]
-x + 1/x - 1 [x, [1, -1, 1], 1/x]
-x + 1/x [x^2, [1, 1], 1/x]
-x^3 + 2 x [x^2, [2, 1], x]
-x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
-x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
-(exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
address@hidden smallexample
-
-The @code{poly} and @code{gpoly} functions accept a third integer argument
-which specifies the largest degree of polynomial that is acceptable.
-If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
-or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
-call will remain in symbolic form. For example, the equation solver
-can handle quartics and smaller polynomials, so it calls
address@hidden(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
-can be treated by its linear, quadratic, cubic, or quartic formulas.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden pdeg
-The @code{pdeg} function computes the degree of a polynomial;
address@hidden(p,x)} is the highest power of @code{x} that appears in
address@hidden This is the same as @samp{vlen(poly(p,x))-1}, but is
-much more efficient. If @code{p} is constant with respect to @code{x},
-then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
-(e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
-It is possible to omit the second argument @code{x}, in which case
address@hidden(p)} returns the highest total degree of any term of the
-polynomial, counting all variables that appear in @code{p}. Note
-that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
-the degree of the constant zero is considered to be @code{-inf}
-(minus infinity).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden plead
-The @code{plead} function finds the leading term of a polynomial.
-Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
-though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
-returns 1024 without expanding out the list of coefficients. The
-value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden pcont
-The @code{pcont} function finds the @dfn{content} of a polynomial. This
-is the greatest common divisor of all the coefficients of the polynomial.
-With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
-to get a list of coefficients, then uses @code{pgcd} (the polynomial
-GCD function) to combine these into an answer. For example,
address@hidden(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
-basically the ``biggest'' polynomial that can be divided into @code{p}
-exactly. The sign of the content is the same as the sign of the leading
-coefficient.
-
-With only one argument, @samp{pcont(p)} computes the numerical
-content of the polynomial, i.e., the @code{gcd} of the numerical
-coefficients of all the terms in the formula. Note that @code{gcd}
-is defined on rational numbers as well as integers; it computes
-the @code{gcd} of the numerators and the @code{lcm} of the
-denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
-Dividing the polynomial by this number will clear all the
-denominators, as well as dividing by any common content in the
-numerators. The numerical content of a polynomial is negative only
-if all the coefficients in the polynomial are negative.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden pprim
-The @code{pprim} function finds the @dfn{primitive part} of a
-polynomial, which is simply the polynomial divided (using @code{pdiv}
-if necessary) by its content. If the input polynomial has rational
-coefficients, the result will have integer coefficients in simplest
-terms.
-
address@hidden Numerical Solutions, Curve Fitting, Solving Equations, Algebra
address@hidden Numerical Solutions
-
address@hidden
-Not all equations can be solved symbolically. The commands in this
-section use numerical algorithms that can find a solution to a specific
-instance of an equation to any desired accuracy. Note that the
-numerical commands are slower than their algebraic cousins; it is a
-good idea to try @kbd{a S} before resorting to these commands.
-
-(@xref{Curve Fitting}, for some other, more specialized, operations
-on numerical data.)
-
address@hidden
-* Root Finding::
-* Minimization::
-* Numerical Systems of Equations::
address@hidden menu
-
address@hidden Root Finding, Minimization, Numerical Solutions, Numerical
Solutions
address@hidden Root Finding
-
address@hidden
address@hidden a R
address@hidden calc-find-root
address@hidden root
address@hidden Newton's method
address@hidden Roots of equations
address@hidden Numerical root-finding
-The @kbd{a R} (@code{calc-find-root}) address@hidden command finds a
-numerical solution (or @dfn{root}) of an equation. (This command treats
-inequalities the same as equations. If the input is any other kind
-of formula, it is interpreted as an equation of the form @expr{X = 0}.)
-
-The @kbd{a R} command requires an initial guess on the top of the
-stack, and a formula in the second-to-top position. It prompts for a
-solution variable, which must appear in the formula. All other variables
-that appear in the formula must have assigned values, i.e., when
-a value is assigned to the solution variable and the formula is
-evaluated with @kbd{=}, it should evaluate to a number. Any assigned
-value for the solution variable itself is ignored and unaffected by
-this command.
-
-When the command completes, the initial guess is replaced on the stack
-by a vector of two numbers: The value of the solution variable that
-solves the equation, and the difference between the lefthand and
-righthand sides of the equation at that value. Ordinarily, the second
-number will be zero or very nearly zero. (Note that Calc uses a
-slightly higher precision while finding the root, and thus the second
-number may be slightly different from the value you would compute from
-the equation yourself.)
-
-The @kbd{v h} (@code{calc-head}) command is a handy way to extract
-the first element of the result vector, discarding the error term.
-
-The initial guess can be a real number, in which case Calc searches
-for a real solution near that number, or a complex number, in which
-case Calc searches the whole complex plane near that number for a
-solution, or it can be an interval form which restricts the search
-to real numbers inside that interval.
-
-Calc tries to use @kbd{a d} to take the derivative of the equation.
-If this succeeds, it uses Newton's method. If the equation is not
-differentiable Calc uses a bisection method. (If Newton's method
-appears to be going astray, Calc switches over to bisection if it
-can, or otherwise gives up. In this case it may help to try again
-with a slightly different initial guess.) If the initial guess is a
-complex number, the function must be differentiable.
-
-If the formula (or the difference between the sides of an equation)
-is negative at one end of the interval you specify and positive at
-the other end, the root finder is guaranteed to find a root.
-Otherwise, Calc subdivides the interval into small parts looking for
-positive and negative values to bracket the root. When your guess is
-an interval, Calc will not look outside that interval for a root.
-
address@hidden H a R
address@hidden wroot
-The @kbd{H a R} address@hidden command is similar to @kbd{a R}, except
-that if the initial guess is an interval for which the function has
-the same sign at both ends, then rather than subdividing the interval
-Calc attempts to widen it to enclose a root. Use this mode if
-you are not sure if the function has a root in your interval.
-
-If the function is not differentiable, and you give a simple number
-instead of an interval as your initial guess, Calc uses this widening
-process even if you did not type the Hyperbolic flag. (If the function
address@hidden differentiable, Calc uses Newton's method which does not
-require a bounding interval in order to work.)
-
-If Calc leaves the @code{root} or @code{wroot} function in symbolic
-form on the stack, it will normally display an explanation for why
-no root was found. If you miss this explanation, press @kbd{w}
-(@code{calc-why}) to get it back.
-
address@hidden Minimization, Numerical Systems of Equations, Root Finding,
Numerical Solutions
address@hidden Minimization
-
address@hidden
address@hidden a N
address@hidden H a N
address@hidden a X
address@hidden H a X
address@hidden calc-find-minimum
address@hidden calc-find-maximum
address@hidden minimize
address@hidden maximize
address@hidden Minimization, numerical
-The @kbd{a N} (@code{calc-find-minimum}) address@hidden command
-finds a minimum value for a formula. It is very similar in operation
-to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
-guess on the stack, and are prompted for the name of a variable. The guess
-may be either a number near the desired minimum, or an interval enclosing
-the desired minimum. The function returns a vector containing the
-value of the variable which minimizes the formula's value, along
-with the minimum value itself.
-
-Note that this command looks for a @emph{local} minimum. Many functions
-have more than one minimum; some, like
address@hidden @math{x \sin x},
address@hidden @expr{x sin(x)},
-have infinitely many. In fact, there is no easy way to define the
-``global'' minimum of
address@hidden @math{x \sin x}
address@hidden @expr{x sin(x)}
-but Calc can still locate any particular local minimum
-for you. Calc basically goes downhill from the initial guess until it
-finds a point at which the function's value is greater both to the left
-and to the right. Calc does not use derivatives when minimizing a function.
-
-If your initial guess is an interval and it looks like the minimum
-occurs at one or the other endpoint of the interval, Calc will return
-that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
-over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
address@hidden(2..3]} would report no minimum found. In general, you should
-use closed intervals to find literally the minimum value in that
-range of @expr{x}, or open intervals to find the local minimum, if
-any, that happens to lie in that range.
-
-Most functions are smooth and flat near their minimum values. Because
-of this flatness, if the current precision is, say, 12 digits, the
-variable can only be determined meaningfully to about six digits. Thus
-you should set the precision to twice as many digits as you need in your
-answer.
-
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden wminimize
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden wmaximize
-The @kbd{H a N} address@hidden command, analogously to @kbd{H a R},
-expands the guess interval to enclose a minimum rather than requiring
-that the minimum lie inside the interval you supply.
-
-The @kbd{a X} (@code{calc-find-maximum}) address@hidden and
address@hidden a X} address@hidden commands effectively minimize the
-negative of the formula you supply.
-
-The formula must evaluate to a real number at all points inside the
-interval (or near the initial guess if the guess is a number). If
-the initial guess is a complex number the variable will be minimized
-over the complex numbers; if it is real or an interval it will
-be minimized over the reals.
-
address@hidden Numerical Systems of Equations, , Minimization, Numerical
Solutions
address@hidden Systems of Equations
-
address@hidden
address@hidden Systems of equations, numerical
-The @kbd{a R} command can also solve systems of equations. In this
-case, the equation should instead be a vector of equations, the
-guess should instead be a vector of numbers (intervals are not
-supported), and the variable should be a vector of variables. You
-can omit the brackets while entering the list of variables. Each
-equation must be differentiable by each variable for this mode to
-work. The result will be a vector of two vectors: The variable
-values that solved the system of equations, and the differences
-between the sides of the equations with those variable values.
-There must be the same number of equations as variables. Since
-only plain numbers are allowed as guesses, the Hyperbolic flag has
-no effect when solving a system of equations.
-
-It is also possible to minimize over many variables with @kbd{a N}
-(or maximize with @kbd{a X}). Once again the variable name should
-be replaced by a vector of variables, and the initial guess should
-be an equal-sized vector of initial guesses. But, unlike the case of
-multidimensional @kbd{a R}, the formula being minimized should
-still be a single formula, @emph{not} a vector. Beware that
-multidimensional minimization is currently @emph{very} slow.
-
address@hidden Curve Fitting, Summations, Numerical Solutions, Algebra
address@hidden Curve Fitting
-
address@hidden
-The @kbd{a F} command fits a set of data to a @dfn{model formula},
-such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
-to be determined. For a typical set of measured data there will be
-no single @expr{m} and @expr{b} that exactly fit the data; in this
-case, Calc chooses values of the parameters that provide the closest
-possible fit. The model formula can be entered in various ways after
-the key sequence @kbd{a F} is pressed.
-
-If the letter @kbd{P} is pressed after @kbd{a F} but before the model
-description is entered, the data as well as the model formula will be
-plotted after the formula is determined. This will be indicated by a
-``P'' in the minibuffer after the help message.
-
address@hidden
-* Linear Fits::
-* Polynomial and Multilinear Fits::
-* Error Estimates for Fits::
-* Standard Nonlinear Models::
-* Curve Fitting Details::
-* Interpolation::
address@hidden menu
-
address@hidden Linear Fits, Polynomial and Multilinear Fits, Curve Fitting,
Curve Fitting
address@hidden Linear Fits
-
address@hidden
address@hidden a F
address@hidden calc-curve-fit
address@hidden fit
address@hidden Linear regression
address@hidden Least-squares fits
-The @kbd{a F} (@code{calc-curve-fit}) address@hidden command attempts
-to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
-straight line, polynomial, or other function of @expr{x}. For the
-moment we will consider only the case of fitting to a line, and we
-will ignore the issue of whether or not the model was in fact a good
-fit for the data.
-
-In a standard linear least-squares fit, we have a set of @expr{(x,y)}
-data points that we wish to fit to the model @expr{y = m x + b}
-by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
-values calculated from the formula be as close as possible to the actual
address@hidden values in the data set. (In a polynomial fit, the model is
-instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
-we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
address@hidden = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
-
-In the model formula, variables like @expr{x} and @expr{x_2} are called
-the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
-variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
-the @dfn{parameters} of the model.
-
-The @kbd{a F} command takes the data set to be fitted from the stack.
-By default, it expects the data in the form of a matrix. For example,
-for a linear or polynomial fit, this would be a
address@hidden @math{2\times N}
address@hidden 2xN
-matrix where the first row is a list of @expr{x} values and the second
-row has the corresponding @expr{y} values. For the multilinear fit
-shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
address@hidden, and @expr{y}, respectively).
-
-If you happen to have an
address@hidden @math{N\times2}
address@hidden Nx2
-matrix instead of a
address@hidden @math{2\times N}
address@hidden 2xN
-matrix, just press @kbd{v t} first to transpose the matrix.
-
-After you type @kbd{a F}, Calc prompts you to select a model. For a
-linear fit, press the digit @kbd{1}.
-
-Calc then prompts for you to name the variables. By default it chooses
-high letters like @expr{x} and @expr{y} for independent variables and
-low letters like @expr{a} and @expr{b} for parameters. (The dependent
-variable doesn't need a name.) The two kinds of variables are separated
-by a semicolon. Since you generally care more about the names of the
-independent variables than of the parameters, Calc also allows you to
-name only those and let the parameters use default names.
-
-For example, suppose the data matrix
-
address@hidden
address@hidden
address@hidden
-[ [ 1, 2, 3, 4, 5 ]
- [ 5, 7, 9, 11, 13 ] ]
address@hidden group
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\turnoffactive
-\beforedisplay
-$$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
- 5 & 7 & 9 & 11 & 13 }
-$$
-\afterdisplay
address@hidden tex
-
address@hidden
-is on the stack and we wish to do a simple linear fit. Type
address@hidden F}, then @kbd{1} for the model, then @key{RET} to use
-the default names. The result will be the formula @expr{3. + 2. x}
-on the stack. Calc has created the model expression @kbd{a + b x},
-then found the optimal values of @expr{a} and @expr{b} to fit the
-data. (In this case, it was able to find an exact fit.) Calc then
-substituted those values for @expr{a} and @expr{b} in the model
-formula.
-
-The @kbd{a F} command puts two entries in the trail. One is, as
-always, a copy of the result that went to the stack; the other is
-a vector of the actual parameter values, written as equations:
address@hidden = 3, b = 2]}, in case you'd rather read them in a list
-than pick them out of the formula. (You can type @kbd{t y}
-to move this vector to the stack; see @ref{Trail Commands}.
-
-Specifying a different independent variable name will affect the
-resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
-Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
-the equations that go into the trail.
-
address@hidden
-\bigskip
address@hidden tex
-
-To see what happens when the fit is not exact, we could change
-the number 13 in the data matrix to 14 and try the fit again.
-The result is:
-
address@hidden
-2.6 + 2.2 x
address@hidden example
-
-Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $
@key{RET}}, shows
-a reasonably close match to the y-values in the data.
-
address@hidden
-[4.8, 7., 9.2, 11.4, 13.6]
address@hidden example
-
-Since there is no line which passes through all the @var{n} data points,
-Calc has chosen a line that best approximates the data points using
-the method of least squares. The idea is to define the @dfn{chi-square}
-error measure
-
address@hidden
address@hidden
-chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
-\afterdisplay
address@hidden tex
-
address@hidden
-which is clearly zero if @expr{a + b x} exactly fits all data points,
-and increases as various @expr{a + b x_i} values fail to match the
-corresponding @expr{y_i} values. There are several reasons why the
-summand is squared, one of them being to ensure that
address@hidden @math{\chi^2 \ge 0}.
address@hidden @expr{chi^2 >= 0}.
-Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
-for which the error
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-is as small as possible.
-
-Other kinds of models do the same thing but with a different model
-formula in place of @expr{a + b x_i}.
-
address@hidden
-\bigskip
address@hidden tex
-
-A numeric prefix argument causes the @kbd{a F} command to take the
-data in some other form than one big matrix. A positive argument @var{n}
-will take @var{N} items from the stack, corresponding to the @var{n} rows
-of a data matrix. In the linear case, @var{n} must be 2 since there
-is always one independent variable and one dependent variable.
-
-A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
-items from the stack, an @var{n}-row matrix of @expr{x} values, and a
-vector of @expr{y} values. If there is only one independent variable,
-the @expr{x} values can be either a one-row matrix or a plain vector,
-in which case the @kbd{C-u} prefix is the same as a @address@hidden 2}} prefix.
-
address@hidden Polynomial and Multilinear Fits, Error Estimates for Fits,
Linear Fits, Curve Fitting
address@hidden Polynomial and Multilinear Fits
-
address@hidden
-To fit the data to higher-order polynomials, just type one of the
-digits @kbd{2} through @kbd{9} when prompted for a model. For example,
-we could fit the original data matrix from the previous section
-(with 13, not 14) to a parabola instead of a line by typing
address@hidden F 2 @key{RET}}.
-
address@hidden
-2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
address@hidden example
-
-Note that since the constant and linear terms are enough to fit the
-data exactly, it's no surprise that Calc chose a tiny contribution
-for @expr{x^2}. (The fact that it's not exactly zero is due only
-to roundoff error. Since our data are exact integers, we could get
-an exact answer by typing @kbd{m f} first to get Fraction mode.
-Then the @expr{x^2} term would vanish altogether. Usually, though,
-the data being fitted will be approximate floats so Fraction mode
-won't help.)
-
-Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
-gives a much larger @expr{x^2} contribution, as Calc bends the
-line slightly to improve the fit.
-
address@hidden
-0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
address@hidden example
-
-An important result from the theory of polynomial fitting is that it
-is always possible to fit @var{n} data points exactly using a polynomial
-of degree @address@hidden, sometimes called an @dfn{interpolating polynomial}.
-Using the modified (14) data matrix, a model number of 4 gives
-a polynomial that exactly matches all five data points:
-
address@hidden
-0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
address@hidden example
-
-The actual coefficients we get with a precision of 12, like
address@hidden, clearly suffer from loss of precision.
-It is a good idea to increase the working precision to several
-digits beyond what you need when you do a fitting operation.
-Or, if your data are exact, use Fraction mode to get exact
-results.
-
-You can type @kbd{i} instead of a digit at the model prompt to fit
-the data exactly to a polynomial. This just counts the number of
-columns of the data matrix to choose the degree of the polynomial
-automatically.
-
-Fitting data ``exactly'' to high-degree polynomials is not always
-a good idea, though. High-degree polynomials have a tendency to
-wiggle uncontrollably in between the fitting data points. Also,
-if the exact-fit polynomial is going to be used to interpolate or
-extrapolate the data, it is numerically better to use the @kbd{a p}
-command described below. @xref{Interpolation}.
-
address@hidden
-\bigskip
address@hidden tex
-
-Another generalization of the linear model is to assume the
address@hidden values are a sum of linear contributions from several
address@hidden values. This is a @dfn{multilinear} fit, and it is also
-selected by the @kbd{1} digit key. (Calc decides whether the fit
-is linear or multilinear by counting the rows in the data matrix.)
-
-Given the data matrix,
-
address@hidden
address@hidden
-[ [ 1, 2, 3, 4, 5 ]
- [ 7, 2, 3, 5, 2 ]
- [ 14.5, 15, 18.5, 22.5, 24 ] ]
address@hidden group
address@hidden example
-
address@hidden
-the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
-second row @expr{y}, and will fit the values in the third row to the
-model @expr{a + b x + c y}.
-
address@hidden
-8. + 3. x + 0.5 y
address@hidden example
-
-Calc can do multilinear fits with any number of independent variables
-(i.e., with any number of data rows).
-
address@hidden
-\bigskip
address@hidden tex
-
-Yet another variation is @dfn{homogeneous} linear models, in which
-the constant term is known to be zero. In the linear case, this
-means the model formula is simply @expr{a x}; in the multilinear
-case, the model might be @expr{a x + b y + c z}; and in the polynomial
-case, the model could be @expr{a x + b x^2 + c x^3}. You can get
-a homogeneous linear or multilinear model by pressing the letter
address@hidden followed by a regular model key, like @kbd{1} or @kbd{2}.
-This will be indicated by an ``h'' in the minibuffer after the help
-message.
-
-It is certainly possible to have other constrained linear models,
-like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
-key to select models like these, a later section shows how to enter
-any desired model by hand. In the first case, for example, you
-would enter @kbd{a F ' 2.3 + a x}.
-
-Another class of models that will work but must be entered by hand
-are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
-
address@hidden Error Estimates for Fits, Standard Nonlinear Models, Polynomial
and Multilinear Fits, Curve Fitting
address@hidden Error Estimates for Fits
-
address@hidden
address@hidden H a F
address@hidden efit
-With the Hyperbolic flag, @kbd{H a F} address@hidden performs the same
-fitting operation as @kbd{a F}, but reports the coefficients as error
-forms instead of plain numbers. Fitting our two data matrices (first
-with 13, then with 14) to a line with @kbd{H a F} gives the results,
-
address@hidden
-3. + 2. x
-2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
address@hidden example
-
-In the first case the estimated errors are zero because the linear
-fit is perfect. In the second case, the errors are nonzero but
-moderately small, because the data are still very close to linear.
-
-It is also possible for the @emph{input} to a fitting operation to
-contain error forms. The data values must either all include errors
-or all be plain numbers. Error forms can go anywhere but generally
-go on the numbers in the last row of the data matrix. If the last
-row contains error forms
address@hidden address@hidden@w{ @tfn{+/-} address@hidden',
address@hidden address@hidden@w{ @tfn{+/-} address@hidden',
-then the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-statistic is now,
-
address@hidden
address@hidden
-chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
address@hidden example
address@hidden ifnottex
address@hidden
-\turnoffactive
-\beforedisplay
-$$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
-\afterdisplay
address@hidden tex
-
address@hidden
-so that data points with larger error estimates contribute less to
-the fitting operation.
-
-If there are error forms on other rows of the data matrix, all the
-errors for a given data point are combined; the square root of the
-sum of the squares of the errors forms the
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
-used for the data point.
-
-Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
-matrix, although if you are concerned about error analysis you will
-probably use @kbd{H a F} so that the output also contains error
-estimates.
-
-If the input contains error forms but all the
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
-values are the same, it is easy to see that the resulting fitted model
-will be the same as if the input did not have error forms at all
address@hidden (@math{\chi^2}
address@hidden (@expr{chi^2}
-is simply scaled uniformly by
address@hidden @math{1 / \sigma^2},
address@hidden @expr{1 / sigma^2},
-which doesn't affect where it has a minimum). But there @emph{will} be
-a difference in the estimated errors of the coefficients reported by
address@hidden a F}.
-
-Consult any text on statistical modeling of data for a discussion
-of where these error estimates come from and how they should be
-interpreted.
-
address@hidden
-\bigskip
address@hidden tex
-
address@hidden I a F
address@hidden xfit
-With the Inverse flag, @kbd{I a F} address@hidden produces even more
-information. The result is a vector of six items:
-
address@hidden
address@hidden
-The model formula with error forms for its coefficients or
-parameters. This is the result that @kbd{H a F} would have
-produced.
-
address@hidden
-A vector of ``raw'' parameter values for the model. These are the
-polynomial coefficients or other parameters as plain numbers, in the
-same order as the parameters appeared in the final prompt of the
address@hidden a F} command. For polynomials of degree @expr{d}, this vector
-will have length @expr{M = d+1} with the constant term first.
-
address@hidden
-The covariance matrix @expr{C} computed from the fit. This is
-an @address@hidden symmetric matrix; the diagonal elements
address@hidden @math{C_{jj}}
address@hidden @expr{C_j_j}
-are the variances
address@hidden @math{\sigma_j^2}
address@hidden @expr{sigma_j^2}
-of the parameters. The other elements are covariances
address@hidden @math{\sigma_{ij}^2}
address@hidden @expr{sigma_i_j^2}
-that describe the correlation between pairs of parameters. (A related
-set of numbers, the @dfn{linear correlation coefficients}
address@hidden @math{r_{ij}},
address@hidden @expr{r_i_j},
-are defined as
address@hidden @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
address@hidden @expr{sigma_i_j^2 / sigma_i sigma_j}.)
-
address@hidden
-A vector of @expr{M} ``parameter filter'' functions whose
-meanings are described below. If no filters are necessary this
-will instead be an empty vector; this is always the case for the
-polynomial and multilinear fits described so far.
-
address@hidden
-The value of
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-for the fit, calculated by the formulas shown above. This gives a
-measure of the quality of the fit; statisticians consider
address@hidden @math{\chi^2 \approx N - M}
address@hidden @expr{chi^2 = N - M}
-to indicate a moderately good fit (where again @expr{N} is the number of
-data points and @expr{M} is the number of parameters).
-
address@hidden
-A measure of goodness of fit expressed as a probability @expr{Q}.
-This is computed from the @code{utpc} probability distribution
-function using
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-with @expr{N - M} degrees of freedom. A
-value of 0.5 implies a good fit; some texts recommend that often
address@hidden = 0.1} or even 0.001 can signify an acceptable fit. In
-particular,
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-statistics assume the errors in your inputs
-follow a normal (Gaussian) distribution; if they don't, you may
-have to accept smaller values of @expr{Q}.
-
-The @expr{Q} value is computed only if the input included error
-estimates. Otherwise, Calc will report the symbol @code{nan}
-for @expr{Q}. The reason is that in this case the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-value has effectively been used to estimate the original errors
-in the input, and thus there is no redundant information left
-over to use for a confidence test.
address@hidden enumerate
-
address@hidden Standard Nonlinear Models, Curve Fitting Details, Error
Estimates for Fits, Curve Fitting
address@hidden Standard Nonlinear Models
-
address@hidden
-The @kbd{a F} command also accepts other kinds of models besides
-lines and polynomials. Some common models have quick single-key
-abbreviations; others must be entered by hand as algebraic formulas.
-
-Here is a complete list of the standard models recognized by @kbd{a F}:
-
address@hidden @kbd
address@hidden 1
-Linear or multilinear. @mathit{a + b x + c y + d z}.
address@hidden 2-9
-Polynomials. @mathit{a + b x + c x^2 + d x^3}.
address@hidden e
-Exponential. @mathit{a} @address@hidden(b x)} @address@hidden(c y)}.
address@hidden E
-Base-10 exponential. @mathit{a} @address@hidden(b x)} @address@hidden(c y)}.
address@hidden x
-Exponential (alternate notation). @address@hidden(a + b x + c y)}.
address@hidden X
-Base-10 exponential (alternate). @address@hidden(a + b x + c y)}.
address@hidden l
-Logarithmic. @mathit{a + b} @address@hidden(x) + c} @address@hidden(y)}.
address@hidden L
-Base-10 logarithmic. @mathit{a + b} @address@hidden(x) + c}
@address@hidden(y)}.
address@hidden ^
-General exponential. @mathit{a b^x c^y}.
address@hidden p
-Power law. @mathit{a x^b y^c}.
address@hidden q
-Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
address@hidden g
-Gaussian.
address@hidden @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x
- c \over b \right)^2 \right)}.
address@hidden @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
address@hidden s
-Logistic @emph{s} curve.
address@hidden @math{a/(1+e^{b(x-c)})}.
address@hidden @mathit{a/(1 + exp(b (x - c)))}.
address@hidden b
-Logistic bell curve.
address@hidden @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}.
address@hidden @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}.
address@hidden o
-Hubbert linearization.
address@hidden @math{{y \over x} = a(1-x/b)}.
address@hidden @mathit{(y/x) = a (1 - x/b)}.
address@hidden table
-
-All of these models are used in the usual way; just press the appropriate
-letter at the model prompt, and choose variable names if you wish. The
-result will be a formula as shown in the above table, with the best-fit
-values of the parameters substituted. (You may find it easier to read
-the parameter values from the vector that is placed in the trail.)
-
-All models except Gaussian, logistics, Hubbert and polynomials can
-generalize as shown to any number of independent variables. Also, all
-the built-in models except for the logistic and Hubbert curves have an
-additive or multiplicative parameter shown as @expr{a} in the above table
-which can be replaced by zero or one, as appropriate, by typing @kbd{h}
-before the model key.
-
-Note that many of these models are essentially equivalent, but express
-the parameters slightly differently. For example, @expr{a b^x} and
-the other two exponential models are all algebraic rearrangements of
-each other. Also, the ``quadratic'' model is just a degree-2 polynomial
-with the parameters expressed differently. Use whichever form best
-matches the problem.
-
-The HP-28/48 calculators support four different models for curve
-fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
-These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
address@hidden exp(b x)}, and @samp{a x^b}, respectively. In each case,
address@hidden is what the HP-48 identifies as the ``intercept,'' and
address@hidden is what it calls the ``slope.''
-
address@hidden
-\bigskip
address@hidden tex
-
-If the model you want doesn't appear on this list, press @kbd{'}
-(the apostrophe key) at the model prompt to enter any algebraic
-formula, such as @kbd{m x - b}, as the model. (Not all models
-will work, though---see the next section for details.)
-
-The model can also be an equation like @expr{y = m x + b}.
-In this case, Calc thinks of all the rows of the data matrix on
-equal terms; this model effectively has two parameters
-(@expr{m} and @expr{b}) and two independent variables (@expr{x}
-and @expr{y}), with no ``dependent'' variables. Model equations
-do not need to take this @expr{y =} form. For example, the
-implicit line equation @expr{a x + b y = 1} works fine as a
-model.
-
-When you enter a model, Calc makes an alphabetical list of all
-the variables that appear in the model. These are used for the
-default parameters, independent variables, and dependent variable
-(in that order). If you enter a plain formula (not an equation),
-Calc assumes the dependent variable does not appear in the formula
-and thus does not need a name.
-
-For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
-and the data matrix has three rows (meaning two independent variables),
-Calc will use @expr{a,mu,sigma} as the default parameters, and the
-data rows will be named @expr{t} and @expr{x}, respectively. If you
-enter an equation instead of a plain formula, Calc will use @expr{a,mu}
-as the parameters, and @expr{sigma,t,x} as the three independent
-variables.
-
-You can, of course, override these choices by entering something
-different at the prompt. If you leave some variables out of the list,
-those variables must have stored values and those stored values will
-be used as constants in the model. (Stored values for the parameters
-and independent variables are ignored by the @kbd{a F} command.)
-If you list only independent variables, all the remaining variables
-in the model formula will become parameters.
-
-If there are @kbd{$} signs in the model you type, they will stand
-for parameters and all other variables (in alphabetical order)
-will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
-another, and so on. Thus @kbd{$ x + $$} is another way to describe
-a linear model.
-
-If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
-Calc will take the model formula from the stack. (The data must then
-appear at the second stack level.) The same conventions are used to
-choose which variables in the formula are independent by default and
-which are parameters.
-
-Models taken from the stack can also be expressed as vectors of
-two or three elements, @address@hidden, @var{vars}]} or
address@hidden@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
-and @var{params} may be either a variable or a vector of variables.
-(If @var{params} is omitted, all variables in @var{model} except
-those listed as @var{vars} are parameters.)
-
-When you enter a model manually with @kbd{'}, Calc puts a 3-vector
-describing the model in the trail so you can get it back if you wish.
-
address@hidden
-\bigskip
address@hidden tex
-
address@hidden Model1
address@hidden Model2
-Finally, you can store a model in one of the Calc variables
address@hidden or @code{Model2}, then use this model by typing
address@hidden F u} or @kbd{a F U} (respectively). The value stored in
-the variable can be any of the formats that @kbd{a F $} would
-accept for a model on the stack.
-
address@hidden
-\bigskip
address@hidden tex
-
-Calc uses the principal values of inverse functions like @code{ln}
-and @code{arcsin} when doing fits. For example, when you enter
-the model @samp{y = sin(a t + b)} Calc actually uses the easier
-form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
-returns results in the range from @mathit{-90} to 90 degrees (or the
-equivalent range in radians). Suppose you had data that you
-believed to represent roughly three oscillations of a sine wave,
-so that the argument of the sine might go from zero to
address@hidden @math{3\times360}
address@hidden @mathit{3*360}
-degrees.
-The above model would appear to be a good way to determine the
-true frequency and phase of the sine wave, but in practice it
-would fail utterly. The righthand side of the actual model
address@hidden(y) = a t + b} will grow smoothly with @expr{t}, but
-the lefthand side will bounce back and forth between @mathit{-90} and 90.
-No values of @expr{a} and @expr{b} can make the two sides match,
-even approximately.
-
-There is no good solution to this problem at present. You could
-restrict your data to small enough ranges so that the above problem
-doesn't occur (i.e., not straddling any peaks in the sine wave).
-Or, in this case, you could use a totally different method such as
-Fourier analysis, which is beyond the scope of the @kbd{a F} command.
-(Unfortunately, Calc does not currently have any facilities for
-taking Fourier and related transforms.)
-
address@hidden Curve Fitting Details, Interpolation, Standard Nonlinear Models,
Curve Fitting
address@hidden Curve Fitting Details
-
address@hidden
-Calc's internal least-squares fitter can only handle multilinear
-models. More precisely, it can handle any model of the form
address@hidden f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
-are the parameters and @expr{x,y,z} are the independent variables
-(of course there can be any number of each, not just three).
-
-In a simple multilinear or polynomial fit, it is easy to see how
-to convert the model into this form. For example, if the model
-is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
-and @expr{h(x) = x^2} are suitable functions.
-
-For most other models, Calc uses a variety of algebraic manipulations
-to try to put the problem into the form
-
address@hidden
-Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)
address@hidden smallexample
-
address@hidden
-where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
address@hidden, @expr{F}, @expr{G}, and @expr{H} for all the data points,
-does a standard linear fit to find the values of @expr{A}, @expr{B},
-and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
-in terms of @expr{A,B,C}.
-
-A remarkable number of models can be cast into this general form.
-We'll look at two examples here to see how it works. The power-law
-model @expr{y = a x^b} with two independent variables and two parameters
-can be rewritten as follows:
-
address@hidden
-y = a x^b
-y = a exp(b ln(x))
-y = exp(ln(a) + b ln(x))
-ln(y) = ln(a) + b ln(x)
address@hidden example
-
address@hidden
-which matches the desired form with
address@hidden @math{Y = \ln(y)},
address@hidden @expr{Y = ln(y)},
address@hidden @math{A = \ln(a)},
address@hidden @expr{A = ln(a)},
address@hidden = 1}, @expr{B = b}, and
address@hidden @math{G = \ln(x)}.
address@hidden @expr{G = ln(x)}.
-Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
-does a linear fit for @expr{A} and @expr{B}, then solves to get
address@hidden @math{a = \exp(A)}
address@hidden @expr{a = exp(A)}
-and @expr{b = B}.
-
-Another interesting example is the ``quadratic'' model, which can
-be handled by expanding according to the distributive law.
-
address@hidden
-y = a + b*(x - c)^2
-y = a + b c^2 - 2 b c x + b x^2
address@hidden example
-
address@hidden
-which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
address@hidden = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as
easily
-have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
address@hidden = x^2}.
-
-The Gaussian model looks quite complicated, but a closer examination
-shows that it's actually similar to the quadratic model but with an
-exponential that can be brought to the top and moved into @expr{Y}.
-
-The logistic models cannot be put into general linear form. For these
-models, and the Hubbert linearization, Calc computes a rough
-approximation for the parameters, then uses the Levenberg-Marquardt
-iterative method to refine the approximations.
-
-Another model that cannot be put into general linear
-form is a Gaussian with a constant background added on, i.e.,
address@hidden + the regular Gaussian formula. If you have a model like
-this, your best bet is to replace enough of your parameters with
-constants to make the model linearizable, then adjust the constants
-manually by doing a series of fits. You can compare the fits by
-graphing them, by examining the goodness-of-fit measures returned by
address@hidden a F}, or by some other method suitable to your application.
-Note that some models can be linearized in several ways. The
address@hidden model can be linearized by setting @expr{d}
-(the background) to a constant, or by setting @expr{b} (the standard
-deviation) and @expr{c} (the mean) to constants.
-
-To fit a model with constants substituted for some parameters, just
-store suitable values in those parameter variables, then omit them
-from the list of parameters when you answer the variables prompt.
-
address@hidden
-\bigskip
address@hidden tex
-
-A last desperate step would be to use the general-purpose
address@hidden function rather than @code{fit}. After all, both
-functions solve the problem of minimizing an expression (the
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-sum) by adjusting certain parameters in the expression. The @kbd{a F}
-command is able to use a vastly more efficient algorithm due to its
-special knowledge about linear chi-square sums, but the @kbd{a N}
-command can do the same thing by brute force.
-
-A compromise would be to pick out a few parameters without which the
-fit is linearizable, and use @code{minimize} on a call to @code{fit}
-which efficiently takes care of the rest of the parameters. The thing
-to be minimized would be the value of
address@hidden @math{\chi^2}
address@hidden @expr{chi^2}
-returned as the fifth result of the @code{xfit} function:
-
address@hidden
-minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
address@hidden smallexample
-
address@hidden
-where @code{gaus} represents the Gaussian model with background,
address@hidden represents the data matrix, and @code{guess} represents
-the initial guess for @expr{d} that @code{minimize} requires.
-This operation will only be, shall we say, extraordinarily slow
-rather than astronomically slow (as would be the case if @code{minimize}
-were used by itself to solve the problem).
-
address@hidden
-\bigskip
address@hidden tex
-
-The @kbd{I a F} address@hidden command is somewhat trickier when
-nonlinear models are used. The second item in the result is the
-vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
-covariance matrix is written in terms of those raw parameters.
-The fifth item is a vector of @dfn{filter} expressions. This
-is the empty vector @samp{[]} if the raw parameters were the same
-as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
-and so on (which is always true if the model is already linear
-in the parameters as written, e.g., for polynomial fits). If the
-parameters had to be rearranged, the fifth item is instead a vector
-of one formula per parameter in the original model. The raw
-parameters are expressed in these ``filter'' formulas as
address@hidden(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
-and so on.
-
-When Calc needs to modify the model to return the result, it replaces
address@hidden(1)} in all the filters with the first item in the raw
-parameters list, and so on for the other raw parameters, then
-evaluates the resulting filter formulas to get the actual parameter
-values to be substituted into the original model. In the case of
address@hidden a F} and @kbd{I a F} where the parameters must be error forms,
-Calc uses the square roots of the diagonal entries of the covariance
-matrix as error values for the raw parameters, then lets Calc's
-standard error-form arithmetic take it from there.
-
-If you use @kbd{I a F} with a nonlinear model, be sure to remember
-that the covariance matrix is in terms of the raw parameters,
address@hidden the actual requested parameters. It's up to you to
-figure out how to interpret the covariances in the presence of
-nontrivial filter functions.
-
-Things are also complicated when the input contains error forms.
-Suppose there are three independent and dependent variables, @expr{x},
address@hidden, and @expr{z}, one or more of which are error forms in the
-data. Calc combines all the error values by taking the square root
-of the sum of the squares of the errors. It then changes @expr{x}
-and @expr{y} to be plain numbers, and makes @expr{z} into an error
-form with this combined error. The @expr{Y(x,y,z)} part of the
-linearized model is evaluated, and the result should be an error
-form. The error part of that result is used for
address@hidden @math{\sigma_i}
address@hidden @expr{sigma_i}
-for the data point. If for some reason @expr{Y(x,y,z)} does not return
-an error form, the combined error from @expr{z} is used directly for
address@hidden @math{\sigma_i}.
address@hidden @expr{sigma_i}.
-Finally, @expr{z} is also stripped of its error
-for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
-the righthand side of the linearized model is computed in regular
-arithmetic with no error forms.
-
-(While these rules may seem complicated, they are designed to do
-the most reasonable thing in the typical case that @expr{Y(x,y,z)}
-depends only on the dependent variable @expr{z}, and in fact is
-often simply equal to @expr{z}. For common cases like polynomials
-and multilinear models, the combined error is simply used as the
address@hidden @math{\sigma}
address@hidden @expr{sigma}
-for the data point with no further ado.)
-
address@hidden
-\bigskip
address@hidden tex
-
address@hidden FitRules
-It may be the case that the model you wish to use is linearizable,
-but Calc's built-in rules are unable to figure it out. Calc uses
-its algebraic rewrite mechanism to linearize a model. The rewrite
-rules are kept in the variable @code{FitRules}. You can edit this
-variable using the @kbd{s e FitRules} command; in fact, there is
-a special @kbd{s F} command just for editing @code{FitRules}.
address@hidden on Variables}.
-
address@hidden Rules}, for a discussion of rewrite rules.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden fitvar
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden fitparam
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitmodel
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitsystem
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden fitdummy
-Calc uses @code{FitRules} as follows. First, it converts the model
-to an equation if necessary and encloses the model equation in a
-call to the function @code{fitmodel} (which is not actually a defined
-function in Calc; it is only used as a placeholder by the rewrite rules).
-Parameter variables are renamed to function calls @samp{fitparam(1)},
address@hidden(2)}, and so on, and independent variables are renamed
-to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
-is the highest-numbered @code{fitvar}. For example, the power law
-model @expr{a x^b} is converted to @expr{y = a x^b}, then to
-
address@hidden
address@hidden
-fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
address@hidden group
address@hidden smallexample
-
-Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
-(The zero prefix means that rewriting should continue until no further
-changes are possible.)
-
-When rewriting is complete, the @code{fitmodel} call should have
-been replaced by a @code{fitsystem} call that looks like this:
-
address@hidden
-fitsystem(@var{Y}, @var{FGH}, @var{abc})
address@hidden example
-
address@hidden
-where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
address@hidden is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
-and @var{abc} is the vector of parameter filters which refer to the
-raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
-for @expr{B}, etc. While the number of raw parameters (the length of
-the @var{FGH} vector) is usually the same as the number of original
-parameters (the length of the @var{abc} vector), this is not required.
-
-The power law model eventually boils down to
-
address@hidden
address@hidden
-fitsystem(ln(fitvar(2)),
- [1, ln(fitvar(1))],
- [exp(fitdummy(1)), fitdummy(2)])
address@hidden group
address@hidden smallexample
-
-The actual implementation of @code{FitRules} is complicated; it
-proceeds in four phases. First, common rearrangements are done
-to try to bring linear terms together and to isolate functions like
address@hidden and @code{ln} either all the way ``out'' (so that they
-can be put into @var{Y}) or all the way ``in'' (so that they can
-be put into @var{abc} or @var{FGH}). In particular, all
-non-constant powers are converted to logs-and-exponentials form,
-and the distributive law is used to expand products of sums.
-Quotients are rewritten to use the @samp{fitinv} function, where
address@hidden(x)} represents @expr{1/x} while the @code{FitRules}
-are operating. (The use of @code{fitinv} makes recognition of
-linear-looking forms easier.) If you modify @code{FitRules}, you
-will probably only need to modify the rules for this phase.
-
-Phase two, whose rules can actually also apply during phases one
-and three, first rewrites @code{fitmodel} to a two-argument
-form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
-initially zero and @var{model} has been changed from @expr{a=b}
-to @expr{a-b} form. It then tries to peel off invertible functions
-from the outside of @var{model} and put them into @var{Y} instead,
-calling the equation solver to invert the functions. Finally, when
-this is no longer possible, the @code{fitmodel} is changed to a
-four-argument @code{fitsystem}, where the fourth argument is
address@hidden and the @var{FGH} and @var{abc} vectors are initially
-empty. (The last vector is really @var{ABC}, corresponding to
-raw parameters, for now.)
-
-Phase three converts a sum of items in the @var{model} to a sum
-of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
-terms @address@hidden@address@hidden of the sum, where @var{a}
-is all factors that do not involve any variables, @var{b} is all
-factors that involve only parameters, and @var{c} is the factors
-that involve only independent variables. (If this decomposition
-is not possible, the rule set will not complete and Calc will
-complain that the model is too complex.) Then @code{fitpart}s
-with equal @var{b} or @var{c} components are merged back together
-using the distributive law in order to minimize the number of
-raw parameters needed.
-
-Phase four moves the @code{fitpart} terms into the @var{FGH} and
address@hidden vectors. Also, some of the algebraic expansions that
-were done in phase 1 are undone now to make the formulas more
-computationally efficient. Finally, it calls the solver one more
-time to convert the @var{ABC} vector to an @var{abc} vector, and
-removes the fourth @var{model} argument (which by now will be zero)
-to obtain the three-argument @code{fitsystem} that the linear
-least-squares solver wants to see.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden hasfitparams
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden @null
address@hidden ignore
address@hidden hasfitvars
-Two functions which are useful in connection with @code{FitRules}
-are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
-whether @expr{x} refers to any parameters or independent variables,
-respectively. Specifically, these functions return ``true'' if the
-argument contains any @code{fitparam} (or @code{fitvar}) function
-calls, and ``false'' otherwise. (Recall that ``true'' means a
-nonzero number, and ``false'' means zero. The actual nonzero number
-returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
-or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
-
address@hidden
-\bigskip
address@hidden tex
-
-The @code{fit} function in algebraic notation normally takes four
-arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
-where @var{model} is the model formula as it would be typed after
address@hidden F '}, @var{vars} is the independent variable or a vector of
-independent variables, @var{params} likewise gives the parameter(s),
-and @var{data} is the data matrix. Note that the length of @var{vars}
-must be equal to the number of rows in @var{data} if @var{model} is
-an equation, or one less than the number of rows if @var{model} is
-a plain formula. (Actually, a name for the dependent variable is
-allowed but will be ignored in the plain-formula case.)
-
-If @var{params} is omitted, the parameters are all variables in
address@hidden except those that appear in @var{vars}. If @var{vars}
-is also omitted, Calc sorts all the variables that appear in
address@hidden alphabetically and uses the higher ones for @var{vars}
-and the lower ones for @var{params}.
-
-Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
-where @var{modelvec} is a 2- or 3-vector describing the model
-and variables, as discussed previously.
-
-If Calc is unable to do the fit, the @code{fit} function is left
-in symbolic form, ordinarily with an explanatory message. The
-message will be ``Model expression is too complex'' if the
-linearizer was unable to put the model into the required form.
-
-The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
-(for @kbd{I a F}) functions are completely analogous.
-
address@hidden Interpolation, , Curve Fitting Details, Curve Fitting
address@hidden Polynomial Interpolation
-
address@hidden a p
address@hidden calc-poly-interp
address@hidden polint
-The @kbd{a p} (@code{calc-poly-interp}) address@hidden command does
-a polynomial interpolation at a particular @expr{x} value. It takes
-two arguments from the stack: A data matrix of the sort used by
address@hidden F}, and a single number which represents the desired @expr{x}
-value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
-then substitutes the @expr{x} value into the result in order to get an
-approximate @expr{y} value based on the fit. (Calc does not actually
-use @kbd{a F i}, however; it uses a direct method which is both more
-efficient and more numerically stable.)
-
-The result of @kbd{a p} is actually a vector of two values: The @expr{y}
-value approximation, and an error measure @expr{dy} that reflects Calc's
-estimation of the probable error of the approximation at that value of
address@hidden If the input @expr{x} is equal to any of the @expr{x} values
-in the data matrix, the output @expr{y} will be the corresponding @expr{y}
-value from the matrix, and the output @expr{dy} will be exactly zero.
-
-A prefix argument of 2 causes @kbd{a p} to take separate x- and
-y-vectors from the stack instead of one data matrix.
-
-If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
-interpolated results for each of those @expr{x} values. (The matrix will
-have two columns, the @expr{y} values and the @expr{dy} values.)
-If @expr{x} is a formula instead of a number, the @code{polint} function
-remains in symbolic form; use the @kbd{a "} command to expand it out to
-a formula that describes the fit in symbolic terms.
-
-In all cases, the @kbd{a p} command leaves the data vectors or matrix
-on the stack. Only the @expr{x} value is replaced by the result.
-
address@hidden H a p
address@hidden ratint
-The @kbd{H a p} address@hidden command does a rational function
-interpolation. It is used exactly like @kbd{a p}, except that it
-uses as its model the quotient of two polynomials. If there are
address@hidden data points, the numerator and denominator polynomials will
-each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
-have degree one higher than the numerator).
-
-Rational approximations have the advantage that they can accurately
-describe functions that have poles (points at which the function's value
-goes to infinity, so that the denominator polynomial of the approximation
-goes to zero). If @expr{x} corresponds to a pole of the fitted rational
-function, then the result will be a division by zero. If Infinite mode
-is enabled, the result will be @samp{[uinf, uinf]}.
-
-There is no way to get the actual coefficients of the rational function
-used by @kbd{H a p}. (The algorithm never generates these coefficients
-explicitly, and quotients of polynomials are beyond @address@hidden F}}'s
-capabilities to fit.)
-
address@hidden Summations, Logical Operations, Curve Fitting, Algebra
address@hidden Summations
-
address@hidden
address@hidden Summation of a series
address@hidden a +
address@hidden calc-summation
address@hidden sum
-The @kbd{a +} (@code{calc-summation}) address@hidden command computes
-the sum of a formula over a certain range of index values. The formula
-is taken from the top of the stack; the command prompts for the
-name of the summation index variable, the lower limit of the
-sum (any formula), and the upper limit of the sum. If you
-enter a blank line at any of these prompts, that prompt and
-any later ones are answered by reading additional elements from
-the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a
+ @key{RET}}
-produces the result 55.
address@hidden
-\turnoffactive
-$$ \sum_{k=1}^5 k^2 = 55 $$
address@hidden tex
-
-The choice of index variable is arbitrary, but it's best not to
-use a variable with a stored value. In particular, while
address@hidden is often a favorite index variable, it should be avoided
-in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
-as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
-be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
-If you really want to use @code{i} as an index variable, use
address@hidden@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
-(@xref{Storing Variables}.)
-
-A numeric prefix argument steps the index by that amount rather
-than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0
@key{RET}}
-yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
-argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
-step value, in which case you can enter any formula or enter
-a blank line to take the step value from the stack. With the
address@hidden prefix, @kbd{a +} can take up to five arguments from
-the stack: The formula, the variable, the lower limit, the
-upper limit, and (at the top of the stack), the step value.
-
-Calc knows how to do certain sums in closed form. For example,
address@hidden(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
-this is possible if the formula being summed is polynomial or
-exponential in the index variable. Sums of logarithms are
-transformed into logarithms of products. Sums of trigonometric
-and hyperbolic functions are transformed to sums of exponentials
-and then done in closed form. Also, of course, sums in which the
-lower and upper limits are both numbers can always be evaluated
-just by grinding them out, although Calc will use closed forms
-whenever it can for the sake of efficiency.
-
-The notation for sums in algebraic formulas is
address@hidden(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
-If @var{step} is omitted, it defaults to one. If @var{high} is
-omitted, @var{low} is actually the upper limit and the lower limit
-is one. If @var{low} is also omitted, the limits are @samp{-inf}
-and @samp{inf}, respectively.
-
-Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
-returns @expr{1}. This is done by evaluating the sum in closed
-form (to @samp{1. - 0.5^n} in this case), then evaluating this
-formula with @code{n} set to @code{inf}. Calc's usual rules
-for ``infinite'' arithmetic can find the answer from there. If
-infinite arithmetic yields a @samp{nan}, or if the sum cannot be
-solved in closed form, Calc leaves the @code{sum} function in
-symbolic form. @xref{Infinities}.
-
-As a special feature, if the limits are infinite (or omitted, as
-described above) but the formula includes vectors subscripted by
-expressions that involve the iteration variable, Calc narrows
-the limits to include only the range of integers which result in
-valid subscripts for the vector. For example, the sum
address@hidden(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
-
-The limits of a sum do not need to be integers. For example,
address@hidden(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
-Calc computes the number of iterations using the formula
address@hidden + (@var{high} - @var{low}) / @var{step}}, which must,
-after simplification as if by @kbd{a s}, evaluate to an integer.
-
-If the number of iterations according to the above formula does
-not come out to an integer, the sum is invalid and will be left
-in symbolic form. However, closed forms are still supplied, and
-you are on your honor not to misuse the resulting formulas by
-substituting mismatched bounds into them. For example,
address@hidden(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
-evaluate the closed form solution for the limits 1 and 10 to get
-the rather dubious answer, 29.25.
-
-If the lower limit is greater than the upper limit (assuming a
-positive step size), the result is generally zero. However,
-Calc only guarantees a zero result when the upper limit is
-exactly one step less than the lower limit, i.e., if the number
-of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
-but the sum from @samp{n} to @samp{n-2} may report a nonzero value
-if Calc used a closed form solution.
-
-Calc's logical predicates like @expr{a < b} return 1 for ``true''
-and 0 for ``false.'' @xref{Logical Operations}. This can be
-used to advantage for building conditional sums. For example,
address@hidden(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
-prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
-its argument is prime and 0 otherwise. You can read this expression
-as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
address@hidden(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
-squared, since the limits default to plus and minus infinity, but
-there are no such sums that Calc's built-in rules can do in
-closed form.
-
-As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
-sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
-one value @expr{k_0}. Slightly more tricky is the summand
address@hidden(k != k_0) / (k - k_0)}, which is an attempt to describe
-the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
-this would be a division by zero. But at @expr{k = k_0}, this
-formula works out to the indeterminate form @expr{0 / 0}, which
-Calc will not assume is zero. Better would be to use
address@hidden(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
-an ``if-then-else'' test: This expression says, ``if
address@hidden @math{k \ne k_0},
address@hidden @expr{k != k_0},
-then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
-will not even be evaluated by Calc when @expr{k = k_0}.
-
address@hidden Alternating sums
address@hidden a -
address@hidden calc-alt-summation
address@hidden asum
-The @kbd{a -} (@code{calc-alt-summation}) address@hidden command
-computes an alternating sum. Successive terms of the sequence
-are given alternating signs, with the first term (corresponding
-to the lower index value) being positive. Alternating sums
-are converted to normal sums with an extra term of the form
address@hidden(-1)^(address@hidden)}. This formula is adjusted appropriately
-if the step value is other than one. For example, the Taylor
-series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
-(Calc cannot evaluate this infinite series, but it can approximate
-it if you replace @code{inf} with any particular odd number.)
-Calc converts this series to a regular sum with a step of one,
-namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
-
address@hidden Product of a sequence
address@hidden a *
address@hidden calc-product
address@hidden prod
-The @kbd{a *} (@code{calc-product}) address@hidden command is
-the analogous way to take a product of many terms. Calc also knows
-some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
-Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
-or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
-
address@hidden a T
address@hidden calc-tabulate
address@hidden table
-The @kbd{a T} (@code{calc-tabulate}) address@hidden command
-evaluates a formula at a series of iterated index values, just
-like @code{sum} and @code{prod}, but its result is simply a
-vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
-produces @samp{[a_1, a_3, a_5, a_7]}.
-
address@hidden Logical Operations, Rewrite Rules, Summations, Algebra
address@hidden Logical Operations
-
address@hidden
-The following commands and algebraic functions return true/false values,
-where 1 represents ``true'' and 0 represents ``false.'' In cases where
-a truth value is required (such as for the condition part of a rewrite
-rule, or as the condition for a @address@hidden [ Z ]}} control structure), any
-nonzero value is accepted to mean ``true.'' (Specifically, anything
-for which @code{dnonzero} returns 1 is ``true,'' and anything for
-which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
-Note that this means that @address@hidden [ Z ]}} will execute the ``then''
-portion if its condition is provably true, but it will execute the
-``else'' portion for any condition like @expr{a = b} that is not
-provably true, even if it might be true. Algebraic functions that
-have conditions as arguments, like @code{? :} and @code{&&}, remain
-unevaluated if the condition is neither provably true nor provably
-false. @xref{Declarations}.)
-
address@hidden a =
address@hidden calc-equal-to
address@hidden eq
address@hidden =
address@hidden ==
-The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
-(which can also be written @samp{a = b} or @samp{a == b} in an algebraic
-formula) is true if @expr{a} and @expr{b} are equal, either because they
-are identical expressions, or because they are numbers which are
-numerically equal. (Thus the integer 1 is considered equal to the float
-1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
-the comparison is left in symbolic form. Note that as a command, this
-operation pops two values from the stack and pushes back either a 1 or
-a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
-
-Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
-For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
-an equation to solve for a given variable. The @kbd{a M}
-(@code{calc-map-equation}) command can be used to apply any
-function to both sides of an equation; for example, @kbd{2 a M *}
-multiplies both sides of the equation by two. Note that just
address@hidden *} would not do the same thing; it would produce the formula
address@hidden (a = b)} which represents 2 if the equality is true or
-zero if not.
-
-The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
-or @samp{a = b = c}) tests if all of its arguments are equal. In
-algebraic notation, the @samp{=} operator is unusual in that it is
-neither left- nor right-associative: @samp{a = b = c} is not the
-same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
-one variable with the 1 or 0 that results from comparing two other
-variables).
-
address@hidden a #
address@hidden calc-not-equal-to
address@hidden neq
address@hidden !=
-The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
address@hidden != b} function, is true if @expr{a} and @expr{b} are not equal.
-This also works with more than two arguments; @samp{a != b != c != d}
-tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
-distinct numbers.
-
address@hidden a <
address@hidden lt
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden a >
address@hidden
address@hidden @null
address@hidden ignore
address@hidden a [
address@hidden
address@hidden @null
address@hidden ignore
address@hidden a ]
address@hidden calc-less-than
address@hidden calc-greater-than
address@hidden calc-less-equal
address@hidden calc-greater-equal
address@hidden
address@hidden @null
address@hidden ignore
address@hidden gt
address@hidden
address@hidden @null
address@hidden ignore
address@hidden leq
address@hidden
address@hidden @null
address@hidden ignore
address@hidden geq
address@hidden
address@hidden @null
address@hidden ignore
address@hidden <
address@hidden
address@hidden @null
address@hidden ignore
address@hidden >
address@hidden
address@hidden @null
address@hidden ignore
address@hidden <=
address@hidden
address@hidden @null
address@hidden ignore
address@hidden >=
-The @kbd{a <} (@code{calc-less-than}) address@hidden(a,b)} or @samp{a < b}]
-operation is true if @expr{a} is less than @expr{b}. Similar functions
-are @kbd{a >} (@code{calc-greater-than}) address@hidden(a,b)} or @samp{a > b}],
address@hidden [} (@code{calc-less-equal}) address@hidden(a,b)} or @samp{a <=
b}], and
address@hidden ]} (@code{calc-greater-equal}) address@hidden(a,b)} or @samp{a
>= b}].
-
-While the inequality functions like @code{lt} do not accept more
-than two arguments, the syntax @address@hidden <= b < c}} is translated to an
-equivalent expression involving intervals: @samp{b in [a .. c)}.
-(See the description of @code{in} below.) All four combinations
-of @samp{<} and @samp{<=} are allowed, or any of the four combinations
-of @samp{>} and @samp{>=}. Four-argument constructions like
address@hidden < b < c < d}, and mixtures like @address@hidden < b = c}} that
-involve both equalities and inequalities, are not allowed.
-
address@hidden a .
address@hidden calc-remove-equal
address@hidden rmeq
-The @kbd{a .} (@code{calc-remove-equal}) address@hidden command extracts
-the righthand side of the equation or inequality on the top of the
-stack. It also works elementwise on vectors. For example, if
address@hidden = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
address@hidden, z / 2]}. As a special case, if the righthand side is a
-variable and the lefthand side is a number (as in @samp{2.34 = x}), then
-Calc keeps the lefthand side instead. Finally, this command works with
-assignments @samp{x := 2.34} as well as equations, always taking the
-righthand side, and for @samp{=>} (evaluates-to) operators, always
-taking the lefthand side.
-
address@hidden a &
address@hidden calc-logical-and
address@hidden land
address@hidden &&
-The @kbd{a &} (@code{calc-logical-and}) address@hidden(a,b)} or @samp{a && b}]
-function is true if both of its arguments are true, i.e., are
-non-zero numbers. In this case, the result will be either @expr{a} or
address@hidden, chosen arbitrarily. If either argument is zero, the result is
-zero. Otherwise, the formula is left in symbolic form.
-
address@hidden a |
address@hidden calc-logical-or
address@hidden lor
address@hidden ||
-The @kbd{a |} (@code{calc-logical-or}) address@hidden(a,b)} or @samp{a || b}]
-function is true if either or both of its arguments are true (nonzero).
-The result is whichever argument was nonzero, choosing arbitrarily if both
-are nonzero. If both @expr{a} and @expr{b} are zero, the result is
-zero.
-
address@hidden a !
address@hidden calc-logical-not
address@hidden lnot
address@hidden !
-The @kbd{a !} (@code{calc-logical-not}) address@hidden(a)} or @samp{!@: a}]
-function is true if @expr{a} is false (zero), or false if @expr{a} is
-true (nonzero). It is left in symbolic form if @expr{a} is not a
-number.
-
address@hidden a :
address@hidden calc-logical-if
address@hidden if
address@hidden
address@hidden ? :
address@hidden ignore
address@hidden ?
address@hidden
address@hidden @null
address@hidden ignore
address@hidden :
address@hidden Arguments, not evaluated
-The @kbd{a :} (@code{calc-logical-if}) address@hidden(a,b,c)} or @samp{a ? b
:@: c}]
-function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
-number or zero, respectively. If @expr{a} is not a number, the test is
-left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
-any way. In algebraic formulas, this is one of the few Calc functions
-whose arguments are not automatically evaluated when the function itself
-is evaluated. The others are @code{lambda}, @code{quote}, and
address@hidden
-
-One minor surprise to watch out for is that the formula @samp{a?3:4}
-will not work because the @samp{3:4} is parsed as a fraction instead of
-as three separate symbols. Type something like @samp{a ? 3 : 4} or
address@hidden(3):4} instead.
-
-As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
-and @expr{c} are evaluated; the result is a vector of the same length
-as @expr{a} whose elements are chosen from corresponding elements of
address@hidden and @expr{c} according to whether each element of @expr{a}
-is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
-vector of the same length as @expr{a}, or a non-vector which is matched
-with all elements of @expr{a}.
-
address@hidden a @{
address@hidden calc-in-set
address@hidden in
-The @kbd{a @{} (@code{calc-in-set}) address@hidden(a,b)}] function is true if
-the number @expr{a} is in the set of numbers represented by @expr{b}.
-If @expr{b} is an interval form, @expr{a} must be one of the values
-encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
-equal to one of the elements of the vector. (If any vector elements are
-intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
-plain number, @expr{a} must be numerically equal to @expr{b}.
address@hidden Operations}, for a group of commands that manipulate sets
-of this sort.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden typeof
-The @samp{typeof(a)} function produces an integer or variable which
-characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
-the result will be one of the following numbers:
-
address@hidden
- 1 Integer
- 2 Fraction
- 3 Floating-point number
- 4 HMS form
- 5 Rectangular complex number
- 6 Polar complex number
- 7 Error form
- 8 Interval form
- 9 Modulo form
-10 Date-only form
-11 Date/time form
-12 Infinity (inf, uinf, or nan)
-100 Variable
-101 Vector (but not a matrix)
-102 Matrix
address@hidden example
-
-Otherwise, @expr{a} is a formula, and the result is a variable which
-represents the name of the top-level function call.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden integer
address@hidden
address@hidden
address@hidden ignore
address@hidden real
address@hidden
address@hidden
address@hidden ignore
address@hidden constant
-The @samp{integer(a)} function returns true if @expr{a} is an integer.
-The @samp{real(a)} function
-is true if @expr{a} is a real number, either integer, fraction, or
-float. The @samp{constant(a)} function returns true if @expr{a} is
-any of the objects for which @code{typeof} would produce an integer
-code result except for variables, and provided that the components of
-an object like a vector or error form are themselves constant.
-Note that infinities do not satisfy any of these tests, nor do
-special constants like @code{pi} and @code{e}.
-
address@hidden, for a set of similar functions that recognize
-formulas as well as actual numbers. For example, @samp{dint(floor(x))}
-is true because @samp{floor(x)} is provably integer-valued, but
address@hidden(floor(x))} does not because @samp{floor(x)} is not
-literally an integer constant.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden refers
-The @samp{refers(a,b)} function is true if the variable (or sub-expression)
address@hidden appears in @expr{a}, or false otherwise. Unlike the other
-tests described here, this function returns a definite ``no'' answer
-even if its arguments are still in symbolic form. The only case where
address@hidden will be left unevaluated is if @expr{a} is a plain
-variable (different from @expr{b}).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden negative
-The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
-because it is a negative number, because it is of the form @expr{-x},
-or because it is a product or quotient with a term that looks negative.
-This is most useful in rewrite rules. Beware that @samp{negative(a)}
-evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
-be stored in a formula if the default simplifications are turned off
-first with @kbd{m O} (or if it appears in an unevaluated context such
-as a rewrite rule condition).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden variable
-The @samp{variable(a)} function is true if @expr{a} is a variable,
-or false if not. If @expr{a} is a function call, this test is left
-in symbolic form. Built-in variables like @code{pi} and @code{inf}
-are considered variables like any others by this test.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden nonvar
-The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
-If its argument is a variable it is left unsimplified; it never
-actually returns zero. However, since Calc's condition-testing
-commands consider ``false'' anything not provably true, this is
-often good enough.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden lin
address@hidden
address@hidden
address@hidden ignore
address@hidden linnt
address@hidden
address@hidden
address@hidden ignore
address@hidden islin
address@hidden
address@hidden
address@hidden ignore
address@hidden islinnt
address@hidden Linearity testing
-The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
-check if an expression is ``linear,'' i.e., can be written in the form
address@hidden + b x} for some constants @expr{a} and @expr{b}, and some
-variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
-if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
-example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
address@hidden(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
-is similar, except that instead of returning 1 it returns the vector
address@hidden, b, x]}. For the above examples, this vector would be
address@hidden, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
address@hidden, y/3, x]}, respectively. Both @code{lin} and @code{islin}
-generally remain unevaluated for expressions which are not linear,
-e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
-argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
-returns true.
-
-The @code{linnt} and @code{islinnt} functions perform a similar check,
-but require a ``non-trivial'' linear form, which means that the
address@hidden coefficient must be non-zero. For example, @samp{lin(2,x)}
-returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
-but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
-(in other words, these formulas are considered to be only ``trivially''
-linear in @expr{x}).
-
-All four linearity-testing functions allow you to omit the second
-argument, in which case the input may be linear in any non-constant
-formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
-trivial, and only constant values for @expr{a} and @expr{b} are
-recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
address@hidden(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
-returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
-first two cases but not the third. Also, neither @code{lin} nor
address@hidden accept plain constants as linear in the one-argument
-case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden istrue
-The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
-number or provably nonzero formula, or 0 if @expr{a} is anything else.
-Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
-used to make sure they are not evaluated prematurely. (Note that
-declarations are used when deciding whether a formula is true;
address@hidden returns 1 when @code{dnonzero} would return 1, and
-it returns 0 when @code{dnonzero} would return 0 or leave itself
-in symbolic form.)
-
address@hidden Rewrite Rules, , Logical Operations, Algebra
address@hidden Rewrite Rules
-
address@hidden
address@hidden Rewrite rules
address@hidden Transformations
address@hidden Pattern matching
address@hidden a r
address@hidden calc-rewrite
address@hidden rewrite
-The @kbd{a r} (@code{calc-rewrite}) address@hidden command makes
-substitutions in a formula according to a specified pattern or patterns
-known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
-matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
-matches only the @code{sin} function applied to the variable @code{x},
-rewrite rules match general kinds of formulas; rewriting using the rule
address@hidden(x) := cos(x)} matches @code{sin} of any argument and replaces
-it with @code{cos} of that same argument. The only significance of the
-name @code{x} is that the same name is used on both sides of the rule.
-
-Rewrite rules rearrange formulas already in Calc's memory.
address@hidden Tables}, to read about @dfn{syntax rules}, which are
-similar to algebraic rewrite rules but operate when new algebraic
-entries are being parsed, converting strings of characters into
-Calc formulas.
-
address@hidden
-* Entering Rewrite Rules::
-* Basic Rewrite Rules::
-* Conditional Rewrite Rules::
-* Algebraic Properties of Rewrite Rules::
-* Other Features of Rewrite Rules::
-* Composing Patterns in Rewrite Rules::
-* Nested Formulas with Rewrite Rules::
-* Multi-Phase Rewrite Rules::
-* Selections with Rewrite Rules::
-* Matching Commands::
-* Automatic Rewrites::
-* Debugging Rewrites::
-* Examples of Rewrite Rules::
address@hidden menu
-
address@hidden Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules,
Rewrite Rules
address@hidden Entering Rewrite Rules
-
address@hidden
-Rewrite rules normally use the ``assignment'' operator
address@hidden@var{old} := @var{new}}.
-This operator is equivalent to the function call @samp{assign(old, new)}.
-The @code{assign} function is undefined by itself in Calc, so an
-assignment formula such as a rewrite rule will be left alone by ordinary
-Calc commands. But certain commands, like the rewrite system, interpret
-assignments in special ways.
-
-For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
-every occurrence of the sine of something, squared, with one minus the
-square of the cosine of that same thing. All by itself as a formula
-on the stack it does nothing, but when given to the @kbd{a r} command
-it turns that command into a sine-squared-to-cosine-squared converter.
-
-To specify a set of rules to be applied all at once, make a vector of
-rules.
-
-When @kbd{a r} prompts you to enter the rewrite rules, you can answer
-in several ways:
-
address@hidden
address@hidden
-With a rule: @kbd{f(x) := g(x) @key{RET}}.
address@hidden
-With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
-(You can omit the enclosing square brackets if you wish.)
address@hidden
-With the name of a variable that contains the rule or rules vector:
address@hidden @key{RET}}.
address@hidden
-With any formula except a rule, a vector, or a variable name; this
-will be interpreted as the @var{old} half of a rewrite rule,
-and you will be prompted a second time for the @var{new} half:
address@hidden(x) @key{RET} g(x) @key{RET}}.
address@hidden
-With a blank line, in which case the rule, rules vector, or variable
-will be taken from the top of the stack (and the formula to be
-rewritten will come from the second-to-top position).
address@hidden enumerate
-
-If you enter the rules directly (as opposed to using rules stored
-in a variable), those rules will be put into the Trail so that you
-can retrieve them later. @xref{Trail Commands}.
-
-It is most convenient to store rules you use often in a variable and
-invoke them by giving the variable name. The @kbd{s e}
-(@code{calc-edit-variable}) command is an easy way to create or edit a
-rule set stored in a variable. You may also wish to use @kbd{s p}
-(@code{calc-permanent-variable}) to save your rules permanently;
address@hidden on Variables}.
-
-Rewrite rules are compiled into a special internal form for faster
-matching. If you enter a rule set directly it must be recompiled
-every time. If you store the rules in a variable and refer to them
-through that variable, they will be compiled once and saved away
-along with the variable for later reference. This is another good
-reason to store your rules in a variable.
-
-Calc also accepts an obsolete notation for rules, as vectors
address@hidden@var{old}, @var{new}]}. But because it is easily confused with a
-vector of two rules, the use of this notation is no longer recommended.
-
address@hidden Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite
Rules, Rewrite Rules
address@hidden Basic Rewrite Rules
-
address@hidden
-To match a particular formula @expr{x} with a particular rewrite rule
address@hidden@var{old} := @var{new}}, Calc compares the structure of @expr{x}
with
-the structure of @var{old}. Variables that appear in @var{old} are
-treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
-may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
-would match the expression @samp{f(12, a+1)} with the meta-variable
address@hidden corresponding to 12 and with @samp{y} corresponding to
address@hidden However, this pattern would not match @samp{f(12)} or
address@hidden(12, a+1)}, since there is no assignment of the meta-variables
-that will make the pattern match these expressions. Notice that if
-the pattern is a single meta-variable, it will match any expression.
-
-If a given meta-variable appears more than once in @var{old}, the
-corresponding sub-formulas of @expr{x} must be identical. Thus
-the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
address@hidden(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
-(@xref{Conditional Rewrite Rules}, for a way to match the latter.)
-
-Things other than variables must match exactly between the pattern
-and the target formula. To match a particular variable exactly, use
-the pseudo-function @samp{quote(v)} in the pattern. For example, the
-pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
address@hidden(a)+y}.
-
-The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
address@hidden, @samp{inf}, @samp{uinf}, and @samp{nan} always match
-literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
address@hidden(d + quote(e) + f)}.
-
-If the @var{old} pattern is found to match a given formula, that
-formula is replaced by @var{new}, where any occurrences in @var{new}
-of meta-variables from the pattern are replaced with the sub-formulas
-that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
-to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
-
-The normal @kbd{a r} command applies rewrite rules over and over
-throughout the target formula until no further changes are possible
-(up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
-change at a time.
-
address@hidden Conditional Rewrite Rules, Algebraic Properties of Rewrite
Rules, Basic Rewrite Rules, Rewrite Rules
address@hidden Conditional Rewrite Rules
-
address@hidden
-A rewrite rule can also be @dfn{conditional}, written in the form
address@hidden@var{old} := @var{new} :: @var{cond}}. (There is also the
obsolete
-form @address@hidden, @var{new}, @var{cond}]}.) If a @var{cond} part
-is present in the
-rule, this is an additional condition that must be satisfied before
-the rule is accepted. Once @var{old} has been successfully matched
-to the target expression, @var{cond} is evaluated (with all the
-meta-variables substituted for the values they matched) and simplified
-with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero
-number or any other object known to be nonzero (@pxref{Declarations}),
-the rule is accepted. If the result is zero or if it is a symbolic
-formula that is not known to be nonzero, the rule is rejected.
address@hidden Operations}, for a number of functions that return
-1 or 0 according to the results of various tests.
-
-For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
-is replaced by a positive or nonpositive number, respectively (or if
address@hidden has been declared to be positive or nonpositive). Thus,
-the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
address@hidden(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
-(assuming no outstanding declarations for @expr{a}). In the case of
address@hidden(-3, 2)}, the condition can be shown not to be satisfied; in
-the case of @samp{f(12, a+1)}, the condition merely cannot be shown
-to be satisfied, but that is enough to reject the rule.
-
-While Calc will use declarations to reason about variables in the
-formula being rewritten, declarations do not apply to meta-variables.
-For example, the rule @samp{f(a) := g(a+1)} will match for any values
-of @samp{a}, such as complex numbers, vectors, or formulas, even if
address@hidden has been declared to be real or scalar. If you want the
-meta-variable @samp{a} to match only literal real numbers, use
address@hidden(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
-reals and formulas which are provably real, use @samp{dreal(a)} as
-the condition.
-
-The @samp{::} operator is a shorthand for the @code{condition}
-function; @address@hidden := @var{new} :: @var{cond}} is equivalent to
-the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
-
-If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
-or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
-
-It is also possible to embed conditions inside the pattern:
address@hidden(x :: x>0, y) := g(y+x, x)}. This is purely a notational
-convenience, though; where a condition appears in a rule has no
-effect on when it is tested. The rewrite-rule compiler automatically
-decides when it is best to test each condition while a rule is being
-matched.
-
-Certain conditions are handled as special cases by the rewrite rule
-system and are tested very efficiently: Where @expr{x} is any
-meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
address@hidden(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
-is either a constant or another meta-variable and @samp{>=} may be
-replaced by any of the six relational operators, and @samp{x % a = b}
-where @expr{a} and @expr{b} are constants. Other conditions, like
address@hidden >= y+1} or @samp{dreal(x)}, will be less efficient to check
-since Calc must bring the whole evaluator and simplifier into play.
-
-An interesting property of @samp{::} is that neither of its arguments
-will be touched by Calc's default simplifications. This is important
-because conditions often are expressions that cannot safely be
-evaluated early. For example, the @code{typeof} function never
-remains in symbolic form; entering @samp{typeof(a)} will put the
-number 100 (the type code for variables like @samp{a}) on the stack.
-But putting the condition @samp{... :: typeof(a) = 6} on the stack
-is safe since @samp{::} prevents the @code{typeof} from being
-evaluated until the condition is actually used by the rewrite system.
-
-Since @samp{::} protects its lefthand side, too, you can use a dummy
-condition to protect a rule that must itself not evaluate early.
-For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
-the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
-where the meta-variable-ness of @code{f} on the righthand side has been
-lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
-the condition @samp{1} is always true (nonzero) so it has no effect on
-the functioning of the rule. (The rewrite compiler will ensure that
-it doesn't even impact the speed of matching the rule.)
-
address@hidden Algebraic Properties of Rewrite Rules, Other Features of Rewrite
Rules, Conditional Rewrite Rules, Rewrite Rules
address@hidden Algebraic Properties of Rewrite Rules
-
address@hidden
-The rewrite mechanism understands the algebraic properties of functions
-like @samp{+} and @samp{*}. In particular, pattern matching takes
-the associativity and commutativity of the following functions into
-account:
-
address@hidden
-+ - * = != && || and or xor vint vunion vxor gcd lcm max min beta
address@hidden smallexample
-
-For example, the rewrite rule:
-
address@hidden
-a x + b x := (a + b) x
address@hidden example
-
address@hidden
-will match formulas of the form,
-
address@hidden
-a x + b x, x a + x b, a x + x b, x a + b x
address@hidden example
-
-Rewrites also understand the relationship between the @samp{+} and @samp{-}
-operators. The above rewrite rule will also match the formulas,
-
address@hidden
-a x - b x, x a - x b, a x - x b, x a - b x
address@hidden example
-
address@hidden
-by matching @samp{b} in the pattern to @samp{-b} from the formula.
-
-Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
-pattern will check all pairs of terms for possible matches. The rewrite
-will take whichever suitable pair it discovers first.
-
-In general, a pattern using an associative operator like @samp{a + b}
-will try @var{2 n} different ways to match a sum of @var{n} terms
-like @samp{x + y + z - w}. First, @samp{a} is matched against each
-of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
-being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
-If none of these succeed, then @samp{b} is matched against each of the
-four terms with @samp{a} matching the remainder. Half-and-half matches,
-like @samp{(x + y) + (z - w)}, are not tried.
-
-Note that @samp{*} is not commutative when applied to matrices, but
-rewrite rules pretend that it is. If you type @kbd{m v} to enable
-Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
-literally, ignoring its usual commutativity property. (In the
-current implementation, the associativity also vanishes---it is as
-if the pattern had been enclosed in a @code{plain} marker; see below.)
-If you are applying rewrites to formulas with matrices, it's best to
-enable Matrix mode first to prevent algebraically incorrect rewrites
-from occurring.
-
-The pattern @samp{-x} will actually match any expression. For example,
-the rule
-
address@hidden
-f(-x) := -f(x)
address@hidden example
-
address@hidden
-will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
-a @code{plain} marker as described below, or add a @samp{negative(x)}
-condition. The @code{negative} function is true if its argument
-``looks'' negative, for example, because it is a negative number or
-because it is a formula like @samp{-x}. The new rule using this
-condition is:
-
address@hidden
-f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
-f(-x) := -f(x) :: negative(-x)
address@hidden example
-
-In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
-by matching @samp{y} to @samp{-b}.
-
-The pattern @samp{a b} will also match the formula @samp{x/y} if
address@hidden is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
-will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
address@hidden(a + 1:2) x}, depending on the current fraction mode).
-
-Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
address@hidden For example, the pattern @samp{f(a b)} will not match
address@hidden(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
-though conceivably these patterns could match with @samp{a = b = x}.
-Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
-constant, even though it could be considered to match with @samp{a = x}
-and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
-because while few mathematical operations are substantively different
-for addition and subtraction, often it is preferable to treat the cases
-of multiplication, division, and integer powers separately.
-
-Even more subtle is the rule set
-
address@hidden
-[ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
address@hidden example
-
address@hidden
-attempting to match @samp{f(x) - f(y)}. You might think that Calc
-will view this subtraction as @samp{f(x) + (-f(y))} and then apply
-the above two rules in turn, but actually this will not work because
-Calc only does this when considering rules for @samp{+} (like the
-first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
-does not match @samp{f(a) + f(b)} for any assignments of the
-meta-variables, and then it will see that @samp{f(x) - f(y)} does
-not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
-tries only one rule at a time, it will not be able to rewrite
address@hidden(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
-rule will have to be added.
-
-Another thing patterns will @emph{not} do is break up complex numbers.
-The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
-involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
-it will not match actual complex numbers like @samp{(3, -4)}. A version
-of the above rule for complex numbers would be
-
address@hidden
-myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
address@hidden example
-
address@hidden
-(Because the @code{re} and @code{im} functions understand the properties
-of the special constant @samp{i}, this rule will also work for
address@hidden - 4 i}. In fact, this particular rule would probably be better
-without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
-righthand side of the rule will still give the correct answer for the
-conjugate of a real number.)
-
-It is also possible to specify optional arguments in patterns. The rule
-
address@hidden
-opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
address@hidden example
-
address@hidden
-will match the formula
-
address@hidden
-5 (x^2 - 4) + 3 x
address@hidden example
-
address@hidden
-in a fairly straightforward manner, but it will also match reduced
-formulas like
-
address@hidden
-x + x^2, 2(x + 1) - x, x + x
address@hidden example
-
address@hidden
-producing, respectively,
-
address@hidden
-f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
address@hidden example
-
-(The latter two formulas can be entered only if default simplifications
-have been turned off with @kbd{m O}.)
-
-The default value for a term of a sum is zero. The default value
-for a part of a product, for a power, or for the denominator of a
-quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
-with @samp{a = -1}.
-
-In particular, the distributive-law rule can be refined to
-
address@hidden
-opt(a) x + opt(b) x := (a + b) x
address@hidden example
-
address@hidden
-so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
-
-The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
-are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
-functions with rewrite conditions to test for this; @pxref{Logical
-Operations}. These functions are not as convenient to use in rewrite
-rules, but they recognize more kinds of formulas as linear:
address@hidden/z} is considered linear with @expr{b = 1/z} by @code{lin},
-but it will not match the above pattern because that pattern calls
-for a multiplication, not a division.
-
-As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
-by 1,
-
address@hidden
-sin(x)^2 + cos(x)^2 := 1
address@hidden example
-
address@hidden
-misses many cases because the sine and cosine may both be multiplied by
-an equal factor. Here's a more successful rule:
-
address@hidden
-opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
address@hidden example
-
-Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
-because one @expr{a} would have ``matched'' 1 while the other matched 6.
-
-Calc automatically converts a rule like
-
address@hidden
-f(x-1, x) := g(x)
address@hidden example
-
address@hidden
-into the form
-
address@hidden
-f(temp, x) := g(x) :: temp = x-1
address@hidden example
-
address@hidden
-(where @code{temp} stands for a new, invented meta-variable that
-doesn't actually have a name). This modified rule will successfully
-match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
-respectively, then verifying that they differ by one even though
address@hidden does not superficially look like @samp{x-1}.
-
-However, Calc does not solve equations to interpret a rule. The
-following rule,
-
address@hidden
-f(x-1, x+1) := g(x)
address@hidden example
-
address@hidden
-will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
-but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
-of a variable by literal matching. If the variable appears ``isolated''
-then Calc is smart enough to use it for literal matching. But in this
-last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
-:= g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
-actual ``something-minus-one'' in the target formula.
-
-A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
-You could make this resemble the original form more closely by using
address@hidden notation, which is described in the next section:
-
address@hidden
-f(xm1, x+1) := g(x) :: let(x := xm1+1)
address@hidden example
-
-Calc does this rewriting or ``conditionalizing'' for any sub-pattern
-which involves only the functions in the following list, operating
-only on constants and meta-variables which have already been matched
-elsewhere in the pattern. When matching a function call, Calc is
-careful to match arguments which are plain variables before arguments
-which are calls to any of the functions below, so that a pattern like
address@hidden(x-1, x)} can be conditionalized even though the isolated
address@hidden comes after the @samp{x-1}.
-
address@hidden
-+ - * / \ % ^ abs sign round rounde roundu trunc floor ceil
-max min re im conj arg
address@hidden smallexample
-
-You can suppress all of the special treatments described in this
-section by surrounding a function call with a @code{plain} marker.
-This marker causes the function call which is its argument to be
-matched literally, without regard to commutativity, associativity,
-negation, or conditionalization. When you use @code{plain}, the
-``deep structure'' of the formula being matched can show through.
-For example,
-
address@hidden
-plain(a - a b) := f(a, b)
address@hidden example
-
address@hidden
-will match only literal subtractions. However, the @code{plain}
-marker does not affect its arguments' arguments. In this case,
-commutativity and associativity is still considered while matching
-the @address@hidden b}} sub-pattern, so the whole pattern will match
address@hidden - y x} as well as @samp{x - x y}. We could go still
-further and use
-
address@hidden
-plain(a - plain(a b)) := f(a, b)
address@hidden example
-
address@hidden
-which would do a completely strict match for the pattern.
-
-By contrast, the @code{quote} marker means that not only the
-function name but also the arguments must be literally the same.
-The above pattern will match @samp{x - x y} but
-
address@hidden
-quote(a - a b) := f(a, b)
address@hidden example
-
address@hidden
-will match only the single formula @samp{a - a b}. Also,
-
address@hidden
-quote(a - quote(a b)) := f(a, b)
address@hidden example
-
address@hidden
-will match only @samp{a - quote(a b)}---probably not the desired
-effect!
-
-A certain amount of algebra is also done when substituting the
-meta-variables on the righthand side of a rule. For example,
-in the rule
-
address@hidden
-a + f(b) := f(a + b)
address@hidden example
-
address@hidden
-matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
-taken literally, but the rewrite mechanism will simplify the
-righthand side to @samp{f(x - y)} automatically. (Of course,
-the default simplifications would do this anyway, so this
-special simplification is only noticeable if you have turned the
-default simplifications off.) This rewriting is done only when
-a meta-variable expands to a ``negative-looking'' expression.
-If this simplification is not desirable, you can use a @code{plain}
-marker on the righthand side:
-
address@hidden
-a + f(b) := f(plain(a + b))
address@hidden example
-
address@hidden
-In this example, we are still allowing the pattern-matcher to
-use all the algebra it can muster, but the righthand side will
-always simplify to a literal addition like @samp{f((-y) + x)}.
-
address@hidden Other Features of Rewrite Rules, Composing Patterns in Rewrite
Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
address@hidden Other Features of Rewrite Rules
-
address@hidden
-Certain ``function names'' serve as markers in rewrite rules.
-Here is a complete list of these markers. First are listed the
-markers that work inside a pattern; then come the markers that
-work in the righthand side of a rule.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden import
-One kind of marker, @samp{import(x)}, takes the place of a whole
-rule. Here @expr{x} is the name of a variable containing another
-rule set; those rules are ``spliced into'' the rule set that
-imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
-f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
-then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
-all three rules. It is possible to modify the imported rules
-slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
-the rule set @expr{x} with all occurrences of
address@hidden @math{v_1},
address@hidden @expr{v1},
-as either a variable name or a function name, replaced with
address@hidden @math{x_1}
address@hidden @expr{x1}
-and so on. (If
address@hidden @math{v_1}
address@hidden @expr{v1}
-is used as a function name, then
address@hidden @math{x_1}
address@hidden @expr{x1}
-must be either a function name itself or a @address@hidden< >}} nameless
-function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
-import(linearF, f, g)]} applies the linearity rules to the function
address@hidden instead of @samp{f}. Imports can be nested, but the
-import-with-renaming feature may fail to rename sub-imports properly.
-
-The special functions allowed in patterns are:
-
address@hidden @samp
address@hidden quote(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden quote
-This pattern matches exactly @expr{x}; variable names in @expr{x} are
-not interpreted as meta-variables. The only flexibility is that
-numbers are compared for numeric equality, so that the pattern
address@hidden(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
-(Numbers are always treated this way by the rewrite mechanism:
-The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
-The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
-as a result in this case.)
-
address@hidden plain(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden plain
-Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
-pattern matches a call to function @expr{f} with the specified
-argument patterns. No special knowledge of the properties of the
-function @expr{f} is used in this case; @samp{+} is not commutative or
-associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
-are treated as patterns. If you wish them to be treated ``plainly''
-as well, you must enclose them with more @code{plain} markers:
address@hidden(plain(@w{-a}) + plain(b c))}.
-
address@hidden opt(x,def)
address@hidden
address@hidden
address@hidden ignore
address@hidden opt
-Here @expr{x} must be a variable name. This must appear as an
-argument to a function or an element of a vector; it specifies that
-the argument or element is optional.
-As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
-or as the second argument to @samp{/} or @samp{^}, the value @var{def}
-may be omitted. The pattern @samp{x + opt(y)} matches a sum by
-binding one summand to @expr{x} and the other to @expr{y}, and it
-matches anything else by binding the whole expression to @expr{x} and
-zero to @expr{y}. The other operators above work similarly.
-
-For general miscellaneous functions, the default value @code{def}
-must be specified. Optional arguments are dropped starting with
-the rightmost one during matching. For example, the pattern
address@hidden(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
-or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
-supplied in this example for the omitted arguments. Note that
-the literal variable @expr{b} will be the default in the latter
-case, @emph{not} the value that matched the meta-variable @expr{b}.
-In other words, the default @var{def} is effectively quoted.
-
address@hidden condition(x,c)
address@hidden
address@hidden
address@hidden ignore
address@hidden condition
address@hidden ::
-This matches the pattern @expr{x}, with the attached condition
address@hidden It is the same as @samp{x :: c}.
-
address@hidden pand(x,y)
address@hidden
address@hidden
address@hidden ignore
address@hidden pand
address@hidden &&&
-This matches anything that matches both pattern @expr{x} and
-pattern @expr{y}. It is the same as @samp{x &&& y}.
address@hidden Patterns in Rewrite Rules}.
-
address@hidden por(x,y)
address@hidden
address@hidden
address@hidden ignore
address@hidden por
address@hidden |||
-This matches anything that matches either pattern @expr{x} or
-pattern @expr{y}. It is the same as @address@hidden ||| y}}.
-
address@hidden pnot(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden pnot
address@hidden !!!
-This matches anything that does not match pattern @expr{x}.
-It is the same as @samp{!!! x}.
-
address@hidden cons(h,t)
address@hidden
address@hidden cons
address@hidden ignore
address@hidden cons (rewrites)
-This matches any vector of one or more elements. The first
-element is matched to @expr{h}; a vector of the remaining
-elements is matched to @expr{t}. Note that vectors of fixed
-length can also be matched as actual vectors: The rule
address@hidden(a,cons(b,[])) := cons(a+b,[])} is equivalent
-to the rule @samp{[a,b] := [a+b]}.
-
address@hidden rcons(t,h)
address@hidden
address@hidden rcons
address@hidden ignore
address@hidden rcons (rewrites)
-This is like @code{cons}, except that the @emph{last} element
-is matched to @expr{h}, with the remaining elements matched
-to @expr{t}.
-
address@hidden apply(f,args)
address@hidden
address@hidden apply
address@hidden ignore
address@hidden apply (rewrites)
-This matches any function call. The name of the function, in
-the form of a variable, is matched to @expr{f}. The arguments
-of the function, as a vector of zero or more objects, are
-matched to @samp{args}. Constants, variables, and vectors
-do @emph{not} match an @code{apply} pattern. For example,
address@hidden(f,x)} matches any function call, @samp{apply(quote(f),x)}
-matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
-matches any function call with exactly two arguments, and
address@hidden(quote(f), cons(a,cons(b,x)))} matches any call
-to the function @samp{f} with two or more arguments. Another
-way to implement the latter, if the rest of the rule does not
-need to refer to the first two arguments of @samp{f} by name,
-would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
-Here's a more interesting sample use of @code{apply}:
-
address@hidden
-apply(f,[x+n]) := n + apply(f,[x])
- :: in(f, [floor,ceil,round,trunc]) :: integer(n)
address@hidden example
-
-Note, however, that this will be slower to match than a rule
-set with four separate rules. The reason is that Calc sorts
-the rules of a rule set according to top-level function name;
-if the top-level function is @code{apply}, Calc must try the
-rule for every single formula and sub-formula. If the top-level
-function in the pattern is, say, @code{floor}, then Calc invokes
-the rule only for sub-formulas which are calls to @code{floor}.
-
-Formulas normally written with operators like @code{+} are still
-considered function calls: @code{apply(f,x)} matches @samp{a+b}
-with @samp{f = add}, @samp{x = [a,b]}.
-
-You must use @code{apply} for meta-variables with function names
-on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
-is @emph{not} correct, because it rewrites @samp{spam(6)} into
address@hidden(7)}. The righthand side should be @samp{apply(f, [x+1])}.
-Also note that you will have to use No-Simplify mode (@kbd{m O})
-when entering this rule so that the @code{apply} isn't
-evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
-Or, use @kbd{s e} to enter the rule without going through the stack,
-or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
address@hidden Rewrite Rules}.
-
address@hidden select(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden select
-This is used for applying rules to formulas with selections;
address@hidden with Rewrite Rules}.
address@hidden table
-
-Special functions for the righthand sides of rules are:
-
address@hidden @samp
address@hidden quote(x)
-The notation @samp{quote(x)} is changed to @samp{x} when the
-righthand side is used. As far as the rewrite rule is concerned,
address@hidden is invisible. However, @code{quote} has the special
-property in Calc that its argument is not evaluated. Thus,
-while it will not work to put the rule @samp{t(a) := typeof(a)}
-on the stack because @samp{typeof(a)} is evaluated immediately
-to produce @samp{t(a) := 100}, you can use @code{quote} to
-protect the righthand side: @samp{t(a) := quote(typeof(a))}.
-(@xref{Conditional Rewrite Rules}, for another trick for
-protecting rules from evaluation.)
-
address@hidden plain(x)
-Special properties of and simplifications for the function call
address@hidden are not used. One interesting case where @code{plain}
-is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
-shorthand notation for the @code{quote} function. This rule will
-not work as shown; instead of replacing @samp{q(foo)} with
address@hidden(foo)}, it will replace it with @samp{foo}! The correct
-rule would be @samp{q(x) := plain(quote(x))}.
-
address@hidden cons(h,t)
-Where @expr{t} is a vector, this is converted into an expanded
-vector during rewrite processing. Note that @code{cons} is a regular
-Calc function which normally does this anyway; the only way @code{cons}
-is treated specially by rewrites is that @code{cons} on the righthand
-side of a rule will be evaluated even if default simplifications
-have been turned off.
-
address@hidden rcons(t,h)
-Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
-the vector @expr{t}.
-
address@hidden apply(f,args)
-Where @expr{f} is a variable and @var{args} is a vector, this
-is converted to a function call. Once again, note that @code{apply}
-is also a regular Calc function.
-
address@hidden eval(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden eval
-The formula @expr{x} is handled in the usual way, then the
-default simplifications are applied to it even if they have
-been turned off normally. This allows you to treat any function
-similarly to the way @code{cons} and @code{apply} are always
-treated. However, there is a slight difference: @samp{cons(2+3, [])}
-with default simplifications off will be converted to @samp{[2+3]},
-whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
-
address@hidden evalsimp(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden evalsimp
-The formula @expr{x} has meta-variables substituted in the usual
-way, then algebraically simplified as if by the @kbd{a s} command.
-
address@hidden evalextsimp(x)
address@hidden
address@hidden
address@hidden ignore
address@hidden evalextsimp
-The formula @expr{x} has meta-variables substituted in the normal
-way, then ``extendedly'' simplified as if by the @kbd{a e} command.
-
address@hidden select(x)
address@hidden with Rewrite Rules}.
address@hidden table
-
-There are also some special functions you can use in conditions.
-
address@hidden @samp
address@hidden let(v := x)
address@hidden
address@hidden
address@hidden ignore
address@hidden let
-The expression @expr{x} is evaluated with meta-variables substituted.
-The @kbd{a s} command's simplifications are @emph{not} applied by
-default, but @expr{x} can include calls to @code{evalsimp} or
address@hidden as described above to invoke higher levels
-of simplification. The
-result of @expr{x} is then bound to the meta-variable @expr{v}. As
-usual, if this meta-variable has already been matched to something
-else the two values must be equal; if the meta-variable is new then
-it is bound to the result of the expression. This variable can then
-appear in later conditions, and on the righthand side of the rule.
-In fact, @expr{v} may be any pattern in which case the result of
-evaluating @expr{x} is matched to that pattern, binding any
-meta-variables that appear in that pattern. Note that @code{let}
-can only appear by itself as a condition, or as one term of an
address@hidden&&} which is a whole condition: It cannot be inside
-an @samp{||} term or otherwise buried.
-
-The alternate, equivalent form @samp{let(v, x)} is also recognized.
-Note that the use of @samp{:=} by @code{let}, while still being
-assignment-like in character, is unrelated to the use of @samp{:=}
-in the main part of a rewrite rule.
-
-As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
-replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
-that inverse exists and is constant. For example, if @samp{a} is a
-singular matrix the operation @samp{1/a} is left unsimplified and
address@hidden(ia)} fails, but if @samp{a} is an invertible matrix
-then the rule succeeds. Without @code{let} there would be no way
-to express this rule that didn't have to invert the matrix twice.
-Note that, because the meta-variable @samp{ia} is otherwise unbound
-in this rule, the @code{let} condition itself always ``succeeds''
-because no matter what @samp{1/a} evaluates to, it can successfully
-be bound to @code{ia}.
-
-Here's another example, for integrating cosines of linear
-terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
-The @code{lin} function returns a 3-vector if its argument is linear,
-or leaves itself unevaluated if not. But an unevaluated @code{lin}
-call will not match the 3-vector on the lefthand side of the @code{let},
-so this @code{let} both verifies that @code{y} is linear, and binds
-the coefficients @code{a} and @code{b} for use elsewhere in the rule.
-(It would have been possible to use @samp{sin(a x + b)/b} for the
-righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
-rearrangement of the argument of the sine.)
-
address@hidden
address@hidden
address@hidden ignore
address@hidden ierf
-Similarly, here is a rule that implements an address@hidden
-function. It uses @code{root} to search for a solution. If
address@hidden succeeds, it will return a vector of two numbers
-where the first number is the desired solution. If no solution
-is found, @code{root} remains in symbolic form. So we use
address@hidden to check that the result was indeed a vector.
-
address@hidden
-ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
address@hidden example
-
address@hidden matches(v,p)
-The meta-variable @var{v}, which must already have been matched
-to something elsewhere in the rule, is compared against pattern
address@hidden Since @code{matches} is a standard Calc function, it
-can appear anywhere in a condition. But if it appears alone or
-as a term of a top-level @samp{&&}, then you get the special
-extra feature that meta-variables which are bound to things
-inside @var{p} can be used elsewhere in the surrounding rewrite
-rule.
-
-The only real difference between @samp{let(p := v)} and
address@hidden(v, p)} is that the former evaluates @samp{v} using
-the default simplifications, while the latter does not.
-
address@hidden remember
address@hidden remember
-This is actually a variable, not a function. If @code{remember}
-appears as a condition in a rule, then when that rule succeeds
-the original expression and rewritten expression are added to the
-front of the rule set that contained the rule. If the rule set
-was not stored in a variable, @code{remember} is ignored. The
-lefthand side is enclosed in @code{quote} in the added rule if it
-contains any variables.
-
-For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
-to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
-of the rule set. The rule set @code{EvalRules} works slightly
-differently: There, the evaluation of @samp{f(6)} will complete before
-the result is added to the rule set, in this case as @samp{f(7) := 5040}.
-Thus @code{remember} is most useful inside @code{EvalRules}.
-
-It is up to you to ensure that the optimization performed by
address@hidden is safe. For example, the rule @samp{foo(n) := n
-:: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
-the function equivalent of the @kbd{=} command); if the variable
address@hidden ever contains 1, rules like @samp{foo(7) := 7} will
-be added to the rule set and will continue to operate even if
address@hidden is later changed to 0.
-
address@hidden remember(c)
address@hidden
address@hidden
address@hidden ignore
address@hidden remember
-Remember the match as described above, but only if condition @expr{c}
-is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
-rule remembers only every fourth result. Note that @samp{remember(1)}
-is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
address@hidden table
-
address@hidden Composing Patterns in Rewrite Rules, Nested Formulas with
Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
address@hidden Composing Patterns in Rewrite Rules
-
address@hidden
-There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
-that combine rewrite patterns to make larger patterns. The
-combinations are ``and,'' ``or,'' and ``not,'' respectively, and
-these operators are the pattern equivalents of @samp{&&}, @samp{||}
-and @samp{!} (which operate on zero-or-nonzero logical values).
-
-Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
-form by all regular Calc features; they have special meaning only in
-the context of rewrite rule patterns.
-
-The pattern @address@hidden &&& @var{p2}} matches anything that
-matches both @var{p1} and @var{p2}. One especially useful case is
-when one of @var{p1} or @var{p2} is a meta-variable. For example,
-here is a rule that operates on error forms:
-
address@hidden
-f(x &&& a +/- b, x) := g(x)
address@hidden example
-
-This does the same thing, but is arguably simpler than, the rule
-
address@hidden
-f(a +/- b, a +/- b) := g(a +/- b)
address@hidden example
-
address@hidden
address@hidden
address@hidden ignore
address@hidden ends
-Here's another interesting example:
-
address@hidden
-ends(cons(a, x) &&& rcons(y, b)) := [a, b]
address@hidden example
-
address@hidden
-which effectively clips out the middle of a vector leaving just
-the first and last elements. This rule will change a one-element
-vector @samp{[a]} to @samp{[a, a]}. The similar rule
-
address@hidden
-ends(cons(a, rcons(y, b))) := [a, b]
address@hidden example
-
address@hidden
-would do the same thing except that it would fail to match a
-one-element vector.
-
address@hidden
-\bigskip
address@hidden tex
-
-The pattern @address@hidden ||| @var{p2}} matches anything that
-matches either @var{p1} or @var{p2}. Calc first tries matching
-against @var{p1}; if that fails, it goes on to try @var{p2}.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden curve
-A simple example of @samp{|||} is
-
address@hidden
-curve(inf ||| -inf) := 0
address@hidden example
-
address@hidden
-which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
-
-Here is a larger example:
-
address@hidden
-log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
address@hidden example
-
-This matches both generalized and natural logarithms in a single rule.
-Note that the @samp{::} term must be enclosed in parentheses because
-that operator has lower precedence than @samp{|||} or @samp{:=}.
-
-(In practice this rule would probably include a third alternative,
-omitted here for brevity, to take care of @code{log10}.)
-
-While Calc generally treats interior conditions exactly the same as
-conditions on the outside of a rule, it does guarantee that if all the
-variables in the condition are special names like @code{e}, or already
-bound in the pattern to which the condition is attached (say, if
address@hidden had appeared in this condition), then Calc will process this
-condition right after matching the pattern to the left of the @samp{::}.
-Thus, we know that @samp{b} will be bound to @samp{e} only if the
address@hidden branch of the @samp{|||} was taken.
-
-Note that this rule was careful to bind the same set of meta-variables
-on both sides of the @samp{|||}. Calc does not check this, but if
-you bind a certain meta-variable only in one branch and then use that
-meta-variable elsewhere in the rule, results are unpredictable:
-
address@hidden
-f(a,b) ||| g(b) := h(a,b)
address@hidden example
-
-Here if the pattern matches @samp{g(17)}, Calc makes no promises about
-the value that will be substituted for @samp{a} on the righthand side.
-
address@hidden
-\bigskip
address@hidden tex
-
-The pattern @samp{!!! @var{pat}} matches anything that does not
-match @var{pat}. Any meta-variables that are bound while matching
address@hidden remain unbound outside of @var{pat}.
-
-For example,
-
address@hidden
-f(x &&& !!! a +/- b, !!![]) := g(x)
address@hidden example
-
address@hidden
-converts @code{f} whose first argument is anything @emph{except} an
-error form, and whose second argument is not the empty vector, into
-a similar call to @code{g} (but without the second argument).
-
-If we know that the second argument will be a vector (empty or not),
-then an equivalent rule would be:
-
address@hidden
-f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
address@hidden example
-
address@hidden
-where of course 7 is the @code{typeof} code for error forms.
-Another final condition, that works for any kind of @samp{y},
-would be @samp{!istrue(y == [])}. (The @code{istrue} function
-returns an explicit 0 if its argument was left in symbolic form;
-plain @samp{!(y == [])} or @samp{y != []} would not work to replace
address@hidden since these would be left unsimplified, and thus cause
-the rule to fail, if @samp{y} was something like a variable name.)
-
-It is possible for a @samp{!!!} to refer to meta-variables bound
-elsewhere in the pattern. For example,
-
address@hidden
-f(a, !!!a) := g(a)
address@hidden example
-
address@hidden
-matches any call to @code{f} with different arguments, changing
-this to @code{g} with only the first argument.
-
-If a function call is to be matched and one of the argument patterns
-contains a @samp{!!!} somewhere inside it, that argument will be
-matched last. Thus
-
address@hidden
-f(!!!a, a) := g(a)
address@hidden example
-
address@hidden
-will be careful to bind @samp{a} to the second argument of @code{f}
-before testing the first argument. If Calc had tried to match the
-first argument of @code{f} first, the results would have been
-disastrous: since @code{a} was unbound so far, the pattern @samp{a}
-would have matched anything at all, and the pattern @samp{!!!a}
-therefore would @emph{not} have matched anything at all!
-
address@hidden Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules,
Composing Patterns in Rewrite Rules, Rewrite Rules
address@hidden Nested Formulas with Rewrite Rules
-
address@hidden
-When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
-the top of the stack and attempts to match any of the specified rules
-to any part of the expression, starting with the whole expression
-and then, if that fails, trying deeper and deeper sub-expressions.
-For each part of the expression, the rules are tried in the order
-they appear in the rules vector. The first rule to match the first
-sub-expression wins; it replaces the matched sub-expression according
-to the @var{new} part of the rule.
-
-Often, the rule set will match and change the formula several times.
-The top-level formula is first matched and substituted repeatedly until
-it no longer matches the pattern; then, sub-formulas are tried, and
-so on. Once every part of the formula has gotten its chance, the
-rewrite mechanism starts over again with the top-level formula
-(in case a substitution of one of its arguments has caused it again
-to match). This continues until no further matches can be made
-anywhere in the formula.
-
-It is possible for a rule set to get into an infinite loop. The
-most obvious case, replacing a formula with itself, is not a problem
-because a rule is not considered to ``succeed'' unless the righthand
-side actually comes out to something different than the original
-formula or sub-formula that was matched. But if you accidentally
-had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
address@hidden(a) + ln(b) := ln(a b)} in your rule set, Calc would
-run forever switching a formula back and forth between the two
-forms.
-
-To avoid disaster, Calc normally stops after 100 changes have been
-made to the formula. This will be enough for most multiple rewrites,
-but it will keep an endless loop of rewrites from locking up the
-computer forever. (On most systems, you can also type @kbd{C-g} to
-halt any Emacs command prematurely.)
-
-To change this limit, give a positive numeric prefix argument.
-In particular, @kbd{M-1 a r} applies only one rewrite at a time,
-useful when you are first testing your rule (or just if repeated
-rewriting is not what is called for by your application).
-
address@hidden
address@hidden
address@hidden ignore
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden iterations
-You can also put a ``function call'' @samp{iterations(@var{n})}
-in place of a rule anywhere in your rules vector (but usually at
-the top). Then, @var{n} will be used instead of 100 as the default
-number of iterations for this rule set. You can use
address@hidden(inf)} if you want no iteration limit by default.
-A prefix argument will override the @code{iterations} limit in the
-rule set.
-
address@hidden
-[ iterations(1),
- f(x) := f(x+1) ]
address@hidden example
-
-More precisely, the limit controls the number of ``iterations,''
-where each iteration is a successful matching of a rule pattern whose
-righthand side, after substituting meta-variables and applying the
-default simplifications, is different from the original sub-formula
-that was matched.
-
-A prefix argument of zero sets the limit to infinity. Use with caution!
-
-Given a negative numeric prefix argument, @kbd{a r} will match and
-substitute the top-level expression up to that many times, but
-will not attempt to match the rules to any sub-expressions.
-
-In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
-does a rewriting operation. Here @var{expr} is the expression
-being rewritten, @var{rules} is the rule, vector of rules, or
-variable containing the rules, and @var{n} is the optional
-iteration limit, which may be a positive integer, a negative
-integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
-the @code{iterations} value from the rule set is used; if both
-are omitted, 100 is used.
-
address@hidden Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested
Formulas with Rewrite Rules, Rewrite Rules
address@hidden Multi-Phase Rewrite Rules
-
address@hidden
-It is possible to separate a rewrite rule set into several @dfn{phases}.
-During each phase, certain rules will be enabled while certain others
-will be disabled. A @dfn{phase schedule} controls the order in which
-phases occur during the rewriting process.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden phase
address@hidden all
-If a call to the marker function @code{phase} appears in the rules
-vector in place of a rule, all rules following that point will be
-members of the phase(s) identified in the arguments to @code{phase}.
-Phases are given integer numbers. The markers @samp{phase()} and
address@hidden(all)} both mean the following rules belong to all phases;
-this is the default at the start of the rule set.
-
-If you do not explicitly schedule the phases, Calc sorts all phase
-numbers that appear in the rule set and executes the phases in
-ascending order. For example, the rule set
-
address@hidden
address@hidden
-[ f0(x) := g0(x),
- phase(1),
- f1(x) := g1(x),
- phase(2),
- f2(x) := g2(x),
- phase(3),
- f3(x) := g3(x),
- phase(1,2),
- f4(x) := g4(x) ]
address@hidden group
address@hidden example
-
address@hidden
-has three phases, 1 through 3. Phase 1 consists of the @code{f0},
address@hidden, and @code{f4} rules (in that order). Phase 2 consists of
address@hidden, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
-and @code{f3}.
-
-When Calc rewrites a formula using this rule set, it first rewrites
-the formula using only the phase 1 rules until no further changes are
-possible. Then it switches to the phase 2 rule set and continues
-until no further changes occur, then finally rewrites with phase 3.
-When no more phase 3 rules apply, rewriting finishes. (This is
-assuming @kbd{a r} with a large enough prefix argument to allow the
-rewriting to run to completion; the sequence just described stops
-early if the number of iterations specified in the prefix argument,
-100 by default, is reached.)
-
-During each phase, Calc descends through the nested levels of the
-formula as described previously. (@xref{Nested Formulas with Rewrite
-Rules}.) Rewriting starts at the top of the formula, then works its
-way down to the parts, then goes back to the top and works down again.
-The phase 2 rules do not begin until no phase 1 rules apply anywhere
-in the formula.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden schedule
-A @code{schedule} marker appearing in the rule set (anywhere, but
-conventionally at the top) changes the default schedule of phases.
-In the simplest case, @code{schedule} has a sequence of phase numbers
-for arguments; each phase number is invoked in turn until the
-arguments to @code{schedule} are exhausted. Thus adding
address@hidden(3,2,1)} at the top of the above rule set would
-reverse the order of the phases; @samp{schedule(1,2,3)} would have
-no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
-would give phase 1 a second chance after phase 2 has completed, before
-moving on to phase 3.
-
-Any argument to @code{schedule} can instead be a vector of phase
-numbers (or even of sub-vectors). Then the sub-sequence of phases
-described by the vector are tried repeatedly until no change occurs
-in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
-tries phase 1, then phase 2, then, if either phase made any changes
-to the formula, repeats these two phases until they can make no
-further progress. Finally, it goes on to phase 3 for finishing
-touches.
-
-Also, items in @code{schedule} can be variable names as well as
-numbers. A variable name is interpreted as the name of a function
-to call on the whole formula. For example, @samp{schedule(1, simplify)}
-says to apply the phase-1 rules (presumably, all of them), then to
-call @code{simplify} which is the function name equivalent of @kbd{a s}.
-Likewise, @samp{schedule([1, simplify])} says to alternate between
-phase 1 and @kbd{a s} until no further changes occur.
-
-Phases can be used purely to improve efficiency; if it is known that
-a certain group of rules will apply only at the beginning of rewriting,
-and a certain other group will apply only at the end, then rewriting
-will be faster if these groups are identified as separate phases.
-Once the phase 1 rules are done, Calc can put them aside and no longer
-spend any time on them while it works on phase 2.
-
-There are also some problems that can only be solved with several
-rewrite phases. For a real-world example of a multi-phase rule set,
-examine the set @code{FitRules}, which is used by the curve-fitting
-command to convert a model expression to linear form.
address@hidden Fitting Details}. This set is divided into four phases.
-The first phase rewrites certain kinds of expressions to be more
-easily linearizable, but less computationally efficient. After the
-linear components have been picked out, the final phase includes the
-opposite rewrites to put each component back into an efficient form.
-If both sets of rules were included in one big phase, Calc could get
-into an infinite loop going back and forth between the two forms.
-
-Elsewhere in @code{FitRules}, the components are first isolated,
-then recombined where possible to reduce the complexity of the linear
-fit, then finally packaged one component at a time into vectors.
-If the packaging rules were allowed to begin before the recombining
-rules were finished, some components might be put away into vectors
-before they had a chance to recombine. By putting these rules in
-two separate phases, this problem is neatly avoided.
-
address@hidden Selections with Rewrite Rules, Matching Commands, Multi-Phase
Rewrite Rules, Rewrite Rules
address@hidden Selections with Rewrite Rules
-
address@hidden
-If a sub-formula of the current formula is selected (as by @kbd{j s};
address@hidden Subformulas}), the @kbd{a r} (@code{calc-rewrite})
-command applies only to that sub-formula. Together with a negative
-prefix argument, you can use this fact to apply a rewrite to one
-specific part of a formula without affecting any other parts.
-
address@hidden j r
address@hidden calc-rewrite-selection
-The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
-sophisticated operations on selections. This command prompts for
-the rules in the same way as @kbd{a r}, but it then applies those
-rules to the whole formula in question even though a sub-formula
-of it has been selected. However, the selected sub-formula will
-first have been surrounded by a @samp{select( )} function call.
-(Calc's evaluator does not understand the function name @code{select};
-this is only a tag used by the @kbd{j r} command.)
-
-For example, suppose the formula on the stack is @samp{2 (a + b)^2}
-and the sub-formula @samp{a + b} is selected. This formula will
-be rewritten to @samp{2 select(a + b)^2} and then the rewrite
-rules will be applied in the usual way. The rewrite rules can
-include references to @code{select} to tell where in the pattern
-the selected sub-formula should appear.
-
-If there is still exactly one @samp{select( )} function call in
-the formula after rewriting is done, it indicates which part of
-the formula should be selected afterwards. Otherwise, the
-formula will be unselected.
-
-You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
-of the rewrite rule with @samp{select()}. However, @kbd{j r}
-allows you to use the current selection in more flexible ways.
-Suppose you wished to make a rule which removed the exponent from
-the selected term; the rule @samp{select(a)^x := select(a)} would
-work. In the above example, it would rewrite @samp{2 select(a + b)^2}
-to @samp{2 select(a + b)}. This would then be returned to the
-stack as @samp{2 (a + b)} with the @samp{a + b} selected.
-
-The @kbd{j r} command uses one iteration by default, unlike
address@hidden r} which defaults to 100 iterations. A numeric prefix
-argument affects @kbd{j r} in the same way as @kbd{a r}.
address@hidden Formulas with Rewrite Rules}.
-
-As with other selection commands, @kbd{j r} operates on the stack
-entry that contains the cursor. (If the cursor is on the top-of-stack
address@hidden marker, it works as if the cursor were on the formula
-at stack level 1.)
-
-If you don't specify a set of rules, the rules are taken from the
-top of the stack, just as with @kbd{a r}. In this case, the
-cursor must indicate stack entry 2 or above as the formula to be
-rewritten (otherwise the same formula would be used as both the
-target and the rewrite rules).
-
-If the indicated formula has no selection, the cursor position within
-the formula temporarily selects a sub-formula for the purposes of this
-command. If the cursor is not on any sub-formula (e.g., it is in
-the line-number area to the left of the formula), the @samp{select( )}
-markers are ignored by the rewrite mechanism and the rules are allowed
-to apply anywhere in the formula.
-
-As a special feature, the normal @kbd{a r} command also ignores
address@hidden( )} calls in rewrite rules. For example, if you used the
-above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
-the rule as if it were @samp{a^x := a}. Thus, you can write general
-purpose rules with @samp{select( )} hints inside them so that they
-will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
-both with and without selections.
-
address@hidden Matching Commands, Automatic Rewrites, Selections with Rewrite
Rules, Rewrite Rules
address@hidden Matching Commands
-
address@hidden
address@hidden a m
address@hidden calc-match
address@hidden match
-The @kbd{a m} (@code{calc-match}) address@hidden function takes a
-vector of formulas and a rewrite-rule-style pattern, and produces
-a vector of all formulas which match the pattern. The command
-prompts you to enter the pattern; as for @kbd{a r}, you can enter
-a single pattern (i.e., a formula with meta-variables), or a
-vector of patterns, or a variable which contains patterns, or
-you can give a blank response in which case the patterns are taken
-from the top of the stack. The pattern set will be compiled once
-and saved if it is stored in a variable. If there are several
-patterns in the set, vector elements are kept if they match any
-of the patterns.
-
-For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
-will return @samp{[x+y, x-y, x+y+z]}.
-
-The @code{import} mechanism is not available for pattern sets.
-
-The @kbd{a m} command can also be used to extract all vector elements
-which satisfy any condition: The pattern @samp{x :: x>0} will select
-all the positive vector elements.
-
address@hidden I a m
address@hidden matchnot
-With the Inverse flag address@hidden, this command extracts all
-vector elements which do @emph{not} match the given pattern.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden matches
-There is also a function @samp{matches(@var{x}, @var{p})} which
-evaluates to 1 if expression @var{x} matches pattern @var{p}, or
-to 0 otherwise. This is sometimes useful for including into the
-conditional clauses of other rewrite rules.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden vmatches
-The function @code{vmatches} is just like @code{matches}, except
-that if the match succeeds it returns a vector of assignments to
-the meta-variables instead of the number 1. For example,
address@hidden(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
-If the match fails, the function returns the number 0.
-
address@hidden Automatic Rewrites, Debugging Rewrites, Matching Commands,
Rewrite Rules
address@hidden Automatic Rewrites
-
address@hidden
address@hidden @code{EvalRules} variable
address@hidden EvalRules
-It is possible to get Calc to apply a set of rewrite rules on all
-results, effectively adding to the built-in set of default
-simplifications. To do this, simply store your rule set in the
-variable @code{EvalRules}. There is a convenient @kbd{s E} command
-for editing @code{EvalRules}; @pxref{Operations on Variables}.
-
-For example, suppose you want @samp{sin(a + b)} to be expanded out
-to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
-similarly for @samp{cos(a + b)}. The corresponding rewrite rule
-set would be,
-
address@hidden
address@hidden
-[ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
- cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
address@hidden group
address@hidden smallexample
-
-To apply these manually, you could put them in a variable called
address@hidden and then use @kbd{a r trigexp} every time you wanted
-to expand trig functions. But if instead you store them in the
-variable @code{EvalRules}, they will automatically be applied to all
-sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
-the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
address@hidden sin(2 x) + 0.7071 cos(2 x)} automatically.
-
-As each level of a formula is evaluated, the rules from
address@hidden are applied before the default simplifications.
-Rewriting continues until no further @code{EvalRules} apply.
-Note that this is different from the usual order of application of
-rewrite rules: @code{EvalRules} works from the bottom up, simplifying
-the arguments to a function before the function itself, while @kbd{a r}
-applies rules from the top down.
-
-Because the @code{EvalRules} are tried first, you can use them to
-override the normal behavior of any built-in Calc function.
-
-It is important not to write a rule that will get into an infinite
-loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
-appears to be a good definition of a factorial function, but it is
-unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
-will continue to subtract 1 from this argument forever without reaching
-zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
-Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
address@hidden(2, 4)}, this would bounce back and forth between that and
address@hidden(4, 2)} forever. If an infinite loop in @code{EvalRules}
-occurs, Emacs will eventually stop with a ``Computation got stuck
-or ran too long'' message.
-
-Another subtle difference between @code{EvalRules} and regular rewrites
-concerns rules that rewrite a formula into an identical formula. For
-example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
-already an integer. But in @code{EvalRules} this case is detected only
-if the righthand side literally becomes the original formula before any
-further simplification. This means that @samp{f(n) := f(floor(n))} will
-get into an infinite loop if it occurs in @code{EvalRules}. Calc will
-replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
address@hidden(6)}, so it will consider the rule to have matched and will
-continue simplifying that formula; first the argument is simplified
-to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
-again, ad infinitum. A much safer rule would check its argument first,
-say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
-
-(What really happens is that the rewrite mechanism substitutes the
-meta-variables in the righthand side of a rule, compares to see if the
-result is the same as the original formula and fails if so, then uses
-the default simplifications to simplify the result and compares again
-(and again fails if the formula has simplified back to its original
-form). The only special wrinkle for the @code{EvalRules} is that the
-same rules will come back into play when the default simplifications
-are used. What Calc wants to do is build @samp{f(floor(6))}, see that
-this is different from the original formula, simplify to @samp{f(6)},
-see that this is the same as the original formula, and thus halt the
-rewriting. But while simplifying, @samp{f(6)} will again trigger
-the same @code{EvalRules} rule and Calc will get into a loop inside
-the rewrite mechanism itself.)
-
-The @code{phase}, @code{schedule}, and @code{iterations} markers do
-not work in @code{EvalRules}. If the rule set is divided into phases,
-only the phase 1 rules are applied, and the schedule is ignored.
-The rules are always repeated as many times as possible.
-
-The @code{EvalRules} are applied to all function calls in a formula,
-but not to numbers (and other number-like objects like error forms),
-nor to vectors or individual variable names. (Though they will apply
-to @emph{components} of vectors and error forms when appropriate.) You
-might try to make a variable @code{phihat} which automatically expands
-to its definition without the need to press @kbd{=} by writing the
-rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
-will not work as part of @code{EvalRules}.
-
-Finally, another limitation is that Calc sometimes calls its built-in
-functions directly rather than going through the default simplifications.
-When it does this, @code{EvalRules} will not be able to override those
-functions. For example, when you take the absolute value of the complex
-number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
-the multiplication, addition, and square root functions directly rather
-than applying the default simplifications to this formula. So an
address@hidden rule that (perversely) rewrites @samp{sqrt(13) := 6}
-would not apply. (However, if you put Calc into Symbolic mode so that
address@hidden(13)} will be left in symbolic form by the built-in square
-root function, your rule will be able to apply. But if the complex
-number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
-then Symbolic mode will not help because @samp{sqrt(25)} can be
-evaluated exactly to 5.)
-
-One subtle restriction that normally only manifests itself with
address@hidden is that while a given rewrite rule is in the process
-of being checked, that same rule cannot be recursively applied. Calc
-effectively removes the rule from its rule set while checking the rule,
-then puts it back once the match succeeds or fails. (The technical
-reason for this is that compiled pattern programs are not reentrant.)
-For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
-attempting to match @samp{foo(8)}. This rule will be inactive while
-the condition @samp{foo(4) > 0} is checked, even though it might be
-an integral part of evaluating that condition. Note that this is not
-a problem for the more usual recursive type of rule, such as
address@hidden(x) := foo(x/2)}, because there the rule has succeeded and
-been reactivated by the time the righthand side is evaluated.
-
-If @code{EvalRules} has no stored value (its default state), or if
-anything but a vector is stored in it, then it is ignored.
-
-Even though Calc's rewrite mechanism is designed to compare rewrite
-rules to formulas as quickly as possible, storing rules in
address@hidden may make Calc run substantially slower. This is
-particularly true of rules where the top-level call is a commonly used
-function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
-only activate the rewrite mechanism for calls to the function @code{f},
-but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
-
address@hidden
-apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
address@hidden smallexample
-
address@hidden
-may seem more ``efficient'' than two separate rules for @code{ln} and
address@hidden, but actually it is vastly less efficient because rules
-with @code{apply} as the top-level pattern must be tested against
address@hidden function call that is simplified.
-
address@hidden @code{AlgSimpRules} variable
address@hidden AlgSimpRules
-Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
-but only when @kbd{a s} is used to simplify the formula. The variable
address@hidden holds rules for this purpose. The @kbd{a s} command
-will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as
-well as all of its built-in simplifications.
-
-Most of the special limitations for @code{EvalRules} don't apply to
address@hidden Calc simply does an @kbd{a r AlgSimpRules}
-command with an infinite repeat count as the first step of @kbd{a s}.
-It then applies its own built-in simplifications throughout the
-formula, and then repeats these two steps (along with applying the
-default simplifications) until no further changes are possible.
-
address@hidden @code{ExtSimpRules} variable
address@hidden @code{UnitSimpRules} variable
address@hidden ExtSimpRules
address@hidden UnitSimpRules
-There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
-that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
-also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
address@hidden contains simplification rules that are used
-only during integration by @kbd{a i}.
-
address@hidden Debugging Rewrites, Examples of Rewrite Rules, Automatic
Rewrites, Rewrite Rules
address@hidden Debugging Rewrites
-
address@hidden
-If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
-record some useful information there as it operates. The original
-formula is written there, as is the result of each successful rewrite,
-and the final result of the rewriting. All phase changes are also
-noted.
-
-Calc always appends to @samp{*Trace*}. You must empty this buffer
-yourself periodically if it is in danger of growing unwieldy.
-
-Note that the rewriting mechanism is substantially slower when the
address@hidden buffer exists, even if the buffer is not visible on
-the screen. Once you are done, you will probably want to kill this
-buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
-existence and forget about it, all your future rewrite commands will
-be needlessly slow.
-
address@hidden Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
address@hidden Examples of Rewrite Rules
-
address@hidden
-Returning to the example of substituting the pattern
address@hidden(x)^2 + cos(x)^2} with 1, we saw that the rule
address@hidden(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
-finding suitable cases. Another solution would be to use the rule
address@hidden(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
-if necessary. This rule will be the most effective way to do the job,
-but at the expense of making some changes that you might not desire.
-
-Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
-To make this work with the @address@hidden r}} command so that it can be
-easily targeted to a particular exponential in a large formula,
-you might wish to write the rule as @samp{select(exp(x+y)) :=
-select(exp(x) exp(y))}. The @samp{select} markers will be
-ignored by the regular @kbd{a r} command
-(@pxref{Selections with Rewrite Rules}).
-
-A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
-This will simplify the formula whenever @expr{b} and/or @expr{c} can
-be made simpler by squaring. For example, applying this rule to
address@hidden / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
-Symbolic mode has been enabled to keep the square root from being
-evaluated to a floating-point approximation). This rule is also
-useful when working with symbolic complex numbers, e.g.,
address@hidden(a + b i) / (c + d i)}.
-
-As another example, we could define our own ``triangular numbers'' function
-with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
-this vector and store it in a variable: @address@hidden t} trirules}. Now,
given
-a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
-to apply these rules repeatedly. After six applications, @kbd{a r} will
-stop with 15 on the stack. Once these rules are debugged, it would probably
-be most useful to add them to @code{EvalRules} so that Calc will evaluate
-the new @code{tri} function automatically. We could then use @kbd{Z K} on
-the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
address@hidden to the value on the top of the stack. @xref{Programming}.
-
address@hidden Quaternions
-The following rule set, contributed by
address@hidden Fran\c cois
address@hidden Francois
-Pinard, implements @dfn{quaternions}, a generalization of the concept of
-complex numbers. Quaternions have four components, and are here
-represented by function calls @samp{quat(@var{w}, address@hidden, @var{y},
address@hidden)} with ``real part'' @var{w} and the three ``imaginary'' parts
-collected into a vector. Various arithmetical operations on quaternions
-are supported. To use these rules, either add them to @code{EvalRules},
-or create a command based on @kbd{a r} for simplifying quaternion
-formulas. A convenient way to enter quaternions would be a command
-defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
address@hidden
-
address@hidden
-[ quat(w, x, y, z) := quat(w, [x, y, z]),
- quat(w, [0, 0, 0]) := w,
- abs(quat(w, v)) := hypot(w, v),
- -quat(w, v) := quat(-w, -v),
- r + quat(w, v) := quat(r + w, v) :: real(r),
- r - quat(w, v) := quat(r - w, -v) :: real(r),
- quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
- r * quat(w, v) := quat(r * w, r * v) :: real(r),
- plain(quat(w1, v1) * quat(w2, v2))
- := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
- quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
- z / quat(w, v) := z * quatinv(quat(w, v)),
- quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
- quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
- quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
- :: integer(k) :: k > 0 :: k % 2 = 0,
- quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
- :: integer(k) :: k > 2,
- quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
address@hidden smallexample
-
-Quaternions, like matrices, have non-commutative multiplication.
-In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
address@hidden and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
-rule above uses @code{plain} to prevent Calc from rearranging the
-product. It may also be wise to add the line @samp{[quat(), matrix]}
-to the @code{Decls} matrix, to ensure that Calc's other algebraic
-operations will not rearrange a quaternion product. @xref{Declarations}.
-
-These rules also accept a four-argument @code{quat} form, converting
-it to the preferred form in the first rule. If you would rather see
-results in the four-argument form, just append the two items
address@hidden(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
-of the rule set. (But remember that multi-phase rule sets don't work
-in @code{EvalRules}.)
-
address@hidden Units, Store and Recall, Algebra, Top
address@hidden Operating on Units
-
address@hidden
-One special interpretation of algebraic formulas is as numbers with units.
-For example, the formula @samp{5 m / s^2} can be read ``five meters
-per second squared.'' The commands in this chapter help you
-manipulate units expressions in this form. Units-related commands
-begin with the @kbd{u} prefix key.
-
address@hidden
-* Basic Operations on Units::
-* The Units Table::
-* Predefined Units::
-* User-Defined Units::
address@hidden menu
-
address@hidden Basic Operations on Units, The Units Table, Units, Units
address@hidden Basic Operations on Units
-
address@hidden
-A @dfn{units expression} is a formula which is basically a number
-multiplied and/or divided by one or more @dfn{unit names}, which may
-optionally be raised to integer powers. Actually, the value part need not
-be a number; any product or quotient involving unit names is a units
-expression. Many of the units commands will also accept any formula,
-where the command applies to all units expressions which appear in the
-formula.
-
-A unit name is a variable whose name appears in the @dfn{unit table},
-or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
-or @samp{u} (for ``micro'') followed by a name in the unit table.
-A substantial table of built-in units is provided with Calc;
address@hidden Units}. You can also define your own unit names;
address@hidden Units}.
-
-Note that if the value part of a units expression is exactly @samp{1},
-it will be removed by the Calculator's automatic algebra routines: The
-formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
-display anomaly, however; @samp{mm} will work just fine as a
-representation of one millimeter.
-
-You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
-with units expressions easier. Otherwise, you will have to remember
-to hit the apostrophe key every time you wish to enter units.
-
address@hidden u s
address@hidden calc-simplify-units
address@hidden
address@hidden address@hidden
address@hidden ignore
address@hidden usimplify
-The @kbd{u s} (@code{calc-simplify-units}) address@hidden command
-simplifies a units
-expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the
-expression first as a regular algebraic formula; it then looks for
-features that can be further simplified by converting one object's units
-to be compatible with another's. For example, @samp{5 m + 23 mm} will
-simplify to @samp{5.023 m}. When different but compatible units are
-added, the righthand term's units are converted to match those of the
-lefthand term. @xref{Simplification Modes}, for a way to have this done
-automatically at all times.
-
-Units simplification also handles quotients of two units with the same
-dimensionality, as in @address@hidden in s/L cm}} to @samp{5.08 s/L};
fractional
-powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
address@hidden(9 acre)} to a quantity in meters; and @code{floor},
address@hidden, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
address@hidden, @code{frac}, @code{abs}, and @code{clean}
-applied to units expressions, in which case
-the operation in question is applied only to the numeric part of the
-expression. Finally, trigonometric functions of quantities with units
-of angle are evaluated, regardless of the current angular mode.
-
address@hidden u c
address@hidden calc-convert-units
-The @kbd{u c} (@code{calc-convert-units}) command converts a units
-expression to new, compatible units. For example, given the units
-expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
address@hidden m/s}. If you have previously converted a units expression
-with the same type of units (in this case, distance over time), you will
-be offered the previous choice of new units as a default. Continuing
-the above example, entering the units expression @samp{100 km/hr} and
-typing @kbd{u c @key{RET}} (without specifying new units) produces
address@hidden m/s}.
-
-While many of Calc's conversion factors are exact, some are necessarily
-approximate. If Calc is in fraction mode (@pxref{Fraction Mode}), then
-unit conversions will try to give exact, rational conversions, but it
-isn't always possible. Given @samp{55 mph} in fraction mode, typing
address@hidden c m/s @key{RET}} produces @samp{15367:625 m/s}, for example,
-while typing @kbd{u c au/yr @key{RET}} produces
address@hidden au/yr}.
-
-If the units you request are inconsistent with the original units, the
-number will be converted into your units times whatever ``remainder''
-units are left over. For example, converting @samp{55 mph} into acres
-produces @samp{6.08e-3 acre / m s}. (Recall that multiplication binds
-more strongly than division in Calc formulas, so the units here are
-acres per meter-second.) Remainder units are expressed in terms of
-``fundamental'' units like @samp{m} and @samp{s}, regardless of the
-input units.
-
-One special exception is that if you specify a single unit name, and
-a compatible unit appears somewhere in the units expression, then
-that compatible unit will be converted to the new unit and the
-remaining units in the expression will be left alone. For example,
-given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
-change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
-The ``remainder unit'' @samp{cm} is left alone rather than being
-changed to the base unit @samp{m}.
-
-You can use explicit unit conversion instead of the @kbd{u s} command
-to gain more control over the units of the result of an expression.
-For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
address@hidden c mm} to express the result in either meters or millimeters.
-(For that matter, you could type @kbd{u c fath} to express the result
-in fathoms, if you preferred!)
-
-In place of a specific set of units, you can also enter one of the
-units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
-For example, @kbd{u c si @key{RET}} converts the expression into
-International System of Units (SI) base units. Also, @kbd{u c base}
-converts to Calc's base units, which are the same as @code{si} units
-except that @code{base} uses @samp{g} as the fundamental unit of mass
-whereas @code{si} uses @samp{kg}.
-
address@hidden Composite units
-The @kbd{u c} command also accepts @dfn{composite units}, which
-are expressed as the sum of several compatible unit names. For
-example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
-feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
-sorts the unit names into order of decreasing relative size.
-It then accounts for as much of the input quantity as it can
-using an integer number times the largest unit, then moves on
-to the next smaller unit, and so on. Only the smallest unit
-may have a non-integer amount attached in the result. A few
-standard unit names exist for common combinations, such as
address@hidden for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
-Composite units are expanded as if by @kbd{a x}, so that
address@hidden(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
-
-If the value on the stack does not contain any units, @kbd{u c} will
-prompt first for the old units which this value should be considered
-to have, then for the new units. Assuming the old and new units you
-give are consistent with each other, the result also will not contain
-any units. For example, @address@hidden c} cm @key{RET} in @key{RET}}
converts the number
-2 on the stack to 5.08.
-
address@hidden u b
address@hidden calc-base-units
-The @kbd{u b} (@code{calc-base-units}) command is shorthand for
address@hidden c base}; it converts the units expression on the top of the
-stack into @code{base} units. If @kbd{u s} does not simplify a
-units expression as far as you would like, try @kbd{u b}.
-
-The @kbd{u c} and @kbd{u b} commands treat temperature units (like
address@hidden and @samp{K}) as relative temperatures. For example,
address@hidden c} converts @samp{10 degC} to @samp{18 degF}: A change of 10
-degrees Celsius corresponds to a change of 18 degrees Fahrenheit.
-
address@hidden u t
address@hidden calc-convert-temperature
address@hidden Temperature conversion
-The @kbd{u t} (@code{calc-convert-temperature}) command converts
-absolute temperatures. The value on the stack must be a simple units
-expression with units of temperature only. This command would convert
address@hidden degC} to @samp{50 degF}, the equivalent temperature on the
-Fahrenheit scale.
-
address@hidden u r
address@hidden calc-remove-units
address@hidden u x
address@hidden calc-extract-units
-The @kbd{u r} (@code{calc-remove-units}) command removes units from the
-formula at the top of the stack. The @kbd{u x}
-(@code{calc-extract-units}) command extracts only the units portion of a
-formula. These commands essentially replace every term of the formula
-that does or doesn't (respectively) look like a unit name by the
-constant 1, then resimplify the formula.
-
address@hidden u a
address@hidden calc-autorange-units
-The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
-mode in which unit prefixes like @code{k} (``kilo'') are automatically
-applied to keep the numeric part of a units expression in a reasonable
-range. This mode affects @kbd{u s} and all units conversion commands
-except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
-will be simplified to @samp{12.345 kHz}. Autoranging is useful for
-some kinds of units (like @code{Hz} and @code{m}), but is probably
-undesirable for non-metric units like @code{ft} and @code{tbsp}.
-(Composite units are more appropriate for those; see above.)
-
-Autoranging always applies the prefix to the leftmost unit name.
-Calc chooses the largest prefix that causes the number to be greater
-than or equal to 1.0. Thus an increasing sequence of adjusted times
-would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
-Generally the rule of thumb is that the number will be adjusted
-to be in the interval @samp{[1 .. 1000)}, although there are several
-exceptions to this rule. First, if the unit has a power then this
-is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
-Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
-but will not apply to other units. The ``deci-,'' ``deka-,'' and
-``hecto-'' prefixes are never used. Thus the allowable interval is
address@hidden .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
-Finally, a prefix will not be added to a unit if the resulting name
-is also the actual name of another unit; @samp{1e-15 t} would normally
-be considered a ``femto-ton,'' but it is written as @samp{1000 at}
-(1000 atto-tons) instead because @code{ft} would be confused with feet.
-
address@hidden The Units Table, Predefined Units, Basic Operations on Units,
Units
address@hidden The Units Table
-
address@hidden
address@hidden u v
address@hidden calc-enter-units-table
-The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
-in another buffer called @code{*Units Table*}. Each entry in this table
-gives the unit name as it would appear in an expression, the definition
-of the unit in terms of simpler units, and a full name or description of
-the unit. Fundamental units are defined as themselves; these are the
-units produced by the @kbd{u b} command. The fundamental units are
-meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
-and steradians.
-
-The Units Table buffer also displays the Unit Prefix Table. Note that
-two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
-prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
-prefix. Whenever a unit name can be interpreted as either a built-in name
-or a prefix followed by another built-in name, the former interpretation
-wins. For example, @samp{2 pt} means two pints, not two pico-tons.
-
-The Units Table buffer, once created, is not rebuilt unless you define
-new units. To force the buffer to be rebuilt, give any numeric prefix
-argument to @kbd{u v}.
-
address@hidden u V
address@hidden calc-view-units-table
-The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
-that the cursor is not moved into the Units Table buffer. You can
-type @kbd{u V} again to remove the Units Table from the display. To
-return from the Units Table buffer after a @kbd{u v}, type @kbd{C-x * c}
-again or use the regular Emacs @address@hidden o}} (@code{other-window})
-command. You can also kill the buffer with @kbd{C-x k} if you wish;
-the actual units table is safely stored inside the Calculator.
-
address@hidden u g
address@hidden calc-get-unit-definition
-The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
-defining expression and pushes it onto the Calculator stack. For example,
address@hidden g in} will produce the expression @samp{2.54 cm}. This is the
-same definition for the unit that would appear in the Units Table buffer.
-Note that this command works only for actual unit names; @kbd{u g km}
-will report that no such unit exists, for example, because @code{km} is
-really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
-definition of a unit in terms of base units, it is easier to push the
-unit name on the stack and then reduce it to base units with @kbd{u b}.
-
address@hidden u e
address@hidden calc-explain-units
-The @kbd{u e} (@code{calc-explain-units}) command displays an English
-description of the units of the expression on the stack. For example,
-for the expression @samp{62 km^2 g / s^2 mol K}, the description is
-``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
-command uses the English descriptions that appear in the righthand
-column of the Units Table.
-
address@hidden Predefined Units, User-Defined Units, The Units Table, Units
address@hidden Predefined Units
-
address@hidden
-Since the exact definitions of many kinds of units have evolved over the
-years, and since certain countries sometimes have local differences in
-their definitions, it is a good idea to examine Calc's definition of a
-unit before depending on its exact value. For example, there are three
-different units for gallons, corresponding to the US (@code{gal}),
-Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
-note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
-ounce, and @code{ozfl} is a fluid ounce.
-
-The temperature units corresponding to degrees Kelvin and Centigrade
-(Celsius) are the same in this table, since most units commands treat
-temperatures as being relative. The @code{calc-convert-temperature}
-command has special rules for handling the different absolute magnitudes
-of the various temperature scales.
-
-The unit of volume ``liters'' can be referred to by either the lower-case
address@hidden or the upper-case @code{L}.
-
-The unit @code{A} stands for Amperes; the name @code{Ang} is used
address@hidden
-for \AA ngstroms.
address@hidden tex
address@hidden
-for Angstroms.
address@hidden ifnottex
-
-The unit @code{pt} stands for pints; the name @code{point} stands for
-a typographical point, defined by @samp{72 point = 1 in}. This is
-slightly different than the point defined by the American Typefounder's
-Association in 1886, but the point used by Calc has become standard
-largely due to its use by the PostScript page description language.
-There is also @code{texpt}, which stands for a printer's point as
-defined by the @TeX{} typesetting system: @samp{72.27 texpt = 1 in}.
-Other units used by @TeX{} are available; they are @code{texpc} (a pica),
address@hidden (a ``big point'', equal to a standard point which is larger
-than the point used by @TeX{}), @code{texdd} (a Didot point),
address@hidden (a Cicero) and @code{texsp} (a scaled @TeX{} point,
-all dimensions representable in @TeX{} are multiples of this value).
-
-The unit @code{e} stands for the elementary (electron) unit of charge;
-because algebra command could mistake this for the special constant
address@hidden, Calc provides the alternate unit name @code{ech} which is
-preferable to @code{e}.
-
-The name @code{g} stands for one gram of mass; there is also @code{gf},
-one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
-Meanwhile, one address@hidden'' of acceleration is denoted @code{ga}.
-
-The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
-a metric ton of @samp{1000 kg}.
-
-The names @code{s} (or @code{sec}) and @code{min} refer to units of
-time; @code{arcsec} and @code{arcmin} are units of angle.
-
-Some ``units'' are really physical constants; for example, @code{c}
-represents the speed of light, and @code{h} represents Planck's
-constant. You can use these just like other units: converting
address@hidden c} to @samp{m/s} expresses one-half the speed of light in
-meters per second. You can also use this merely as a handy reference;
-the @kbd{u g} command gets the definition of one of these constants
-in its normal terms, and @kbd{u b} expresses the definition in base
-units.
-
-Two units, @code{pi} and @code{alpha} (the fine structure constant,
-approximately @mathit{1/137}) are dimensionless. The units simplification
-commands simply treat these names as equivalent to their corresponding
-values. However you can, for example, use @kbd{u c} to convert a pure
-number into multiples of the fine structure constant, or @kbd{u b} to
-convert this back into a pure number. (When @kbd{u c} prompts for the
-``old units,'' just enter a blank line to signify that the value
-really is unitless.)
-
address@hidden Describe angular units, luminosity vs. steradians problem.
-
address@hidden User-Defined Units, , Predefined Units, Units
address@hidden User-Defined Units
-
address@hidden
-Calc provides ways to get quick access to your selected ``favorite''
-units, as well as ways to define your own new units.
-
address@hidden u 0-9
address@hidden calc-quick-units
address@hidden Units
address@hidden @code{Units} variable
address@hidden Quick units
-To select your favorite units, store a vector of unit names or
-expressions in the Calc variable @code{Units}. The @kbd{u 1}
-through @kbd{u 9} commands (@code{calc-quick-units}) provide access
-to these units. If the value on the top of the stack is a plain
-number (with no units attached), then @kbd{u 1} gives it the
-specified units. (Basically, it multiplies the number by the
-first item in the @code{Units} vector.) If the number on the
-stack @emph{does} have units, then @kbd{u 1} converts that number
-to the new units. For example, suppose the vector @samp{[in, ft]}
-is stored in @code{Units}. Then @kbd{30 u 1} will create the
-expression @samp{30 in}, and @kbd{u 2} will convert that expression
-to @samp{2.5 ft}.
-
-The @kbd{u 0} command accesses the tenth element of @code{Units}.
-Only ten quick units may be defined at a time. If the @code{Units}
-variable has no stored value (the default), or if its value is not
-a vector, then the quick-units commands will not function. The
address@hidden U} command is a convenient way to edit the @code{Units}
-variable; @pxref{Operations on Variables}.
-
address@hidden u d
address@hidden calc-define-unit
address@hidden User-defined units
-The @kbd{u d} (@code{calc-define-unit}) command records the units
-expression on the top of the stack as the definition for a new,
-user-defined unit. For example, putting @samp{16.5 ft} on the stack and
-typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
-16.5 feet. The unit conversion and simplification commands will now
-treat @code{rod} just like any other unit of length. You will also be
-prompted for an optional English description of the unit, which will
-appear in the Units Table.
-
address@hidden u u
address@hidden calc-undefine-unit
-The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
-unit. It is not possible to remove one of the predefined units,
-however.
-
-If you define a unit with an existing unit name, your new definition
-will replace the original definition of that unit. If the unit was a
-predefined unit, the old definition will not be replaced, only
-``shadowed.'' The built-in definition will reappear if you later use
address@hidden u} to remove the shadowing definition.
-
-To create a new fundamental unit, use either 1 or the unit name itself
-as the defining expression. Otherwise the expression can involve any
-other units that you like (except for composite units like @samp{mfi}).
-You can create a new composite unit with a sum of other units as the
-defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
-will rebuild the internal unit table incorporating your modifications.
-Note that erroneous definitions (such as two units defined in terms of
-each other) will not be detected until the unit table is next rebuilt;
address@hidden v} is a convenient way to force this to happen.
-
-Temperature units are treated specially inside the Calculator; it is not
-possible to create user-defined temperature units.
-
address@hidden u p
address@hidden calc-permanent-units
address@hidden Calc init file, user-defined units
-The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
-units in your Calc init file (the file given by the variable
address@hidden, typically @file{~/.calc.el}), so that the
-units will still be available in subsequent Emacs sessions. If there
-was already a set of user-defined units in your Calc init file, it
-is replaced by the new set. (@xref{General Mode Commands}, for a way to
-tell Calc to use a different file for the Calc init file.)
-
address@hidden Store and Recall, Graphics, Units, Top
address@hidden Storing and Recalling
-
address@hidden
-Calculator variables are really just Lisp variables that contain numbers
-or formulas in a form that Calc can understand. The commands in this
-section allow you to manipulate variables conveniently. Commands related
-to variables use the @kbd{s} prefix key.
-
address@hidden
-* Storing Variables::
-* Recalling Variables::
-* Operations on Variables::
-* Let Command::
-* Evaluates-To Operator::
address@hidden menu
-
address@hidden Storing Variables, Recalling Variables, Store and Recall, Store
and Recall
address@hidden Storing Variables
-
address@hidden
address@hidden s s
address@hidden calc-store
address@hidden Storing variables
address@hidden Quick variables
address@hidden q0
address@hidden q9
-The @kbd{s s} (@code{calc-store}) command stores the value at the top of
-the stack into a specified variable. It prompts you to enter the
-name of the variable. If you press a single digit, the value is stored
-immediately in one of the ``quick'' variables @code{q0} through
address@hidden Or you can enter any variable name.
-
address@hidden s t
address@hidden calc-store-into
-The @kbd{s s} command leaves the stored value on the stack. There is
-also an @kbd{s t} (@code{calc-store-into}) command, which removes a
-value from the stack and stores it in a variable.
-
-If the top of stack value is an equation @samp{a = 7} or assignment
address@hidden := 7} with a variable on the lefthand side, then Calc will
-assign that variable with that value by default, i.e., if you type
address@hidden s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
-value 7 would be stored in the variable @samp{a}. (If you do type
-a variable name at the prompt, the top-of-stack value is stored in
-its entirety, even if it is an equation: @samp{s s b @key{RET}}
-with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
-
-In fact, the top of stack value can be a vector of equations or
-assignments with different variables on their lefthand sides; the
-default will be to store all the variables with their corresponding
-righthand sides simultaneously.
-
-It is also possible to type an equation or assignment directly at
-the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
-In this case the expression to the right of the @kbd{=} or @kbd{:=}
-symbol is evaluated as if by the @kbd{=} command, and that value is
-stored in the variable. No value is taken from the stack; @kbd{s s}
-and @kbd{s t} are equivalent when used in this way.
-
address@hidden s 0-9
address@hidden t 0-9
-The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
-digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
-equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
-for trail and time/date commands.)
-
address@hidden s +
address@hidden s -
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden s *
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s /
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s ^
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s |
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s n
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s &
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s [
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s ]
address@hidden calc-store-plus
address@hidden calc-store-minus
address@hidden calc-store-times
address@hidden calc-store-div
address@hidden calc-store-power
address@hidden calc-store-concat
address@hidden calc-store-neg
address@hidden calc-store-inv
address@hidden calc-store-decr
address@hidden calc-store-incr
-There are also several ``arithmetic store'' commands. For example,
address@hidden +} removes a value from the stack and adds it to the specified
-variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
address@hidden ^}, and @address@hidden |}} (vector concatenation), plus @kbd{s
n} and
address@hidden &} which negate or invert the value in a variable, and
@address@hidden [}}
-and @kbd{s ]} which decrease or increase a variable by one.
-
-All the arithmetic stores accept the Inverse prefix to reverse the
-order of the operands. If @expr{v} represents the contents of the
-variable, and @expr{a} is the value drawn from the stack, then regular
address@hidden@kbd{s -}} assigns
address@hidden @math{v \coloneq v - a},
address@hidden @expr{v := v - a},
-but @kbd{I s -} assigns
address@hidden @math{v \coloneq a - v}.
address@hidden @expr{v := a - v}.
-While @kbd{I s *} might seem pointless, it is
-useful if matrix multiplication is involved. Actually, all the
-arithmetic stores use formulas designed to behave usefully both
-forwards and backwards:
-
address@hidden
address@hidden
-s + v := v + a v := a + v
-s - v := v - a v := a - v
-s * v := v * a v := a * v
-s / v := v / a v := a / v
-s ^ v := v ^ a v := a ^ v
-s | v := v | a v := a | v
-s n v := v / (-1) v := (-1) / v
-s & v := v ^ (-1) v := (-1) ^ v
-s [ v := v - 1 v := 1 - v
-s ] v := v - (-1) v := (-1) - v
address@hidden group
address@hidden example
-
-In the last four cases, a numeric prefix argument will be used in
-place of the number one. (For example, @kbd{M-2 s ]} increases
-a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
-minus-two minus the variable.
-
-The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
-etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
-arithmetic stores that don't remove the value @expr{a} from the stack.
-
-All arithmetic stores report the new value of the variable in the
-Trail for your information. They signal an error if the variable
-previously had no stored value. If default simplifications have been
-turned off, the arithmetic stores temporarily turn them on for numeric
-arguments only (i.e., they temporarily do an @kbd{m N} command).
address@hidden Modes}. Large vectors put in the trail by
-these commands always use abbreviated (@kbd{t .}) mode.
-
address@hidden s m
address@hidden calc-store-map
-The @kbd{s m} command is a general way to adjust a variable's value
-using any Calc function. It is a ``mapping'' command analogous to
address@hidden M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
-how to specify a function for a mapping command. Basically,
-all you do is type the Calc command key that would invoke that
-function normally. For example, @kbd{s m n} applies the @kbd{n}
-key to negate the contents of the variable, so @kbd{s m n} is
-equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
-of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
-reverse the vector stored in the variable, and @kbd{s m H I S}
-takes the hyperbolic arcsine of the variable contents.
-
-If the mapping function takes two or more arguments, the additional
-arguments are taken from the stack; the old value of the variable
-is provided as the first argument. Thus @kbd{s m -} with @expr{a}
-on the stack computes @expr{v - a}, just like @kbd{s -}. With the
-Inverse prefix, the variable's original value becomes the @emph{last}
-argument instead of the first. Thus @kbd{I s m -} is also
-equivalent to @kbd{I s -}.
-
address@hidden s x
address@hidden calc-store-exchange
-The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
-of a variable with the value on the top of the stack. Naturally, the
-variable must already have a stored value for this to work.
-
-You can type an equation or assignment at the @kbd{s x} prompt. The
-command @kbd{s x a=6} takes no values from the stack; instead, it
-pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
-
address@hidden s u
address@hidden calc-unstore
address@hidden Void variables
address@hidden Un-storing variables
-Until you store something in them, most variables are ``void,'' that is,
-they contain no value at all. If they appear in an algebraic formula
-they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
-The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
-void state.
-
address@hidden s c
address@hidden calc-copy-variable
-The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
-value of one variable to another. One way it differs from a simple
address@hidden r} followed by an @kbd{s t} (aside from saving keystrokes) is
-that the value never goes on the stack and thus is never rounded,
-evaluated, or simplified in any way; it is not even rounded down to the
-current precision.
-
-The only variables with predefined values are the ``special constants''
address@hidden, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
-to unstore these variables or to store new values into them if you like,
-although some of the algebraic-manipulation functions may assume these
-variables represent their standard values. Calc displays a warning if
-you change the value of one of these variables, or of one of the other
-special variables @code{inf}, @code{uinf}, and @code{nan} (which are
-normally void).
-
-Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
-but rather a special magic value that evaluates to @cpi{} at the current
-precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
-according to the current precision or polar mode. If you recall a value
-from @code{pi} and store it back, this magic property will be lost. The
-magic property is preserved, however, when a variable is copied with
address@hidden c}.
-
address@hidden s k
address@hidden calc-copy-special-constant
-If one of the ``special constants'' is redefined (or undefined) so that
-it no longer has its magic property, the property can be restored with
address@hidden k} (@code{calc-copy-special-constant}). This command will prompt
-for a special constant and a variable to store it in, and so a special
-constant can be stored in any variable. Here, the special constant that
-you enter doesn't depend on the value of the corresponding variable;
address@hidden will represent address@hidden regardless of what is currently
-stored in the Calc variable @code{pi}. If one of the other special
-variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
-original behavior can be restored by voiding it with @kbd{s u}.
-
address@hidden Recalling Variables, Operations on Variables, Storing Variables,
Store and Recall
address@hidden Recalling Variables
-
address@hidden
address@hidden s r
address@hidden calc-recall
address@hidden Recalling variables
-The most straightforward way to extract the stored value from a variable
-is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
-for a variable name (similarly to @code{calc-store}), looks up the value
-of the specified variable, and pushes that value onto the stack. It is
-an error to try to recall a void variable.
-
-It is also possible to recall the value from a variable by evaluating a
-formula containing that variable. For example, @kbd{' a @key{RET} =} is
-the same as @kbd{s r a @key{RET}} except that if the variable is void, the
-former will simply leave the formula @samp{a} on the stack whereas the
-latter will produce an error message.
-
address@hidden r 0-9
-The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
-equivalent to @kbd{s r 9}. (The @kbd{r} prefix is otherwise unused
-in the current version of Calc.)
-
address@hidden Operations on Variables, Let Command, Recalling Variables, Store
and Recall
address@hidden Other Operations on Variables
-
address@hidden
address@hidden s e
address@hidden calc-edit-variable
-The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
-value of a variable without ever putting that value on the stack
-or simplifying or evaluating the value. It prompts for the name of
-the variable to edit. If the variable has no stored value, the
-editing buffer will start out empty. If the editing buffer is
-empty when you press @kbd{C-c C-c} to finish, the variable will
-be made void. @xref{Editing Stack Entries}, for a general
-description of editing.
-
-The @kbd{s e} command is especially useful for creating and editing
-rewrite rules which are stored in variables. Sometimes these rules
-contain formulas which must not be evaluated until the rules are
-actually used. (For example, they may refer to @samp{deriv(x,y)},
-where @code{x} will someday become some expression involving @code{y};
-if you let Calc evaluate the rule while you are defining it, Calc will
-replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
-not itself refer to @code{y}.) By contrast, recalling the variable,
-editing with @kbd{`}, and storing will evaluate the variable's value
-as a side effect of putting the value on the stack.
-
address@hidden s A
address@hidden s D
address@hidden
address@hidden @idots
address@hidden ignore
address@hidden s E
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s F
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s G
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s H
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s I
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s L
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s P
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s R
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s T
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s U
address@hidden
address@hidden @null
address@hidden ignore
address@hidden s X
address@hidden calc-store-AlgSimpRules
address@hidden calc-store-Decls
address@hidden calc-store-EvalRules
address@hidden calc-store-FitRules
address@hidden calc-store-GenCount
address@hidden calc-store-Holidays
address@hidden calc-store-IntegLimit
address@hidden calc-store-LineStyles
address@hidden calc-store-PointStyles
address@hidden calc-store-PlotRejects
address@hidden calc-store-TimeZone
address@hidden calc-store-Units
address@hidden calc-store-ExtSimpRules
-There are several special-purpose variable-editing commands that
-use the @kbd{s} prefix followed by a shifted letter:
-
address@hidden @kbd
address@hidden s A
-Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
address@hidden s D
-Edit @code{Decls}. @xref{Declarations}.
address@hidden s E
-Edit @code{EvalRules}. @xref{Default Simplifications}.
address@hidden s F
-Edit @code{FitRules}. @xref{Curve Fitting}.
address@hidden s G
-Edit @code{GenCount}. @xref{Solving Equations}.
address@hidden s H
-Edit @code{Holidays}. @xref{Business Days}.
address@hidden s I
-Edit @code{IntegLimit}. @xref{Calculus}.
address@hidden s L
-Edit @code{LineStyles}. @xref{Graphics}.
address@hidden s P
-Edit @code{PointStyles}. @xref{Graphics}.
address@hidden s R
-Edit @code{PlotRejects}. @xref{Graphics}.
address@hidden s T
-Edit @code{TimeZone}. @xref{Time Zones}.
address@hidden s U
-Edit @code{Units}. @xref{User-Defined Units}.
address@hidden s X
-Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
address@hidden table
-
-These commands are just versions of @kbd{s e} that use fixed variable
-names rather than prompting for the variable name.
-
address@hidden s p
address@hidden calc-permanent-variable
address@hidden Storing variables
address@hidden Permanent variables
address@hidden Calc init file, variables
-The @kbd{s p} (@code{calc-permanent-variable}) command saves a
-variable's value permanently in your Calc init file (the file given by
-the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
-that its value will still be available in future Emacs sessions. You
-can re-execute @address@hidden p}} later on to update the saved value, but the
-only way to remove a saved variable is to edit your calc init file
-by hand. (@xref{General Mode Commands}, for a way to tell Calc to
-use a different file for the Calc init file.)
-
-If you do not specify the name of a variable to save (i.e.,
address@hidden p @key{RET}}), all Calc variables with defined values
-are saved except for the special constants @code{pi}, @code{e},
address@hidden, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
-and @code{PlotRejects};
address@hidden, @code{DistribRules}, and other built-in rewrite
-rules; and @address@hidden variables generated
-by the graphics commands. (You can still save these variables by
-explicitly naming them in an @kbd{s p} command.)
-
address@hidden s i
address@hidden calc-insert-variables
-The @kbd{s i} (@code{calc-insert-variables}) command writes
-the values of all Calc variables into a specified buffer.
-The variables are written with the prefix @code{var-} in the form of
-Lisp @code{setq} commands
-which store the values in string form. You can place these commands
-in your Calc init file (or @file{.emacs}) if you wish, though in this case it
-would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
-omits the same set of variables as @address@hidden p @key{RET}}}; the
difference
-is that @kbd{s i} will store the variables in any buffer, and it also
-stores in a more human-readable format.)
-
address@hidden Let Command, Evaluates-To Operator, Operations on Variables,
Store and Recall
address@hidden The Let Command
-
address@hidden
address@hidden s l
address@hidden calc-let
address@hidden Variables, temporary assignment
address@hidden Temporary assignment to variables
-If you have an expression like @samp{a+b^2} on the stack and you wish to
-compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
-then press @kbd{=} to reevaluate the formula. This has the side-effect
-of leaving the stored value of 3 in @expr{b} for future operations.
-
-The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
address@hidden assignment of a variable. It stores the value on the
-top of the stack into the specified variable, then evaluates the
-second-to-top stack entry, then restores the original value (or lack of one)
-in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
-the stack will contain the formula @samp{a + 9}. The subsequent command
address@hidden@w{5 s l a} @key{RET}} will replace this formula with the number
14.
-The variables @samp{a} and @samp{b} are not permanently affected in any way
-by these commands.
-
-The value on the top of the stack may be an equation or assignment, or
-a vector of equations or assignments, in which case the default will be
-analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
-
-Also, you can answer the variable-name prompt with an equation or
-assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
-and typing @kbd{s l b @key{RET}}.
-
-The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
-a variable with a value in a formula. It does an actual substitution
-rather than temporarily assigning the variable and evaluating. For
-example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
-produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
-since the evaluation step will also evaluate @code{pi}.
-
address@hidden Evaluates-To Operator, , Let Command, Store and Recall
address@hidden The Evaluates-To Operator
-
address@hidden
address@hidden evalto
address@hidden =>
address@hidden Evaluates-to operator
address@hidden @samp{=>} operator
-The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
-operator}. (It will show up as an @code{evalto} function call in
-other language modes like Pascal and address@hidden) This is a binary
-operator, that is, it has a lefthand and a righthand argument,
-although it can be entered with the righthand argument omitted.
-
-A formula like @address@hidden => @var{b}} is evaluated by Calc as
-follows: First, @var{a} is not simplified or modified in any
-way. The previous value of argument @var{b} is thrown away; the
-formula @var{a} is then copied and evaluated as if by the @kbd{=}
-command according to all current modes and stored variable values,
-and the result is installed as the new value of @var{b}.
-
-For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
-The number 17 is ignored, and the lefthand argument is left in its
-unevaluated form; the result is the formula @samp{2 + 3 => 5}.
-
address@hidden s =
address@hidden calc-evalto
-You can enter an @samp{=>} formula either directly using algebraic
-entry (in which case the righthand side may be omitted since it is
-going to be replaced right away anyhow), or by using the @kbd{s =}
-(@code{calc-evalto}) command, which takes @var{a} from the stack
-and replaces it with @address@hidden => @var{b}}.
-
-Calc keeps track of all @samp{=>} operators on the stack, and
-recomputes them whenever anything changes that might affect their
-values, i.e., a mode setting or variable value. This occurs only
-if the @samp{=>} operator is at the top level of the formula, or
-if it is part of a top-level vector. In other words, pushing
address@hidden + (a => 17)} will change the 17 to the actual value of
address@hidden when you enter the formula, but the result will not be
-dynamically updated when @samp{a} is changed later because the
address@hidden>} operator is buried inside a sum. However, a vector
-of @samp{=>} operators will be recomputed, since it is convenient
-to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
-make a concise display of all the variables in your problem.
-(Another way to do this would be to use @samp{[a, b, c] =>},
-which provides a slightly different format of display. You
-can use whichever you find easiest to read.)
-
address@hidden m C
address@hidden calc-auto-recompute
-The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
-turn this automatic recomputation on or off. If you turn
-recomputation off, you must explicitly recompute an @samp{=>}
-operator on the stack in one of the usual ways, such as by
-pressing @kbd{=}. Turning recomputation off temporarily can save
-a lot of time if you will be changing several modes or variables
-before you look at the @samp{=>} entries again.
-
-Most commands are not especially useful with @samp{=>} operators
-as arguments. For example, given @samp{x + 2 => 17}, it won't
-work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
-to operate on the lefthand side of the @samp{=>} operator on
-the top of the stack, type @kbd{j 1} (that's the digit ``one'')
-to select the lefthand side, execute your commands, then type
address@hidden u} to unselect.
-
-All current modes apply when an @samp{=>} operator is computed,
-including the current simplification mode. Recall that the
-formula @samp{x + y + x} is not handled by Calc's default
-simplifications, but the @kbd{a s} command will reduce it to
-the simpler form @samp{y + 2 x}. You can also type @kbd{m A}
-to enable an Algebraic Simplification mode in which the
-equivalent of @kbd{a s} is used on all of Calc's results.
-If you enter @samp{x + y + x =>} normally, the result will
-be @samp{x + y + x => x + y + x}. If you change to
-Algebraic Simplification mode, the result will be
address@hidden + y + x => y + 2 x}. However, just pressing @kbd{a s}
-once will have no effect on @samp{x + y + x => x + y + x},
-because the righthand side depends only on the lefthand side
-and the current mode settings, and the lefthand side is not
-affected by commands like @kbd{a s}.
-
-The ``let'' command (@kbd{s l}) has an interesting interaction
-with the @samp{=>} operator. The @kbd{s l} command evaluates the
-second-to-top stack entry with the top stack entry supplying
-a temporary value for a given variable. As you might expect,
-if that stack entry is an @samp{=>} operator its righthand
-side will temporarily show this value for the variable. In
-fact, all @samp{=>}s on the stack will be updated if they refer
-to that variable. But this change is temporary in the sense
-that the next command that causes Calc to look at those stack
-entries will make them revert to the old variable value.
-
address@hidden
address@hidden
-2: a => a 2: a => 17 2: a => a
-1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
- . . .
-
- 17 s l a @key{RET} p 8 @key{RET}
address@hidden group
address@hidden smallexample
-
-Here the @kbd{p 8} command changes the current precision,
-thus causing the @samp{=>} forms to be recomputed after the
-influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
-(@code{calc-refresh}) is a handy way to force the @samp{=>}
-operators on the stack to be recomputed without any other
-side effects.
-
address@hidden s :
address@hidden calc-assign
address@hidden assign
address@hidden :=
-Embedded mode also uses @samp{=>} operators. In Embedded mode,
-the lefthand side of an @samp{=>} operator can refer to variables
-assigned elsewhere in the file by @samp{:=} operators. The
-assignment operator @samp{a := 17} does not actually do anything
-by itself. But Embedded mode recognizes it and marks it as a sort
-of file-local definition of the variable. You can enter @samp{:=}
-operators in Algebraic mode, or by using the @kbd{s :}
-(@code{calc-assign}) address@hidden command which takes a variable
-and value from the stack and replaces them with an assignment.
-
address@hidden and LaTeX Language Modes}, for the way @samp{=>} appears in
address@hidden language output. The @dfn{eqn} mode gives similar
-treatment to @samp{=>}.
-
address@hidden Graphics, Kill and Yank, Store and Recall, Top
address@hidden Graphics
-
address@hidden
-The commands for graphing data begin with the @kbd{g} prefix key. Calc
-uses GNUPLOT 2.0 or later to do graphics. These commands will only work
-if GNUPLOT is available on your system. (While GNUPLOT sounds like
-a relative of GNU Emacs, it is actually completely unrelated.
-However, it is free software. It can be obtained from
address@hidden://www.gnuplot.info}.)
-
address@hidden calc-gnuplot-name
-If you have GNUPLOT installed on your system but Calc is unable to
-find it, you may need to set the @code{calc-gnuplot-name} variable
-in your Calc init file or @file{.emacs}. You may also need to set some Lisp
-variables to show Calc how to run GNUPLOT on your system; these
-are described under @kbd{g D} and @kbd{g O} below. If you are
-using the X window system, Calc will configure GNUPLOT for you
-automatically. If you have GNUPLOT 3.0 or later and you are not using X,
-Calc will configure GNUPLOT to display graphs using simple character
-graphics that will work on any terminal.
-
address@hidden
-* Basic Graphics::
-* Three Dimensional Graphics::
-* Managing Curves::
-* Graphics Options::
-* Devices::
address@hidden menu
-
address@hidden Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
address@hidden Basic Graphics
-
address@hidden
address@hidden g f
address@hidden calc-graph-fast
-The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
-This command takes two vectors of equal length from the stack.
-The vector at the top of the stack represents the ``y'' values of
-the various data points. The vector in the second-to-top position
-represents the corresponding ``x'' values. This command runs
-GNUPLOT (if it has not already been started by previous graphing
-commands) and displays the set of data points. The points will
-be connected by lines, and there will also be some kind of symbol
-to indicate the points themselves.
-
-The ``x'' entry may instead be an interval form, in which case suitable
-``x'' values are interpolated between the minimum and maximum values of
-the interval (whether the interval is open or closed is ignored).
-
-The ``x'' entry may also be a number, in which case Calc uses the
-sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
-(Generally the number 0 or 1 would be used for @expr{x} in this case.)
-
-The ``y'' entry may be any formula instead of a vector. Calc effectively
-uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
-the result of this must be a formula in a single (unassigned) variable.
-The formula is plotted with this variable taking on the various ``x''
-values. Graphs of formulas by default use lines without symbols at the
-computed data points. Note that if neither ``x'' nor ``y'' is a vector,
-Calc guesses at a reasonable number of data points to use. See the
address@hidden N} command below. (The ``x'' values must be either a vector
-or an interval if ``y'' is a formula.)
-
address@hidden
address@hidden
address@hidden ignore
address@hidden xy
-If ``y'' is (or evaluates to) a formula of the form
address@hidden(@var{x}, @var{y})} then the result is a
-parametric plot. The two arguments of the fictitious @code{xy} function
-are used as the ``x'' and ``y'' coordinates of the curve, respectively.
-In this case the ``x'' vector or interval you specified is not directly
-visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
-and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
-will be a circle.
-
-Also, ``x'' and ``y'' may each be variable names, in which case Calc
-looks for suitable vectors, intervals, or formulas stored in those
-variables.
-
-The ``x'' and ``y'' values for the data points (as pulled from the vectors,
-calculated from the formulas, or interpolated from the intervals) should
-be real numbers (integers, fractions, or floats). One exception to this
-is that the ``y'' entry can consist of a vector of numbers combined with
-error forms, in which case the points will be plotted with the
-appropriate error bars. Other than this, if either the ``x''
-value or the ``y'' value of a given data point is not a real number, that
-data point will be omitted from the graph. The points on either side
-of the invalid point will @emph{not} be connected by a line.
-
-See the documentation for @kbd{g a} below for a description of the way
-numeric prefix arguments affect @kbd{g f}.
-
address@hidden @code{PlotRejects} variable
address@hidden PlotRejects
-If you store an empty vector in the variable @code{PlotRejects}
-(i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
-this vector for every data point which was rejected because its
-``x'' or ``y'' values were not real numbers. The result will be
-a matrix where each row holds the curve number, data point number,
-``x'' value, and ``y'' value for a rejected data point.
address@hidden Operator}, for a handy way to keep tabs on the
-current value of @code{PlotRejects}. @xref{Operations on Variables},
-for the @kbd{s R} command which is another easy way to examine
address@hidden
-
address@hidden g c
address@hidden calc-graph-clear
-To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
-If the GNUPLOT output device is an X window, the window will go away.
-Effects on other kinds of output devices will vary. You don't need
-to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
-or @kbd{g p} command later on, it will reuse the existing graphics
-window if there is one.
-
address@hidden Three Dimensional Graphics, Managing Curves, Basic Graphics,
Graphics
address@hidden Three-Dimensional Graphics
-
address@hidden g F
address@hidden calc-graph-fast-3d
-The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
-graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
-you will see a GNUPLOT error message if you try this command.
-
-The @kbd{g F} command takes three values from the stack, called ``x'',
-``y'', and ``z'', respectively. As was the case for 2D graphs, there
-are several options for these values.
-
-In the first case, ``x'' and ``y'' are each vectors (not necessarily of
-the same length); either or both may instead be interval forms. The
-``z'' value must be a matrix with the same number of rows as elements
-in ``x'', and the same number of columns as elements in ``y''. The
-result is a surface plot where
address@hidden @math{z_{ij}}
address@hidden @expr{z_ij}
-is the height of the point
-at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
-be displayed from a certain default viewpoint; you can change this
-viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
-buffer as described later. See the GNUPLOT documentation for a
-description of the @samp{set view} command.
-
-Each point in the matrix will be displayed as a dot in the graph,
-and these points will be connected by a grid of lines (@dfn{isolines}).
-
-In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
-length. The resulting graph displays a 3D line instead of a surface,
-where the coordinates of points along the line are successive triplets
-of values from the input vectors.
-
-In the third case, ``x'' and ``y'' are vectors or interval forms, and
-``z'' is any formula involving two variables (not counting variables
-with assigned values). These variables are sorted into alphabetical
-order; the first takes on values from ``x'' and the second takes on
-values from ``y'' to form a matrix of results that are graphed as a
-3D surface.
-
address@hidden
address@hidden
address@hidden ignore
address@hidden xyz
-If the ``z'' formula evaluates to a call to the fictitious function
address@hidden(@var{x}, @var{y}, @var{z})}, then the result is a
-``parametric surface.'' In this case, the axes of the graph are
-taken from the @var{x} and @var{y} values in these calls, and the
-``x'' and ``y'' values from the input vectors or intervals are used only
-to specify the range of inputs to the formula. For example, plotting
address@hidden, [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
-will draw a sphere. (Since the default resolution for 3D plots is
-5 steps in each of ``x'' and ``y'', this will draw a very crude
-sphere. You could use the @kbd{g N} command, described below, to
-increase this resolution, or specify the ``x'' and ``y'' values as
-vectors with more than 5 elements.
-
-It is also possible to have a function in a regular @kbd{g f} plot
-evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
-a surface, the result will be a 3D parametric line. For example,
address@hidden, xyz(sin(x), cos(x), x)]} will plot two turns of a
-helix (a three-dimensional spiral).
-
-As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
-variables containing the relevant data.
-
address@hidden Managing Curves, Graphics Options, Three Dimensional Graphics,
Graphics
address@hidden Managing Curves
-
address@hidden
-The @kbd{g f} command is really shorthand for the following commands:
address@hidden g d g a g p}. Likewise, @address@hidden F}} is shorthand for
address@hidden g d g A g p}. You can gain more control over your graph
-by using these commands directly.
-
address@hidden g a
address@hidden calc-graph-add
-The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
-represented by the two values on the top of the stack to the current
-graph. You can have any number of curves in the same graph. When
-you give the @kbd{g p} command, all the curves will be drawn superimposed
-on the same axes.
-
-The @kbd{g a} command (and many others that affect the current graph)
-will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
-in another window. This buffer is a template of the commands that will
-be sent to GNUPLOT when it is time to draw the graph. The first
address@hidden a} command adds a @code{plot} command to this buffer. Succeeding
address@hidden a} commands add extra curves onto that @code{plot} command.
-Other graph-related commands put other GNUPLOT commands into this
-buffer. In normal usage you never need to work with this buffer
-directly, but you can if you wish. The only constraint is that there
-must be only one @code{plot} command, and it must be the last command
-in the buffer. If you want to save and later restore a complete graph
-configuration, you can use regular Emacs commands to save and restore
-the contents of the @samp{*Gnuplot Commands*} buffer.
-
address@hidden PlotData1
address@hidden PlotData2
-If the values on the stack are not variable names, @kbd{g a} will invent
-variable names for them (of the form @address@hidden) and store
-the values in those variables. The ``x'' and ``y'' variables are what
-go into the @code{plot} command in the template. If you add a curve
-that uses a certain variable and then later change that variable, you
-can replot the graph without having to delete and re-add the curve.
-That's because the variable name, not the vector, interval or formula
-itself, is what was added by @kbd{g a}.
-
-A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
-stack entries are interpreted as curves. With a positive prefix
-argument @expr{n}, the top @expr{n} stack entries are ``y'' values
-for @expr{n} different curves which share a common ``x'' value in
-the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
-argument is equivalent to @kbd{C-u 1 g a}.)
-
-A prefix of zero or plain @kbd{C-u} means to take two stack entries,
-``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
-``y'' values for several curves that share a common ``x''.
-
-A negative prefix argument tells Calc to read @expr{n} vectors from
-the stack; each vector @expr{[x, y]} describes an independent curve.
-This is the only form of @kbd{g a} that creates several curves at once
-that don't have common ``x'' values. (Of course, the range of ``x''
-values covered by all the curves ought to be roughly the same if
-they are to look nice on the same graph.)
-
-For example, to plot
address@hidden @math{\sin n x}
address@hidden @expr{sin(n x)}
-for integers @expr{n}
-from 1 to 5, you could use @kbd{v x} to create a vector of integers
-(@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
-across this vector. The resulting vector of formulas is suitable
-for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
-command.
-
address@hidden g A
address@hidden calc-graph-add-3d
-The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
-to the graph. It is not valid to intermix 2D and 3D curves in a
-single graph. This command takes three arguments, ``x'', ``y'',
-and ``z'', from the stack. With a positive prefix @expr{n}, it
-takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
-separate ``z''s). With a zero prefix, it takes three stack entries
-but the ``z'' entry is a vector of curve values. With a negative
-prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
-The @kbd{g A} command works by adding a @code{splot} (surface-plot)
-command to the @samp{*Gnuplot Commands*} buffer.
-
-(Although @kbd{g a} adds a 2D @code{plot} command to the
address@hidden Commands*} buffer, Calc changes this to @code{splot}
-before sending it to GNUPLOT if it notices that the data points are
-evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
address@hidden a} curves in a single graph, although Calc does not currently
-check for this.)
-
address@hidden g d
address@hidden calc-graph-delete
-The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
-recently added curve from the graph. It has no effect if there are
-no curves in the graph. With a numeric prefix argument of any kind,
-it deletes all of the curves from the graph.
-
address@hidden g H
address@hidden calc-graph-hide
-The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
-the most recently added curve. A hidden curve will not appear in
-the actual plot, but information about it such as its name and line and
-point styles will be retained.
-
address@hidden g j
address@hidden calc-graph-juggle
-The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
-at the end of the list (the ``most recently added curve'') to the
-front of the list. The next-most-recent curve is thus exposed for
address@hidden@kbd{g d}} or similar commands to use. With @kbd{g j} you can
work
-with any curve in the graph even though curve-related commands only
-affect the last curve in the list.
-
address@hidden g p
address@hidden calc-graph-plot
-The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
-the graph described in the @samp{*Gnuplot Commands*} buffer. Any
-GNUPLOT parameters which are not defined by commands in this buffer
-are reset to their default values. The variables named in the @code{plot}
-command are written to a temporary data file and the variable names
-are then replaced by the file name in the template. The resulting
-plotting commands are fed to the GNUPLOT program. See the documentation
-for the GNUPLOT program for more specific information. All temporary
-files are removed when Emacs or GNUPLOT exits.
-
-If you give a formula for ``y'', Calc will remember all the values that
-it calculates for the formula so that later plots can reuse these values.
-Calc throws out these saved values when you change any circumstances
-that may affect the data, such as switching from Degrees to Radians
-mode, or changing the value of a parameter in the formula. You can
-force Calc to recompute the data from scratch by giving a negative
-numeric prefix argument to @kbd{g p}.
-
-Calc uses a fairly rough step size when graphing formulas over intervals.
-This is to ensure quick response. You can ``refine'' a plot by giving
-a positive numeric prefix argument to @kbd{g p}. Calc goes through
-the data points it has computed and saved from previous plots of the
-function, and computes and inserts a new data point midway between
-each of the existing points. You can refine a plot any number of times,
-but beware that the amount of calculation involved doubles each time.
-
-Calc does not remember computed values for 3D graphs. This means the
-numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
-the current graph is three-dimensional.
-
address@hidden g P
address@hidden calc-graph-print
-The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
-except that it sends the output to a printer instead of to the
-screen. More precisely, @kbd{g p} looks for @samp{set terminal}
-or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
-lacking these it uses the default settings. However, @kbd{g P}
-ignores @samp{set terminal} and @samp{set output} commands and
-uses a different set of default values. All of these values are
-controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
-Provided everything is set up properly, @kbd{g p} will plot to
-the screen unless you have specified otherwise and @kbd{g P} will
-always plot to the printer.
-
address@hidden Graphics Options, Devices, Managing Curves, Graphics
address@hidden Graphics Options
-
address@hidden
address@hidden g g
address@hidden calc-graph-grid
-The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
-on and off. It is off by default; tick marks appear only at the
-edges of the graph. With the grid turned on, dotted lines appear
-across the graph at each tick mark. Note that this command only
-changes the setting in @samp{*Gnuplot Commands*}; to see the effects
-of the change you must give another @kbd{g p} command.
-
address@hidden g b
address@hidden calc-graph-border
-The @kbd{g b} (@code{calc-graph-border}) command turns the border
-(the box that surrounds the graph) on and off. It is on by default.
-This command will only work with GNUPLOT 3.0 and later versions.
-
address@hidden g k
address@hidden calc-graph-key
-The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
-on and off. The key is a chart in the corner of the graph that
-shows the correspondence between curves and line styles. It is
-off by default, and is only really useful if you have several
-curves on the same graph.
-
address@hidden g N
address@hidden calc-graph-num-points
-The @kbd{g N} (@code{calc-graph-num-points}) command allows you
-to select the number of data points in the graph. This only affects
-curves where neither ``x'' nor ``y'' is specified as a vector.
-Enter a blank line to revert to the default value (initially 15).
-With no prefix argument, this command affects only the current graph.
-With a positive prefix argument this command changes or, if you enter
-a blank line, displays the default number of points used for all
-graphs created by @kbd{g a} that don't specify the resolution explicitly.
-With a negative prefix argument, this command changes or displays
-the default value (initially 5) used for 3D graphs created by @kbd{g A}.
-Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
-will be computed for the surface.
-
-Data values in the graph of a function are normally computed to a
-precision of five digits, regardless of the current precision at the
-time. This is usually more than adequate, but there are cases where
-it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
-interval @samp{[0 ..@: 1e-6]} will round all the data points down
-to 1.0! Putting the command @samp{set precision @var{n}} in the
address@hidden Commands*} buffer will cause the data to be computed
-at precision @var{n} instead of 5. Since this is such a rare case,
-there is no keystroke-based command to set the precision.
-
address@hidden g h
address@hidden calc-graph-header
-The @kbd{g h} (@code{calc-graph-header}) command sets the title
-for the graph. This will show up centered above the graph.
-The default title is blank (no title).
-
address@hidden g n
address@hidden calc-graph-name
-The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
-individual curve. Like the other curve-manipulating commands, it
-affects the most recently added curve, i.e., the last curve on the
-list in the @samp{*Gnuplot Commands*} buffer. To set the title of
-the other curves you must first juggle them to the end of the list
-with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
-Curve titles appear in the key; if the key is turned off they are
-not used.
-
address@hidden g t
address@hidden g T
address@hidden calc-graph-title-x
address@hidden calc-graph-title-y
-The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
-(@code{calc-graph-title-y}) commands set the titles on the ``x''
-and ``y'' axes, respectively. These titles appear next to the
-tick marks on the left and bottom edges of the graph, respectively.
-Calc does not have commands to control the tick marks themselves,
-but you can edit them into the @samp{*Gnuplot Commands*} buffer if
-you wish. See the GNUPLOT documentation for details.
-
address@hidden g r
address@hidden g R
address@hidden calc-graph-range-x
address@hidden calc-graph-range-y
-The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
-(@code{calc-graph-range-y}) commands set the range of values on the
-``x'' and ``y'' axes, respectively. You are prompted to enter a
-suitable range. This should be either a pair of numbers of the
-form, @address@hidden:@var{max}}, or a blank line to revert to the
-default behavior of setting the range based on the range of values
-in the data, or @samp{$} to take the range from the top of the stack.
-Ranges on the stack can be represented as either interval forms or
-vectors: @address@hidden ..@: @var{max}]} or @address@hidden, @var{max}]}.
-
address@hidden g l
address@hidden g L
address@hidden calc-graph-log-x
address@hidden calc-graph-log-y
-The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
-commands allow you to set either or both of the axes of the graph to
-be logarithmic instead of linear.
-
address@hidden g C-l
address@hidden g C-r
address@hidden g C-t
address@hidden calc-graph-log-z
address@hidden calc-graph-range-z
address@hidden calc-graph-title-z
-For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
-letters with the Control key held down) are the corresponding commands
-for the ``z'' axis.
-
address@hidden g z
address@hidden g Z
address@hidden calc-graph-zero-x
address@hidden calc-graph-zero-y
-The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
-(@code{calc-graph-zero-y}) commands control whether a dotted line is
-drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
-dotted lines that would be drawn there anyway if you used @kbd{g g} to
-turn the ``grid'' feature on.) Zero-axis lines are on by default, and
-may be turned off only in GNUPLOT 3.0 and later versions. They are
-not available for 3D plots.
-
address@hidden g s
address@hidden calc-graph-line-style
-The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
-lines on or off for the most recently added curve, and optionally selects
-the style of lines to be used for that curve. Plain @kbd{g s} simply
-toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
-turns lines on and sets a particular line style. Line style numbers
-start at one and their meanings vary depending on the output device.
-GNUPLOT guarantees that there will be at least six different line styles
-available for any device.
-
address@hidden g S
address@hidden calc-graph-point-style
-The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
-the symbols at the data points on or off, or sets the point style.
-If you turn both lines and points off, the data points will show as
-tiny dots. If the ``y'' values being plotted contain error forms and
-the connecting lines are turned off, then this command will also turn
-the error bars on or off.
-
address@hidden @code{LineStyles} variable
address@hidden @code{PointStyles} variable
address@hidden LineStyles
address@hidden PointStyles
-Another way to specify curve styles is with the @code{LineStyles} and
address@hidden variables. These variables initially have no stored
-values, but if you store a vector of integers in one of these variables,
-the @kbd{g a} and @kbd{g f} commands will use those style numbers
-instead of the defaults for new curves that are added to the graph.
-An entry should be a positive integer for a specific style, or 0 to let
-the style be chosen automatically, or @mathit{-1} to turn off lines or points
-altogether. If there are more curves than elements in the vector, the
-last few curves will continue to have the default styles. Of course,
-you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
-
-For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
-to have lines in style number 2, the second curve to have no connecting
-lines, and the third curve to have lines in style 3. Point styles will
-still be assigned automatically, but you could store another vector in
address@hidden to define them, too.
-
address@hidden Devices, , Graphics Options, Graphics
address@hidden Graphical Devices
-
address@hidden
address@hidden g D
address@hidden calc-graph-device
-The @kbd{g D} (@code{calc-graph-device}) command sets the device name
-(or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
-on this graph. It does not affect the permanent default device name.
-If you enter a blank name, the device name reverts to the default.
-Enter @samp{?} to see a list of supported devices.
-
-With a positive numeric prefix argument, @kbd{g D} instead sets
-the default device name, used by all plots in the future which do
-not override it with a plain @kbd{g D} command. If you enter a
-blank line this command shows you the current default. The special
-name @code{default} signifies that Calc should choose @code{x11} if
-the X window system is in use (as indicated by the presence of a
address@hidden environment variable), or otherwise @code{dumb} under
-GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
-This is the initial default value.
-
-The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
-terminals with no special graphics facilities. It writes a crude
-picture of the graph composed of characters like @code{-} and @code{|}
-to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
-The graph is made the same size as the Emacs screen, which on most
-dumb terminals will be
address@hidden @math{80\times24}
address@hidden 80x24
-characters. The graph is displayed in
-an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
-the recursive edit and return to Calc. Note that the @code{dumb}
-device is present only in GNUPLOT 3.0 and later versions.
-
-The word @code{dumb} may be followed by two numbers separated by
-spaces. These are the desired width and height of the graph in
-characters. Also, the device name @code{big} is like @code{dumb}
-but creates a graph four times the width and height of the Emacs
-screen. You will then have to scroll around to view the entire
-graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
address@hidden<}, and @kbd{>} are defined to scroll by one screenful in each
-of the four directions.
-
-With a negative numeric prefix argument, @kbd{g D} sets or displays
-the device name used by @kbd{g P} (@code{calc-graph-print}). This
-is initially @code{postscript}. If you don't have a PostScript
-printer, you may decide once again to use @code{dumb} to create a
-plot on any text-only printer.
-
address@hidden g O
address@hidden calc-graph-output
-The @kbd{g O} (@code{calc-graph-output}) command sets the name of
-the output file used by GNUPLOT. For some devices, notably @code{x11},
-there is no output file and this information is not used. Many other
-``devices'' are really file formats like @code{postscript}; in these
-cases the output in the desired format goes into the file you name
-with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
-to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
-This is the default setting.
-
-Another special output name is @code{tty}, which means that GNUPLOT
-is going to write graphics commands directly to its standard output,
-which you wish Emacs to pass through to your terminal. Tektronix
-graphics terminals, among other devices, operate this way. Calc does
-this by telling GNUPLOT to write to a temporary file, then running a
-sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
-typical Unix systems, this will copy the temporary file directly to
-the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
-to Emacs afterwards to refresh the screen.
-
-Once again, @kbd{g O} with a positive or negative prefix argument
-sets the default or printer output file names, respectively. In each
-case you can specify @code{auto}, which causes Calc to invent a temporary
-file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
-will be deleted once it has been displayed or printed. If the output file
-name is not @code{auto}, the file is not automatically deleted.
-
-The default and printer devices and output files can be saved
-permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
-default number of data points (see @kbd{g N}) and the X geometry
-(see @kbd{g X}) are also saved. Other graph information is @emph{not}
-saved; you can save a graph's configuration simply by saving the contents
-of the @samp{*Gnuplot Commands*} buffer.
-
address@hidden calc-gnuplot-plot-command
address@hidden calc-gnuplot-default-device
address@hidden calc-gnuplot-default-output
address@hidden calc-gnuplot-print-command
address@hidden calc-gnuplot-print-device
address@hidden calc-gnuplot-print-output
-You may wish to configure the default and
-printer devices and output files for the whole system. The relevant
-Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
-and @code{calc-gnuplot-print-device} and @code{-output}. The output
-file names must be either strings as described above, or Lisp
-expressions which are evaluated on the fly to get the output file names.
-
-Other important Lisp variables are @code{calc-gnuplot-plot-command} and
address@hidden, which give the system commands to
-display or print the output of GNUPLOT, respectively. These may be
address@hidden if no command is necessary, or strings which can include
address@hidden to signify the name of the file to be displayed or printed.
-Or, these variables may contain Lisp expressions which are evaluated
-to display or print the output. These variables are customizable
-(@pxref{Customizing Calc}).
-
address@hidden g x
address@hidden calc-graph-display
-The @kbd{g x} (@code{calc-graph-display}) command lets you specify
-on which X window system display your graphs should be drawn. Enter
-a blank line to see the current display name. This command has no
-effect unless the current device is @code{x11}.
-
address@hidden g X
address@hidden calc-graph-geometry
-The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
-command for specifying the position and size of the X window.
-The normal value is @code{default}, which generally means your
-window manager will let you place the window interactively.
-Entering @samp{800x500+0+0} would create an 800-by-500 pixel
-window in the upper-left corner of the screen.
-
-The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
-session with GNUPLOT. This shows the commands Calc has ``typed'' to
-GNUPLOT and the responses it has received. Calc tries to notice when an
-error message has appeared here and display the buffer for you when
-this happens. You can check this buffer yourself if you suspect
-something has gone wrong.
-
address@hidden g C
address@hidden calc-graph-command
-The @kbd{g C} (@code{calc-graph-command}) command prompts you to
-enter any line of text, then simply sends that line to the current
-GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
-like a Shell buffer but you can't type commands in it yourself.
-Instead, you must use @kbd{g C} for this purpose.
-
address@hidden g v
address@hidden g V
address@hidden calc-graph-view-commands
address@hidden calc-graph-view-trail
-The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
-(@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
-and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
-This happens automatically when Calc thinks there is something you
-will want to see in either of these buffers. If you type @kbd{g v}
-or @kbd{g V} when the relevant buffer is already displayed, the
-buffer is hidden again.
-
-One reason to use @kbd{g v} is to add your own commands to the
address@hidden Commands*} buffer. Press @kbd{g v}, then use
address@hidden o} to switch into that window. For example, GNUPLOT has
address@hidden label} and @samp{set arrow} commands that allow you to
-annotate your plots. Since Calc doesn't understand these commands,
-you have to add them to the @samp{*Gnuplot Commands*} buffer
-yourself, then use @address@hidden p}} to replot using these new commands.
Note
-that your commands must appear @emph{before} the @code{plot} command.
-To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
-You may have to type @kbd{g C @key{RET}} a few times to clear the
-``press return for more'' or ``subtopic of @dots{}'' requests.
-Note that Calc always sends commands (like @samp{set nolabel}) to
-reset all plotting parameters to the defaults before each plot, so
-to delete a label all you need to do is delete the @samp{set label}
-line you added (or comment it out with @samp{#}) and then replot
-with @kbd{g p}.
-
address@hidden g q
address@hidden calc-graph-quit
-You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
-process that is running. The next graphing command you give will
-start a fresh GNUPLOT process. The word @samp{Graph} appears in
-the Calc window's mode line whenever a GNUPLOT process is currently
-running. The GNUPLOT process is automatically killed when you
-exit Emacs if you haven't killed it manually by then.
-
address@hidden g K
address@hidden calc-graph-kill
-The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
-except that it also views the @samp{*Gnuplot Trail*} buffer so that
-you can see the process being killed. This is better if you are
-killing GNUPLOT because you think it has gotten stuck.
-
address@hidden Kill and Yank, Keypad Mode, Graphics, Top
address@hidden Kill and Yank Functions
-
address@hidden
-The commands in this chapter move information between the Calculator and
-other Emacs editing buffers.
-
-In many cases Embedded mode is an easier and more natural way to
-work with Calc from a regular editing buffer. @xref{Embedded Mode}.
-
address@hidden
-* Killing From Stack::
-* Yanking Into Stack::
-* Grabbing From Buffers::
-* Yanking Into Buffers::
-* X Cut and Paste::
address@hidden menu
-
address@hidden Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and
Yank
address@hidden Killing from the Stack
-
address@hidden
address@hidden C-k
address@hidden calc-kill
address@hidden M-k
address@hidden calc-copy-as-kill
address@hidden C-w
address@hidden calc-kill-region
address@hidden M-w
address@hidden calc-copy-region-as-kill
address@hidden Kill ring
address@hidden commands are Emacs commands that insert text into the
-``kill ring,'' from which it can later be ``yanked'' by a @kbd{C-y}
-command. Three common kill commands in normal Emacs are @kbd{C-k}, which
-kills one line, @kbd{C-w}, which kills the region between mark and point,
-and @kbd{M-w}, which puts the region into the kill ring without actually
-deleting it. All of these commands work in the Calculator, too. Also,
address@hidden has been provided to complete the set; it puts the current line
-into the kill ring without deleting anything.
-
-The kill commands are unusual in that they pay attention to the location
-of the cursor in the Calculator buffer. If the cursor is on or below the
-bottom line, the kill commands operate on the top of the stack. Otherwise,
-they operate on whatever stack element the cursor is on. Calc's kill
-commands always operate on whole stack entries. (They act the same as their
-standard Emacs cousins except they ``round up'' the specified region to
-encompass full lines.) The text is copied into the kill ring exactly as
-it appears on the screen, including line numbers if they are enabled.
-
-A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
-of lines killed. A positive argument kills the current line and @expr{n-1}
-lines below it. A negative argument kills the @expr{-n} lines above the
-current line. Again this mirrors the behavior of the standard Emacs
address@hidden command. Although a whole line is always deleted, @kbd{C-k}
-with no argument copies only the number itself into the kill ring, whereas
address@hidden with a prefix argument of 1 copies the number with its trailing
-newline.
-
address@hidden Yanking Into Stack, Grabbing From Buffers, Killing From Stack,
Kill and Yank
address@hidden Yanking into the Stack
-
address@hidden
address@hidden C-y
address@hidden calc-yank
-The @kbd{C-y} command yanks the most recently killed text back into the
-Calculator. It pushes this value onto the top of the stack regardless of
-the cursor position. In general it re-parses the killed text as a number
-or formula (or a list of these separated by commas or newlines). However if
-the thing being yanked is something that was just killed from the Calculator
-itself, its full internal structure is yanked. For example, if you have
-set the floating-point display mode to show only four significant digits,
-then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
-full 3.14159, even though yanking it into any other buffer would yank the
-number in its displayed form, 3.142. (Since the default display modes
-show all objects to their full precision, this feature normally makes no
-difference.)
-
address@hidden Grabbing From Buffers, Yanking Into Buffers, Yanking Into Stack,
Kill and Yank
address@hidden Grabbing from Other Buffers
-
address@hidden
address@hidden C-x * g
address@hidden calc-grab-region
-The @kbd{C-x * g} (@code{calc-grab-region}) command takes the text between
-point and mark in the current buffer and attempts to parse it as a
-vector of values. Basically, it wraps the text in vector brackets
address@hidden ]} unless the text already is enclosed in vector brackets,
-then reads the text as if it were an algebraic entry. The contents
-of the vector may be numbers, formulas, or any other Calc objects.
-If the @kbd{C-x * g} command works successfully, it does an automatic
address@hidden * c} to enter the Calculator buffer.
-
-A numeric prefix argument grabs the specified number of lines around
-point, ignoring the mark. A positive prefix grabs from point to the
address@hidden following newline (so that @kbd{M-1 C-x * g} grabs from point
-to the end of the current line); a negative prefix grabs from point
-back to the @expr{n+1}st preceding newline. In these cases the text
-that is grabbed is exactly the same as the text that @kbd{C-k} would
-delete given that prefix argument.
-
-A prefix of zero grabs the current line; point may be anywhere on the
-line.
-
-A plain @kbd{C-u} prefix interprets the region between point and mark
-as a single number or formula rather than a vector. For example,
address@hidden * g} on the text @samp{2 a b} produces the vector of three
-values @samp{[2, a, b]}, but @kbd{C-u C-x * g} on the same region
-reads a formula which is a product of three things: @samp{2 a b}.
-(The text @samp{a + b}, on the other hand, will be grabbed as a
-vector of one element by plain @kbd{C-x * g} because the interpretation
address@hidden, +, b]} would be a syntax error.)
-
-If a different language has been specified (@pxref{Language Modes}),
-the grabbed text will be interpreted according to that language.
-
address@hidden C-x * r
address@hidden calc-grab-rectangle
-The @kbd{C-x * r} (@code{calc-grab-rectangle}) command takes the text between
-point and mark and attempts to parse it as a matrix. If point and mark
-are both in the leftmost column, the lines in between are parsed in their
-entirety. Otherwise, point and mark define the corners of a rectangle
-whose contents are parsed.
-
-Each line of the grabbed area becomes a row of the matrix. The result
-will actually be a vector of vectors, which Calc will treat as a matrix
-only if every row contains the same number of values.
-
-If a line contains a portion surrounded by square brackets (or curly
-braces), that portion is interpreted as a vector which becomes a row
-of the matrix. Any text surrounding the bracketed portion on the line
-is ignored.
-
-Otherwise, the entire line is interpreted as a row vector as if it
-were surrounded by square brackets. Leading line numbers (in the
-format used in the Calc stack buffer) are ignored. If you wish to
-force this interpretation (even if the line contains bracketed
-portions), give a negative numeric prefix argument to the
address@hidden * r} command.
-
-If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
-line is instead interpreted as a single formula which is converted into
-a one-element vector. Thus the result of @kbd{C-u C-x * r} will be a
-one-column matrix. For example, suppose one line of the data is the
-expression @samp{2 a}. A plain @address@hidden * r}} will interpret this as
address@hidden a]}, which in turn is read as a two-element vector that forms
-one row of the matrix. But a @kbd{C-u C-x * r} will interpret this row
-as @samp{[2*a]}.
-
-If you give a positive numeric prefix argument @var{n}, then each line
-will be split up into columns of width @var{n}; each column is parsed
-separately as a matrix element. If a line contained
address@hidden@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
-would correctly split the line into two error forms.
-
address@hidden Functions}, to see how to pull the matrix apart into its
-constituent rows and columns. (If it is a
address@hidden @math{1\times1}
address@hidden 1x1
-matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
-
address@hidden C-x * :
address@hidden C-x * _
address@hidden calc-grab-sum-across
address@hidden calc-grab-sum-down
address@hidden Summing rows and columns of data
-The @kbd{C-x * :} (@code{calc-grab-sum-down}) command is a handy way to
-grab a rectangle of data and sum its columns. It is equivalent to
-typing @kbd{C-x * r}, followed by @kbd{V R : +} (the vector reduction
-command that sums the columns of a matrix; @pxref{Reducing}). The
-result of the command will be a vector of numbers, one for each column
-in the input data. The @kbd{C-x * _} (@code{calc-grab-sum-across}) command
-similarly grabs a rectangle and sums its rows by executing @address@hidden R _
+}}.
-
-As well as being more convenient, @kbd{C-x * :} and @kbd{C-x * _} are also
-much faster because they don't actually place the grabbed vector on
-the stack. In a @kbd{C-x * r V R : +} sequence, formatting the vector
-for display on the stack takes a large fraction of the total time
-(unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
-
-For example, suppose we have a column of numbers in a file which we
-wish to sum. Go to one corner of the column and press @kbd{C-@@} to
-set the mark; go to the other corner and type @kbd{C-x * :}. Since there
-is only one column, the result will be a vector of one number, the sum.
-(You can type @kbd{v u} to unpack this vector into a plain number if
-you want to do further arithmetic with it.)
-
-To compute the product of the column of numbers, we would have to do
-it ``by hand'' since there's no special grab-and-multiply command.
-Use @kbd{C-x * r} to grab the column of numbers into the calculator in
-the form of a column matrix. The statistics command @kbd{u *} is a
-handy way to find the product of a vector or matrix of numbers.
address@hidden Operations}. Another approach would be to use
-an explicit column reduction command, @kbd{V R : *}.
-
address@hidden Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers,
Kill and Yank
address@hidden Yanking into Other Buffers
-
address@hidden
address@hidden y
address@hidden calc-copy-to-buffer
-The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
-at the top of the stack into the most recently used normal editing buffer.
-(More specifically, this is the most recently used buffer which is displayed
-in a window and whose name does not begin with @samp{*}. If there is no
-such buffer, this is the most recently used buffer except for Calculator
-and Calc Trail buffers.) The number is inserted exactly as it appears and
-without a newline. (If line-numbering is enabled, the line number is
-normally not included.) The number is @emph{not} removed from the stack.
-
-With a prefix argument, @kbd{y} inserts several numbers, one per line.
-A positive argument inserts the specified number of values from the top
-of the stack. A negative argument inserts the @expr{n}th value from the
-top of the stack. An argument of zero inserts the entire stack. Note
-that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
-with no argument; the former always copies full lines, whereas the
-latter strips off the trailing newline.
-
-With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
-region in the other buffer with the yanked text, then quits the
-Calculator, leaving you in that buffer. A typical use would be to use
address@hidden * g} to read a region of data into the Calculator, operate on the
-data to produce a new matrix, then type @kbd{C-u y} to replace the
-original data with the new data. One might wish to alter the matrix
-display style (@pxref{Vector and Matrix Formats}) or change the current
-display language (@pxref{Language Modes}) before doing this. Also, note
-that this command replaces a linear region of text (as grabbed by
address@hidden * g}), not a rectangle (as grabbed by @kbd{C-x * r}).
-
-If the editing buffer is in overwrite (as opposed to insert) mode,
-and the @kbd{C-u} prefix was not used, then the yanked number will
-overwrite the characters following point rather than being inserted
-before those characters. The usual conventions of overwrite mode
-are observed; for example, characters will be inserted at the end of
-a line rather than overflowing onto the next line. Yanking a multi-line
-object such as a matrix in overwrite mode overwrites the next @var{n}
-lines in the buffer, lengthening or shortening each line as necessary.
-Finally, if the thing being yanked is a simple integer or floating-point
-number (like @samp{-1.2345e-3}) and the characters following point also
-make up such a number, then Calc will replace that number with the new
-number, lengthening or shortening as necessary. The concept of
-``overwrite mode'' has thus been generalized from overwriting characters
-to overwriting one complete number with another.
-
address@hidden C-x * y
-The @kbd{C-x * y} key sequence is equivalent to @kbd{y} except that
-it can be typed anywhere, not just in Calc. This provides an easy
-way to guarantee that Calc knows which editing buffer you want to use!
-
address@hidden X Cut and Paste, , Yanking Into Buffers, Kill and Yank
address@hidden X Cut and Paste
-
address@hidden
-If you are using Emacs with the X window system, there is an easier
-way to move small amounts of data into and out of the calculator:
-Use the mouse-oriented cut and paste facilities of X.
-
-The default bindings for a three-button mouse cause the left button
-to move the Emacs cursor to the given place, the right button to
-select the text between the cursor and the clicked location, and
-the middle button to yank the selection into the buffer at the
-clicked location. So, if you have a Calc window and an editing
-window on your Emacs screen, you can use left-click/right-click
-to select a number, vector, or formula from one window, then
-middle-click to paste that value into the other window. When you
-paste text into the Calc window, Calc interprets it as an algebraic
-entry. It doesn't matter where you click in the Calc window; the
-new value is always pushed onto the top of the stack.
-
-The @code{xterm} program that is typically used for general-purpose
-shell windows in X interprets the mouse buttons in the same way.
-So you can use the mouse to move data between Calc and any other
-Unix program. One nice feature of @code{xterm} is that a double
-left-click selects one word, and a triple left-click selects a
-whole line. So you can usually transfer a single number into Calc
-just by double-clicking on it in the shell, then middle-clicking
-in the Calc window.
-
address@hidden Keypad Mode, Embedded Mode, Kill and Yank, Top
address@hidden Keypad Mode
-
address@hidden
address@hidden C-x * k
address@hidden calc-keypad
-The @kbd{C-x * k} (@code{calc-keypad}) command starts the Calculator
-and displays a picture of a calculator-style keypad. If you are using
-the X window system, you can click on any of the ``keys'' in the
-keypad using the left mouse button to operate the calculator.
-The original window remains the selected window; in Keypad mode
-you can type in your file while simultaneously performing
-calculations with the mouse.
-
address@hidden full-calc-keypad
-If you have used @kbd{C-x * b} first, @kbd{C-x * k} instead invokes
-the @code{full-calc-keypad} command, which takes over the whole
-Emacs screen and displays the keypad, the Calc stack, and the Calc
-trail all at once. This mode would normally be used when running
-Calc standalone (@pxref{Standalone Operation}).
-
-If you aren't using the X window system, you must switch into
-the @samp{*Calc Keypad*} window, place the cursor on the desired
-``key,'' and type @key{SPC} or @key{RET}. If you think this
-is easier than using Calc normally, go right ahead.
-
-Calc commands are more or less the same in Keypad mode. Certain
-keypad keys differ slightly from the corresponding normal Calc
-keystrokes; all such deviations are described below.
-
-Keypad mode includes many more commands than will fit on the keypad
-at once. Click the right mouse button address@hidden
-to switch to the next menu. The bottom five rows of the keypad
-stay the same; the top three rows change to a new set of commands.
-To return to earlier menus, click the middle mouse button
address@hidden or simply advance through the menus
-until you wrap around. Typing @key{TAB} inside the keypad window
-is equivalent to clicking the right mouse button there.
-
-You can always click the @key{EXEC} button and type any normal
-Calc key sequence. This is equivalent to switching into the
-Calc buffer, typing the keys, then switching back to your
-original buffer.
-
address@hidden
-* Keypad Main Menu::
-* Keypad Functions Menu::
-* Keypad Binary Menu::
-* Keypad Vectors Menu::
-* Keypad Modes Menu::
address@hidden menu
-
address@hidden Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
address@hidden Main Menu
-
address@hidden
address@hidden
-|----+-----Calc 2.1------+----1
-|FLR |CEIL|RND |TRNC|CLN2|FLT |
-|----+----+----+----+----+----|
-| LN |EXP | |ABS |IDIV|MOD |
-|----+----+----+----+----+----|
-|SIN |COS |TAN |SQRT|y^x |1/x |
-|----+----+----+----+----+----|
-| ENTER |+/- |EEX |UNDO| <- |
-|-----+---+-+--+--+-+---++----|
-| INV | 7 | 8 | 9 | / |
-|-----+-----+-----+-----+-----|
-| HYP | 4 | 5 | 6 | * |
-|-----+-----+-----+-----+-----|
-|EXEC | 1 | 2 | 3 | - |
-|-----+-----+-----+-----+-----|
-| OFF | 0 | . | PI | + |
-|-----+-----+-----+-----+-----+
address@hidden group
address@hidden smallexample
-
address@hidden
-This is the menu that appears the first time you start Keypad mode.
-It will show up in a vertical window on the right side of your screen.
-Above this menu is the traditional Calc stack display. On a 24-line
-screen you will be able to see the top three stack entries.
-
-The ten digit keys, decimal point, and @key{EEX} key are used for
-entering numbers in the obvious way. @key{EEX} begins entry of an
-exponent in scientific notation. Just as with regular Calc, the
-number is pushed onto the stack as soon as you press @key{ENTER}
-or any other function key.
-
-The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
-numeric entry it changes the sign of the number or of the exponent.
-At other times it changes the sign of the number on the top of the
-stack.
-
-The @key{INV} and @key{HYP} keys modify other keys. As well as
-having the effects described elsewhere in this manual, Keypad mode
-defines several other ``inverse'' operations. These are described
-below and in the following sections.
-
-The @key{ENTER} key finishes the current numeric entry, or otherwise
-duplicates the top entry on the stack.
-
-The @key{UNDO} key undoes the most recent Calc operation.
address@hidden UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
-``last arguments'' (@address@hidden).
-
-The @key{<-} key acts as a ``backspace'' during numeric entry.
-At other times it removes the top stack entry. @kbd{INV <-}
-clears the entire stack. @kbd{HYP <-} takes an integer from
-the stack, then removes that many additional stack elements.
-
-The @key{EXEC} key prompts you to enter any keystroke sequence
-that would normally work in Calc mode. This can include a
-numeric prefix if you wish. It is also possible simply to
-switch into the Calc window and type commands in it; there is
-nothing ``magic'' about this window when Keypad mode is active.
-
-The other keys in this display perform their obvious calculator
-functions. @key{CLN2} rounds the top-of-stack by temporarily
-reducing the precision by 2 digits. @key{FLT} converts an
-integer or fraction on the top of the stack to floating-point.
-
-The @key{INV} and @key{HYP} keys combined with several of these keys
-give you access to some common functions even if the appropriate menu
-is not displayed. Obviously you don't need to learn these keys
-unless you find yourself wasting time switching among the menus.
-
address@hidden @kbd
address@hidden INV +/-
-is the same as @key{1/x}.
address@hidden INV +
-is the same as @key{SQRT}.
address@hidden INV -
-is the same as @key{CONJ}.
address@hidden INV *
-is the same as @key{y^x}.
address@hidden INV /
-is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
address@hidden HYP/INV 1
-are the same as @key{SIN} / @kbd{INV SIN}.
address@hidden HYP/INV 2
-are the same as @key{COS} / @kbd{INV COS}.
address@hidden HYP/INV 3
-are the same as @key{TAN} / @kbd{INV TAN}.
address@hidden INV/HYP 4
-are the same as @key{LN} / @kbd{HYP LN}.
address@hidden INV/HYP 5
-are the same as @key{EXP} / @kbd{HYP EXP}.
address@hidden INV 6
-is the same as @key{ABS}.
address@hidden INV 7
-is the same as @key{RND} (@code{calc-round}).
address@hidden INV 8
-is the same as @key{CLN2}.
address@hidden INV 9
-is the same as @key{FLT} (@code{calc-float}).
address@hidden INV 0
-is the same as @key{IMAG}.
address@hidden INV .
-is the same as @key{PREC}.
address@hidden INV ENTER
-is the same as @key{SWAP}.
address@hidden HYP ENTER
-is the same as @key{RLL3}.
address@hidden INV HYP ENTER
-is the same as @key{OVER}.
address@hidden HYP +/-
-packs the top two stack entries as an error form.
address@hidden HYP EEX
-packs the top two stack entries as a modulo form.
address@hidden INV EEX
-creates an interval form; this removes an integer which is one
-of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
-by the two limits of the interval.
address@hidden table
-
-The @kbd{OFF} key turns Calc off; typing @kbd{C-x * k} or @kbd{C-x * *}
-again has the same effect. This is analogous to typing @kbd{q} or
-hitting @kbd{C-x * c} again in the normal calculator. If Calc is
-running standalone (the @code{full-calc-keypad} command appeared in the
-command line that started Emacs), then @kbd{OFF} is replaced with
address@hidden; clicking on this actually exits Emacs itself.
-
address@hidden Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu,
Keypad Mode
address@hidden Functions Menu
-
address@hidden
address@hidden
-|----+----+----+----+----+----2
-|IGAM|BETA|IBET|ERF |BESJ|BESY|
-|----+----+----+----+----+----|
-|IMAG|CONJ| RE |ATN2|RAND|RAGN|
-|----+----+----+----+----+----|
-|GCD |FACT|DFCT|BNOM|PERM|NXTP|
-|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
-
address@hidden
-This menu provides various operations from the @kbd{f} and @kbd{k}
-prefix keys.
-
address@hidden multiplies the number on the stack by the imaginary
-number @expr{i = (0, 1)}.
-
address@hidden extracts the real part a complex number. @kbd{INV RE}
-extracts the imaginary part.
-
address@hidden takes a number from the top of the stack and computes
-a random number greater than or equal to zero but less than that
-number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
-again'' command; it computes another random number using the
-same limit as last time.
-
address@hidden GCD} computes the LCM (least common multiple) function.
-
address@hidden FACT} is the gamma function.
address@hidden @math{\Gamma(x) = (x-1)!}.
address@hidden @expr{gamma(x) = (x-1)!}.
-
address@hidden is the number-of-permutations function, which is on the
address@hidden k c} key in normal Calc.
-
address@hidden finds the next prime after a number. @kbd{INV NXTP}
-finds the previous prime.
-
address@hidden Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu,
Keypad Mode
address@hidden Binary Menu
-
address@hidden
address@hidden
-|----+----+----+----+----+----3
-|AND | OR |XOR |NOT |LSH |RSH |
-|----+----+----+----+----+----|
-|DEC |HEX |OCT |BIN |WSIZ|ARSH|
-|----+----+----+----+----+----|
-| A | B | C | D | E | F |
-|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
-
address@hidden
-The keys in this menu perform operations on binary integers.
-Note that both logical and arithmetic right-shifts are provided.
address@hidden LSH} rotates one bit to the left.
-
-The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
-The ``clip'' function (normally on @address@hidden c}}) is on @key{INV NOT}.
-
-The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
-current radix for display and entry of numbers: Decimal, hexadecimal,
-octal, or binary. The six letter keys @key{A} through @key{F} are used
-for entering hexadecimal numbers.
-
-The @key{WSIZ} key displays the current word size for binary operations
-and allows you to enter a new word size. You can respond to the prompt
-using either the keyboard or the digits and @key{ENTER} from the keypad.
-The initial word size is 32 bits.
-
address@hidden Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu,
Keypad Mode
address@hidden Vectors Menu
-
address@hidden
address@hidden
-|----+----+----+----+----+----4
-|SUM |PROD|MAX |MAP*|MAP^|MAP$|
-|----+----+----+----+----+----|
-|MINV|MDET|MTRN|IDNT|CRSS|"x" |
-|----+----+----+----+----+----|
-|PACK|UNPK|INDX|BLD |LEN |... |
-|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
-
address@hidden
-The keys in this menu operate on vectors and matrices.
-
address@hidden removes an integer @var{n} from the top of the stack;
-the next @var{n} stack elements are removed and packed into a vector,
-which is replaced onto the stack. Thus the sequence
address@hidden ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
address@hidden, 3, 5]} onto the stack. To enter a matrix, build each row
-on the stack as a vector, then use a final @key{PACK} to collect the
-rows into a matrix.
-
address@hidden unpacks the vector on the stack, pushing each of its
-components separately.
-
address@hidden removes an integer @var{n}, then builds a vector of
-integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
-from the stack: The vector size @var{n}, the starting number,
-and the increment. @kbd{BLD} takes an integer @var{n} and any
-value @var{x} and builds a vector of @var{n} copies of @var{x}.
-
address@hidden removes an integer @var{n}, then builds an @address@hidden
-identity matrix.
-
address@hidden replaces a vector by its length, an integer.
-
address@hidden turns on or off ``abbreviated'' display mode for large vectors.
-
address@hidden, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
-inverse, determinant, and transpose, and vector cross product.
-
address@hidden replaces a vector by the sum of its elements. It is
-equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
address@hidden computes the product of the elements of a vector, and
address@hidden computes the maximum of all the elements of a vector.
-
address@hidden SUM} computes the alternating sum of the first element
-minus the second, plus the third, minus the fourth, and so on.
address@hidden MAX} computes the minimum of the vector elements.
-
address@hidden SUM} computes the mean of the vector elements.
address@hidden PROD} computes the sample standard deviation.
address@hidden MAX} computes the median.
-
address@hidden multiplies two vectors elementwise. It is equivalent
-to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
-The arguments must be vectors of equal length, or one must be a vector
-and the other must be a plain number. For example, @kbd{2 MAP^} squares
-all the elements of a vector.
-
address@hidden maps the formula on the top of the stack across the
-vector in the second-to-top position. If the formula contains
-several variables, Calc takes that many vectors starting at the
-second-to-top position and matches them to the variables in
-alphabetical order. The result is a vector of the same size as
-the input vectors, whose elements are the formula evaluated with
-the variables set to the various sets of numbers in those vectors.
-For example, you could simulate @key{MAP^} using @key{MAP$} with
-the formula @samp{x^y}.
-
-The @kbd{"x"} key pushes the variable name @expr{x} onto the
-stack. To build the formula @expr{x^2 + 6}, you would use the
-key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
-suitable for use with the @key{MAP$} key described above.
-With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
address@hidden"x"} key pushes the variable names @expr{y}, @expr{z}, and
address@hidden, respectively.
-
address@hidden Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
address@hidden Modes Menu
-
address@hidden
address@hidden
-|----+----+----+----+----+----5
-|FLT |FIX |SCI |ENG |GRP | |
-|----+----+----+----+----+----|
-|RAD |DEG |FRAC|POLR|SYMB|PREC|
-|----+----+----+----+----+----|
-|SWAP|RLL3|RLL4|OVER|STO |RCL |
-|----+----+----+----+----+----|
address@hidden group
address@hidden smallexample
-
address@hidden
-The keys in this menu manipulate modes, variables, and the stack.
-
-The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
-floating-point, fixed-point, scientific, or engineering notation.
address@hidden displays two digits after the decimal by default; the
-others display full precision. With the @key{INV} prefix, these
-keys pop a number-of-digits argument from the stack.
-
-The @key{GRP} key turns grouping of digits with commas on or off.
address@hidden GRP} enables grouping to the right of the decimal point as
-well as to the left.
-
-The @key{RAD} and @key{DEG} keys switch between radians and degrees
-for trigonometric functions.
-
-The @key{FRAC} key turns Fraction mode on or off. This affects
-whether commands like @kbd{/} with integer arguments produce
-fractional or floating-point results.
-
-The @key{POLR} key turns Polar mode on or off, determining whether
-polar or rectangular complex numbers are used by default.
-
-The @key{SYMB} key turns Symbolic mode on or off, in which
-operations that would produce inexact floating-point results
-are left unevaluated as algebraic formulas.
-
-The @key{PREC} key selects the current precision. Answer with
-the keyboard or with the keypad digit and @key{ENTER} keys.
-
-The @key{SWAP} key exchanges the top two stack elements.
-The @key{RLL3} key rotates the top three stack elements upwards.
-The @key{RLL4} key rotates the top four stack elements upwards.
-The @key{OVER} key duplicates the second-to-top stack element.
-
-The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
address@hidden r} in regular Calc. @xref{Store and Recall}. Click the
address@hidden or @key{RCL} key, then one of the ten digits. (Named
-variables are not available in Keypad mode.) You can also use,
-for example, @kbd{STO + 3} to add to register 3.
-
address@hidden Embedded Mode, Programming, Keypad Mode, Top
address@hidden Embedded Mode
-
address@hidden
-Embedded mode in Calc provides an alternative to copying numbers
-and formulas back and forth between editing buffers and the Calc
-stack. In Embedded mode, your editing buffer becomes temporarily
-linked to the stack and this copying is taken care of automatically.
-
address@hidden
-* Basic Embedded Mode::
-* More About Embedded Mode::
-* Assignments in Embedded Mode::
-* Mode Settings in Embedded Mode::
-* Customizing Embedded Mode::
address@hidden menu
-
address@hidden Basic Embedded Mode, More About Embedded Mode, Embedded Mode,
Embedded Mode
address@hidden Basic Embedded Mode
-
address@hidden
address@hidden C-x * e
address@hidden calc-embedded
-To enter Embedded mode, position the Emacs point (cursor) on a
-formula in any buffer and press @kbd{C-x * e} (@code{calc-embedded}).
-Note that @kbd{C-x * e} is not to be used in the Calc stack buffer
-like most Calc commands, but rather in regular editing buffers that
-are visiting your own files.
-
-Calc will try to guess an appropriate language based on the major mode
-of the editing buffer. (@xref{Language Modes}.) If the current buffer is
-in @code{latex-mode}, for example, Calc will set its language to address@hidden
-Similarly, Calc will use @TeX{} language for @code{tex-mode},
address@hidden and @code{context-mode}, C language for
address@hidden and @code{c++-mode}, FORTRAN language for
address@hidden and @code{f90-mode}, Pascal for @code{pascal-mode},
-and eqn for @code{nroff-mode} (@pxref{Customizing Calc}).
-These can be overridden with Calc's mode
-changing commands (@pxref{Mode Settings in Embedded Mode}). If no
-suitable language is available, Calc will continue with its current language.
-
-Calc normally scans backward and forward in the buffer for the
-nearest opening and closing @dfn{formula delimiters}. The simplest
-delimiters are blank lines. Other delimiters that Embedded mode
-understands are:
-
address@hidden
address@hidden
-The @TeX{} and address@hidden math delimiters @samp{$ $}, @samp{$$ $$},
address@hidden \]}, and @samp{\( \)};
address@hidden
-Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
address@hidden
-Lines beginning with @samp{@@} (Texinfo delimiters).
address@hidden
-Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
address@hidden
-Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
address@hidden enumerate
-
address@hidden Embedded Mode}, to see how to make Calc recognize
-your own favorite delimiters. Delimiters like @samp{$ $} can appear
-on their own separate lines or in-line with the formula.
-
-If you give a positive or negative numeric prefix argument, Calc
-instead uses the current point as one end of the formula, and includes
-that many lines forward or backward (respectively, including the current
-line). Explicit delimiters are not necessary in this case.
-
-With a prefix argument of zero, Calc uses the current region (delimited
-by point and mark) instead of formula delimiters. With a prefix
-argument of @kbd{C-u} only, Calc uses the current line as the formula.
-
address@hidden C-x * w
address@hidden calc-embedded-word
-The @kbd{C-x * w} (@code{calc-embedded-word}) command will start Embedded
-mode on the current ``word''; in this case Calc will scan for the first
-non-numeric character (i.e., the first character that is not a digit,
-sign, decimal point, or upper- or lower-case @samp{e}) forward and
-backward to delimit the formula.
-
-When you enable Embedded mode for a formula, Calc reads the text
-between the delimiters and tries to interpret it as a Calc formula.
-Calc can generally identify @TeX{} formulas and
-Big-style formulas even if the language mode is wrong. If Calc
-can't make sense of the formula, it beeps and refuses to enter
-Embedded mode. But if the current language is wrong, Calc can
-sometimes parse the formula successfully (but incorrectly);
-for example, the C expression @samp{atan(a[1])} can be parsed
-in Normal language mode, but the @code{atan} won't correspond to
-the built-in @code{arctan} function, and the @samp{a[1]} will be
-interpreted as @samp{a} times the vector @samp{[1]}!
-
-If you press @kbd{C-x * e} or @kbd{C-x * w} to activate an embedded
-formula which is blank, say with the cursor on the space between
-the two delimiters @samp{$ $}, Calc will immediately prompt for
-an algebraic entry.
-
-Only one formula in one buffer can be enabled at a time. If you
-move to another area of the current buffer and give Calc commands,
-Calc turns Embedded mode off for the old formula and then tries
-to restart Embedded mode at the new position. Other buffers are
-not affected by Embedded mode.
-
-When Embedded mode begins, Calc pushes the current formula onto
-the stack. No Calc stack window is created; however, Calc copies
-the top-of-stack position into the original buffer at all times.
-You can create a Calc window by hand with @kbd{C-x * o} if you
-find you need to see the entire stack.
-
-For example, typing @kbd{C-x * e} while somewhere in the formula
address@hidden>2} in the following line enables Embedded mode on that
-inequality:
-
address@hidden
-We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
address@hidden example
-
address@hidden
-The formula @expr{n>2} will be pushed onto the Calc stack, and
-the top of stack will be copied back into the editing buffer.
-This means that spaces will appear around the @samp{>} symbol
-to match Calc's usual display style:
-
address@hidden
-We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
address@hidden example
-
address@hidden
-No spaces have appeared around the @samp{+} sign because it's
-in a different formula, one which we have not yet touched with
-Embedded mode.
-
-Now that Embedded mode is enabled, keys you type in this buffer
-are interpreted as Calc commands. At this point we might use
-the ``commute'' command @kbd{j C} to reverse the inequality.
-This is a selection-based command for which we first need to
-move the cursor onto the operator (@samp{>} in this case) that
-needs to be commuted.
-
address@hidden
-We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
address@hidden example
-
-The @kbd{C-x * o} command is a useful way to open a Calc window
-without actually selecting that window. Giving this command
-verifies that @samp{2 < n} is also on the Calc stack. Typing
address@hidden @key{RET}} would produce:
-
address@hidden
-We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
address@hidden example
-
address@hidden
-with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
-at this point will exchange the two stack values and restore
address@hidden < n} to the embedded formula. Even though you can't
-normally see the stack in Embedded mode, it is still there and
-it still operates in the same way. But, as with old-fashioned
-RPN calculators, you can only see the value at the top of the
-stack at any given time (unless you use @kbd{C-x * o}).
-
-Typing @kbd{C-x * e} again turns Embedded mode off. The Calc
-window reveals that the formula @address@hidden < n}} is automatically
-removed from the stack, but the @samp{17} is not. Entering
-Embedded mode always pushes one thing onto the stack, and
-leaving Embedded mode always removes one thing. Anything else
-that happens on the stack is entirely your business as far as
-Embedded mode is concerned.
-
-If you press @kbd{C-x * e} in the wrong place by accident, it is
-possible that Calc will be able to parse the nearby text as a
-formula and will mangle that text in an attempt to redisplay it
-``properly'' in the current language mode. If this happens,
-press @kbd{C-x * e} again to exit Embedded mode, then give the
-regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
-the text back the way it was before Calc edited it. Note that Calc's
-own Undo command (typed before you turn Embedded mode back off)
-will not do you any good, because as far as Calc is concerned
-you haven't done anything with this formula yet.
-
address@hidden More About Embedded Mode, Assignments in Embedded Mode, Basic
Embedded Mode, Embedded Mode
address@hidden More About Embedded Mode
-
address@hidden
-When Embedded mode ``activates'' a formula, i.e., when it examines
-the formula for the first time since the buffer was created or
-loaded, Calc tries to sense the language in which the formula was
-written. If the formula contains any address@hidden @samp{\} sequences,
-it is parsed (i.e., read) in address@hidden mode. If the formula appears to
-be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
-it is parsed according to the current language mode.
-
-Note that Calc does not change the current language mode according
-the formula it reads in. Even though it can read a address@hidden formula when
-not in address@hidden mode, it will immediately rewrite this formula using
-whatever language mode is in effect.
-
address@hidden
-\bigskip
address@hidden tex
-
address@hidden d p
address@hidden calc-show-plain
-Calc's parser is unable to read certain kinds of formulas. For
-example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
-specify matrix display styles which the parser is unable to
-recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
-command turns on a mode in which a ``plain'' version of a
-formula is placed in front of the fully-formatted version.
-When Calc reads a formula that has such a plain version in
-front, it reads the plain version and ignores the formatted
-version.
-
-Plain formulas are preceded and followed by @samp{%%%} signs
-by default. This notation has the advantage that the @samp{%}
-character begins a comment in @TeX{} and address@hidden, so if your formula is
-embedded in a @TeX{} or address@hidden document its plain version will be
-invisible in the final printed copy. Certain major modes have different
-delimiters to ensure that the ``plain'' version will be
-in a comment for those modes, also.
-See @ref{Customizing Embedded Mode} to see how to change the ``plain''
-formula delimiters.
-
-There are several notations which Calc's parser for ``big''
-formatted formulas can't yet recognize. In particular, it can't
-read the large symbols for @code{sum}, @code{prod}, and @code{integ},
-and it can't handle @samp{=>} with the righthand argument omitted.
-Also, Calc won't recognize special formats you have defined with
-the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
-these cases it is important to use ``plain'' mode to make sure
-Calc will be able to read your formula later.
-
-Another example where ``plain'' mode is important is if you have
-specified a float mode with few digits of precision. Normally
-any digits that are computed but not displayed will simply be
-lost when you save and re-load your embedded buffer, but ``plain''
-mode allows you to make sure that the complete number is present
-in the file as well as the rounded-down number.
-
address@hidden
-\bigskip
address@hidden tex
-
-Embedded buffers remember active formulas for as long as they
-exist in Emacs memory. Suppose you have an embedded formula
-which is @cpi{} to the normal 12 decimal places, and then
-type @address@hidden 5 d n}} to display only five decimal places.
-If you then type @kbd{d n}, all 12 places reappear because the
-full number is still there on the Calc stack. More surprisingly,
-even if you exit Embedded mode and later re-enter it for that
-formula, typing @kbd{d n} will restore all 12 places because
-each buffer remembers all its active formulas. However, if you
-save the buffer in a file and reload it in a new Emacs session,
-all non-displayed digits will have been lost unless you used
-``plain'' mode.
-
address@hidden
-\bigskip
address@hidden tex
-
-In some applications of Embedded mode, you will want to have a
-sequence of copies of a formula that show its evolution as you
-work on it. For example, you might want to have a sequence
-like this in your file (elaborating here on the example from
-the ``Getting Started'' chapter):
-
address@hidden
-The derivative of
-
- ln(ln(x))
-
-is
-
- @r{(the derivative of }ln(ln(x))@r{)}
-
-whose value at x = 2 is
-
- @r{(the value)}
-
-and at x = 3 is
-
- @r{(the value)}
address@hidden smallexample
-
address@hidden C-x * d
address@hidden calc-embedded-duplicate
-The @kbd{C-x * d} (@code{calc-embedded-duplicate}) command is a
-handy way to make sequences like this. If you type @kbd{C-x * d},
-the formula under the cursor (which may or may not have Embedded
-mode enabled for it at the time) is copied immediately below and
-Embedded mode is then enabled for that copy.
-
-For this example, you would start with just
-
address@hidden
-The derivative of
-
- ln(ln(x))
address@hidden smallexample
-
address@hidden
-and press @kbd{C-x * d} with the cursor on this formula. The result
-is
-
address@hidden
-The derivative of
-
- ln(ln(x))
-
-
- ln(ln(x))
address@hidden smallexample
-
address@hidden
-with the second copy of the formula enabled in Embedded mode.
-You can now press @kbd{a d x @key{RET}} to take the derivative, and
address@hidden * d C-x * d} to make two more copies of the derivative.
-To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
-the last formula, then move up to the second-to-last formula
-and type @kbd{2 s l x @key{RET}}.
-
-Finally, you would want to press @kbd{C-x * e} to exit Embedded
-mode, then go up and insert the necessary text in between the
-various formulas and numbers.
-
address@hidden
-\bigskip
address@hidden tex
-
address@hidden C-x * f
address@hidden C-x * '
address@hidden calc-embedded-new-formula
-The @kbd{C-x * f} (@code{calc-embedded-new-formula}) command
-creates a new embedded formula at the current point. It inserts
-some default delimiters, which are usually just blank lines,
-and then does an algebraic entry to get the formula (which is
-then enabled for Embedded mode). This is just shorthand for
-typing the delimiters yourself, positioning the cursor between
-the new delimiters, and pressing @kbd{C-x * e}. The key sequence
address@hidden * '} is equivalent to @kbd{C-x * f}.
-
address@hidden C-x * n
address@hidden C-x * p
address@hidden calc-embedded-next
address@hidden calc-embedded-previous
-The @kbd{C-x * n} (@code{calc-embedded-next}) and @kbd{C-x * p}
-(@code{calc-embedded-previous}) commands move the cursor to the
-next or previous active embedded formula in the buffer. They
-can take positive or negative prefix arguments to move by several
-formulas. Note that these commands do not actually examine the
-text of the buffer looking for formulas; they only see formulas
-which have previously been activated in Embedded mode. In fact,
address@hidden * n} and @kbd{C-x * p} are a useful way to tell which
-embedded formulas are currently active. Also, note that these
-commands do not enable Embedded mode on the next or previous
-formula, they just move the cursor.
-
address@hidden C-x * `
address@hidden calc-embedded-edit
-The @kbd{C-x * `} (@code{calc-embedded-edit}) command edits the
-embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
-Embedded mode does not have to be enabled for this to work. Press
address@hidden C-c} to finish the edit, or @kbd{C-x k} to cancel.
-
address@hidden Assignments in Embedded Mode, Mode Settings in Embedded Mode,
More About Embedded Mode, Embedded Mode
address@hidden Assignments in Embedded Mode
-
address@hidden
-The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
-are especially useful in Embedded mode. They allow you to make
-a definition in one formula, then refer to that definition in
-other formulas embedded in the same buffer.
-
-An embedded formula which is an assignment to a variable, as in
-
address@hidden
-foo := 5
address@hidden example
-
address@hidden
-records @expr{5} as the stored value of @code{foo} for the
-purposes of Embedded mode operations in the current buffer. It
-does @emph{not} actually store @expr{5} as the ``global'' value
-of @code{foo}, however. Regular Calc operations, and Embedded
-formulas in other buffers, will not see this assignment.
-
-One way to use this assigned value is simply to create an
-Embedded formula elsewhere that refers to @code{foo}, and to press
address@hidden in that formula. However, this permanently replaces the
address@hidden in the formula with its current value. More interesting
-is to use @samp{=>} elsewhere:
-
address@hidden
-foo + 7 => 12
address@hidden example
-
address@hidden Operator}, for a general discussion of @samp{=>}.
-
-If you move back and change the assignment to @code{foo}, any
address@hidden>} formulas which refer to it are automatically updated.
-
address@hidden
-foo := 17
-
-foo + 7 => 24
address@hidden example
-
-The obvious question then is, @emph{how} can one easily change the
-assignment to @code{foo}? If you simply select the formula in
-Embedded mode and type 17, the assignment itself will be replaced
-by the 17. The effect on the other formula will be that the
-variable @code{foo} becomes unassigned:
-
address@hidden
-17
-
-foo + 7 => foo + 7
address@hidden example
-
-The right thing to do is first to use a selection command (@kbd{j 2}
-will do the trick) to select the righthand side of the assignment.
-Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place
(@pxref{Selecting
-Subformulas}, to see how this works).
-
address@hidden C-x * j
address@hidden calc-embedded-select
-The @kbd{C-x * j} (@code{calc-embedded-select}) command provides an
-easy way to operate on assignments. It is just like @kbd{C-x * e},
-except that if the enabled formula is an assignment, it uses
address@hidden 2} to select the righthand side. If the enabled formula
-is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
-A formula can also be a combination of both:
-
address@hidden
-bar := foo + 3 => 20
address@hidden example
-
address@hidden
-in which case @kbd{C-x * j} will select the middle part (@samp{foo + 3}).
-
-The formula is automatically deselected when you leave Embedded
-mode.
-
address@hidden C-x * u
address@hidden calc-embedded-update-formula
-Another way to change the assignment to @code{foo} would simply be
-to edit the number using regular Emacs editing rather than Embedded
-mode. Then, we have to find a way to get Embedded mode to notice
-the change. The @kbd{C-x * u} (@code{calc-embedded-update-formula})
-command is a convenient way to do this.
-
address@hidden
-foo := 6
-
-foo + 7 => 13
address@hidden example
-
-Pressing @kbd{C-x * u} is much like pressing @kbd{C-x * e = C-x * e}, that
-is, temporarily enabling Embedded mode for the formula under the
-cursor and then evaluating it with @kbd{=}. But @kbd{C-x * u} does
-not actually use @kbd{C-x * e}, and in fact another formula somewhere
-else can be enabled in Embedded mode while you use @kbd{C-x * u} and
-that formula will not be disturbed.
-
-With a numeric prefix argument, @kbd{C-x * u} updates all active
address@hidden>} formulas in the buffer. Formulas which have not yet
-been activated in Embedded mode, and formulas which do not have
address@hidden>} as their top-level operator, are not affected by this.
-(This is useful only if you have used @kbd{m C}; see below.)
-
-With a plain @kbd{C-u} prefix, @kbd{C-u C-x * u} updates only in the
-region between mark and point rather than in the whole buffer.
-
address@hidden * u} is also a handy way to activate a formula, such as an
address@hidden>} formula that has freshly been typed in or loaded from a
-file.
-
address@hidden C-x * a
address@hidden calc-embedded-activate
-The @kbd{C-x * a} (@code{calc-embedded-activate}) command scans
-through the current buffer and activates all embedded formulas
-that contain @samp{:=} or @samp{=>} symbols. This does not mean
-that Embedded mode is actually turned on, but only that the
-formulas' positions are registered with Embedded mode so that
-the @samp{=>} values can be properly updated as assignments are
-changed.
-
-It is a good idea to type @kbd{C-x * a} right after loading a file
-that uses embedded @samp{=>} operators. Emacs includes a nifty
-``buffer-local variables'' feature that you can use to do this
-automatically. The idea is to place near the end of your file
-a few lines that look like this:
-
address@hidden
---- Local Variables: ---
---- eval:(calc-embedded-activate) ---
---- End: ---
address@hidden example
-
address@hidden
-where the leading and trailing @samp{---} can be replaced by
-any suitable strings (which must be the same on all three lines)
-or omitted altogether; in a @TeX{} or address@hidden file, @samp{%} would be a
good
-leading string and no trailing string would be necessary. In a
-C program, @samp{/*} and @samp{*/} would be good leading and
-trailing strings.
-
-When Emacs loads a file into memory, it checks for a Local Variables
-section like this one at the end of the file. If it finds this
-section, it does the specified things (in this case, running
address@hidden * a} automatically) before editing of the file begins.
-The Local Variables section must be within 3000 characters of the
-end of the file for Emacs to find it, and it must be in the last
-page of the file if the file has any page separators.
address@hidden Variables, , Local Variables in Files, emacs, the
-Emacs manual}.
-
-Note that @kbd{C-x * a} does not update the formulas it finds.
-To do this, type, say, @kbd{M-1 C-x * u} after @address@hidden * a}}.
-Generally this should not be a problem, though, because the
-formulas will have been up-to-date already when the file was
-saved.
-
-Normally, @kbd{C-x * a} activates all the formulas it finds, but
-any previous active formulas remain active as well. With a
-positive numeric prefix argument, @kbd{C-x * a} first deactivates
-all current active formulas, then actives the ones it finds in
-its scan of the buffer. With a negative prefix argument,
address@hidden * a} simply deactivates all formulas.
-
-Embedded mode has two symbols, @samp{Active} and @samp{~Active},
-which it puts next to the major mode name in a buffer's mode line.
-It puts @samp{Active} if it has reason to believe that all
-formulas in the buffer are active, because you have typed @kbd{C-x * a}
-and Calc has not since had to deactivate any formulas (which can
-happen if Calc goes to update an @samp{=>} formula somewhere because
-a variable changed, and finds that the formula is no longer there
-due to some kind of editing outside of Embedded mode). Calc puts
address@hidden in the mode line if some, but probably not all,
-formulas in the buffer are active. This happens if you activate
-a few formulas one at a time but never use @kbd{C-x * a}, or if you
-used @kbd{C-x * a} but then Calc had to deactivate a formula
-because it lost track of it. If neither of these symbols appears
-in the mode line, no embedded formulas are active in the buffer
-(e.g., before Embedded mode has been used, or after a @kbd{M-- C-x * a}).
-
-Embedded formulas can refer to assignments both before and after them
-in the buffer. If there are several assignments to a variable, the
-nearest preceding assignment is used if there is one, otherwise the
-following assignment is used.
-
address@hidden
-x => 1
-
-x := 1
-
-x => 1
-
-x := 2
-
-x => 2
address@hidden example
-
-As well as simple variables, you can also assign to subscript
-expressions of the form @address@hidden@var{number}} (as in
address@hidden), or @address@hidden@var{var}} (as in @code{x_max}).
-Assignments to other kinds of objects can be represented by Calc,
-but the automatic linkage between assignments and references works
-only for plain variables and these two kinds of subscript expressions.
-
-If there are no assignments to a given variable, the global
-stored value for the variable is used (@pxref{Storing Variables}),
-or, if no value is stored, the variable is left in symbolic form.
-Note that global stored values will be lost when the file is saved
-and loaded in a later Emacs session, unless you have used the
address@hidden p} (@code{calc-permanent-variable}) command to save them;
address@hidden on Variables}.
-
-The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
-recomputation of @samp{=>} forms on and off. If you turn automatic
-recomputation off, you will have to use @kbd{C-x * u} to update these
-formulas manually after an assignment has been changed. If you
-plan to change several assignments at once, it may be more efficient
-to type @kbd{m C}, change all the assignments, then use @kbd{M-1 C-x * u}
-to update the entire buffer afterwards. The @kbd{m C} command also
-controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
-Operator}. When you turn automatic recomputation back on, the
-stack will be updated but the Embedded buffer will not; you must
-use @kbd{C-x * u} to update the buffer by hand.
-
address@hidden Mode Settings in Embedded Mode, Customizing Embedded Mode,
Assignments in Embedded Mode, Embedded Mode
address@hidden Mode Settings in Embedded Mode
-
address@hidden m e
address@hidden calc-embedded-preserve-modes
address@hidden
-The mode settings can be changed while Calc is in embedded mode, but
-by default they will revert to their original values when embedded mode
-is ended. However, the modes saved when the mode-recording mode is
address@hidden (see below) and the modes in effect when the @kbd{m e}
-(@code{calc-embedded-preserve-modes}) command is given
-will be preserved when embedded mode is ended.
-
-Embedded mode has a rather complicated mechanism for handling mode
-settings in Embedded formulas. It is possible to put annotations
-in the file that specify mode settings either global to the entire
-file or local to a particular formula or formulas. In the latter
-case, different modes can be specified for use when a formula
-is the enabled Embedded mode formula.
-
-When you give any mode-setting command, like @kbd{m f} (for Fraction
-mode) or @kbd{d s} (for scientific notation), Embedded mode adds
-a line like the following one to the file just before the opening
-delimiter of the formula.
-
address@hidden
-% [calc-mode: fractions: t]
-% [calc-mode: float-format: (sci 0)]
address@hidden example
-
-When Calc interprets an embedded formula, it scans the text before
-the formula for mode-setting annotations like these and sets the
-Calc buffer to match these modes. Modes not explicitly described
-in the file are not changed. Calc scans all the way to the top of
-the file, or up to a line of the form
-
address@hidden
-% [calc-defaults]
address@hidden example
-
address@hidden
-which you can insert at strategic places in the file if this backward
-scan is getting too slow, or just to provide a barrier between one
-``zone'' of mode settings and another.
-
-If the file contains several annotations for the same mode, the
-closest one before the formula is used. Annotations after the
-formula are never used (except for global annotations, described
-below).
-
-The scan does not look for the leading @samp{% }, only for the
-square brackets and the text they enclose. In fact, the leading
-characters are different for different major modes. You can edit the
-mode annotations to a style that works better in context if you wish.
address@hidden Embedded Mode}, to see how to change the style
-that Calc uses when it generates the annotations. You can write
-mode annotations into the file yourself if you know the syntax;
-the easiest way to find the syntax for a given mode is to let
-Calc write the annotation for it once and see what it does.
-
-If you give a mode-changing command for a mode that already has
-a suitable annotation just above the current formula, Calc will
-modify that annotation rather than generating a new, conflicting
-one.
-
-Mode annotations have three parts, separated by colons. (Spaces
-after the colons are optional.) The first identifies the kind
-of mode setting, the second is a name for the mode itself, and
-the third is the value in the form of a Lisp symbol, number,
-or list. Annotations with unrecognizable text in the first or
-second parts are ignored. The third part is not checked to make
-sure the value is of a valid type or range; if you write an
-annotation by hand, be sure to give a proper value or results
-will be unpredictable. Mode-setting annotations are case-sensitive.
-
-While Embedded mode is enabled, the word @code{Local} appears in
-the mode line. This is to show that mode setting commands generate
-annotations that are ``local'' to the current formula or set of
-formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
-causes Calc to generate different kinds of annotations. Pressing
address@hidden R} repeatedly cycles through the possible modes.
-
address@hidden and @code{LocPerm} modes generate annotations
-that look like this, respectively:
-
address@hidden
-% [calc-edit-mode: float-format: (sci 0)]
-% [calc-perm-mode: float-format: (sci 5)]
address@hidden example
-
-The first kind of annotation will be used only while a formula
-is enabled in Embedded mode. The second kind will be used only
-when the formula is @emph{not} enabled. (Whether the formula
-is ``active'' or not, i.e., whether Calc has seen this formula
-yet, is not relevant here.)
-
address@hidden mode generates an annotation like this at the end
-of the file:
-
address@hidden
-% [calc-global-mode: fractions t]
address@hidden example
-
-Global mode annotations affect all formulas throughout the file,
-and may appear anywhere in the file. This allows you to tuck your
-mode annotations somewhere out of the way, say, on a new page of
-the file, as long as those mode settings are suitable for all
-formulas in the file.
-
-Enabling a formula with @kbd{C-x * e} causes a fresh scan for local
-mode annotations; you will have to use this after adding annotations
-above a formula by hand to get the formula to notice them. Updating
-a formula with @kbd{C-x * u} will also re-scan the local modes, but
-global modes are only re-scanned by @kbd{C-x * a}.
-
-Another way that modes can get out of date is if you add a local
-mode annotation to a formula that has another formula after it.
-In this example, we have used the @kbd{d s} command while the
-first of the two embedded formulas is active. But the second
-formula has not changed its style to match, even though by the
-rules of reading annotations the @samp{(sci 0)} applies to it, too.
-
address@hidden
-% [calc-mode: float-format: (sci 0)]
-1.23e2
-
-456.
address@hidden example
-
-We would have to go down to the other formula and press @kbd{C-x * u}
-on it in order to get it to notice the new annotation.
-
-Two more mode-recording modes selectable by @kbd{m R} are available
-which are also available outside of Embedded mode.
-(@pxref{General Mode Commands}.) They are @code{Save}, in which mode
-settings are recorded permanently in your Calc init file (the file given
-by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
-rather than by annotating the current document, and no-recording
-mode (where there is no symbol like @code{Save} or @code{Local} in
-the mode line), in which mode-changing commands do not leave any
-annotations at all.
-
-When Embedded mode is not enabled, mode-recording modes except
-for @code{Save} have no effect.
-
address@hidden Customizing Embedded Mode, , Mode Settings in Embedded Mode,
Embedded Mode
address@hidden Customizing Embedded Mode
-
address@hidden
-You can modify Embedded mode's behavior by setting various Lisp
-variables described here. These variables are customizable
-(@pxref{Customizing Calc}), or you can use @kbd{M-x set-variable}
-or @kbd{M-x edit-options} to adjust a variable on the fly.
-(Another possibility would be to use a file-local variable annotation at
-the end of the file;
address@hidden Variables, , Local Variables in Files, emacs, the Emacs manual}.)
-Many of the variables given mentioned here can be set to depend on the
-major mode of the editing buffer (@pxref{Customizing Calc}).
-
address@hidden calc-embedded-open-formula
-The @code{calc-embedded-open-formula} variable holds a regular
-expression for the opening delimiter of a formula. @xref{Regexp Search,
-, Regular Expression Search, emacs, the Emacs manual}, to see
-how regular expressions work. Basically, a regular expression is a
-pattern that Calc can search for. A regular expression that considers
-blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
address@hidden"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
-regular expression is not completely plain, let's go through it
-in detail.
-
-The surrounding @samp{" "} marks quote the text between them as a
-Lisp string. If you left them off, @code{set-variable} or
address@hidden would try to read the regular expression as a
-Lisp program.
-
-The most obvious property of this regular expression is that it
-contains indecently many backslashes. There are actually two levels
-of backslash usage going on here. First, when Lisp reads a quoted
-string, all pairs of characters beginning with a backslash are
-interpreted as special characters. Here, @code{\n} changes to a
-new-line character, and @code{\\} changes to a single backslash.
-So the actual regular expression seen by Calc is
address@hidden|^ @r{(newline)} \|\$\$?}.
-
-Regular expressions also consider pairs beginning with backslash
-to have special meanings. Sometimes the backslash is used to quote
-a character that otherwise would have a special meaning in a regular
-expression, like @samp{$}, which normally means ``end-of-line,''
-or @samp{?}, which means that the preceding item is optional. So
address@hidden matches either one or two dollar signs.
-
-The other codes in this regular expression are @samp{^}, which matches
-``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
-which matches ``beginning-of-buffer.'' So the whole pattern means
-that a formula begins at the beginning of the buffer, or on a newline
-that occurs at the beginning of a line (i.e., a blank line), or at
-one or two dollar signs.
-
-The default value of @code{calc-embedded-open-formula} looks just
-like this example, with several more alternatives added on to
-recognize various other common kinds of delimiters.
-
-By the way, the reason to use @samp{^\n} rather than @samp{^$}
-or @samp{\n\n}, which also would appear to match blank lines,
-is that the former expression actually ``consumes'' only one
-newline character as @emph{part of} the delimiter, whereas the
-latter expressions consume zero or two newlines, respectively.
-The former choice gives the most natural behavior when Calc
-must operate on a whole formula including its delimiters.
-
-See the Emacs manual for complete details on regular expressions.
-But just for your convenience, here is a list of all characters
-which must be quoted with backslash (like @samp{\$}) to avoid
-some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
-the backslash in this list; for example, to match @samp{\[} you
-must use @code{"\\\\\\["}. An exercise for the reader is to
-account for each of these six backslashes!)
-
address@hidden calc-embedded-close-formula
-The @code{calc-embedded-close-formula} variable holds a regular
-expression for the closing delimiter of a formula. A closing
-regular expression to match the above example would be
address@hidden"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
-other one, except it now uses @samp{\'} (``end-of-buffer'') and
address@hidden (newline occurring at end of line, yet another way
-of describing a blank line that is more appropriate for this
-case).
-
address@hidden calc-embedded-open-word
address@hidden calc-embedded-close-word
-The @code{calc-embedded-open-word} and @code{calc-embedded-close-word}
-variables are similar expressions used when you type @kbd{C-x * w}
-instead of @kbd{C-x * e} to enable Embedded mode.
-
address@hidden calc-embedded-open-plain
-The @code{calc-embedded-open-plain} variable is a string which
-begins a ``plain'' formula written in front of the formatted
-formula when @kbd{d p} mode is turned on. Note that this is an
-actual string, not a regular expression, because Calc must be able
-to write this string into a buffer as well as to recognize it.
-The default string is @code{"%%% "} (note the trailing space), but may
-be different for certain major modes.
-
address@hidden calc-embedded-close-plain
-The @code{calc-embedded-close-plain} variable is a string which
-ends a ``plain'' formula. The default is @code{" %%%\n"}, but may be
-different for different major modes. Without
-the trailing newline here, the first line of a Big mode formula
-that followed might be shifted over with respect to the other lines.
-
address@hidden calc-embedded-open-new-formula
-The @code{calc-embedded-open-new-formula} variable is a string
-which is inserted at the front of a new formula when you type
address@hidden * f}. Its default value is @code{"\n\n"}. If this
-string begins with a newline character and the @kbd{C-x * f} is
-typed at the beginning of a line, @kbd{C-x * f} will skip this
-first newline to avoid introducing unnecessary blank lines in
-the file.
-
address@hidden calc-embedded-close-new-formula
-The @code{calc-embedded-close-new-formula} variable is the corresponding
-string which is inserted at the end of a new formula. Its default
-value is also @code{"\n\n"}. The final newline is omitted by
address@hidden@kbd{C-x * f}} if typed at the end of a line. (It follows that if
address@hidden * f} is typed on a blank line, both a leading opening
-newline and a trailing closing newline are omitted.)
-
address@hidden calc-embedded-announce-formula
-The @code{calc-embedded-announce-formula} variable is a regular
-expression which is sure to be followed by an embedded formula.
-The @kbd{C-x * a} command searches for this pattern as well as for
address@hidden>} and @samp{:=} operators. Note that @kbd{C-x * a} will
-not activate just anything surrounded by formula delimiters; after
-all, blank lines are considered formula delimiters by default!
-But if your language includes a delimiter which can only occur
-actually in front of a formula, you can take advantage of it here.
-The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, but may be
-different for different major modes.
-This pattern will check for @samp{%Embed} followed by any number of
-lines beginning with @samp{%} and a space. This last is important to
-make Calc consider mode annotations part of the pattern, so that the
-formula's opening delimiter really is sure to follow the pattern.
-
address@hidden calc-embedded-open-mode
-The @code{calc-embedded-open-mode} variable is a string (not a
-regular expression) which should precede a mode annotation.
-Calc never scans for this string; Calc always looks for the
-annotation itself. But this is the string that is inserted before
-the opening bracket when Calc adds an annotation on its own.
-The default is @code{"% "}, but may be different for different major
-modes.
-
address@hidden calc-embedded-close-mode
-The @code{calc-embedded-close-mode} variable is a string which
-follows a mode annotation written by Calc. Its default value
-is simply a newline, @code{"\n"}, but may be different for different
-major modes. If you change this, it is a good idea still to end with a
-newline so that mode annotations will appear on lines by themselves.
-
address@hidden Programming, Copying, Embedded Mode, Top
address@hidden Programming
-
address@hidden
-There are several ways to ``program'' the Emacs Calculator, depending
-on the nature of the problem you need to solve.
-
address@hidden
address@hidden
address@hidden macros} allow you to record a sequence of keystrokes
-and play them back at a later time. This is just the standard Emacs
-keyboard macro mechanism, dressed up with a few more features such
-as loops and conditionals.
-
address@hidden
address@hidden definitions} allow you to use any formula to define a
-new function. This function can then be used in algebraic formulas or
-as an interactive command.
-
address@hidden
address@hidden rules} are discussed in the section on algebra commands.
address@hidden Rules}. If you put your rewrite rules in the variable
address@hidden, they will be applied automatically to all Calc
-results in just the same way as an internal ``rule'' is applied to
-evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
-
address@hidden
address@hidden is the programming language that Calc (and most of Emacs)
-is written in. If the above techniques aren't powerful enough, you
-can write Lisp functions to do anything that built-in Calc commands
-can do. Lisp code is also somewhat faster than keyboard macros or
-rewrite rules.
address@hidden enumerate
-
address@hidden z
-Programming features are available through the @kbd{z} and @kbd{Z}
-prefix keys. New commands that you define are two-key sequences
-beginning with @kbd{z}. Commands for managing these definitions
-use the address@hidden prefix. (The @kbd{Z T} (@code{calc-timing})
-command is described elsewhere; @pxref{Troubleshooting Commands}.
-The @kbd{Z C} (@code{calc-user-define-composition}) command is also
-described elsewhere; @pxref{User-Defined Compositions}.)
-
address@hidden
-* Creating User Keys::
-* Keyboard Macros::
-* Invocation Macros::
-* Algebraic Definitions::
-* Lisp Definitions::
address@hidden menu
-
address@hidden Creating User Keys, Keyboard Macros, Programming, Programming
address@hidden Creating User Keys
-
address@hidden
address@hidden Z D
address@hidden calc-user-define
-Any Calculator command may be bound to a key using the @kbd{Z D}
-(@code{calc-user-define}) command. Actually, it is bound to a two-key
-sequence beginning with the lower-case @kbd{z} prefix.
-
-The @kbd{Z D} command first prompts for the key to define. For example,
-press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
-prompted for the name of the Calculator command that this key should
-run. For example, the @code{calc-sincos} command is not normally
-available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
address@hidden s} key sequence to run @code{calc-sincos}. This definition will
remain
-in effect for the rest of this Emacs session, or until you redefine
address@hidden s} to be something else.
-
-You can actually bind any Emacs command to a @kbd{z} key sequence by
-backspacing over the @samp{calc-} when you are prompted for the command name.
-
-As with any other prefix key, you can type @kbd{z ?} to see a list of
-all the two-key sequences you have defined that start with @kbd{z}.
-Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
-
-User keys are typically letters, but may in fact be any key.
-(@key{META}-keys are not permitted, nor are a terminal's special
-function keys which generate multi-character sequences when pressed.)
-You can define different commands on the shifted and unshifted versions
-of a letter if you wish.
-
address@hidden Z U
address@hidden calc-user-undefine
-The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
-For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
-key we defined above.
-
address@hidden Z P
address@hidden calc-user-define-permanent
address@hidden Storing user definitions
address@hidden Permanent user definitions
address@hidden Calc init file, user-defined commands
-The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
-binding permanent so that it will remain in effect even in future Emacs
-sessions. (It does this by adding a suitable bit of Lisp code into
-your Calc init file; that is, the file given by the variable
address@hidden, typically @file{~/.calc.el}.) For example,
address@hidden P s} would register our @code{sincos} command permanently. If
-you later wish to unregister this command you must edit your Calc init
-file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
-use a different file for the Calc init file.)
-
-The @kbd{Z P} command also saves the user definition, if any, for the
-command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
-key could invoke a command, which in turn calls an algebraic function,
-which might have one or more special display formats. A single @kbd{Z P}
-command will save all of these definitions.
-To save an algebraic function, type @kbd{'} (the apostrophe)
-when prompted for a key, and type the function name. To save a command
-without its key binding, type @kbd{M-x} and enter a function name. (The
address@hidden prefix will automatically be inserted for you.)
-(If the command you give implies a function, the function will be saved,
-and if the function has any display formats, those will be saved, but
-not the other way around: Saving a function will not save any commands
-or key bindings associated with the function.)
-
address@hidden Z E
address@hidden calc-user-define-edit
address@hidden Editing user definitions
-The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
-of a user key. This works for keys that have been defined by either
-keyboard macros or formulas; further details are contained in the relevant
-following sections.
-
address@hidden Keyboard Macros, Invocation Macros, Creating User Keys,
Programming
address@hidden Programming with Keyboard Macros
-
address@hidden
address@hidden X
address@hidden Programming with keyboard macros
address@hidden Keyboard macros
-The easiest way to ``program'' the Emacs Calculator is to use standard
-keyboard macros. Press @address@hidden (}} to begin recording a macro. From
-this point on, keystrokes you type will be saved away as well as
-performing their usual functions. Press @kbd{C-x )} to end recording.
-Press address@hidden (or the standard Emacs key sequence @kbd{C-x e}) to
-execute your keyboard macro by replaying the recorded keystrokes.
address@hidden Macros, , , emacs, the Emacs Manual}, for further
-information.
-
-When you use @kbd{X} to invoke a keyboard macro, the entire macro is
-treated as a single command by the undo and trail features. The stack
-display buffer is not updated during macro execution, but is instead
-fixed up once the macro completes. Thus, commands defined with keyboard
-macros are convenient and efficient. The @kbd{C-x e} command, on the
-other hand, invokes the keyboard macro with no special treatment: Each
-command in the macro will record its own undo information and trail entry,
-and update the stack buffer accordingly. If your macro uses features
-outside of Calc's control to operate on the contents of the Calc stack
-buffer, or if it includes Undo, Redo, or last-arguments commands, you
-must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
-at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
-instead of @address@hidden (@code{calc-last-args}).
-
-Calc extends the standard Emacs keyboard macros in several ways.
-Keyboard macros can be used to create user-defined commands. Keyboard
-macros can include conditional and iteration structures, somewhat
-analogous to those provided by a traditional programmable calculator.
-
address@hidden
-* Naming Keyboard Macros::
-* Conditionals in Macros::
-* Loops in Macros::
-* Local Values in Macros::
-* Queries in Macros::
address@hidden menu
-
address@hidden Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros,
Keyboard Macros
address@hidden Naming Keyboard Macros
-
address@hidden
address@hidden Z K
address@hidden calc-user-define-kbd-macro
-Once you have defined a keyboard macro, you can bind it to a @kbd{z}
-key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
-This command prompts first for a key, then for a command name. For
-example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
-define a keyboard macro which negates the top two numbers on the stack
-(@key{TAB} swaps the top two stack elements). Now you can type
address@hidden K n @key{RET}} to define this keyboard macro onto the @kbd{z n}
key
-sequence. The default command name (if you answer the second prompt with
-just the @key{RET} key as in this example) will be something like
address@hidden The keyboard macro will now be available as both
address@hidden n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
-descriptive command name if you wish.
-
-Macros defined by @kbd{Z K} act like single commands; they are executed
-in the same way as by the @kbd{X} key. If you wish to define the macro
-as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
-give a negative prefix argument to @kbd{Z K}.
-
-Once you have bound your keyboard macro to a key, you can use
address@hidden P} to register it permanently with Emacs. @xref{Creating User
Keys}.
-
address@hidden Keyboard macros, editing
-The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
-been defined by a keyboard macro tries to use the @code{edmacro} package
-edit the macro. Type @kbd{C-c C-c} to finish editing and update
-the definition stored on the key, or, to cancel the edit, kill the
-buffer with @kbd{C-x k}.
-The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
address@hidden, and @code{NUL} must be entered as these three character
-sequences, written in all uppercase, as must the prefixes @code{C-} and
address@hidden Spaces and line breaks are ignored. Other characters are
-copied verbatim into the keyboard macro. Basically, the notation is the
-same as is used in all of this manual's examples, except that the manual
-takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
-we take it for granted that it is clear we really mean
address@hidden' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
-
address@hidden C-x * m
address@hidden read-kbd-macro
-The @kbd{C-x * m} (@code{read-kbd-macro}) command reads an Emacs ``region''
-of spelled-out keystrokes and defines it as the current keyboard macro.
-It is a convenient way to define a keyboard macro that has been stored
-in a file, or to define a macro without executing it at the same time.
-
address@hidden Conditionals in Macros, Loops in Macros, Naming Keyboard Macros,
Keyboard Macros
address@hidden Conditionals in Keyboard Macros
-
address@hidden
address@hidden Z [
address@hidden Z ]
address@hidden calc-kbd-if
address@hidden calc-kbd-else
address@hidden calc-kbd-else-if
address@hidden calc-kbd-end-if
address@hidden Conditional structures
-The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
-commands allow you to put simple tests in a keyboard macro. When Calc
-sees the @kbd{Z [}, it pops an object from the stack and, if the object is
-a non-zero value, continues executing keystrokes. But if the object is
-zero, or if it is not provably nonzero, Calc skips ahead to the matching
address@hidden ]} keystroke. @xref{Logical Operations}, for a set of commands
for
-performing tests which conveniently produce 1 for true and 0 for false.
-
-For example, @address@hidden 0 a < Z [ n Z ]} implements an absolute-value
-function in the form of a keyboard macro. This macro duplicates the
-number on the top of the stack, pushes zero and compares using @kbd{a <}
-(@code{calc-less-than}), then, if the number was less than zero,
-executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
-command is skipped.
-
-To program this macro, type @kbd{C-x (}, type the above sequence of
-keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
-executed while you are making the definition as well as when you later
-re-execute the macro by typing @kbd{X}. Thus you should make sure a
-suitable number is on the stack before defining the macro so that you
-don't get a stack-underflow error during the definition process.
-
-Conditionals can be nested arbitrarily. However, there should be exactly
-one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
-
address@hidden Z :
-The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
-two keystroke sequences. The general format is @address@hidden Z [
address@hidden Z : @var{else-part} Z ]}. If @var{cond} is true
-(i.e., if the top of stack contains a non-zero number after @var{cond}
-has been executed), the @var{then-part} will be executed and the
address@hidden will be skipped. Otherwise, the @var{then-part} will
-be skipped and the @var{else-part} will be executed.
-
address@hidden Z |
-The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
-between any number of alternatives. For example,
address@hidden@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
address@hidden Z ]} will execute @var{part1} if @var{cond1} is true,
-otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
-it will execute @var{part3}.
-
-More precisely, @kbd{Z [} pops a number and conditionally skips to the
-next matching @kbd{Z :} or @kbd{Z ]} key. @address@hidden ]}} has no effect
when
-actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
address@hidden |} pops a number and conditionally skips to the next matching
address@hidden :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
-equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
-does not.
-
-Calc's conditional and looping constructs work by scanning the
-keyboard macro for occurrences of character sequences like @samp{Z:}
-and @samp{Z]}. One side-effect of this is that if you use these
-constructs you must be careful that these character pairs do not
-occur by accident in other parts of the macros. Since Calc rarely
-uses address@hidden for any purpose except as a prefix character, this
-is not likely to be a problem. Another side-effect is that it will
-not work to define your own custom key bindings for these commands.
-Only the standard address@hidden bindings will work correctly.
-
address@hidden Z C-g
-If Calc gets stuck while skipping characters during the definition of a
-macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
-actually adds a @kbd{C-g} keystroke to the macro.)
-
address@hidden Loops in Macros, Local Values in Macros, Conditionals in Macros,
Keyboard Macros
address@hidden Loops in Keyboard Macros
-
address@hidden
address@hidden Z <
address@hidden Z >
address@hidden calc-kbd-repeat
address@hidden calc-kbd-end-repeat
address@hidden Looping structures
address@hidden Iterative structures
-The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
-(@code{calc-kbd-end-repeat}) commands pop a number from the stack,
-which must be an integer, then repeat the keystrokes between the brackets
-the specified number of times. If the integer is zero or negative, the
-body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
-computes two to a nonnegative integer power. First, we push 1 on the
-stack and then swap the integer argument back to the top. The @kbd{Z <}
-pops that argument leaving the 1 back on top of the stack. Then, we
-repeat a multiply-by-two step however many times.
-
-Once again, the keyboard macro is executed as it is being entered.
-In this case it is especially important to set up reasonable initial
-conditions before making the definition: Suppose the integer 1000 just
-happened to be sitting on the stack before we typed the above definition!
-Another approach is to enter a harmless dummy definition for the macro,
-then go back and edit in the real one with a @kbd{Z E} command. Yet
-another approach is to type the macro as written-out keystroke names
-in a buffer, then use @kbd{C-x * m} (@code{read-kbd-macro}) to read the
-macro.
-
address@hidden Z /
address@hidden calc-break
-The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
-of a keyboard macro loop prematurely. It pops an object from the stack;
-if that object is true (a non-zero number), control jumps out of the
-innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
-after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
-effect. Thus @address@hidden Z /} is similar to @samp{if (@var{cond}) break;}
-in the C language.
-
address@hidden Z (
address@hidden Z )
address@hidden calc-kbd-for
address@hidden calc-kbd-end-for
-The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
-commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
-value of the counter available inside the loop. The general layout is
address@hidden@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The
@kbd{Z (}
-command pops initial and final values from the stack. It then creates
-a temporary internal counter and initializes it with the value @var{init}.
-The @kbd{Z (} command then repeatedly pushes the counter value onto the
-stack and executes @var{body} and @var{step}, adding @var{step} to the
-counter each time until the loop finishes.
-
address@hidden Summations (by keyboard macros)
-By default, the loop finishes when the counter becomes greater than (or
-less than) @var{final}, assuming @var{initial} is less than (greater
-than) @var{final}. If @var{initial} is equal to @var{final}, the body
-executes exactly once. The body of the loop always executes at least
-once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
-squares of the integers from 1 to 10, in steps of 1.
-
-If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
-forced to use upward-counting conventions. In this case, if @var{initial}
-is greater than @var{final} the body will not be executed at all.
-Note that @var{step} may still be negative in this loop; the prefix
-argument merely constrains the loop-finished test. Likewise, a prefix
-argument of @mathit{-1} forces downward-counting conventions.
-
address@hidden Z @{
address@hidden Z @}
address@hidden calc-kbd-loop
address@hidden calc-kbd-end-loop
-The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
-(@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
address@hidden >}, except that they do not pop a count from the stack---they
-effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
-loop ought to include at least one @kbd{Z /} to make sure the loop
-doesn't run forever. (If any error message occurs which causes Emacs
-to beep, the keyboard macro will also be halted; this is a standard
-feature of Emacs. You can also generally press @kbd{C-g} to halt a
-running keyboard macro, although not all versions of Unix support
-this feature.)
-
-The conditional and looping constructs are not actually tied to
-keyboard macros, but they are most often used in that context.
-For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
-ten copies of 23 onto the stack. This can be typed ``live'' just
-as easily as in a macro definition.
-
address@hidden in Macros}, for some additional notes about
-conditional and looping commands.
-
address@hidden Local Values in Macros, Queries in Macros, Loops in Macros,
Keyboard Macros
address@hidden Local Values in Macros
-
address@hidden
address@hidden Local variables
address@hidden Restoring saved modes
-Keyboard macros sometimes want to operate under known conditions
-without affecting surrounding conditions. For example, a keyboard
-macro may wish to turn on Fraction mode, or set a particular
-precision, independent of the user's normal setting for those
-modes.
-
address@hidden Z `
address@hidden Z '
address@hidden calc-kbd-push
address@hidden calc-kbd-pop
-Macros also sometimes need to use local variables. Assignments to
-local variables inside the macro should not affect any variables
-outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
-(@code{calc-kbd-pop}) commands give you both of these capabilities.
-
-When you type @kbd{Z `} (with a backquote or accent grave character),
-the values of various mode settings are saved away. The ten ``quick''
-variables @code{q0} through @code{q9} are also saved. When
-you type @address@hidden '}} (with an apostrophe), these values are restored.
-Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
-
-If a keyboard macro halts due to an error in between a @kbd{Z `} and
-a @kbd{Z '}, the saved values will be restored correctly even though
-the macro never reaches the @kbd{Z '} command. Thus you can use
address@hidden `} and @kbd{Z '} without having to worry about what happens
-in exceptional conditions.
-
-If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
-you into a ``recursive edit.'' You can tell you are in a recursive
-edit because there will be extra square brackets in the mode line,
-as in @samp{[(Calculator)]}. These brackets will go away when you
-type the matching @kbd{Z '} command. The modes and quick variables
-will be saved and restored in just the same way as if actual keyboard
-macros were involved.
-
-The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
-and binary word size, the angular mode (Deg, Rad, or HMS), the
-simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
-Matrix or Scalar mode, Fraction mode, and the current complex mode
-(Polar or Rectangular). The ten ``quick'' variables' values (or lack
-thereof) are also saved.
-
-Most mode-setting commands act as toggles, but with a numeric prefix
-they force the mode either on (positive prefix) or off (negative
-or zero prefix). Since you don't know what the environment might
-be when you invoke your macro, it's best to use prefix arguments
-for all mode-setting commands inside the macro.
-
-In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
-listed above to their default values. As usual, the matching @kbd{Z '}
-will restore the modes to their settings from before the @kbd{C-u Z `}.
-Also, @address@hidden `}} with a negative prefix argument resets the algebraic
mode
-to its default (off) but leaves the other modes the same as they were
-outside the construct.
-
-The contents of the stack and trail, values of non-quick variables, and
-other settings such as the language mode and the various display modes,
-are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
-
address@hidden Queries in Macros, , Local Values in Macros, Keyboard Macros
address@hidden Queries in Keyboard Macros
-
address@hidden @noindent
address@hidden @kindex Z =
address@hidden @pindex calc-kbd-report
address@hidden The @kbd{Z =} (@code{calc-kbd-report}) command displays an
informative
address@hidden message including the value on the top of the stack. You are
prompted
address@hidden to enter a string. That string, along with the top-of-stack
value,
address@hidden is displayed unless @kbd{m w} (@code{calc-working}) has been used
address@hidden to turn such messages off.
-
address@hidden
address@hidden Z #
address@hidden calc-kbd-query
-The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
-entry which takes its input from the keyboard, even during macro
-execution. All the normal conventions of algebraic input, including the
-use of @kbd{$} characters, are supported. The prompt message itself is
-taken from the top of the stack, and so must be entered (as a string)
-before the @kbd{Z #} command. (Recall, as a string it can be entered by
-pressing the @kbd{"} key and will appear as a vector when it is put on
-the stack. The prompt message is only put on the stack to provide a
-prompt for the @kbd{Z #} command; it will not play any role in any
-subsequent calculations.) This command allows your keyboard macros to
-accept numbers or formulas as interactive input.
-
-As an example,
address@hidden @key{RET} "Power: " @key{RET} Z # 3 @key{RET} ^} will prompt for
-input with ``Power: '' in the minibuffer, then return 2 to the provided
-power. (The response to the prompt that's given, 3 in this example,
-will not be part of the macro.)
-
address@hidden Macro Query, , , emacs, the Emacs Manual}, for a description of
address@hidden q} (@code{kbd-macro-query}), the standard Emacs way to accept
-keyboard input during a keyboard macro. In particular, you can use
address@hidden q} to enter a recursive edit, which allows the user to perform
-any Calculator operations interactively before pressing @kbd{C-M-c} to
-return control to the keyboard macro.
-
address@hidden Invocation Macros, Algebraic Definitions, Keyboard Macros,
Programming
address@hidden Invocation Macros
-
address@hidden C-x * z
address@hidden Z I
address@hidden calc-user-invocation
address@hidden calc-user-define-invocation
-Calc provides one special keyboard macro, called up by @kbd{C-x * z}
-(@code{calc-user-invocation}), that is intended to allow you to define
-your own special way of starting Calc. To define this ``invocation
-macro,'' create the macro in the usual way with @kbd{C-x (} and
address@hidden )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
-There is only one invocation macro, so you don't need to type any
-additional letters after @kbd{Z I}. From now on, you can type
address@hidden * z} at any time to execute your invocation macro.
-
-For example, suppose you find yourself often grabbing rectangles of
-numbers into Calc and multiplying their columns. You can do this
-by typing @kbd{C-x * r} to grab, and @kbd{V R : *} to multiply columns.
-To make this into an invocation macro, just type @kbd{C-x ( C-x * r
-V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
-just mark the data in its buffer in the usual way and type @kbd{C-x * z}.
-
-Invocation macros are treated like regular Emacs keyboard macros;
-all the special features described above for @kbd{Z K}-style macros
-do not apply. @kbd{C-x * z} is just like @kbd{C-x e}, except that it
-uses the macro that was last stored by @kbd{Z I}. (In fact, the
-macro does not even have to have anything to do with Calc!)
-
-The @kbd{m m} command saves the last invocation macro defined by
address@hidden I} along with all the other Calc mode settings.
address@hidden Mode Commands}.
-
address@hidden Algebraic Definitions, Lisp Definitions, Invocation Macros,
Programming
address@hidden Programming with Formulas
-
address@hidden
address@hidden Z F
address@hidden calc-user-define-formula
address@hidden Programming with algebraic formulas
-Another way to create a new Calculator command uses algebraic formulas.
-The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
-formula at the top of the stack as the definition for a key. This
-command prompts for five things: The key, the command name, the function
-name, the argument list, and the behavior of the command when given
-non-numeric arguments.
-
-For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
address@hidden + 2*b} onto the stack. We now type @kbd{Z F m} to define this
-formula on the @kbd{z m} key sequence. The next prompt is for a command
-name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
-for the new command. If you simply press @key{RET}, a default name like
address@hidden will be constructed. In our example, suppose we enter
address@hidden @key{RET}} to define the new command as @code{calc-spam}.
-
-If you want to give the formula a long-style name only, you can press
address@hidden or @key{RET} when asked which single key to use. For example
address@hidden F @key{RET} spam @key{RET}} defines the new command as
address@hidden calc-spam}, with no keyboard equivalent.
-
-The third prompt is for an algebraic function name. The default is to
-use the same name as the command name but without the @samp{calc-}
-prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
-it won't be taken for a minus sign in algebraic formulas.)
-This is the name you will use if you want to enter your
-new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
-Then the new function can be invoked by pushing two numbers on the
-stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
-formula @samp{yow(x,y)}.
-
-The fourth prompt is for the function's argument list. This is used to
-associate values on the stack with the variables that appear in the formula.
-The default is a list of all variables which appear in the formula, sorted
-into alphabetical order. In our case, the default would be @samp{(a b)}.
-This means that, when the user types @kbd{z m}, the Calculator will remove
-two numbers from the stack, substitute these numbers for @samp{a} and
address@hidden (respectively) in the formula, then simplify the formula and
-push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
-would replace the 10 and 100 on the stack with the number 210, which is
address@hidden + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
address@hidden(10, 100)} will be evaluated by substituting @expr{a=10} and
address@hidden in the definition.
-
-You can rearrange the order of the names before pressing @key{RET} to
-control which stack positions go to which variables in the formula. If
-you remove a variable from the argument list, that variable will be left
-in symbolic form by the command. Thus using an argument list of @samp{(b)}
-for our function would cause @kbd{10 z m} to replace the 10 on the stack
-with the formula @samp{a + 20}. If we had used an argument list of
address@hidden(b a)}, the result with inputs 10 and 100 would have been 120.
-
-You can also put a nameless function on the stack instead of just a
-formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
-In this example, the command will be defined by the formula @samp{a + 2 b}
-using the argument list @samp{(a b)}.
-
-The final prompt is a y-or-n question concerning what to do if symbolic
-arguments are given to your function. If you answer @kbd{y}, then
-executing @kbd{z m} (using the original argument list @samp{(a b)}) with
-arguments @expr{10} and @expr{x} will leave the function in symbolic
-form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
-then the formula will always be expanded, even for non-constant
-arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
-formulas to your new function, it doesn't matter how you answer this
-question.
-
-If you answered @kbd{y} to this question you can still cause a function
-call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
-Also, Calc will expand the function if necessary when you take a
-derivative or integral or solve an equation involving the function.
-
address@hidden Z G
address@hidden calc-get-user-defn
-Once you have defined a formula on a key, you can retrieve this formula
-with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
-key, and this command pushes the formula that was used to define that
-key onto the stack. Actually, it pushes a nameless function that
-specifies both the argument list and the defining formula. You will get
-an error message if the key is undefined, or if the key was not defined
-by a @kbd{Z F} command.
-
-The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
-been defined by a formula uses a variant of the @code{calc-edit} command
-to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
-store the new formula back in the definition, or kill the buffer with
address@hidden k} to
-cancel the edit. (The argument list and other properties of the
-definition are unchanged; to adjust the argument list, you can use
address@hidden G} to grab the function onto the stack, edit with @kbd{`}, and
-then re-execute the @kbd{Z F} command.)
-
-As usual, the @kbd{Z P} command records your definition permanently.
-In this case it will permanently record all three of the relevant
-definitions: the key, the command, and the function.
-
-You may find it useful to turn off the default simplifications with
address@hidden O} (@code{calc-no-simplify-mode}) when entering a formula to be
-used as a function definition. For example, the formula @samp{deriv(a^2,v)}
-which might be used to define a new function @samp{dsqr(a,v)} will be
-``simplified'' to 0 immediately upon entry since @code{deriv} considers
address@hidden to be constant with respect to @expr{v}. Turning off
-default simplifications cures this problem: The definition will be stored
-in symbolic form without ever activating the @code{deriv} function. Press
address@hidden D} to turn the default simplifications back on afterwards.
-
address@hidden Lisp Definitions, , Algebraic Definitions, Programming
address@hidden Programming with Lisp
-
address@hidden
-The Calculator can be programmed quite extensively in Lisp. All you
-do is write a normal Lisp function definition, but with @code{defmath}
-in place of @code{defun}. This has the same form as @code{defun}, but it
-automagically replaces calls to standard Lisp functions like @code{+} and
address@hidden with calls to the corresponding functions in Calc's own library.
-Thus you can write natural-looking Lisp code which operates on all of the
-standard Calculator data types. You can then use @kbd{Z D} if you wish to
-bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
-will not edit a Lisp-based definition.
-
-Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
-assumes a familiarity with Lisp programming concepts; if you do not know
-Lisp, you may find keyboard macros or rewrite rules to be an easier way
-to program the Calculator.
-
-This section first discusses ways to write commands, functions, or
-small programs to be executed inside of Calc. Then it discusses how
-your own separate programs are able to call Calc from the outside.
-Finally, there is a list of internal Calc functions and data structures
-for the true Lisp enthusiast.
-
address@hidden
-* Defining Functions::
-* Defining Simple Commands::
-* Defining Stack Commands::
-* Argument Qualifiers::
-* Example Definitions::
-
-* Calling Calc from Your Programs::
-* Internals::
address@hidden menu
-
address@hidden Defining Functions, Defining Simple Commands, Lisp Definitions,
Lisp Definitions
address@hidden Defining New Functions
-
address@hidden
address@hidden defmath
-The @code{defmath} function (actually a Lisp macro) is like @code{defun}
-except that code in the body of the definition can make use of the full
-range of Calculator data types. The prefix @samp{calcFunc-} is added
-to the specified name to get the actual Lisp function name. As a simple
-example,
-
address@hidden
-(defmath myfact (n)
- (if (> n 0)
- (* n (myfact (1- n)))
- 1))
address@hidden example
-
address@hidden
-This actually expands to the code,
-
address@hidden
-(defun calcFunc-myfact (n)
- (if (math-posp n)
- (math-mul n (calcFunc-myfact (math-add n -1)))
- 1))
address@hidden example
-
address@hidden
-This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
-
-The @samp{myfact} function as it is defined above has the bug that an
-expression @samp{myfact(a+b)} will be simplified to 1 because the
-formula @samp{a+b} is not considered to be @code{posp}. A robust
-factorial function would be written along the following lines:
-
address@hidden
-(defmath myfact (n)
- (if (> n 0)
- (* n (myfact (1- n)))
- (if (= n 0)
- 1
- nil))) ; this could be simplified as: (and (= n 0) 1)
address@hidden smallexample
-
-If a function returns @code{nil}, it is left unsimplified by the Calculator
-(except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
-will be simplified to @samp{myfact(a+3)} but no further. Beware that every
-time the Calculator reexamines this formula it will attempt to resimplify
-it, so your function ought to detect the address@hidden case as
-efficiently as possible.
-
-The following standard Lisp functions are treated by @code{defmath}:
address@hidden, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
address@hidden, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
address@hidden/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior},
@code{logxor},
address@hidden, @code{lognot}. Also, @code{~=} is an abbreviation for
address@hidden, which is useful in implementing Taylor series.
-
-For other functions @var{func}, if a function by the name
address@hidden@var{func}} exists it is used, otherwise if a function by the
-name @address@hidden exists it is used, otherwise if @var{func} itself
-is defined as a function it is used, otherwise @address@hidden is
-used on the assumption that this is a to-be-defined math function. Also, if
-the function name is quoted as in @samp{('integerp a)} the function name is
-always used exactly as written (but not quoted).
-
-Variable names have @samp{var-} prepended to them unless they appear in
-the function's argument list or in an enclosing @code{let}, @code{let*},
address@hidden, or @code{foreach} form,
-or their names already contain a @samp{-} character. Thus a reference to
address@hidden is the same as a reference to @samp{var-foo}.
-
-A few other Lisp extensions are available in @code{defmath} definitions:
-
address@hidden @bullet
address@hidden
-The @code{elt} function accepts any number of index variables.
-Note that Calc vectors are stored as Lisp lists whose first
-element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
-the second element of vector @code{v}, and @samp{(elt m i j)}
-yields one element of a Calc matrix.
-
address@hidden
-The @code{setq} function has been extended to act like the Common
-Lisp @code{setf} function. (The name @code{setf} is recognized as
-a synonym of @code{setq}.) Specifically, the first argument of
address@hidden can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
-in which case the effect is to store into the specified
-element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
-into one element of a matrix.
-
address@hidden
-A @code{for} looping construct is available. For example,
address@hidden(for ((i 0 10)) body)} executes @code{body} once for each
-binding of @expr{i} from zero to 10. This is like a @code{let}
-form in that @expr{i} is temporarily bound to the loop count
-without disturbing its value outside the @code{for} construct.
-Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
-are also available. For each value of @expr{i} from zero to 10,
address@hidden counts from 0 to @expr{i-1} in steps of two. Note that
address@hidden has the same general outline as @code{let*}, except
-that each element of the header is a list of three or four
-things, not just two.
-
address@hidden
-The @code{foreach} construct loops over elements of a list.
-For example, @samp{(foreach ((x (cdr v))) body)} executes
address@hidden with @expr{x} bound to each element of Calc vector
address@hidden in turn. The purpose of @code{cdr} here is to skip over
-the initial @code{vec} symbol in the vector.
-
address@hidden
-The @code{break} function breaks out of the innermost enclosing
address@hidden, @code{for}, or @code{foreach} loop. If given a
-value, as in @samp{(break x)}, this value is returned by the
-loop. (Lisp loops otherwise always return @code{nil}.)
-
address@hidden
-The @code{return} function prematurely returns from the enclosing
-function. For example, @samp{(return (+ x y))} returns @expr{x+y}
-as the value of a function. You can use @code{return} anywhere
-inside the body of the function.
address@hidden itemize
-
-Non-integer numbers (and extremely large integers) cannot be included
-directly into a @code{defmath} definition. This is because the Lisp
-reader will fail to parse them long before @code{defmath} ever gets control.
-Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
-formula can go between the quotes. For example,
-
address@hidden
-(defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
- (and (numberp x)
- (exp :"x * 0.5")))
address@hidden smallexample
-
-expands to
-
address@hidden
-(defun calcFunc-sqexp (x)
- (and (math-numberp x)
- (calcFunc-exp (math-mul x '(float 5 -1)))))
address@hidden smallexample
-
-Note the use of @code{numberp} as a guard to ensure that the argument is
-a number first, returning @code{nil} if not. The exponential function
-could itself have been included in the expression, if we had preferred:
address@hidden:"exp(x * 0.5)"}. As another example, the
multiplication-and-recursion
-step of @code{myfact} could have been written
-
address@hidden
-:"n * myfact(n-1)"
address@hidden example
-
-A good place to put your @code{defmath} commands is your Calc init file
-(the file given by @code{calc-settings-file}, typically
address@hidden/.calc.el}), which will not be loaded until Calc starts.
-If a file named @file{.emacs} exists in your home directory, Emacs reads
-and executes the Lisp forms in this file as it starts up. While it may
-seem reasonable to put your favorite @code{defmath} commands there,
-this has the unfortunate side-effect that parts of the Calculator must be
-loaded in to process the @code{defmath} commands whether or not you will
-actually use the Calculator! If you want to put the @code{defmath}
-commands there (for example, if you redefine @code{calc-settings-file}
-to be @file{.emacs}), a better effect can be had by writing
-
address@hidden
-(put 'calc-define 'thing '(progn
- (defmath ... )
- (defmath ... )
-))
address@hidden example
-
address@hidden
address@hidden calc-define
-The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
-symbol has a list of properties associated with it. Here we add a
-property with a name of @code{thing} and a @samp{(progn ...)} form as
-its value. When Calc starts up, and at the start of every Calc command,
-the property list for the symbol @code{calc-define} is checked and the
-values of any properties found are evaluated as Lisp forms. The
-properties are removed as they are evaluated. The property names
-(like @code{thing}) are not used; you should choose something like the
-name of your project so as not to conflict with other properties.
-
-The net effect is that you can put the above code in your @file{.emacs}
-file and it will not be executed until Calc is loaded. Or, you can put
-that same code in another file which you load by hand either before or
-after Calc itself is loaded.
-
-The properties of @code{calc-define} are evaluated in the same order
-that they were added. They can assume that the Calc modules @file{calc.el},
address@hidden, and @file{calc-macs.el} have been fully loaded, and
-that the @samp{*Calculator*} buffer will be the current buffer.
-
-If your @code{calc-define} property only defines algebraic functions,
-you can be sure that it will have been evaluated before Calc tries to
-call your function, even if the file defining the property is loaded
-after Calc is loaded. But if the property defines commands or key
-sequences, it may not be evaluated soon enough. (Suppose it defines the
-new command @code{tweak-calc}; the user can load your file, then type
address@hidden tweak-calc} before Calc has had chance to do anything.) To
-protect against this situation, you can put
-
address@hidden
-(run-hooks 'calc-check-defines)
address@hidden example
-
address@hidden calc-check-defines
address@hidden
-at the end of your file. The @code{calc-check-defines} function is what
-looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
-has the advantage that it is quietly ignored if @code{calc-check-defines}
-is not yet defined because Calc has not yet been loaded.
-
-Examples of things that ought to be enclosed in a @code{calc-define}
-property are @code{defmath} calls, @code{define-key} calls that modify
-the Calc key map, and any calls that redefine things defined inside Calc.
-Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
-
address@hidden Defining Simple Commands, Defining Stack Commands, Defining
Functions, Lisp Definitions
address@hidden Defining New Simple Commands
-
address@hidden
address@hidden interactive
-If a @code{defmath} form contains an @code{interactive} clause, it defines
-a Calculator command. Actually such a @code{defmath} results in @emph{two}
-function definitions: One, a @samp{calcFunc-} function as was just described,
-with the @code{interactive} clause removed. Two, a @samp{calc-} function
-with a suitable @code{interactive} clause and some sort of wrapper to make
-the command work in the Calc environment.
-
-In the simple case, the @code{interactive} clause has the same form as
-for normal Emacs Lisp commands:
-
address@hidden
-(defmath increase-precision (delta)
- "Increase precision by DELTA." ; This is the "documentation string"
- (interactive "p") ; Register this as a M-x-able command
- (setq calc-internal-prec (+ calc-internal-prec delta)))
address@hidden smallexample
-
-This expands to the pair of definitions,
-
address@hidden
-(defun calc-increase-precision (delta)
- "Increase precision by DELTA."
- (interactive "p")
- (calc-wrapper
- (setq calc-internal-prec (math-add calc-internal-prec delta))))
-
-(defun calcFunc-increase-precision (delta)
- "Increase precision by DELTA."
- (setq calc-internal-prec (math-add calc-internal-prec delta)))
address@hidden smallexample
-
address@hidden
-where in this case the latter function would never really be used! Note
-that since the Calculator stores small integers as plain Lisp integers,
-the @code{math-add} function will work just as well as the native
address@hidden even when the intent is to operate on native Lisp integers.
-
address@hidden calc-wrapper
-The @samp{calc-wrapper} call invokes a macro which surrounds the body of
-the function with code that looks roughly like this:
-
address@hidden
-(let ((calc-command-flags nil))
- (unwind-protect
- (save-excursion
- (calc-select-buffer)
- @emph{body of function}
- @emph{renumber stack}
- @emph{clear} Working @emph{message})
- @emph{realign cursor and window}
- @emph{clear Inverse, Hyperbolic, and Keep Args flags}
- @emph{update Emacs mode line}))
address@hidden smallexample
-
address@hidden calc-select-buffer
-The @code{calc-select-buffer} function selects the @samp{*Calculator*}
-buffer if necessary, say, because the command was invoked from inside
-the @samp{*Calc Trail*} window.
-
address@hidden calc-set-command-flag
-You can call, for example, @code{(calc-set-command-flag 'no-align)} to
-set the above-mentioned command flags. Calc routines recognize the
-following command flags:
-
address@hidden @code
address@hidden renum-stack
-Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
-after this command completes. This is set by routines like
address@hidden
-
address@hidden clear-message
-Calc should call @samp{(message "")} if this command completes normally
-(to clear a address@hidden'' message out of the echo area).
-
address@hidden no-align
-Do not move the cursor back to the @samp{.} top-of-stack marker.
-
address@hidden position-point
-Use the variables @code{calc-position-point-line} and
address@hidden to position the cursor after
-this command finishes.
-
address@hidden keep-flags
-Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
-and @code{calc-keep-args-flag} at the end of this command.
-
address@hidden do-edit
-Switch to buffer @samp{*Calc Edit*} after this command.
-
address@hidden hold-trail
-Do not move trail pointer to end of trail when something is recorded
-there.
address@hidden table
-
address@hidden Y
address@hidden Y ?
address@hidden calc-Y-help-msgs
-Calc reserves a special prefix key, address@hidden, for user-written
-extensions to Calc. There are no built-in commands that work with
-this prefix key; you must call @code{define-key} from Lisp (probably
-from inside a @code{calc-define} property) to add to it. Initially only
address@hidden ?} is defined; it takes help messages from a list of strings
-(initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
-other undefined keys except for @kbd{Y} are reserved for use by
-future versions of Calc.
-
-If you are writing a Calc enhancement which you expect to give to
-others, it is best to minimize the number of @kbd{Y}-key sequences
-you use. In fact, if you have more than one key sequence you should
-consider defining three-key sequences with a @kbd{Y}, then a key that
-stands for your package, then a third key for the particular command
-within your package.
-
-Users may wish to install several Calc enhancements, and it is possible
-that several enhancements will choose to use the same key. In the
-example below, a variable @code{inc-prec-base-key} has been defined
-to contain the key that identifies the @code{inc-prec} package. Its
-value is initially @code{"P"}, but a user can change this variable
-if necessary without having to modify the file.
-
-Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
-command that increases the precision, and a @kbd{Y P D} command that
-decreases the precision.
-
address@hidden
-;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
-;; (Include copyright or copyleft stuff here.)
-
-(defvar inc-prec-base-key "P"
- "Base key for inc-prec.el commands.")
-
-(put 'calc-define 'inc-prec '(progn
-
-(define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
- 'increase-precision)
-(define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
- 'decrease-precision)
-
-(setq calc-Y-help-msgs
- (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
- calc-Y-help-msgs))
-
-(defmath increase-precision (delta)
- "Increase precision by DELTA."
- (interactive "p")
- (setq calc-internal-prec (+ calc-internal-prec delta)))
-
-(defmath decrease-precision (delta)
- "Decrease precision by DELTA."
- (interactive "p")
- (setq calc-internal-prec (- calc-internal-prec delta)))
-
-)) ; end of calc-define property
-
-(run-hooks 'calc-check-defines)
address@hidden smallexample
-
address@hidden Defining Stack Commands, Argument Qualifiers, Defining Simple
Commands, Lisp Definitions
address@hidden Defining New Stack-Based Commands
-
address@hidden
-To define a new computational command which takes and/or leaves arguments
-on the stack, a special form of @code{interactive} clause is used.
-
address@hidden
-(interactive @var{num} @var{tag})
address@hidden example
-
address@hidden
-where @var{num} is an integer, and @var{tag} is a string. The effect is
-to pop @var{num} values off the stack, resimplify them by calling
address@hidden, and hand them to your function according to the
-function's argument list. Your function may include @code{&optional} and
address@hidden&rest} parameters, so long as calling the function with @var{num}
-parameters is valid.
-
-Your function must return either a number or a formula in a form
-acceptable to Calc, or a list of such numbers or formulas. These value(s)
-are pushed onto the stack when the function completes. They are also
-recorded in the Calc Trail buffer on a line beginning with @var{tag},
-a string of (normally) four characters or less. If you omit @var{tag}
-or use @code{nil} as a tag, the result is not recorded in the trail.
-
-As an example, the definition
-
address@hidden
-(defmath myfact (n)
- "Compute the factorial of the integer at the top of the stack."
- (interactive 1 "fact")
- (if (> n 0)
- (* n (myfact (1- n)))
- (and (= n 0) 1)))
address@hidden smallexample
-
address@hidden
-is a version of the factorial function shown previously which can be used
-as a command as well as an algebraic function. It expands to
-
address@hidden
-(defun calc-myfact ()
- "Compute the factorial of the integer at the top of the stack."
- (interactive)
- (calc-slow-wrapper
- (calc-enter-result 1 "fact"
- (cons 'calcFunc-myfact (calc-top-list-n 1)))))
-
-(defun calcFunc-myfact (n)
- "Compute the factorial of the integer at the top of the stack."
- (if (math-posp n)
- (math-mul n (calcFunc-myfact (math-add n -1)))
- (and (math-zerop n) 1)))
address@hidden smallexample
-
address@hidden calc-slow-wrapper
-The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
-that automatically puts up a @samp{Working...} message before the
-computation begins. (This message can be turned off by the user
-with an @kbd{m w} (@code{calc-working}) command.)
-
address@hidden calc-top-list-n
-The @code{calc-top-list-n} function returns a list of the specified number
-of values from the top of the stack. It resimplifies each value by
-calling @code{calc-normalize}. If its argument is zero it returns an
-empty list. It does not actually remove these values from the stack.
-
address@hidden calc-enter-result
-The @code{calc-enter-result} function takes an integer @var{num} and string
address@hidden as described above, plus a third argument which is either a
-Calculator data object or a list of such objects. These objects are
-resimplified and pushed onto the stack after popping the specified number
-of values from the stack. If @var{tag} is address@hidden, the values
-being pushed are also recorded in the trail.
-
-Note that if @code{calcFunc-myfact} returns @code{nil} this represents
-``leave the function in symbolic form.'' To return an actual empty list,
-in the sense that @code{calc-enter-result} will push zero elements back
-onto the stack, you should return the special value @samp{'(nil)}, a list
-containing the single symbol @code{nil}.
-
-The @code{interactive} declaration can actually contain a limited
-Emacs-style code string as well which comes just before @var{num} and
address@hidden Currently the only Emacs code supported is @samp{"p"}, as in
-
address@hidden
-(defmath foo (a b &optional c)
- (interactive "p" 2 "foo")
- @var{body})
address@hidden example
-
-In this example, the command @code{calc-foo} will evaluate the expression
address@hidden(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
-executed with a numeric prefix argument of @expr{n}.
-
-The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
-code as used with @code{defun}). It uses the numeric prefix argument as the
-number of objects to remove from the stack and pass to the function.
-In this case, the integer @var{num} serves as a default number of
-arguments to be used when no prefix is supplied.
-
address@hidden Argument Qualifiers, Example Definitions, Defining Stack
Commands, Lisp Definitions
address@hidden Argument Qualifiers
-
address@hidden
-Anywhere a parameter name can appear in the parameter list you can also use
-an @dfn{argument qualifier}. Thus the general form of a definition is:
-
address@hidden
-(defmath @var{name} (@var{param} @var{param...}
- &optional @var{param} @var{param...}
- &rest @var{param})
- @var{body})
address@hidden example
-
address@hidden
-where each @var{param} is either a symbol or a list of the form
-
address@hidden
-(@var{qual} @var{param})
address@hidden example
-
-The following qualifiers are recognized:
-
address@hidden @samp
address@hidden complete
address@hidden complete
-The argument must not be an incomplete vector, interval, or complex number.
-(This is rarely needed since the Calculator itself will never call your
-function with an incomplete argument. But there is nothing stopping your
-own Lisp code from calling your function with an incomplete argument.)
-
address@hidden integer
address@hidden integer
-The argument must be an integer. If it is an integer-valued float
-it will be accepted but converted to integer form. Non-integers and
-formulas are rejected.
-
address@hidden natnum
address@hidden natnum
-Like @samp{integer}, but the argument must be non-negative.
-
address@hidden fixnum
address@hidden fixnum
-Like @samp{integer}, but the argument must fit into a native Lisp integer,
-which on most systems means less than 2^23 in absolute value. The
-argument is converted into Lisp-integer form if necessary.
-
address@hidden float
address@hidden float
-The argument is converted to floating-point format if it is a number or
-vector. If it is a formula it is left alone. (The argument is never
-actually rejected by this qualifier.)
-
address@hidden @var{pred}
-The argument must satisfy predicate @var{pred}, which is one of the
-standard Calculator predicates. @xref{Predicates}.
-
address@hidden address@hidden
-The argument must @emph{not} satisfy predicate @var{pred}.
address@hidden table
-
-For example,
-
address@hidden
-(defmath foo (a (constp (not-matrixp b)) &optional (float c)
- &rest (integer d))
- @var{body})
address@hidden example
-
address@hidden
-expands to
-
address@hidden
-(defun calcFunc-foo (a b &optional c &rest d)
- (and (math-matrixp b)
- (math-reject-arg b 'not-matrixp))
- (or (math-constp b)
- (math-reject-arg b 'constp))
- (and c (setq c (math-check-float c)))
- (setq d (mapcar 'math-check-integer d))
- @var{body})
address@hidden example
-
address@hidden
-which performs the necessary checks and conversions before executing the
-body of the function.
-
address@hidden Example Definitions, Calling Calc from Your Programs, Argument
Qualifiers, Lisp Definitions
address@hidden Example Definitions
-
address@hidden
-This section includes some Lisp programming examples on a larger scale.
-These programs make use of some of the Calculator's internal functions;
address@hidden
-
address@hidden
-* Bit Counting Example::
-* Sine Example::
address@hidden menu
-
address@hidden Bit Counting Example, Sine Example, Example Definitions, Example
Definitions
address@hidden Bit-Counting
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden bcount
-Calc does not include a built-in function for counting the number of
-``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
-to convert the integer to a set, and @kbd{V #} to count the elements of
-that set; let's write a function that counts the bits without having to
-create an intermediate set.
-
address@hidden
-(defmath bcount ((natnum n))
- (interactive 1 "bcnt")
- (let ((count 0))
- (while (> n 0)
- (if (oddp n)
- (setq count (1+ count)))
- (setq n (lsh n -1)))
- count))
address@hidden smallexample
-
address@hidden
-When this is expanded by @code{defmath}, it will become the following
-Emacs Lisp function:
-
address@hidden
-(defun calcFunc-bcount (n)
- (setq n (math-check-natnum n))
- (let ((count 0))
- (while (math-posp n)
- (if (math-oddp n)
- (setq count (math-add count 1)))
- (setq n (calcFunc-lsh n -1)))
- count))
address@hidden smallexample
-
-If the input numbers are large, this function involves a fair amount
-of arithmetic. A binary right shift is essentially a division by two;
-recall that Calc stores integers in decimal form so bit shifts must
-involve actual division.
-
-To gain a bit more efficiency, we could divide the integer into
address@hidden chunks, each of which can be handled quickly because
-they fit into Lisp integers. It turns out that Calc's arithmetic
-routines are especially fast when dividing by an integer less than
-1000, so we can set @var{n = 9} bits and use repeated division by 512:
-
address@hidden
-(defmath bcount ((natnum n))
- (interactive 1 "bcnt")
- (let ((count 0))
- (while (not (fixnump n))
- (let ((qr (idivmod n 512)))
- (setq count (+ count (bcount-fixnum (cdr qr)))
- n (car qr))))
- (+ count (bcount-fixnum n))))
-
-(defun bcount-fixnum (n)
- (let ((count 0))
- (while (> n 0)
- (setq count (+ count (logand n 1))
- n (lsh n -1)))
- count))
address@hidden smallexample
-
address@hidden
-Note that the second function uses @code{defun}, not @code{defmath}.
-Because this function deals only with native Lisp integers (``fixnums''),
-it can use the actual Emacs @code{+} and related functions rather
-than the slower but more general Calc equivalents which @code{defmath}
-uses.
-
-The @code{idivmod} function does an integer division, returning both
-the quotient and the remainder at once. Again, note that while it
-might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
-more efficient ways to split off the bottom nine bits of @code{n},
-actually they are less efficient because each operation is really
-a division by 512 in disguise; @code{idivmod} allows us to do the
-same thing with a single division by 512.
-
address@hidden Sine Example, , Bit Counting Example, Example Definitions
address@hidden The Sine Function
-
address@hidden
address@hidden
address@hidden
address@hidden ignore
address@hidden mysin
-A somewhat limited sine function could be defined as follows, using the
-well-known Taylor series expansion for
address@hidden @math{\sin x}:
address@hidden @samp{sin(x)}:
-
address@hidden
-(defmath mysin ((float (anglep x)))
- (interactive 1 "mysn")
- (setq x (to-radians x)) ; Convert from current angular mode.
- (let ((sum x) ; Initial term of Taylor expansion of sin.
- newsum
- (nfact 1) ; "nfact" equals "n" factorial at all times.
- (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
- (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
- (working "mysin" sum) ; Display "Working" message, if enabled.
- (setq nfact (* nfact (1- n) n)
- x (* x xnegsqr)
- newsum (+ sum (/ x nfact)))
- (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
- (break)) ; then we are done.
- (setq sum newsum))
- sum))
address@hidden smallexample
-
-The actual @code{sin} function in Calc works by first reducing the problem
-to a sine or cosine of a nonnegative number less than @cpiover{4}. This
-ensures that the Taylor series will converge quickly. Also, the calculation
-is carried out with two extra digits of precision to guard against cumulative
-round-off in @samp{sum}. Finally, complex arguments are allowed and handled
-by a separate algorithm.
-
address@hidden
-(defmath mysin ((float (scalarp x)))
- (interactive 1 "mysn")
- (setq x (to-radians x)) ; Convert from current angular mode.
- (with-extra-prec 2 ; Evaluate with extra precision.
- (cond ((complexp x)
- (mysin-complex x))
- ((< x 0)
- (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
- (t (mysin-raw x))))))
-
-(defmath mysin-raw (x)
- (cond ((>= x 7)
- (mysin-raw (% x (two-pi)))) ; Now x < 7.
- ((> x (pi-over-2))
- (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
- ((> x (pi-over-4))
- (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
- ((< x (- (pi-over-4)))
- (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
- (t (mysin-series x)))) ; so the series will be efficient.
address@hidden smallexample
-
address@hidden
-where @code{mysin-complex} is an appropriate function to handle complex
-numbers, @code{mysin-series} is the routine to compute the sine Taylor
-series as before, and @code{mycos-raw} is a function analogous to
address@hidden for cosines.
-
-The strategy is to ensure that @expr{x} is nonnegative before calling
address@hidden This function then recursively reduces its argument
-to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
-test, and particularly the first comparison against 7, is designed so
-that small roundoff errors cannot produce an infinite loop. (Suppose
-we compared with @samp{(two-pi)} instead; if due to roundoff problems
-the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
-recursion could result!) We use modulo only for arguments that will
-clearly get reduced, knowing that the next rule will catch any reductions
-that this rule misses.
-
-If a program is being written for general use, it is important to code
-it carefully as shown in this second example. For quick-and-dirty programs,
-when you know that your own use of the sine function will never encounter
-a large argument, a simpler program like the first one shown is fine.
-
address@hidden Calling Calc from Your Programs, Internals, Example Definitions,
Lisp Definitions
address@hidden Calling Calc from Your Lisp Programs
-
address@hidden
-A later section (@pxref{Internals}) gives a full description of
-Calc's internal Lisp functions. It's not hard to call Calc from
-inside your programs, but the number of these functions can be daunting.
-So Calc provides one special ``programmer-friendly'' function called
address@hidden that can be made to do just about everything you
-need. It's not as fast as the low-level Calc functions, but it's
-much simpler to use!
-
-It may seem that @code{calc-eval} itself has a daunting number of
-options, but they all stem from one simple operation.
-
-In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
-string @code{"1+2"} as if it were a Calc algebraic entry and returns
-the result formatted as a string: @code{"3"}.
-
-Since @code{calc-eval} is on the list of recommended @code{autoload}
-functions, you don't need to make any special preparations to load
-Calc before calling @code{calc-eval} the first time. Calc will be
-loaded and initialized for you.
-
-All the Calc modes that are currently in effect will be used when
-evaluating the expression and formatting the result.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Additional Arguments to @code{calc-eval}
-
address@hidden
-If the input string parses to a list of expressions, Calc returns
-the results separated by @code{", "}. You can specify a different
-separator by giving a second string argument to @code{calc-eval}:
address@hidden(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
-
-The ``separator'' can also be any of several Lisp symbols which
-request other behaviors from @code{calc-eval}. These are discussed
-one by one below.
-
-You can give additional arguments to be substituted for
address@hidden, @samp{$$}, and so on in the main expression. For
-example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
-expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
-(assuming Fraction mode is not in effect). Note the @code{nil}
-used as a placeholder for the item-separator argument.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Error Handling
-
address@hidden
-If @code{calc-eval} encounters an error, it returns a list containing
-the character position of the error, plus a suitable message as a
-string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
-standards; it simply returns the string @code{"1 / 0"} which is the
-division left in symbolic form. But @samp{(calc-eval "1/")} will
-return the list @samp{(2 "Expected a number")}.
-
-If you bind the variable @code{calc-eval-error} to @code{t}
-using a @code{let} form surrounding the call to @code{calc-eval},
-errors instead call the Emacs @code{error} function which aborts
-to the Emacs command loop with a beep and an error message.
-
-If you bind this variable to the symbol @code{string}, error messages
-are returned as strings instead of lists. The character position is
-ignored.
-
-As a courtesy to other Lisp code which may be using Calc, be sure
-to bind @code{calc-eval-error} using @code{let} rather than changing
-it permanently with @code{setq}.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Numbers Only
-
address@hidden
-Sometimes it is preferable to treat @samp{1 / 0} as an error
-rather than returning a symbolic result. If you pass the symbol
address@hidden as the second argument to @code{calc-eval}, results
-that are not constants are treated as errors. The error message
-reported is the first @code{calc-why} message if there is one,
-or otherwise ``Number expected.''
-
-A result is ``constant'' if it is a number, vector, or other
-object that does not include variables or function calls. If it
-is a vector, the components must themselves be constants.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Default Modes
-
address@hidden
-If the first argument to @code{calc-eval} is a list whose first
-element is a formula string, then @code{calc-eval} sets all the
-various Calc modes to their default values while the formula is
-evaluated and formatted. For example, the precision is set to 12
-digits, digit grouping is turned off, and the Normal language
-mode is used.
-
-This same principle applies to the other options discussed below.
-If the first argument would normally be @var{x}, then it can also
-be the list @samp{(@var{x})} to use the default mode settings.
-
-If there are other elements in the list, they are taken as
-variable-name/value pairs which override the default mode
-settings. Look at the documentation at the front of the
address@hidden file to find the names of the Lisp variables for
-the various modes. The mode settings are restored to their
-original values when @code{calc-eval} is done.
-
-For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
-computes the sum of two numbers, requiring a numeric result, and
-using default mode settings except that the precision is 8 instead
-of the default of 12.
-
-It's usually best to use this form of @code{calc-eval} unless your
-program actually considers the interaction with Calc's mode settings
-to be a feature. This will avoid all sorts of potential ``gotchas'';
-consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
-when the user has left Calc in Symbolic mode or No-Simplify mode.
-
-As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
-checks if the number in string @expr{a} is less than the one in
-string @expr{b}. Without using a list, the integer 1 might
-come out in a variety of formats which would be hard to test for
-conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
-see ``Predicates'' mode, below.)
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Raw Numbers
-
address@hidden
-Normally all input and output for @code{calc-eval} is done with strings.
-You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
-in place of @samp{(+ a b)}, but this is very inefficient since the
-numbers must be converted to and from string format as they are passed
-from one @code{calc-eval} to the next.
-
-If the separator is the symbol @code{raw}, the result will be returned
-as a raw Calc data structure rather than a string. You can read about
-how these objects look in the following sections, but usually you can
-treat them as ``black box'' objects with no important internal
-structure.
-
-There is also a @code{rawnum} symbol, which is a combination of
address@hidden (returning a raw Calc object) and @code{num} (signaling
-an error if that object is not a constant).
-
-You can pass a raw Calc object to @code{calc-eval} in place of a
-string, either as the formula itself or as one of the @samp{$}
-arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
-addition function that operates on raw Calc objects. Of course
-in this case it would be easier to call the low-level @code{math-add}
-function in Calc, if you can remember its name.
-
-In particular, note that a plain Lisp integer is acceptable to Calc
-as a raw object. (All Lisp integers are accepted on input, but
-integers of more than six decimal digits are converted to ``big-integer''
-form for output. @xref{Data Type Formats}.)
-
-When it comes time to display the object, just use @samp{(calc-eval a)}
-to format it as a string.
-
-It is an error if the input expression evaluates to a list of
-values. The separator symbol @code{list} is like @code{raw}
-except that it returns a list of one or more raw Calc objects.
-
-Note that a Lisp string is not a valid Calc object, nor is a list
-containing a string. Thus you can still safely distinguish all the
-various kinds of error returns discussed above.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Predicates
-
address@hidden
-If the separator symbol is @code{pred}, the result of the formula is
-treated as a true/false value; @code{calc-eval} returns @code{t} or
address@hidden, respectively. A value is considered ``true'' if it is a
-non-zero number, or false if it is zero or if it is not a number.
-
-For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
-one value is less than another.
-
-As usual, it is also possible for @code{calc-eval} to return one of
-the error indicators described above. Lisp will interpret such an
-indicator as ``true'' if you don't check for it explicitly. If you
-wish to have an error register as ``false'', use something like
address@hidden(eq (calc-eval ...) t)}.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Variable Values
-
address@hidden
-Variables in the formula passed to @code{calc-eval} are not normally
-replaced by their values. If you wish this, you can use the
address@hidden function (@pxref{Algebraic Manipulation}). For example,
-if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
address@hidden), then @samp{(calc-eval "a+pi")} will return the
-formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
-will return @code{"7.14159265359"}.
-
-To store in a Calc variable, just use @code{setq} to store in the
-corresponding Lisp variable. (This is obtained by prepending
address@hidden to the Calc variable name.) Calc routines will
-understand either string or raw form values stored in variables,
-although raw data objects are much more efficient. For example,
-to increment the Calc variable @code{a}:
-
address@hidden
-(setq var-a (calc-eval "evalv(a+1)" 'raw))
address@hidden example
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Stack Access
-
address@hidden
-If the separator symbol is @code{push}, the formula argument is
-evaluated (with possible @samp{$} expansions, as usual). The
-result is pushed onto the Calc stack. The return value is @code{nil}
-(unless there is an error from evaluating the formula, in which
-case the return value depends on @code{calc-eval-error} in the
-usual way).
-
-If the separator symbol is @code{pop}, the first argument to
address@hidden must be an integer instead of a string. That
-many values are popped from the stack and thrown away. A negative
-argument deletes the entry at that stack level. The return value
-is the number of elements remaining in the stack after popping;
address@hidden(calc-eval 0 'pop)} is a good way to measure the size of
-the stack.
-
-If the separator symbol is @code{top}, the first argument to
address@hidden must again be an integer. The value at that
-stack level is formatted as a string and returned. Thus
address@hidden(calc-eval 1 'top)} returns the top-of-stack value. If the
-integer is out of range, @code{nil} is returned.
-
-The separator symbol @code{rawtop} is just like @code{top} except
-that the stack entry is returned as a raw Calc object instead of
-as a string.
-
-In all of these cases the first argument can be made a list in
-order to force the default mode settings, as described above.
-Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
-second-to-top stack entry, formatted as a string using the default
-instead of current display modes, except that the radix is
-hexadecimal instead of decimal.
-
-It is, of course, polite to put the Calc stack back the way you
-found it when you are done, unless the user of your program is
-actually expecting it to affect the stack.
-
-Note that you do not actually have to switch into the @samp{*Calculator*}
-buffer in order to use @code{calc-eval}; it temporarily switches into
-the stack buffer if necessary.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Keyboard Macros
-
address@hidden
-If the separator symbol is @code{macro}, the first argument must be a
-string of characters which Calc can execute as a sequence of keystrokes.
-This switches into the Calc buffer for the duration of the macro.
-For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
-vector @samp{[1,2,3,4,5]} on the stack and then replaces it
-with the sum of those numbers. Note that @samp{\r} is the Lisp
-notation for the carriage-return, @key{RET}, character.
-
-If your keyboard macro wishes to pop the stack, @samp{\C-d} is
-safer than @samp{\177} (the @key{DEL} character) because some
-installations may have switched the meanings of @key{DEL} and
address@hidden Calc always interprets @kbd{C-d} as a synonym for
-``pop-stack'' regardless of key mapping.
-
-If you provide a third argument to @code{calc-eval}, evaluation
-of the keyboard macro will leave a record in the Trail using
-that argument as a tag string. Normally the Trail is unaffected.
-
-The return value in this case is always @code{nil}.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Lisp Evaluation
-
address@hidden
-Finally, if the separator symbol is @code{eval}, then the Lisp
address@hidden function is called on the first argument, which must
-be a Lisp expression rather than a Calc formula. Remember to
-quote the expression so that it is not evaluated until inside
address@hidden
-
-The difference from plain @code{eval} is that @code{calc-eval}
-switches to the Calc buffer before evaluating the expression.
-For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
-will correctly affect the buffer-local Calc precision variable.
-
-An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
-This is evaluating a call to the function that is normally invoked
-by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
-Note that this function will leave a message in the echo area as
-a side effect. Also, all Calc functions switch to the Calc buffer
-automatically if not invoked from there, so the above call is
-also equivalent to @samp{(calc-precision 17)} by itself.
-In all cases, Calc uses @code{save-excursion} to switch back to
-your original buffer when it is done.
-
-As usual the first argument can be a list that begins with a Lisp
-expression to use default instead of current mode settings.
-
-The result of @code{calc-eval} in this usage is just the result
-returned by the evaluated Lisp expression.
-
address@hidden
address@hidden
-
address@hidden example
address@hidden ifinfo
address@hidden Example
-
address@hidden
address@hidden convert-temp
-Here is a sample Emacs command that uses @code{calc-eval}. Suppose
-you have a document with lots of references to temperatures on the
-Fahrenheit scale, say ``98.6 F'', and you wish to convert these
-references to Centigrade. The following command does this conversion.
-Place the Emacs cursor right after the letter ``F'' and invoke the
-command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
-already in Centigrade form, the command changes it back to Fahrenheit.
-
address@hidden
-(defun convert-temp ()
- (interactive)
- (save-excursion
- (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
- (let* ((top1 (match-beginning 1))
- (bot1 (match-end 1))
- (number (buffer-substring top1 bot1))
- (top2 (match-beginning 2))
- (bot2 (match-end 2))
- (type (buffer-substring top2 bot2)))
- (if (equal type "F")
- (setq type "C"
- number (calc-eval "($ - 32)*5/9" nil number))
- (setq type "F"
- number (calc-eval "$*9/5 + 32" nil number)))
- (goto-char top2)
- (delete-region top2 bot2)
- (insert-before-markers type)
- (goto-char top1)
- (delete-region top1 bot1)
- (if (string-match "\\.$" number) ; change "37." to "37"
- (setq number (substring number 0 -1)))
- (insert number))))
address@hidden example
-
-Note the use of @code{insert-before-markers} when changing between
-``F'' and ``C'', so that the character winds up before the cursor
-instead of after it.
-
address@hidden Internals, , Calling Calc from Your Programs, Lisp Definitions
address@hidden Calculator Internals
-
address@hidden
-This section describes the Lisp functions defined by the Calculator that
-may be of use to user-written Calculator programs (as described in the
-rest of this chapter). These functions are shown by their names as they
-conventionally appear in @code{defmath}. Their full Lisp names are
-generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
-apparent names. (Names that begin with @samp{calc-} are already in
-their full Lisp form.) You can use the actual full names instead if you
-prefer them, or if you are calling these functions from regular Lisp.
-
-The functions described here are scattered throughout the various
-Calc component files. Note that @file{calc.el} includes @code{autoload}s
-for only a few component files; when Calc wants to call an advanced
-function it calls @samp{(calc-extensions)} first; this function
-autoloads @file{calc-ext.el}, which in turn autoloads all the functions
-in the remaining component files.
-
-Because @code{defmath} itself uses the extensions, user-written code
-generally always executes with the extensions already loaded, so
-normally you can use any Calc function and be confident that it will
-be autoloaded for you when necessary. If you are doing something
-special, check carefully to make sure each function you are using is
-from @file{calc.el} or its components, and call @samp{(calc-extensions)}
-before using any function based in @file{calc-ext.el} if you can't
-prove this file will already be loaded.
-
address@hidden
-* Data Type Formats::
-* Interactive Lisp Functions::
-* Stack Lisp Functions::
-* Predicates::
-* Computational Lisp Functions::
-* Vector Lisp Functions::
-* Symbolic Lisp Functions::
-* Formatting Lisp Functions::
-* Hooks::
address@hidden menu
-
address@hidden Data Type Formats, Interactive Lisp Functions, Internals,
Internals
address@hidden Data Type Formats
-
address@hidden
-Integers are stored in either of two ways, depending on their magnitude.
-Integers less than one million in absolute value are stored as standard
-Lisp integers. This is the only storage format for Calc data objects
-which is not a Lisp list.
-
-Large integers are stored as lists of the form @samp{(bigpos @var{d0}
address@hidden @var{d2} @dots{})} for positive integers 1000000 or more, or
address@hidden(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
address@hidden or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
-from 0 to 999. The least significant digit is @var{d0}; the last digit,
address@hidden, which is always nonzero, is the most significant digit. For
-example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345
12)}.
-
-The distinction between small and large integers is entirely hidden from
-the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
-returns true for either kind of integer, and in general both big and small
-integers are accepted anywhere the word ``integer'' is used in this manual.
-If the distinction must be made, native Lisp integers are called @dfn{fixnums}
-and large integers are called @dfn{bignums}.
-
-Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
-where @var{n} is an integer (big or small) numerator, @var{d} is an
-integer denominator greater than one, and @var{n} and @var{d} are relatively
-prime. Note that fractions where @var{d} is one are automatically converted
-to plain integers by all math routines; fractions where @var{d} is negative
-are normalized by negating the numerator and denominator.
-
-Floating-point numbers are stored in the form, @samp{(float @var{mant}
address@hidden)}, where @var{mant} (the ``mantissa'') is an integer less than
address@hidden@var{p}} in absolute value (@var{p} represents the current
-precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
-the float is @address@hidden * address@hidden For example, the number
address@hidden is stored as @samp{(float -314 -2) = -314*10^-2}. Other
constraints
-are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
-except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
-always nonzero. (If the rightmost digit is zero, the number is
-rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
-
-Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
address@hidden)}, where @var{re} and @var{im} are each real numbers, either
-integers, fractions, or floats. The value is @address@hidden + @var{im}i}.
-The @var{im} part is nonzero; complex numbers with zero imaginary
-components are converted to real numbers automatically.
-
-Polar complex numbers are stored in the form @samp{(polar @var{r}
address@hidden)}, where @var{r} is a positive real value and @var{theta}
-is a real value or HMS form representing an angle. This angle is
-usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
-or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
-If the angle is 0 the value is converted to a real number automatically.
-(If the angle is 180 degrees, the value is usually also converted to a
-negative real number.)
-
-Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
address@hidden)}, where @var{h} is an integer or an integer-valued float (i.e.,
-a float with @address@hidden >= 0}), @var{m} is an integer or integer-valued
-float in the range @address@hidden ..@: 60)}}, and @var{s} is any real number
-in the range @samp{[0 ..@: 60)}.
-
-Date forms are stored as @samp{(date @var{n})}, where @var{n} is
-a real number that counts days since midnight on the morning of
-January 1, 1 AD. If @var{n} is an integer, this is a pure date
-form. If @var{n} is a fraction or float, this is a date/time form.
-
-Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
-positive real number or HMS form, and @var{n} is a real number or HMS
-form in the range @samp{[0 ..@: @var{m})}.
-
-Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
-is the mean value and @var{sigma} is the standard deviation. Each
-component is either a number, an HMS form, or a symbolic object
-(a variable or function call). If @var{sigma} is zero, the value is
-converted to a plain real number. If @var{sigma} is negative or
-complex, it is automatically normalized to be a positive real.
-
-Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
-where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
address@hidden are real numbers, HMS forms, or symbolic objects. The @var{mask}
-is a binary integer where 1 represents the fact that the interval is
-closed on the high end, and 2 represents the fact that it is closed on
-the low end. (Thus 3 represents a fully closed interval.) The interval
address@hidden@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number
@var{x};
-intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
-represent empty intervals. If @var{hi} is less than @var{lo}, the interval
-is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
-
-Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
-is the first element of the vector, @var{v2} is the second, and so on.
-An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
-where all @var{v}'s are themselves vectors of equal lengths. Note that
-Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
-generally unused by Calc data structures.
-
-Variables are stored as @samp{(var @var{name} @var{sym})}, where
address@hidden is a Lisp symbol whose print name is used as the visible name
-of the variable, and @var{sym} is a Lisp symbol in which the variable's
-value is actually stored. Thus, @samp{(var pi var-pi)} represents the
-special constant @samp{pi}. Almost always, the form is @samp{(var
address@hidden address@hidden)}. If the variable name was entered with @code{#}
-signs (which are converted to hyphens internally), the form is
address@hidden(var @var{u} @var{v})}, where @var{u} is a symbol whose name
-contains @code{#} characters, and @var{v} is a symbol that contains
address@hidden characters instead. The value of a variable is the Calc
-object stored in its @var{sym} symbol's value cell. If the symbol's
-value cell is void or if it contains @code{nil}, the variable has no
-value. Special constants have the form @samp{(special-const
address@hidden)} stored in their value cell, where @var{value} is a formula
-which is evaluated when the constant's value is requested. Variables
-which represent units are not stored in any special way; they are units
-only because their names appear in the units table. If the value
-cell contains a string, it is parsed to get the variable's value when
-the variable is used.
-
-A Lisp list with any other symbol as the first element is a function call.
-The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
-and @code{|} represent special binary operators; these lists are always
-of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
-sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
-right. The symbol @code{neg} represents unary negation; this list is always
-of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
-function that would be displayed in function-call notation; the symbol
address@hidden is in general always of the form @address@hidden
-The function cell of the symbol @var{func} should contain a Lisp function
-for evaluating a call to @var{func}. This function is passed the remaining
-elements of the list (themselves already evaluated) as arguments; such
-functions should return @code{nil} or call @code{reject-arg} to signify
-that they should be left in symbolic form, or they should return a Calc
-object which represents their value, or a list of such objects if they
-wish to return multiple values. (The latter case is allowed only for
-functions which are the outer-level call in an expression whose value is
-about to be pushed on the stack; this feature is considered obsolete
-and is not used by any built-in Calc functions.)
-
address@hidden Interactive Lisp Functions, Stack Lisp Functions, Data Type
Formats, Internals
address@hidden Interactive Functions
-
address@hidden
-The functions described here are used in implementing interactive Calc
-commands. Note that this list is not exhaustive! If there is an
-existing command that behaves similarly to the one you want to define,
-you may find helpful tricks by checking the source code for that command.
-
address@hidden calc-set-command-flag flag
-Set the command flag @var{flag}. This is generally a Lisp symbol, but
-may in fact be anything. The effect is to add @var{flag} to the list
-stored in the variable @code{calc-command-flags}, unless it is already
-there. @xref{Defining Simple Commands}.
address@hidden defun
-
address@hidden calc-clear-command-flag flag
-If @var{flag} appears among the list of currently-set command flags,
-remove it from that list.
address@hidden defun
-
address@hidden calc-record-undo rec
-Add the ``undo record'' @var{rec} to the list of steps to take if the
-current operation should need to be undone. Stack push and pop functions
-automatically call @code{calc-record-undo}, so the kinds of undo records
-you might need to create take the form @samp{(set @var{sym} @var{value})},
-which says that the Lisp variable @var{sym} was changed and had previously
-contained @var{value}; @samp{(store @var{var} @var{value})} which says that
-the Calc variable @var{var} (a string which is the name of the symbol that
-contains the variable's value) was stored and its previous value was
address@hidden (either a Calc data object, or @code{nil} if the variable was
-previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
-which means that to undo requires calling the function @samp{(@var{undo}
address@hidden @dots{})} and, if the undo is later redone, calling
address@hidden(@var{redo} @var{args} @dots{})}.
address@hidden defun
-
address@hidden calc-record-why msg args
-Record the error or warning message @var{msg}, which is normally a string.
-This message will be replayed if the user types @kbd{w} (@code{calc-why});
-if the message string begins with a @samp{*}, it is considered important
-enough to display even if the user doesn't type @kbd{w}. If one or more
address@hidden are present, the displayed message will be of the form,
address@hidden@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments
are
-formatted on the assumption that they are either strings or Calc objects of
-some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
-(such as @code{integerp} or @code{numvecp}) which the arguments did not
-satisfy; it is expanded to a suitable string such as ``Expected an
-integer.'' The @code{reject-arg} function calls @code{calc-record-why}
-automatically; @pxref{Predicates}.
address@hidden defun
-
address@hidden calc-is-inverse
-This predicate returns true if the current command is inverse,
-i.e., if the Inverse (@kbd{I} key) flag was set.
address@hidden defun
-
address@hidden calc-is-hyperbolic
-This predicate is the analogous function for the @kbd{H} key.
address@hidden defun
-
address@hidden Stack Lisp Functions, Predicates, Interactive Lisp Functions,
Internals
address@hidden Stack-Oriented Functions
-
address@hidden
-The functions described here perform various operations on the Calc
-stack and trail. They are to be used in interactive Calc commands.
-
address@hidden calc-push-list vals n
-Push the Calc objects in list @var{vals} onto the stack at stack level
address@hidden If @var{n} is omitted it defaults to 1, so that the elements
-are pushed at the top of the stack. If @var{n} is greater than 1, the
-elements will be inserted into the stack so that the last element will
-end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
-The elements of @var{vals} are assumed to be valid Calc objects, and
-are not evaluated, rounded, or renormalized in any way. If @var{vals}
-is an empty list, nothing happens.
-
-The stack elements are pushed without any sub-formula selections.
-You can give an optional third argument to this function, which must
-be a list the same size as @var{vals} of selections. Each selection
-must be @code{eq} to some sub-formula of the corresponding formula
-in @var{vals}, or @code{nil} if that formula should have no selection.
address@hidden defun
-
address@hidden calc-top-list n m
-Return a list of the @var{n} objects starting at level @var{m} of the
-stack. If @var{m} is omitted it defaults to 1, so that the elements are
-taken from the top of the stack. If @var{n} is omitted, it also
-defaults to 1, so that the top stack element (in the form of a
-one-element list) is returned. If @var{m} is greater than 1, the
address@hidden stack element will be at the end of the list, the @var{m}+1st
-element will be next-to-last, etc. If @var{n} or @var{m} are out of
-range, the command is aborted with a suitable error message. If @var{n}
-is zero, the function returns an empty list. The stack elements are not
-evaluated, rounded, or renormalized.
-
-If any stack elements contain selections, and selections have not
-been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
-this function returns the selected portions rather than the entire
-stack elements. It can be given a third ``selection-mode'' argument
-which selects other behaviors. If it is the symbol @code{t}, then
-a selection in any of the requested stack elements produces an
-``invalid operation on selections'' error. If it is the symbol @code{full},
-the whole stack entry is always returned regardless of selections.
-If it is the symbol @code{sel}, the selected portion is always returned,
-or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
-command.) If the symbol is @code{entry}, the complete stack entry in
-list form is returned; the first element of this list will be the whole
-formula, and the third element will be the selection (or @code{nil}).
address@hidden defun
-
address@hidden calc-pop-stack n m
-Remove the specified elements from the stack. The parameters @var{n}
-and @var{m} are defined the same as for @code{calc-top-list}. The return
-value of @code{calc-pop-stack} is uninteresting.
-
-If there are any selected sub-formulas among the popped elements, and
address@hidden e} has not been used to disable selections, this produces an
-error without changing the stack. If you supply an optional third
-argument of @code{t}, the stack elements are popped even if they
-contain selections.
address@hidden defun
-
address@hidden calc-record-list vals tag
-This function records one or more results in the trail. The @var{vals}
-are a list of strings or Calc objects. The @var{tag} is the four-character
-tag string to identify the values. If @var{tag} is omitted, a blank tag
-will be used.
address@hidden defun
-
address@hidden calc-normalize n
-This function takes a Calc object and ``normalizes'' it. At the very
-least this involves re-rounding floating-point values according to the
-current precision and other similar jobs. Also, unless the user has
-selected No-Simplify mode (@pxref{Simplification Modes}), this involves
-actually evaluating a formula object by executing the function calls
-it contains, and possibly also doing algebraic simplification, etc.
address@hidden defun
-
address@hidden calc-top-list-n n m
-This function is identical to @code{calc-top-list}, except that it calls
address@hidden on the values that it takes from the stack. They
-are also passed through @code{check-complete}, so that incomplete
-objects will be rejected with an error message. All computational
-commands should use this in preference to @code{calc-top-list}; the only
-standard Calc commands that operate on the stack without normalizing
-are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
-This function accepts the same optional selection-mode argument as
address@hidden
address@hidden defun
-
address@hidden calc-top-n m
-This function is a convenient form of @code{calc-top-list-n} in which only
-a single element of the stack is taken and returned, rather than a list
-of elements. This also accepts an optional selection-mode argument.
address@hidden defun
-
address@hidden calc-enter-result n tag vals
-This function is a convenient interface to most of the above functions.
-The @var{vals} argument should be either a single Calc object, or a list
-of Calc objects; the object or objects are normalized, and the top @var{n}
-stack entries are replaced by the normalized objects. If @var{tag} is
address@hidden, the normalized objects are also recorded in the trail.
-A typical stack-based computational command would take the form,
-
address@hidden
-(calc-enter-result @var{n} @var{tag} (cons 'address@hidden
- (calc-top-list-n @var{n})))
address@hidden smallexample
-
-If any of the @var{n} stack elements replaced contain sub-formula
-selections, and selections have not been disabled by @kbd{j e},
-this function takes one of two courses of action. If @var{n} is
-equal to the number of elements in @var{vals}, then each element of
address@hidden is spliced into the corresponding selection; this is what
-happens when you use the @key{TAB} key, or when you use a unary
-arithmetic operation like @code{sqrt}. If @var{vals} has only one
-element but @var{n} is greater than one, there must be only one
-selection among the top @var{n} stack elements; the element from
address@hidden is spliced into that selection. This is what happens when
-you use a binary arithmetic operation like @kbd{+}. Any other
-combination of @var{n} and @var{vals} is an error when selections
-are present.
address@hidden defun
-
address@hidden calc-unary-op tag func arg
-This function implements a unary operator that allows a numeric prefix
-argument to apply the operator over many stack entries. If the prefix
-argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
-as outlined above. Otherwise, it maps the function over several stack
-elements; @pxref{Prefix Arguments}. For example,
-
address@hidden
-(defun calc-zeta (arg)
- (interactive "P")
- (calc-unary-op "zeta" 'calcFunc-zeta arg))
address@hidden smallexample
address@hidden defun
-
address@hidden calc-binary-op tag func arg ident unary
-This function implements a binary operator, analogously to
address@hidden The optional @var{ident} and @var{unary}
-arguments specify the behavior when the prefix argument is zero or
-one, respectively. If the prefix is zero, the value @var{ident}
-is pushed onto the stack, if specified, otherwise an error message
-is displayed. If the prefix is one, the unary function @var{unary}
-is applied to the top stack element, or, if @var{unary} is not
-specified, nothing happens. When the argument is two or more,
-the binary function @var{func} is reduced across the top @var{arg}
-stack elements; when the argument is negative, the function is
-mapped between the next-to-top @address@hidden stack elements and the
-top element.
address@hidden defun
-
address@hidden calc-stack-size
-Return the number of elements on the stack as an integer. This count
-does not include elements that have been temporarily hidden by stack
-truncation; @pxref{Truncating the Stack}.
address@hidden defun
-
address@hidden calc-cursor-stack-index n
-Move the point to the @var{n}th stack entry. If @var{n} is zero, this
-will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
-this will be the beginning of the first line of that stack entry's display.
-If line numbers are enabled, this will move to the first character of the
-line number, not the stack entry itself.
address@hidden defun
-
address@hidden calc-substack-height n
-Return the number of lines between the beginning of the @var{n}th stack
-entry and the bottom of the buffer. If @var{n} is zero, this
-will be one (assuming no stack truncation). If all stack entries are
-one line long (i.e., no matrices are displayed), the return value will
-be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
-mode, the return value includes the blank lines that separate stack
-entries.)
address@hidden defun
-
address@hidden calc-refresh
-Erase the @code{*Calculator*} buffer and reformat its contents from memory.
-This must be called after changing any parameter, such as the current
-display radix, which might change the appearance of existing stack
-entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
-is suppressed, but a flag is set so that the entire stack will be refreshed
-rather than just the top few elements when the macro finishes.)
address@hidden defun
-
address@hidden Predicates, Computational Lisp Functions, Stack Lisp Functions,
Internals
address@hidden Predicates
-
address@hidden
-The functions described here are predicates, that is, they return a
-true/false value where @code{nil} means false and anything else means
-true. These predicates are expanded by @code{defmath}, for example,
-from @code{zerop} to @code{math-zerop}. In many cases they correspond
-to native Lisp functions by the same name, but are extended to cover
-the full range of Calc data types.
-
address@hidden zerop x
-Returns true if @var{x} is numerically zero, in any of the Calc data
-types. (Note that for some types, such as error forms and intervals,
-it never makes sense to return true.) In @code{defmath}, the expression
address@hidden(= x 0)} will automatically be converted to @samp{(math-zerop x)},
-and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
address@hidden defun
-
address@hidden negp x
-Returns true if @var{x} is negative. This accepts negative real numbers
-of various types, negative HMS and date forms, and intervals in which
-all included values are negative. In @code{defmath}, the expression
address@hidden(< x 0)} will automatically be converted to @samp{(math-negp x)},
-and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
address@hidden defun
-
address@hidden posp x
-Returns true if @var{x} is positive (and non-zero). For complex
-numbers, none of these three predicates will return true.
address@hidden defun
-
address@hidden looks-negp x
-Returns true if @var{x} is ``negative-looking.'' This returns true if
address@hidden is a negative number, or a formula with a leading minus sign
-such as @samp{-a/b}. In other words, this is an object which can be
-made simpler by calling @code{(- @var{x})}.
address@hidden defun
-
address@hidden integerp x
-Returns true if @var{x} is an integer of any size.
address@hidden defun
-
address@hidden fixnump x
-Returns true if @var{x} is a native Lisp integer.
address@hidden defun
-
address@hidden natnump x
-Returns true if @var{x} is a nonnegative integer of any size.
address@hidden defun
-
address@hidden fixnatnump x
-Returns true if @var{x} is a nonnegative Lisp integer.
address@hidden defun
-
address@hidden num-integerp x
-Returns true if @var{x} is numerically an integer, i.e., either a
-true integer or a float with no significant digits to the right of
-the decimal point.
address@hidden defun
-
address@hidden messy-integerp x
-Returns true if @var{x} is numerically, but not literally, an integer.
-A value is @code{num-integerp} if it is @code{integerp} or
address@hidden (but it is never both at once).
address@hidden defun
-
address@hidden num-natnump x
-Returns true if @var{x} is numerically a nonnegative integer.
address@hidden defun
-
address@hidden evenp x
-Returns true if @var{x} is an even integer.
address@hidden defun
-
address@hidden looks-evenp x
-Returns true if @var{x} is an even integer, or a formula with a leading
-multiplicative coefficient which is an even integer.
address@hidden defun
-
address@hidden oddp x
-Returns true if @var{x} is an odd integer.
address@hidden defun
-
address@hidden ratp x
-Returns true if @var{x} is a rational number, i.e., an integer or a
-fraction.
address@hidden defun
-
address@hidden realp x
-Returns true if @var{x} is a real number, i.e., an integer, fraction,
-or floating-point number.
address@hidden defun
-
address@hidden anglep x
-Returns true if @var{x} is a real number or HMS form.
address@hidden defun
-
address@hidden floatp x
-Returns true if @var{x} is a float, or a complex number, error form,
-interval, date form, or modulo form in which at least one component
-is a float.
address@hidden defun
-
address@hidden complexp x
-Returns true if @var{x} is a rectangular or polar complex number
-(but not a real number).
address@hidden defun
-
address@hidden rect-complexp x
-Returns true if @var{x} is a rectangular complex number.
address@hidden defun
-
address@hidden polar-complexp x
-Returns true if @var{x} is a polar complex number.
address@hidden defun
-
address@hidden numberp x
-Returns true if @var{x} is a real number or a complex number.
address@hidden defun
-
address@hidden scalarp x
-Returns true if @var{x} is a real or complex number or an HMS form.
address@hidden defun
-
address@hidden vectorp x
-Returns true if @var{x} is a vector (this simply checks if its argument
-is a list whose first element is the symbol @code{vec}).
address@hidden defun
-
address@hidden numvecp x
-Returns true if @var{x} is a number or vector.
address@hidden defun
-
address@hidden matrixp x
-Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
-all of the same size.
address@hidden defun
-
address@hidden square-matrixp x
-Returns true if @var{x} is a square matrix.
address@hidden defun
-
address@hidden objectp x
-Returns true if @var{x} is any numeric Calc object, including real and
-complex numbers, HMS forms, date forms, error forms, intervals, and
-modulo forms. (Note that error forms and intervals may include formulas
-as their components; see @code{constp} below.)
address@hidden defun
-
address@hidden objvecp x
-Returns true if @var{x} is an object or a vector. This also accepts
-incomplete objects, but it rejects variables and formulas (except as
-mentioned above for @code{objectp}).
address@hidden defun
-
address@hidden primp x
-Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
-i.e., one whose components cannot be regarded as sub-formulas. This
-includes variables, and all @code{objectp} types except error forms
-and intervals.
address@hidden defun
-
address@hidden constp x
-Returns true if @var{x} is constant, i.e., a real or complex number,
-HMS form, date form, or error form, interval, or vector all of whose
-components are @code{constp}.
address@hidden defun
-
address@hidden lessp x y
-Returns true if @var{x} is numerically less than @var{y}. Returns false
-if @var{x} is greater than or equal to @var{y}, or if the order is
-undefined or cannot be determined. Generally speaking, this works
-by checking whether @address@hidden - @var{y}} is @code{negp}. In
address@hidden, the expression @samp{(< x y)} will automatically be
-converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
-and @code{>=} are similarly converted in terms of @code{lessp}.
address@hidden defun
-
address@hidden beforep x y
-Returns true if @var{x} comes before @var{y} in a canonical ordering
-of Calc objects. If @var{x} and @var{y} are both real numbers, this
-will be the same as @code{lessp}. But whereas @code{lessp} considers
-other types of objects to be unordered, @code{beforep} puts any two
-objects into a definite, consistent order. The @code{beforep}
-function is used by the @kbd{V S} vector-sorting command, and also
-by @kbd{a s} to put the terms of a product into canonical order:
-This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}.
address@hidden defun
-
address@hidden equal x y
-This is the standard Lisp @code{equal} predicate; it returns true if
address@hidden and @var{y} are structurally identical. This is the usual way
-to compare numbers for equality, but note that @code{equal} will treat
-0 and 0.0 as different.
address@hidden defun
-
address@hidden math-equal x y
-Returns true if @var{x} and @var{y} are numerically equal, either because
-they are @code{equal}, or because their difference is @code{zerop}. In
address@hidden, the expression @samp{(= x y)} will automatically be
-converted to @samp{(math-equal x y)}.
address@hidden defun
-
address@hidden equal-int x n
-Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
-is a fixnum which is not a multiple of 10. This will automatically be
-used by @code{defmath} in place of the more general @code{math-equal}
-whenever possible.
address@hidden defun
-
address@hidden nearly-equal x y
-Returns true if @var{x} and @var{y}, as floating-point numbers, are
-equal except possibly in the last decimal place. For example,
-314.159 and 314.166 are considered nearly equal if the current
-precision is 6 (since they differ by 7 units), but not if the current
-precision is 7 (since they differ by 70 units). Most functions which
-use series expansions use @code{with-extra-prec} to evaluate the
-series with 2 extra digits of precision, then use @code{nearly-equal}
-to decide when the series has converged; this guards against cumulative
-error in the series evaluation without doing extra work which would be
-lost when the result is rounded back down to the current precision.
-In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
-The @var{x} and @var{y} can be numbers of any kind, including complex.
address@hidden defun
-
address@hidden nearly-zerop x y
-Returns true if @var{x} is nearly zero, compared to @var{y}. This
-checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
-to @var{y} itself, to within the current precision, in other words,
-if adding @var{x} to @var{y} would have a negligible effect on @var{y}
-due to roundoff error. @var{X} may be a real or complex number, but
address@hidden must be real.
address@hidden defun
-
address@hidden is-true x
-Return true if the formula @var{x} represents a true value in
-Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
-or a provably non-zero formula.
address@hidden defun
-
address@hidden reject-arg val pred
-Abort the current function evaluation due to unacceptable argument values.
-This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
-Lisp error which @code{normalize} will trap. The net effect is that the
-function call which led here will be left in symbolic form.
address@hidden defun
-
address@hidden inexact-value
-If Symbolic mode is enabled, this will signal an error that causes
address@hidden to leave the formula in symbolic form, with the message
-``Inexact result.'' (This function has no effect when not in Symbolic mode.)
-Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
address@hidden function will call @code{inexact-value}, which will cause your
-function to be left unsimplified. You may instead wish to call
address@hidden(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
-return the formula @samp{sin(5)} to your function.
address@hidden defun
-
address@hidden overflow
-This signals an error that will be reported as a floating-point overflow.
address@hidden defun
-
address@hidden underflow
-This signals a floating-point underflow.
address@hidden defun
-
address@hidden Computational Lisp Functions, Vector Lisp Functions, Predicates,
Internals
address@hidden Computational Functions
-
address@hidden
-The functions described here do the actual computational work of the
-Calculator. In addition to these, note that any function described in
-the main body of this manual may be called from Lisp; for example, if
-the documentation refers to the @code{calc-sqrt} address@hidden command,
-this means @code{calc-sqrt} is an interactive stack-based square-root
-command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
-is the actual Lisp function for taking square roots.
-
-The functions @code{math-add}, @code{math-sub}, @code{math-mul},
address@hidden, @code{math-mod}, and @code{math-neg} are not included
-in this list, since @code{defmath} allows you to write native Lisp
address@hidden, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
-respectively, instead.
-
address@hidden normalize val
-(Full form: @code{math-normalize}.)
-Reduce the value @var{val} to standard form. For example, if @var{val}
-is a fixnum, it will be converted to a bignum if it is too large, and
-if @var{val} is a bignum it will be normalized by clipping off trailing
-(i.e., most-significant) zero digits and converting to a fixnum if it is
-small. All the various data types are similarly converted to their standard
-forms. Variables are left alone, but function calls are actually evaluated
-in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
-return 6.
-
-If a function call fails, because the function is void or has the wrong
-number of parameters, or because it returns @code{nil} or calls
address@hidden or @code{inexact-result}, @code{normalize} returns
-the formula still in symbolic form.
-
-If the current simplification mode is ``none'' or ``numeric arguments
-only,'' @code{normalize} will act appropriately. However, the more
-powerful simplification modes (like Algebraic Simplification) are
-not handled by @code{normalize}. They are handled by @code{calc-normalize},
-which calls @code{normalize} and possibly some other routines, such
-as @code{simplify} or @code{simplify-units}. Programs generally will
-never call @code{calc-normalize} except when popping or pushing values
-on the stack.
address@hidden defun
-
address@hidden evaluate-expr expr
-Replace all variables in @var{expr} that have values with their values,
-then use @code{normalize} to simplify the result. This is what happens
-when you press the @kbd{=} key interactively.
address@hidden defun
-
address@hidden with-extra-prec n body
-Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
-digits. This is a macro which expands to
-
address@hidden
-(math-normalize
- (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
- @var{body}))
address@hidden smallexample
-
-The surrounding call to @code{math-normalize} causes a floating-point
-result to be rounded down to the original precision afterwards. This
-is important because some arithmetic operations assume a number's
-mantissa contains no more digits than the current precision allows.
address@hidden defmac
-
address@hidden make-frac n d
-Build a fraction @address@hidden:@var{d}}. This is equivalent to calling
address@hidden(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
address@hidden defun
-
address@hidden make-float mant exp
-Build a floating-point value out of @var{mant} and @var{exp}, both
-of which are arbitrary integers. This function will return a
-properly normalized float value, or signal an overflow or underflow
-if @var{exp} is out of range.
address@hidden defun
-
address@hidden make-sdev x sigma
-Build an error form out of @var{x} and the absolute value of @var{sigma}.
-If @var{sigma} is zero, the result is the number @var{x} directly.
-If @var{sigma} is negative or complex, its absolute value is used.
-If @var{x} or @var{sigma} is not a valid type of object for use in
-error forms, this calls @code{reject-arg}.
address@hidden defun
-
address@hidden make-intv mask lo hi
-Build an interval form out of @var{mask} (which is assumed to be an
-integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
address@hidden is greater than @var{hi}, an empty interval form is returned.
-This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
address@hidden defun
-
address@hidden sort-intv mask lo hi
-Build an interval form, similar to @code{make-intv}, except that if
address@hidden is less than @var{hi} they are simply exchanged, and the
-bits of @var{mask} are swapped accordingly.
address@hidden defun
-
address@hidden make-mod n m
-Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
-forms do not allow formulas as their components, if @var{n} or @var{m}
-is not a real number or HMS form the result will be a formula which
-is a call to @code{makemod}, the algebraic version of this function.
address@hidden defun
-
address@hidden float x
-Convert @var{x} to floating-point form. Integers and fractions are
-converted to numerically equivalent floats; components of complex
-numbers, vectors, HMS forms, date forms, error forms, intervals, and
-modulo forms are recursively floated. If the argument is a variable
-or formula, this calls @code{reject-arg}.
address@hidden defun
-
address@hidden compare x y
-Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
address@hidden(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
-0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
-undefined or cannot be determined.
address@hidden defun
-
address@hidden numdigs n
-Return the number of digits of integer @var{n}, effectively
address@hidden(log10(@var{n}))}, but much more efficient. Zero is
-considered to have zero digits.
address@hidden defun
-
address@hidden scale-int x n
-Shift integer @var{x} left @var{n} decimal digits, or right @address@hidden
-digits with truncation toward zero.
address@hidden defun
-
address@hidden scale-rounding x n
-Like @code{scale-int}, except that a right shift rounds to the nearest
-integer rather than truncating.
address@hidden defun
-
address@hidden fixnum n
-Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
-If @var{n} is outside the permissible range for Lisp integers (usually
-24 binary bits) the result is undefined.
address@hidden defun
-
address@hidden sqr x
-Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
address@hidden defun
-
address@hidden quotient x y
-Divide integer @var{x} by integer @var{y}; return an integer quotient
-and discard the remainder. If @var{x} or @var{y} is negative, the
-direction of rounding is undefined.
address@hidden defun
-
address@hidden idiv x y
-Perform an integer division; if @var{x} and @var{y} are both nonnegative
-integers, this uses the @code{quotient} function, otherwise it computes
address@hidden(@var{x}/@var{y})}. Thus the result is well-defined but
-slower than for @code{quotient}.
address@hidden defun
-
address@hidden imod x y
-Divide integer @var{x} by integer @var{y}; return the integer remainder
-and discard the quotient. Like @code{quotient}, this works only for
-integer arguments and is not well-defined for negative arguments.
-For a more well-defined result, use @samp{(% @var{x} @var{y})}.
address@hidden defun
-
address@hidden idivmod x y
-Divide integer @var{x} by integer @var{y}; return a cons cell whose
address@hidden is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
-is @samp{(imod @var{x} @var{y})}.
address@hidden defun
-
address@hidden pow x y
-Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
-also be written @samp{(^ @var{x} @var{y})} or
address@hidden@samp{(expt @var{x} @var{y})}}.
address@hidden defun
-
address@hidden abs-approx x
-Compute a fast approximation to the absolute value of @var{x}. For
-example, for a rectangular complex number the result is the sum of
-the absolute values of the components.
address@hidden defun
-
address@hidden e
address@hidden gamma-const
address@hidden ln-2
address@hidden ln-10
address@hidden phi
address@hidden pi-over-2
address@hidden pi-over-4
address@hidden pi-over-180
address@hidden sqrt-two-pi
address@hidden sqrt-e
address@hidden two-pi
address@hidden pi
-The function @samp{(pi)} computes @samp{pi} to the current precision.
-Other related constant-generating functions are @code{two-pi},
address@hidden, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
address@hidden, @code{sqrt-e}, @code{ln-2}, @code{ln-10}, @code{phi} and
address@hidden Each function returns a floating-point value in the
-current precision, and each uses caching so that all calls after the
-first are essentially free.
address@hidden defun
-
address@hidden math-defcache @var{func} @var{initial} @var{form}
-This macro, usually used as a top-level call like @code{defun} or
address@hidden, defines a new cached constant analogous to @code{pi}, etc.
-It defines a function @code{func} which returns the requested value;
-if @var{initial} is address@hidden it must be a @samp{(float @dots{})}
-form which serves as an initial value for the cache. If @var{func}
-is called when the cache is empty or does not have enough digits to
-satisfy the current precision, the Lisp expression @var{form} is evaluated
-with the current precision increased by four, and the result minus its
-two least significant digits is stored in the cache. For example,
-calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
-digits, rounds it down to 32 digits for future use, then rounds it
-again to 30 digits for use in the present request.
address@hidden defmac
-
address@hidden half-circle
address@hidden quarter-circle
address@hidden full-circle symb
-If the current angular mode is Degrees or HMS, this function returns the
-integer 360. In Radians mode, this function returns either the
-corresponding value in radians to the current precision, or the formula
address@hidden, depending on the Symbolic mode. There are also similar
-function @code{half-circle} and @code{quarter-circle}.
address@hidden defun
-
address@hidden power-of-2 n
-Compute two to the integer power @var{n}, as a (potentially very large)
-integer. Powers of two are cached, so only the first call for a
-particular @var{n} is expensive.
address@hidden defun
-
address@hidden integer-log2 n
-Compute the base-2 logarithm of @var{n}, which must be an integer which
-is a power of two. If @var{n} is not a power of two, this function will
-return @code{nil}.
address@hidden defun
-
address@hidden div-mod a b m
-Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
-there is no solution, or if any of the arguments are not integers.
address@hidden defun
-
address@hidden pow-mod a b m
-Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
address@hidden, and @var{m} are integers, this uses an especially efficient
-algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
address@hidden defun
-
address@hidden isqrt n
-Compute the integer square root of @var{n}. This is the square root
-of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
-If @var{n} is itself an integer, the computation is especially efficient.
address@hidden defun
-
address@hidden to-hms a ang
-Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
-it is the angular mode in which to interpret @var{a}, either @code{deg}
-or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
-is already an HMS form it is returned as-is.
address@hidden defun
-
address@hidden from-hms a ang
-Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
-it is the angular mode in which to express the result, otherwise the
-current angular mode is used. If @var{a} is already a real number, it
-is returned as-is.
address@hidden defun
-
address@hidden to-radians a
-Convert the number or HMS form @var{a} to radians from the current
-angular mode.
address@hidden defun
-
address@hidden from-radians a
-Convert the number @var{a} from radians to the current angular mode.
-If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
address@hidden defun
-
address@hidden to-radians-2 a
-Like @code{to-radians}, except that in Symbolic mode a degrees to
-radians conversion yields a formula like @address@hidden/180}.
address@hidden defun
-
address@hidden from-radians-2 a
-Like @code{from-radians}, except that in Symbolic mode a radians to
-degrees conversion yields a formula like @address@hidden/pi}.
address@hidden defun
-
address@hidden random-digit
-Produce a random base-1000 digit in the range 0 to 999.
address@hidden defun
-
address@hidden random-digits n
-Produce a random @var{n}-digit integer; this will be an integer
-in the interval @samp{[0, address@hidden)}.
address@hidden defun
-
address@hidden random-float
-Produce a random float in the interval @samp{[0, 1)}.
address@hidden defun
-
address@hidden prime-test n iters
-Determine whether the integer @var{n} is prime. Return a list which has
-one of these forms: @samp{(nil @var{f})} means the number is non-prime
-because it was found to be divisible by @var{f}; @samp{(nil)} means it
-was found to be non-prime by table look-up (so no factors are known);
address@hidden(nil unknown)} means it is definitely non-prime but no factors
-are known because @var{n} was large enough that Fermat's probabilistic
-test had to be used; @samp{(t)} means the number is definitely prime;
-and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
-iterations, is @var{p} percent sure that the number is prime. The
address@hidden parameter is the number of Fermat iterations to use, in the
-case that this is necessary. If @code{prime-test} returns ``maybe,''
-you can call it again with the same @var{n} to get a greater certainty;
address@hidden remembers where it left off.
address@hidden defun
-
address@hidden to-simple-fraction f
-If @var{f} is a floating-point number which can be represented exactly
-as a small rational number. return that number, else return @var{f}.
-For example, 0.75 would be converted to 3:4. This function is very
-fast.
address@hidden defun
-
address@hidden to-fraction f tol
-Find a rational approximation to floating-point number @var{f} to within
-a specified tolerance @var{tol}; this corresponds to the algebraic
-function @code{frac}, and can be rather slow.
address@hidden defun
-
address@hidden quarter-integer n
-If @var{n} is an integer or integer-valued float, this function
-returns zero. If @var{n} is a half-integer (i.e., an integer plus
address@hidden:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
-it returns 1 or 3. If @var{n} is anything else, this function
-returns @code{nil}.
address@hidden defun
-
address@hidden Vector Lisp Functions, Symbolic Lisp Functions, Computational
Lisp Functions, Internals
address@hidden Vector Functions
-
address@hidden
-The functions described here perform various operations on vectors and
-matrices.
-
address@hidden math-concat x y
-Do a vector concatenation; this operation is written @address@hidden | @var{y}}
-in a symbolic formula. @xref{Building Vectors}.
address@hidden defun
-
address@hidden vec-length v
-Return the length of vector @var{v}. If @var{v} is not a vector, the
-result is zero. If @var{v} is a matrix, this returns the number of
-rows in the matrix.
address@hidden defun
-
address@hidden mat-dimens m
-Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
-a vector, the result is an empty list. If @var{m} is a plain vector
-but not a matrix, the result is a one-element list containing the length
-of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
-the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
-produce lists of more than two dimensions. Note that the object
address@hidden, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
-and is treated by this and other Calc routines as a plain vector of two
-elements.
address@hidden defun
-
address@hidden dimension-error
-Abort the current function with a message of ``Dimension error.''
-The Calculator will leave the function being evaluated in symbolic
-form; this is really just a special case of @code{reject-arg}.
address@hidden defun
-
address@hidden build-vector args
-Return a Calc vector with @var{args} as elements.
-For example, @samp{(build-vector 1 2 3)} returns the Calc vector
address@hidden, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
address@hidden defun
-
address@hidden make-vec obj dims
-Return a Calc vector or matrix all of whose elements are equal to
address@hidden For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
-filled with 27's.
address@hidden defun
-
address@hidden row-matrix v
-If @var{v} is a plain vector, convert it into a row matrix, i.e.,
-a matrix whose single row is @var{v}. If @var{v} is already a matrix,
-leave it alone.
address@hidden defun
-
address@hidden col-matrix v
-If @var{v} is a plain vector, convert it into a column matrix, i.e., a
-matrix with each element of @var{v} as a separate row. If @var{v} is
-already a matrix, leave it alone.
address@hidden defun
-
address@hidden map-vec f v
-Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
address@hidden(map-vec 'math-floor v)} returns a vector of the floored
components
-of vector @var{v}.
address@hidden defun
-
address@hidden map-vec-2 f a b
-Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
-If @var{a} and @var{b} are vectors of equal length, the result is a
-vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
-for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
address@hidden is a scalar, it is matched with each value of the other vector.
-For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
-with each element increased by one. Note that using @samp{'+} would not
-work here, since @code{defmath} does not expand function names everywhere,
-just where they are in the function position of a Lisp expression.
address@hidden defun
-
address@hidden reduce-vec f v
-Reduce the function @var{f} over the vector @var{v}. For example, if
address@hidden is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30)
40)}.
-If @var{v} is a matrix, this reduces over the rows of @var{v}.
address@hidden defun
-
address@hidden reduce-cols f m
-Reduce the function @var{f} over the columns of matrix @var{m}. For
-example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
-is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
address@hidden defun
-
address@hidden mat-row m n
-Return the @var{n}th row of matrix @var{m}. This is equivalent to
address@hidden(elt m n)}. For a slower but safer version, use @code{mrow}.
-(@xref{Extracting Elements}.)
address@hidden defun
-
address@hidden mat-col m n
-Return the @var{n}th column of matrix @var{m}, in the form of a vector.
-The arguments are not checked for correctness.
address@hidden defun
-
address@hidden mat-less-row m n
-Return a copy of matrix @var{m} with its @var{n}th row deleted. The
-number @var{n} must be in range from 1 to the number of rows in @var{m}.
address@hidden defun
-
address@hidden mat-less-col m n
-Return a copy of matrix @var{m} with its @var{n}th column deleted.
address@hidden defun
-
address@hidden transpose m
-Return the transpose of matrix @var{m}.
address@hidden defun
-
address@hidden flatten-vector v
-Flatten nested vector @var{v} into a vector of scalars. For example,
-if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
address@hidden defun
-
address@hidden copy-matrix m
-If @var{m} is a matrix, return a copy of @var{m}. This maps
address@hidden over the rows of @var{m}; in Lisp terms, each
-element of the result matrix will be @code{eq} to the corresponding
-element of @var{m}, but none of the @code{cons} cells that make up
-the structure of the matrix will be @code{eq}. If @var{m} is a plain
-vector, this is the same as @code{copy-sequence}.
address@hidden defun
-
address@hidden swap-rows m r1 r2
-Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
-other words, unlike most of the other functions described here, this
-function changes @var{m} itself rather than building up a new result
-matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
-is true, with the side effect of exchanging the first two rows of
address@hidden
address@hidden defun
-
address@hidden Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp
Functions, Internals
address@hidden Symbolic Functions
-
address@hidden
-The functions described here operate on symbolic formulas in the
-Calculator.
-
address@hidden calc-prepare-selection num
-Prepare a stack entry for selection operations. If @var{num} is
-omitted, the stack entry containing the cursor is used; otherwise,
-it is the number of the stack entry to use. This function stores
-useful information about the current stack entry into a set of
-variables. @code{calc-selection-cache-num} contains the number of
-the stack entry involved (equal to @var{num} if you specified it);
address@hidden contains the stack entry as a
-list (such as @code{calc-top-list} would return with @code{entry}
-as the selection mode); and @code{calc-selection-cache-comp} contains
-a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
-which allows Calc to relate cursor positions in the buffer with
-their corresponding sub-formulas.
-
-A slight complication arises in the selection mechanism because
-formulas may contain small integers. For example, in the vector
address@hidden, 2, 1]} the first and last elements are @code{eq} to each
-other; selections are recorded as the actual Lisp object that
-appears somewhere in the tree of the whole formula, but storing
address@hidden would falsely select both @code{1}'s in the vector. So
address@hidden also checks the stack entry and
-replaces any plain integers with ``complex number'' lists of the form
address@hidden(cplx @var{n} 0)}. This list will be displayed the same as a
-plain @var{n} and the change will be completely invisible to the
-user, but it will guarantee that no two sub-formulas of the stack
-entry will be @code{eq} to each other. Next time the stack entry
-is involved in a computation, @code{calc-normalize} will replace
-these lists with plain numbers again, again invisibly to the user.
address@hidden defun
-
address@hidden calc-encase-atoms x
-This modifies the formula @var{x} to ensure that each part of the
-formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
-described above. This function may use @code{setcar} to modify
-the formula in-place.
address@hidden defun
-
address@hidden calc-find-selected-part
-Find the smallest sub-formula of the current formula that contains
-the cursor. This assumes @code{calc-prepare-selection} has been
-called already. If the cursor is not actually on any part of the
-formula, this returns @code{nil}.
address@hidden defun
-
address@hidden calc-change-current-selection selection
-Change the currently prepared stack element's selection to
address@hidden, which should be @code{eq} to some sub-formula
-of the stack element, or @code{nil} to unselect the formula.
-The stack element's appearance in the Calc buffer is adjusted
-to reflect the new selection.
address@hidden defun
-
address@hidden calc-find-nth-part expr n
-Return the @var{n}th sub-formula of @var{expr}. This function is used
-by the selection commands, and (unless @kbd{j b} has been used) treats
-sums and products as flat many-element formulas. Thus if @var{expr}
-is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
address@hidden equal to four will return @samp{d}.
address@hidden defun
-
address@hidden calc-find-parent-formula expr part
-Return the sub-formula of @var{expr} which immediately contains
address@hidden If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
-is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
-will return @samp{(c+1)*d}. If @var{part} turns out not to be a
-sub-formula of @var{expr}, the function returns @code{nil}. If
address@hidden is @code{eq} to @var{expr}, the function returns @code{t}.
-This function does not take associativity into account.
address@hidden defun
-
address@hidden calc-find-assoc-parent-formula expr part
-This is the same as @code{calc-find-parent-formula}, except that
-(unless @kbd{j b} has been used) it continues widening the selection
-to contain a complete level of the formula. Given @samp{a} from
address@hidden((a + b) - c) + d}, @code{calc-find-parent-formula} will
-return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
-return the whole expression.
address@hidden defun
-
address@hidden calc-grow-assoc-formula expr part
-This expands sub-formula @var{part} of @var{expr} to encompass a
-complete level of the formula. If @var{part} and its immediate
-parent are not compatible associative operators, or if @kbd{j b}
-has been used, this simply returns @var{part}.
address@hidden defun
-
address@hidden calc-find-sub-formula expr part
-This finds the immediate sub-formula of @var{expr} which contains
address@hidden It returns an index @var{n} such that
address@hidden(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
-If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
-If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
-function does not take associativity into account.
address@hidden defun
-
address@hidden calc-replace-sub-formula expr old new
-This function returns a copy of formula @var{expr}, with the
-sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
address@hidden defun
-
address@hidden simplify expr
-Simplify the expression @var{expr} by applying various algebraic rules.
-This is what the @address@hidden s}} (@code{calc-simplify}) command uses. This
-always returns a copy of the expression; the structure @var{expr} points
-to remains unchanged in memory.
-
-More precisely, here is what @code{simplify} does: The expression is
-first normalized and evaluated by calling @code{normalize}. If any
address@hidden have been defined, they are then applied. Then
-the expression is traversed in a depth-first, bottom-up fashion; at
-each level, any simplifications that can be made are made until no
-further changes are possible. Once the entire formula has been
-traversed in this way, it is compared with the original formula (from
-before the call to @code{normalize}) and, if it has changed,
-the entire procedure is repeated (starting with @code{normalize})
-until no further changes occur. Usually only two iterations are
-needed:@: one to simplify the formula, and another to verify that no
-further simplifications were possible.
address@hidden defun
-
address@hidden simplify-extended expr
-Simplify the expression @var{expr}, with additional rules enabled that
-help do a more thorough job, while not being entirely ``safe'' in all
-circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
-to @samp{x}, which is only valid when @var{x} is positive.) This is
-implemented by temporarily binding the variable @code{math-living-dangerously}
-to @code{t} (using a @code{let} form) and calling @code{simplify}.
-Dangerous simplification rules are written to check this variable
-before taking any action.
address@hidden defun
-
address@hidden simplify-units expr
-Simplify the expression @var{expr}, treating variable names as units
-whenever possible. This works by binding the variable
address@hidden to @code{t} while calling @code{simplify}.
address@hidden defun
-
address@hidden math-defsimplify funcs body
-Register a new simplification rule; this is normally called as a top-level
-form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
-(like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
-applied to the formulas which are calls to the specified function. Or,
address@hidden can be a list of such symbols; the rule applies to all
-functions on the list. The @var{body} is written like the body of a
-function with a single argument called @code{expr}. The body will be
-executed with @code{expr} bound to a formula which is a call to one of
-the functions @var{funcs}. If the function body returns @code{nil}, or
-if it returns a result @code{equal} to the original @code{expr}, it is
-ignored and Calc goes on to try the next simplification rule that applies.
-If the function body returns something different, that new formula is
-substituted for @var{expr} in the original formula.
-
-At each point in the formula, rules are tried in the order of the
-original calls to @code{math-defsimplify}; the search stops after the
-first rule that makes a change. Thus later rules for that same
-function will not have a chance to trigger until the next iteration
-of the main @code{simplify} loop.
-
-Note that, since @code{defmath} is not being used here, @var{body} must
-be written in true Lisp code without the conveniences that @code{defmath}
-provides. If you prefer, you can have @var{body} simply call another
-function (defined with @code{defmath}) which does the real work.
-
-The arguments of a function call will already have been simplified
-before any rules for the call itself are invoked. Since a new argument
-list is consed up when this happens, this means that the rule's body is
-allowed to rearrange the function's arguments destructively if that is
-convenient. Here is a typical example of a simplification rule:
-
address@hidden
-(math-defsimplify calcFunc-arcsinh
- (or (and (math-looks-negp (nth 1 expr))
- (math-neg (list 'calcFunc-arcsinh
- (math-neg (nth 1 expr)))))
- (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
- (or math-living-dangerously
- (math-known-realp (nth 1 (nth 1 expr))))
- (nth 1 (nth 1 expr)))))
address@hidden smallexample
-
-This is really a pair of rules written with one @code{math-defsimplify}
-for convenience; the first replaces @samp{arcsinh(-x)} with
address@hidden(x)}, and the second, which is safe only for real @samp{x},
-replaces @samp{arcsinh(sinh(x))} with @samp{x}.
address@hidden defmac
-
address@hidden common-constant-factor expr
-Check @var{expr} to see if it is a sum of terms all multiplied by the
-same rational value. If so, return this value. If not, return @code{nil}.
-For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
-3 is a common factor of all the terms.
address@hidden defun
-
address@hidden cancel-common-factor expr factor
-Assuming @var{expr} is a sum with @var{factor} as a common factor,
-divide each term of the sum by @var{factor}. This is done by
-destructively modifying parts of @var{expr}, on the assumption that
-it is being used by a simplification rule (where such things are
-allowed; see above). For example, consider this built-in rule for
-square roots:
-
address@hidden
-(math-defsimplify calcFunc-sqrt
- (let ((fac (math-common-constant-factor (nth 1 expr))))
- (and fac (not (eq fac 1))
- (math-mul (math-normalize (list 'calcFunc-sqrt fac))
- (math-normalize
- (list 'calcFunc-sqrt
- (math-cancel-common-factor
- (nth 1 expr) fac)))))))
address@hidden smallexample
address@hidden defun
-
address@hidden frac-gcd a b
-Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
-rational numbers. This is the fraction composed of the GCD of the
-numerators of @var{a} and @var{b}, over the GCD of the denominators.
-It is used by @code{common-constant-factor}. Note that the standard
address@hidden function uses the LCM to combine the denominators.
address@hidden defun
-
address@hidden map-tree func expr many
-Try applying Lisp function @var{func} to various sub-expressions of
address@hidden Initially, call @var{func} with @var{expr} itself as an
-argument. If this returns an expression which is not @code{equal} to
address@hidden, apply @var{func} again until eventually it does return
address@hidden with no changes. Then, if @var{expr} is a function call,
-recursively apply @var{func} to each of the arguments. This keeps going
-until no changes occur anywhere in the expression; this final expression
-is returned by @code{map-tree}. Note that, unlike simplification rules,
address@hidden functions may @emph{not} make destructive changes to
address@hidden If a third argument @var{many} is provided, it is an
-integer which says how many times @var{func} may be applied; the
-default, as described above, is infinitely many times.
address@hidden defun
-
address@hidden compile-rewrites rules
-Compile the rewrite rule set specified by @var{rules}, which should
-be a formula that is either a vector or a variable name. If the latter,
-the compiled rules are saved so that later @code{compile-rules} calls
-for that same variable can return immediately. If there are problems
-with the rules, this function calls @code{error} with a suitable
-message.
address@hidden defun
-
address@hidden apply-rewrites expr crules heads
-Apply the compiled rewrite rule set @var{crules} to the expression
address@hidden This will make only one rewrite and only checks at the
-top level of the expression. The result @code{nil} if no rules
-matched, or if the only rules that matched did not actually change
-the expression. The @var{heads} argument is optional; if is given,
-it should be a list of all function names that (may) appear in
address@hidden The rewrite compiler tags each rule with the
-rarest-looking function name in the rule; if you specify @var{heads},
address@hidden can use this information to narrow its search
-down to just a few rules in the rule set.
address@hidden defun
-
address@hidden rewrite-heads expr
-Compute a @var{heads} list for @var{expr} suitable for use with
address@hidden, as discussed above.
address@hidden defun
-
address@hidden rewrite expr rules many
-This is an all-in-one rewrite function. It compiles the rule set
-specified by @var{rules}, then uses @code{map-tree} to apply the
-rules throughout @var{expr} up to @var{many} (default infinity)
-times.
address@hidden defun
-
address@hidden match-patterns pat vec not-flag
-Given a Calc vector @var{vec} and an uncompiled pattern set or
-pattern set variable @var{pat}, this function returns a new vector
-of all elements of @var{vec} which do (or don't, if @var{not-flag} is
address@hidden) match any of the patterns in @var{pat}.
address@hidden defun
-
address@hidden deriv expr var value symb
-Compute the derivative of @var{expr} with respect to variable @var{var}
-(which may actually be any sub-expression). If @var{value} is specified,
-the derivative is evaluated at the value of @var{var}; otherwise, the
-derivative is left in terms of @var{var}. If the expression contains
-functions for which no derivative formula is known, new derivative
-functions are invented by adding primes to the names; @pxref{Calculus}.
-However, if @var{symb} is address@hidden, the presence of undifferentiable
-functions in @var{expr} instead cancels the whole differentiation, and
address@hidden returns @code{nil} instead.
-
-Derivatives of an @var{n}-argument function can be defined by
-adding a @address@hidden property to the property list
-of the symbol for the function's derivative, which will be the
-function name followed by an apostrophe. The value of the property
-should be a Lisp function; it is called with the same arguments as the
-original function call that is being differentiated. It should return
-a formula for the derivative. For example, the derivative of @code{ln}
-is defined by
-
address@hidden
-(put 'calcFunc-ln\' 'math-derivative-1
- (function (lambda (u) (math-div 1 u))))
address@hidden smallexample
-
-The two-argument @code{log} function has two derivatives,
address@hidden
-(put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
- (function (lambda (x b) ... )))
-(put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
- (function (lambda (x b) ... )))
address@hidden smallexample
address@hidden defun
-
address@hidden tderiv expr var value symb
-Compute the total derivative of @var{expr}. This is the same as
address@hidden, except that variables other than @var{var} are not
-assumed to be constant with respect to @var{var}.
address@hidden defun
-
address@hidden integ expr var low high
-Compute the integral of @var{expr} with respect to @var{var}.
address@hidden, for further details.
address@hidden defun
-
address@hidden math-defintegral funcs body
-Define a rule for integrating a function or functions of one argument;
-this macro is very similar in format to @code{math-defsimplify}.
-The main difference is that here @var{body} is the body of a function
-with a single argument @code{u} which is bound to the argument to the
-function being integrated, not the function call itself. Also, the
-variable of integration is available as @code{math-integ-var}. If
-evaluation of the integral requires doing further integrals, the body
-should call @samp{(math-integral @var{x})} to find the integral of
address@hidden with respect to @code{math-integ-var}; this function returns
address@hidden if the integral could not be done. Some examples:
-
address@hidden
-(math-defintegral calcFunc-conj
- (let ((int (math-integral u)))
- (and int
- (list 'calcFunc-conj int))))
-
-(math-defintegral calcFunc-cos
- (and (equal u math-integ-var)
- (math-from-radians-2 (list 'calcFunc-sin u))))
address@hidden smallexample
-
-In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
-relying on the general integration-by-substitution facility to handle
-cosines of more complicated arguments. An integration rule should return
address@hidden if it can't do the integral; if several rules are defined for
-the same function, they are tried in order until one returns a address@hidden
-result.
address@hidden defmac
-
address@hidden math-defintegral-2 funcs body
-Define a rule for integrating a function or functions of two arguments.
-This is exactly analogous to @code{math-defintegral}, except that @var{body}
-is written as the body of a function with two arguments, @var{u} and
address@hidden
address@hidden defmac
-
address@hidden solve-for lhs rhs var full
-Attempt to solve the equation @address@hidden = @var{rhs}} by isolating
-the variable @var{var} on the lefthand side; return the resulting righthand
-side, or @code{nil} if the equation cannot be solved. The variable
address@hidden must appear at least once in @var{lhs} or @var{rhs}. Note that
-the return value is a formula which does not contain @var{var}; this is
-different from the user-level @code{solve} and @code{finv} functions,
-which return a rearranged equation or a functional inverse, respectively.
-If @var{full} is address@hidden, a full solution including dummy signs
-and dummy integers will be produced. User-defined inverses are provided
-as properties in a manner similar to derivatives:
-
address@hidden
-(put 'calcFunc-ln 'math-inverse
- (function (lambda (x) (list 'calcFunc-exp x))))
address@hidden smallexample
-
-This function can call @samp{(math-solve-get-sign @var{x})} to create
-a new arbitrary sign variable, returning @var{x} times that sign, and
address@hidden(math-solve-get-int @var{x})} to create a new arbitrary integer
-variable multiplied by @var{x}. These functions simply return @var{x}
-if the caller requested a non-``full'' solution.
address@hidden defun
-
address@hidden solve-eqn expr var full
-This version of @code{solve-for} takes an expression which will
-typically be an equation or inequality. (If it is not, it will be
-interpreted as the equation @address@hidden = 0}.) It returns an
-equation or inequality, or @code{nil} if no solution could be found.
address@hidden defun
-
address@hidden solve-system exprs vars full
-This function solves a system of equations. Generally, @var{exprs}
-and @var{vars} will be vectors of equal length.
address@hidden Systems of Equations}, for other options.
address@hidden defun
-
address@hidden expr-contains expr var
-Returns a address@hidden value if @var{var} occurs as a subexpression
-of @var{expr}.
-
-This function might seem at first to be identical to
address@hidden The key difference is that
address@hidden uses @code{equal} to test for matches, whereas
address@hidden uses @code{eq}. In the formula
address@hidden(a, a)}, the two @samp{a}s will be @code{equal} but not
address@hidden to each other.
address@hidden defun
-
address@hidden expr-contains-count expr var
-Returns the number of occurrences of @var{var} as a subexpression
-of @var{expr}, or @code{nil} if there are no occurrences.
address@hidden defun
-
address@hidden expr-depends expr var
-Returns true if @var{expr} refers to any variable the occurs in @var{var}.
-In other words, it checks if @var{expr} and @var{var} have any variables
-in common.
address@hidden defun
-
address@hidden expr-contains-vars expr
-Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
-contains only constants and functions with constant arguments.
address@hidden defun
-
address@hidden expr-subst expr old new
-Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
-by @var{new}. This treats @code{lambda} forms specially with respect
-to the dummy argument variables, so that the effect is always to return
address@hidden evaluated at @var{old} = @var{new}.
address@hidden defun
-
address@hidden multi-subst expr old new
-This is like @code{expr-subst}, except that @var{old} and @var{new}
-are lists of expressions to be substituted simultaneously. If one
-list is shorter than the other, trailing elements of the longer list
-are ignored.
address@hidden defun
-
address@hidden expr-weight expr
-Returns the ``weight'' of @var{expr}, basically a count of the total
-number of objects and function calls that appear in @var{expr}. For
-``primitive'' objects, this will be one.
address@hidden defun
-
address@hidden expr-height expr
-Returns the ``height'' of @var{expr}, which is the deepest level to
-which function calls are nested. (Note that @address@hidden + @var{b}}
-counts as a function call.) For primitive objects, this returns zero.
address@hidden defun
-
address@hidden polynomial-p expr var
-Check if @var{expr} is a polynomial in variable (or sub-expression)
address@hidden If so, return the degree of the polynomial, that is, the
-highest power of @var{var} that appears in @var{expr}. For example,
-for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
address@hidden unless @var{expr}, when expanded out by @kbd{a x}
-(@code{calc-expand}), would consist of a sum of terms in which @var{var}
-appears only raised to nonnegative integer powers. Note that if
address@hidden does not occur in @var{expr}, then @var{expr} is considered
-a polynomial of degree 0.
address@hidden defun
-
address@hidden is-polynomial expr var degree loose
-Check if @var{expr} is a polynomial in variable or sub-expression
address@hidden, and, if so, return a list representation of the polynomial
-where the elements of the list are coefficients of successive powers of
address@hidden: @address@hidden + @var{b} x + @var{c} x^3} would produce the
-list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
-produce the list @samp{(1 2 1)}. The highest element of the list will
-be non-zero, with the special exception that if @var{expr} is the
-constant zero, the returned value will be @samp{(0)}. Return @code{nil}
-if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
-specified, this will not consider polynomials of degree higher than that
-value. This is a good precaution because otherwise an input of
address@hidden(x+1)^1000} will cause a huge coefficient list to be built. If
address@hidden is address@hidden, then a looser definition of a polynomial
-is used in which coefficients are no longer required not to depend on
address@hidden, but are only required not to take the form of polynomials
-themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
-polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
-x))}. The result will never be @code{nil} in loose mode, since any
-expression can be interpreted as a ``constant'' loose polynomial.
address@hidden defun
-
address@hidden polynomial-base expr pred
-Check if @var{expr} is a polynomial in any variable that occurs in it;
-if so, return that variable. (If @var{expr} is a multivariate polynomial,
-this chooses one variable arbitrarily.) If @var{pred} is specified, it should
-be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
-and which should return true if @code{mpb-top-expr} (a global name for
-the original @var{expr}) is a suitable polynomial in @var{subexpr}.
-The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
-you can use @var{pred} to specify additional conditions. Or, you could
-have @var{pred} build up a list of every suitable @var{subexpr} that
-is found.
address@hidden defun
-
address@hidden poly-simplify poly
-Simplify polynomial coefficient list @var{poly} by (destructively)
-clipping off trailing zeros.
address@hidden defun
-
address@hidden poly-mix a ac b bc
-Mix two polynomial lists @var{a} and @var{b} (in the form returned by
address@hidden) in a linear combination with coefficient expressions
address@hidden and @var{bc}. The result is a (not necessarily simplified)
-polynomial list representing @address@hidden @var{a} + @var{bc} @var{b}}.
address@hidden defun
-
address@hidden poly-mul a b
-Multiply two polynomial coefficient lists @var{a} and @var{b}. The
-result will be in simplified form if the inputs were simplified.
address@hidden defun
-
address@hidden build-polynomial-expr poly var
-Construct a Calc formula which represents the polynomial coefficient
-list @var{poly} applied to variable @var{var}. The @kbd{a c}
-(@code{calc-collect}) command uses @code{is-polynomial} to turn an
-expression into a coefficient list, then @code{build-polynomial-expr}
-to turn the list back into an expression in regular form.
address@hidden defun
-
address@hidden check-unit-name var
-Check if @var{var} is a variable which can be interpreted as a unit
-name. If so, return the units table entry for that unit. This
-will be a list whose first element is the unit name (not counting
-prefix characters) as a symbol and whose second element is the
-Calc expression which defines the unit. (Refer to the Calc sources
-for details on the remaining elements of this list.) If @var{var}
-is not a variable or is not a unit name, return @code{nil}.
address@hidden defun
-
address@hidden units-in-expr-p expr sub-exprs
-Return true if @var{expr} contains any variables which can be
-interpreted as units. If @var{sub-exprs} is @code{t}, the entire
-expression is searched. If @var{sub-exprs} is @code{nil}, this
-checks whether @var{expr} is directly a units expression.
address@hidden defun
-
address@hidden single-units-in-expr-p expr
-Check whether @var{expr} contains exactly one units variable. If so,
-return the units table entry for the variable. If @var{expr} does
-not contain any units, return @code{nil}. If @var{expr} contains
-two or more units, return the symbol @code{wrong}.
address@hidden defun
-
address@hidden to-standard-units expr which
-Convert units expression @var{expr} to base units. If @var{which}
-is @code{nil}, use Calc's native base units. Otherwise, @var{which}
-can specify a units system, which is a list of two-element lists,
-where the first element is a Calc base symbol name and the second
-is an expression to substitute for it.
address@hidden defun
-
address@hidden remove-units expr
-Return a copy of @var{expr} with all units variables replaced by ones.
-This expression is generally normalized before use.
address@hidden defun
-
address@hidden extract-units expr
-Return a copy of @var{expr} with everything but units variables replaced
-by ones.
address@hidden defun
-
address@hidden Formatting Lisp Functions, Hooks, Symbolic Lisp Functions,
Internals
address@hidden I/O and Formatting Functions
-
address@hidden
-The functions described here are responsible for parsing and formatting
-Calc numbers and formulas.
-
address@hidden calc-eval str sep arg1 arg2 @dots{}
-This is the simplest interface to the Calculator from another Lisp program.
address@hidden Calc from Your Programs}.
address@hidden defun
-
address@hidden read-number str
-If string @var{str} contains a valid Calc number, either integer,
-fraction, float, or HMS form, this function parses and returns that
-number. Otherwise, it returns @code{nil}.
address@hidden defun
-
address@hidden read-expr str
-Read an algebraic expression from string @var{str}. If @var{str} does
-not have the form of a valid expression, return a list of the form
address@hidden(error @var{pos} @var{msg})} where @var{pos} is an integer index
-into @var{str} of the general location of the error, and @var{msg} is
-a string describing the problem.
address@hidden defun
-
address@hidden read-exprs str
-Read a list of expressions separated by commas, and return it as a
-Lisp list. If an error occurs in any expressions, an error list as
-shown above is returned instead.
address@hidden defun
-
address@hidden calc-do-alg-entry initial prompt no-norm
-Read an algebraic formula or formulas using the minibuffer. All
-conventions of regular algebraic entry are observed. The return value
-is a list of Calc formulas; there will be more than one if the user
-entered a list of values separated by commas. The result is @code{nil}
-if the user presses Return with a blank line. If @var{initial} is
-given, it is a string which the minibuffer will initially contain.
-If @var{prompt} is given, it is the prompt string to use; the default
-is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
-be returned exactly as parsed; otherwise, they will be passed through
address@hidden first.
-
-To support the use of @kbd{$} characters in the algebraic entry, use
address@hidden to bind @code{calc-dollar-values} to a list of the values
-to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
address@hidden to 0. Upon return, @code{calc-dollar-used}
-will have been changed to the highest number of consecutive @kbd{$}s
-that actually appeared in the input.
address@hidden defun
-
address@hidden format-number a
-Convert the real or complex number or HMS form @var{a} to string form.
address@hidden defun
-
address@hidden format-flat-expr a prec
-Convert the arbitrary Calc number or formula @var{a} to string form,
-in the style used by the trail buffer and the @code{calc-edit} command.
-This is a simple format designed
-mostly to guarantee the string is of a form that can be re-parsed by
address@hidden Most formatting modes, such as digit grouping,
-complex number format, and point character, are ignored to ensure the
-result will be re-readable. The @var{prec} parameter is normally 0; if
-you pass a large integer like 1000 instead, the expression will be
-surrounded by parentheses unless it is a plain number or variable name.
address@hidden defun
-
address@hidden format-nice-expr a width
-This is like @code{format-flat-expr} (with @var{prec} equal to 0),
-except that newlines will be inserted to keep lines down to the
-specified @var{width}, and vectors that look like matrices or rewrite
-rules are written in a pseudo-matrix format. The @code{calc-edit}
-command uses this when only one stack entry is being edited.
address@hidden defun
-
address@hidden format-value a width
-Convert the Calc number or formula @var{a} to string form, using the
-format seen in the stack buffer. Beware the string returned may
-not be re-readable by @code{read-expr}, for example, because of digit
-grouping. Multi-line objects like matrices produce strings that
-contain newline characters to separate the lines. The @var{w}
-parameter, if given, is the target window size for which to format
-the expressions. If @var{w} is omitted, the width of the Calculator
-window is used.
address@hidden defun
-
address@hidden compose-expr a prec
-Format the Calc number or formula @var{a} according to the current
-language mode, returning a ``composition.'' To learn about the
-structure of compositions, see the comments in the Calc source code.
-You can specify the format of a given type of function call by putting
-a @address@hidden property on the function's symbol,
-whose value is a Lisp function that takes @var{a} and @var{prec} as
-arguments and returns a composition. Here @var{lang} is a language
-mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
address@hidden, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
-In Big mode, Calc actually tries @code{math-compose-big} first, then
-tries @code{math-compose-normal}. If this property does not exist,
-or if the function returns @code{nil}, the function is written in the
-normal function-call notation for that language.
address@hidden defun
-
address@hidden composition-to-string c w
-Convert a composition structure returned by @code{compose-expr} into
-a string. Multi-line compositions convert to strings containing
-newline characters. The target window size is given by @var{w}.
-The @code{format-value} function basically calls @code{compose-expr}
-followed by @code{composition-to-string}.
address@hidden defun
-
address@hidden comp-width c
-Compute the width in characters of composition @var{c}.
address@hidden defun
-
address@hidden comp-height c
-Compute the height in lines of composition @var{c}.
address@hidden defun
-
address@hidden comp-ascent c
-Compute the portion of the height of composition @var{c} which is on or
-above the baseline. For a one-line composition, this will be one.
address@hidden defun
-
address@hidden comp-descent c
-Compute the portion of the height of composition @var{c} which is below
-the baseline. For a one-line composition, this will be zero.
address@hidden defun
-
address@hidden comp-first-char c
-If composition @var{c} is a ``flat'' composition, return the first
-(leftmost) character of the composition as an integer. Otherwise,
-return @code{nil}.
address@hidden defun
-
address@hidden comp-last-char c
-If composition @var{c} is a ``flat'' composition, return the last
-(rightmost) character, otherwise return @code{nil}.
address@hidden defun
-
address@hidden @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
address@hidden @subsubsection Lisp Variables
address@hidden
address@hidden @noindent
address@hidden (This section is currently unfinished.)
-
address@hidden Hooks, , Formatting Lisp Functions, Internals
address@hidden Hooks
-
address@hidden
-Hooks are variables which contain Lisp functions (or lists of functions)
-which are called at various times. Calc defines a number of hooks
-that help you to customize it in various ways. Calc uses the Lisp
-function @code{run-hooks} to invoke the hooks shown below. Several
-other customization-related variables are also described here.
-
address@hidden calc-load-hook
-This hook is called at the end of @file{calc.el}, after the file has
-been loaded, before any functions in it have been called, but after
address@hidden and similar variables have been set up.
address@hidden defvar
-
address@hidden calc-ext-load-hook
-This hook is called at the end of @file{calc-ext.el}.
address@hidden defvar
-
address@hidden calc-start-hook
-This hook is called as the last step in a @kbd{M-x calc} command.
-At this point, the Calc buffer has been created and initialized if
-necessary, the Calc window and trail window have been created,
-and the ``Welcome to Calc'' message has been displayed.
address@hidden defvar
-
address@hidden calc-mode-hook
-This hook is called when the Calc buffer is being created. Usually
-this will only happen once per Emacs session. The hook is called
-after Emacs has switched to the new buffer, the mode-settings file
-has been read if necessary, and all other buffer-local variables
-have been set up. After this hook returns, Calc will perform a
address@hidden operation, set up the mode line display, then
-evaluate any deferred @code{calc-define} properties that have not
-been evaluated yet.
address@hidden defvar
-
address@hidden calc-trail-mode-hook
-This hook is called when the Calc Trail buffer is being created.
-It is called as the very last step of setting up the Trail buffer.
-Like @code{calc-mode-hook}, this will normally happen only once
-per Emacs session.
address@hidden defvar
-
address@hidden calc-end-hook
-This hook is called by @code{calc-quit}, generally because the user
-presses @kbd{q} or @kbd{C-x * c} while in Calc. The Calc buffer will
-be the current buffer. The hook is called as the very first
-step, before the Calc window is destroyed.
address@hidden defvar
-
address@hidden calc-window-hook
-If this hook is address@hidden, it is called to create the Calc window.
-Upon return, this new Calc window should be the current window.
-(The Calc buffer will already be the current buffer when the
-hook is called.) If the hook is not defined, Calc will
-generally use @code{split-window}, @code{set-window-buffer},
-and @code{select-window} to create the Calc window.
address@hidden defvar
-
address@hidden calc-trail-window-hook
-If this hook is address@hidden, it is called to create the Calc Trail
-window. The variable @code{calc-trail-buffer} will contain the buffer
-which the window should use. Unlike @code{calc-window-hook}, this hook
-must @emph{not} switch into the new window.
address@hidden defvar
-
address@hidden calc-embedded-mode-hook
-This hook is called the first time that Embedded mode is entered.
address@hidden defvar
-
address@hidden calc-embedded-new-buffer-hook
-This hook is called each time that Embedded mode is entered in a
-new buffer.
address@hidden defvar
-
address@hidden calc-embedded-new-formula-hook
-This hook is called each time that Embedded mode is enabled for a
-new formula.
address@hidden defvar
-
address@hidden calc-edit-mode-hook
-This hook is called by @code{calc-edit} (and the other ``edit''
-commands) when the temporary editing buffer is being created.
-The buffer will have been selected and set up to be in
address@hidden, but will not yet have been filled with
-text. (In fact it may still have leftover text from a previous
address@hidden command.)
address@hidden defvar
-
address@hidden calc-mode-save-hook
-This hook is called by the @code{calc-save-modes} command,
-after Calc's own mode features have been inserted into the
-Calc init file and just before the ``End of mode settings''
-message is inserted.
address@hidden defvar
-
address@hidden calc-reset-hook
-This hook is called after @kbd{C-x * 0} (@code{calc-reset}) has
-reset all modes. The Calc buffer will be the current buffer.
address@hidden defvar
-
address@hidden calc-other-modes
-This variable contains a list of strings. The strings are
-concatenated at the end of the modes portion of the Calc
-mode line (after standard modes such as ``Deg'', ``Inv'' and
-``Hyp''). Each string should be a short, single word followed
-by a space. The variable is @code{nil} by default.
address@hidden defvar
-
address@hidden calc-mode-map
-This is the keymap that is used by Calc mode. The best time
-to adjust it is probably in a @code{calc-mode-hook}. If the
-Calc extensions package (@file{calc-ext.el}) has not yet been
-loaded, many of these keys will be bound to @code{calc-missing-key},
-which is a command that loads the extensions package and
-``retypes'' the key. If your @code{calc-mode-hook} rebinds
-one of these keys, it will probably be overridden when the
-extensions are loaded.
address@hidden defvar
-
address@hidden calc-digit-map
-This is the keymap that is used during numeric entry. Numeric
-entry uses the minibuffer, but this map binds every non-numeric
-key to @code{calcDigit-nondigit} which generally calls
address@hidden and ``retypes'' the key.
address@hidden defvar
-
address@hidden calc-alg-ent-map
-This is the keymap that is used during algebraic entry. This is
-mostly a copy of @code{minibuffer-local-map}.
address@hidden defvar
-
address@hidden calc-store-var-map
-This is the keymap that is used during entry of variable names for
-commands like @code{calc-store} and @code{calc-recall}. This is
-mostly a copy of @code{minibuffer-local-completion-map}.
address@hidden defvar
-
address@hidden calc-edit-mode-map
-This is the (sparse) keymap used by @code{calc-edit} and other
-temporary editing commands. It binds @key{RET}, @key{LFD},
-and @kbd{C-c C-c} to @code{calc-edit-finish}.
address@hidden defvar
-
address@hidden calc-mode-var-list
-This is a list of variables which are saved by @code{calc-save-modes}.
-Each entry is a list of two items, the variable (as a Lisp symbol)
-and its default value. When modes are being saved, each variable
-is compared with its default value (using @code{equal}) and any
-non-default variables are written out.
address@hidden defvar
-
address@hidden calc-local-var-list
-This is a list of variables which should be buffer-local to the
-Calc buffer. Each entry is a variable name (as a Lisp symbol).
-These variables also have their default values manipulated by
-the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
-Since @code{calc-mode-hook} is called after this list has been
-used the first time, your hook should add a variable to the
-list and also call @code{make-local-variable} itself.
address@hidden defvar
-
address@hidden Copying, GNU Free Documentation License, Programming, Top
address@hidden GNU GENERAL PUBLIC LICENSE
address@hidden gpl.texi
-
address@hidden GNU Free Documentation License, Customizing Calc, Copying, Top
address@hidden GNU Free Documentation License
address@hidden doclicense.texi
-
address@hidden Customizing Calc, Reporting Bugs, GNU Free Documentation
License, Top
address@hidden Customizing Calc
-
-The usual prefix for Calc is the key sequence @kbd{C-x *}. If you wish
-to use a different prefix, you can put
-
address@hidden
-(global-set-key "NEWPREFIX" 'calc-dispatch)
address@hidden example
-
address@hidden
-in your .emacs file.
-(@xref{Key Bindings,,Customizing Key Bindings,emacs,
-The GNU Emacs Manual}, for more information on binding keys.)
-A convenient way to start Calc is with @kbd{C-x * *}; to make it equally
-convenient for users who use a different prefix, the prefix can be
-followed by @kbd{=}, @kbd{&}, @kbd{#}, @kbd{\}, @kbd{/}, @kbd{+} or
address@hidden as well as @kbd{*} to start Calc, and so in many cases the last
-character of the prefix can simply be typed twice.
-
-Calc is controlled by many variables, most of which can be reset
-from within Calc. Some variables are less involved with actual
-calculation, and can be set outside of Calc using Emacs's
-customization facilities. These variables are listed below.
-Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
-will bring up a buffer in which the variable's value can be redefined.
-Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
-contains all of Calc's customizable variables. (These variables can
-also be reset by putting the appropriate lines in your .emacs file;
address@hidden File, ,Init File, emacs, The GNU Emacs Manual}.)
-
-Some of the customizable variables are regular expressions. A regular
-expression is basically a pattern that Calc can search for.
-See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs
Manual}
-to see how regular expressions work.
-
address@hidden calc-settings-file
-The variable @code{calc-settings-file} holds the file name in
-which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
-definitions.
-If @code{calc-settings-file} is not your user init file (typically
address@hidden/.emacs}) and if the variable @code{calc-loaded-settings-file} is
address@hidden, then Calc will automatically load your settings file (if it
-exists) the first time Calc is invoked.
-
-The default value for this variable is @code{"~/.calc.el"}.
address@hidden defvar
-
address@hidden calc-gnuplot-name
-See @address@hidden
-The variable @code{calc-gnuplot-name} should be the name of the
-GNUPLOT program (a string). If you have GNUPLOT installed on your
-system but Calc is unable to find it, you may need to set this
-variable. You may also need to set some Lisp variables to show Calc how
-to run GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} .
-The default value of @code{calc-gnuplot-name} is @code{"gnuplot"}.
address@hidden defvar
-
address@hidden calc-gnuplot-plot-command
address@hidden calc-gnuplot-print-command
-See @ref{Devices, ,Graphical address@hidden
-The variables @code{calc-gnuplot-plot-command} and
address@hidden represent system commands to
-display and print the output of GNUPLOT, respectively. These may be
address@hidden if no command is necessary, or strings which can include
address@hidden to signify the name of the file to be displayed or printed.
-Or, these variables may contain Lisp expressions which are evaluated
-to display or print the output.
-
-The default value of @code{calc-gnuplot-plot-command} is @code{nil},
-and the default value of @code{calc-gnuplot-print-command} is
address@hidden"lp %s"}.
address@hidden defvar
-
address@hidden calc-language-alist
-See @ref{Basic Embedded address@hidden
-The variable @code{calc-language-alist} controls the languages that
-Calc will associate with major modes. When Calc embedded mode is
-enabled, it will try to use the current major mode to
-determine what language should be used. (This can be overridden using
-Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
-The variable @code{calc-language-alist} consists of a list of pairs of
-the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
address@hidden(latex-mode . latex)} is one such pair. If Calc embedded is
-activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
-to use the language @var{LANGUAGE}.
-
-The default value of @code{calc-language-alist} is
address@hidden
- ((latex-mode . latex)
- (tex-mode . tex)
- (plain-tex-mode . tex)
- (context-mode . tex)
- (nroff-mode . eqn)
- (pascal-mode . pascal)
- (c-mode . c)
- (c++-mode . c)
- (fortran-mode . fortran)
- (f90-mode . fortran))
address@hidden example
address@hidden defvar
-
address@hidden calc-embedded-announce-formula
address@hidden calc-embedded-announce-formula-alist
-See @ref{Customizing Embedded address@hidden
-The variable @code{calc-embedded-announce-formula} helps determine
-what formulas @kbd{C-x * a} will activate in a buffer. It is a
-regular expression, and when activating embedded formulas with
address@hidden * a}, it will tell Calc that what follows is a formula to be
-activated. (Calc also uses other patterns to find formulas, such as
address@hidden>} and @samp{:=}.)
-
-The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
-for @samp{%Embed} followed by any number of lines beginning with
address@hidden and a space.
-
-The variable @code{calc-embedded-announce-formula-alist} is used to
-set @code{calc-embedded-announce-formula} to different regular
-expressions depending on the major mode of the editing buffer.
-It consists of a list of pairs of the form @code{(@var{MAJOR-MODE} .
address@hidden)}, and its default value is
address@hidden
- ((c++-mode . "//Embed\n\\(// .*\n\\)*")
- (c-mode . "/\\*Embed\\*/\n\\(/\\* .*\\*/\n\\)*")
- (f90-mode . "!Embed\n\\(! .*\n\\)*")
- (fortran-mode . "C Embed\n\\(C .*\n\\)*")
- (html-helper-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
- (html-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
- (nroff-mode . "\\\\\"Embed\n\\(\\\\\" .*\n\\)*")
- (pascal-mode . "@address@hidden(@address@hidden)*")
- (sgml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
- (xml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
- (texinfo-mode . "@@c Embed\n\\(@@c .*\n\\)*"))
address@hidden example
-Any major modes added to @code{calc-embedded-announce-formula-alist}
-should also be added to @code{calc-embedded-open-close-plain-alist}
-and @code{calc-embedded-open-close-mode-alist}.
address@hidden defvar
-
address@hidden calc-embedded-open-formula
address@hidden calc-embedded-close-formula
address@hidden calc-embedded-open-close-formula-alist
-See @ref{Customizing Embedded address@hidden
-The variables @code{calc-embedded-open-formula} and
address@hidden control the region that Calc will
-activate as a formula when Embedded mode is entered with @kbd{C-x * e}.
-They are regular expressions;
-Calc normally scans backward and forward in the buffer for the
-nearest text matching these regular expressions to be the ``formula
-delimiters''.
-
-The simplest delimiters are blank lines. Other delimiters that
-Embedded mode understands by default are:
address@hidden
address@hidden
-The @TeX{} and address@hidden math delimiters @samp{$ $}, @samp{$$ $$},
address@hidden \]}, and @samp{\( \)};
address@hidden
-Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
address@hidden
-Lines beginning with @samp{@@} (Texinfo delimiters).
address@hidden
-Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
address@hidden
-Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
address@hidden enumerate
-
-The variable @code{calc-embedded-open-close-formula-alist} is used to
-set @code{calc-embedded-open-formula} and
address@hidden to different regular
-expressions depending on the major mode of the editing buffer.
-It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-FORMULA-REGEXP}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
-
address@hidden calc-embedded-open-word
address@hidden calc-embedded-close-word
address@hidden calc-embedded-open-close-word-alist
-See @ref{Customizing Embedded address@hidden
-The variables @code{calc-embedded-open-word} and
address@hidden control the region that Calc will
-activate when Embedded mode is entered with @kbd{C-x * w}. They are
-regular expressions.
-
-The default values of @code{calc-embedded-open-word} and
address@hidden are @code{"^\\|[^-+0-9.eE]"} and
address@hidden"$\\|[^-+0-9.eE]"} respectively.
-
-The variable @code{calc-embedded-open-close-word-alist} is used to
-set @code{calc-embedded-open-word} and
address@hidden to different regular
-expressions depending on the major mode of the editing buffer.
-It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-WORD-REGEXP}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
-
address@hidden calc-embedded-open-plain
address@hidden calc-embedded-close-plain
address@hidden calc-embedded-open-close-plain-alist
-See @ref{Customizing Embedded address@hidden
-The variables @code{calc-embedded-open-plain} and
address@hidden are used to delimit ``plain''
-formulas. Note that these are actual strings, not regular
-expressions, because Calc must be able to write these string into a
-buffer as well as to recognize them.
-
-The default string for @code{calc-embedded-open-plain} is
address@hidden"%%% "}, note the trailing space. The default string for
address@hidden is @code{" %%%\n"}, without
-the trailing newline here, the first line of a Big mode formula
-that followed might be shifted over with respect to the other lines.
-
-The variable @code{calc-embedded-open-close-plain-alist} is used to
-set @code{calc-embedded-open-plain} and
address@hidden to different strings
-depending on the major mode of the editing buffer.
-It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-PLAIN-STRING}
address@hidden)}, and its default value is
address@hidden
- ((c++-mode "// %% " " %%\n")
- (c-mode "/* %% " " %% */\n")
- (f90-mode "! %% " " %%\n")
- (fortran-mode "C %% " " %%\n")
- (html-helper-mode "<!-- %% " " %% -->\n")
- (html-mode "<!-- %% " " %% -->\n")
- (nroff-mode "\\\" %% " " %%\n")
- (pascal-mode "@{%% " " address@hidden")
- (sgml-mode "<!-- %% " " %% -->\n")
- (xml-mode "<!-- %% " " %% -->\n")
- (texinfo-mode "@@c %% " " %%\n"))
address@hidden example
-Any major modes added to @code{calc-embedded-open-close-plain-alist}
-should also be added to @code{calc-embedded-announce-formula-alist}
-and @code{calc-embedded-open-close-mode-alist}.
address@hidden defvar
-
address@hidden calc-embedded-open-new-formula
address@hidden calc-embedded-close-new-formula
address@hidden calc-embedded-open-close-new-formula-alist
-See @ref{Customizing Embedded address@hidden
-The variables @code{calc-embedded-open-new-formula} and
address@hidden are strings which are
-inserted before and after a new formula when you type @kbd{C-x * f}.
-
-The default value of @code{calc-embedded-open-new-formula} is
address@hidden"\n\n"}. If this string begins with a newline character and the
address@hidden * f} is typed at the beginning of a line, @kbd{C-x * f} will skip
-this first newline to avoid introducing unnecessary blank lines in the
-file. The default value of @code{calc-embedded-close-new-formula} is
-also @code{"\n\n"}. The final newline is omitted by @address@hidden * f}}
-if typed at the end of a line. (It follows that if @kbd{C-x * f} is
-typed on a blank line, both a leading opening newline and a trailing
-closing newline are omitted.)
-
-The variable @code{calc-embedded-open-close-new-formula-alist} is used to
-set @code{calc-embedded-open-new-formula} and
address@hidden to different strings
-depending on the major mode of the editing buffer.
-It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-NEW-FORMULA-STRING}
address@hidden)}, and its default value is
address@hidden
address@hidden defvar
-
address@hidden calc-embedded-open-mode
address@hidden calc-embedded-close-mode
address@hidden calc-embedded-open-close-mode-alist
-See @ref{Customizing Embedded address@hidden
-The variables @code{calc-embedded-open-mode} and
address@hidden are strings which Calc will place before
-and after any mode annotations that it inserts. Calc never scans for
-these strings; Calc always looks for the annotation itself, so it is not
-necessary to add them to user-written annotations.
-
-The default value of @code{calc-embedded-open-mode} is @code{"% "}
-and the default value of @code{calc-embedded-close-mode} is
address@hidden"\n"}.
-If you change the value of @code{calc-embedded-close-mode}, it is a good
-idea still to end with a newline so that mode annotations will appear on
-lines by themselves.
-
-The variable @code{calc-embedded-open-close-mode-alist} is used to
-set @code{calc-embedded-open-mode} and
address@hidden to different strings
-expressions depending on the major mode of the editing buffer.
-It consists of a list of lists of the form
address@hidden(@var{MAJOR-MODE} @var{OPEN-MODE-STRING}
address@hidden)}, and its default value is
address@hidden
- ((c++-mode "// " "\n")
- (c-mode "/* " " */\n")
- (f90-mode "! " "\n")
- (fortran-mode "C " "\n")
- (html-helper-mode "<!-- " " -->\n")
- (html-mode "<!-- " " -->\n")
- (nroff-mode "\\\" " "\n")
- (pascal-mode "@{ " " @}\n")
- (sgml-mode "<!-- " " -->\n")
- (xml-mode "<!-- " " -->\n")
- (texinfo-mode "@@c " "\n"))
address@hidden example
-Any major modes added to @code{calc-embedded-open-close-mode-alist}
-should also be added to @code{calc-embedded-announce-formula-alist}
-and @code{calc-embedded-open-close-plain-alist}.
address@hidden defvar
-
address@hidden calc-multiplication-has-precedence
-The variable @code{calc-multiplication-has-precedence} determines
-whether multiplication has precedence over division in algebraic formulas
-in normal language modes. If @code{calc-multiplication-has-precedence}
-is address@hidden, then multiplication has precedence, and so for
-example @samp{a/b*c} will be interpreted as @samp{a/(b*c)}. If
address@hidden is @code{nil}, then
-multiplication has the same precedence as division, and so for example
address@hidden/b*c} will be interpreted as @samp{(a/b)*c}. The default value
-of @code{calc-multiplication-has-precedence} is @code{t}.
address@hidden defvar
-
address@hidden Reporting Bugs, Summary, Customizing Calc, Top
address@hidden Reporting Bugs
-
address@hidden
-If you find a bug in Calc, send e-mail to Jay Belanger,
-
address@hidden
-jay.p.belanger@@gmail.com
address@hidden example
-
address@hidden
-There is an automatic command @kbd{M-x report-calc-bug} which helps
-you to report bugs. This command prompts you for a brief subject
-line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
-send your mail. Make sure your subject line indicates that you are
-reporting a Calc bug; this command sends mail to the maintainer's
-regular mailbox.
-
-If you have suggestions for additional features for Calc, please send
-them. Some have dared to suggest that Calc is already top-heavy with
-features; this obviously cannot be the case, so if you have ideas, send
-them right in.
-
-At the front of the source file, @file{calc.el}, is a list of ideas for
-future work. If any enthusiastic souls wish to take it upon themselves
-to work on these, please send a message (using @kbd{M-x report-calc-bug})
-so any efforts can be coordinated.
-
-The latest version of Calc is available from Savannah, in the Emacs
-CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
-
address@hidden [summary]
address@hidden Summary, Key Index, Reporting Bugs, Top
address@hidden Calc Summary
-
address@hidden
-This section includes a complete list of Calc 2.1 keystroke commands.
-Each line lists the stack entries used by the command (top-of-stack
-last), the keystrokes themselves, the prompts asked by the command,
-and the result of the command (also with top-of-stack last).
-The result is expressed using the equivalent algebraic function.
-Commands which put no results on the stack show the full @kbd{M-x}
-command name in that position. Numbers preceding the result or
-command name refer to notes at the end.
-
-Algebraic functions and @kbd{M-x} commands that don't have corresponding
-keystrokes are not listed in this summary.
address@hidden Index}. @xref{Function Index}.
-
address@hidden
address@hidden
address@hidden
-\vskip-2\baselineskip \null
-\gdef\sumrow#1{\sumrowx#1\relax}%
-\gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
-\leavevmode%
-{\smallfonts
-\hbox to5em{\sl\hss#1}%
-\hbox to5em{\tt#2\hss}%
-\hbox to4em{\sl#3\hss}%
-\hbox to5em{\rm\hss#4}%
-\thinspace%
-{\tt#5}%
-{\sl#6}%
-}}%
-\gdef\sumlpar{{\rm(}}%
-\gdef\sumrpar{{\rm)}}%
-\gdef\sumcomma{{\rm,\thinspace}}%
-\gdef\sumexcl{{\rm!}}%
-\gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
-\gdef\minus#1{{\tt-}}%
address@hidden tex
address@hidden@:address@hidden
address@hidden@address@hidden
address@hidden@(address@hidden @let(address@hidden
address@hidden@)address@hidden @let)address@hidden
address@hidden@,address@hidden @let,address@hidden
address@hidden@address@hidden @address@hidden
address@hidden iftex
address@hidden
address@hidden
address@hidden@baselineskip-2.5pt
address@hidden@address@hidden
address@hidden iftex
address@hidden @: C-x * a @: @: 33
@:calc-embedded-activate@:}
address@hidden @: C-x * b @: @:
@:calc-big-or-small@:}
address@hidden @: C-x * c @: @: @:calc@:}
address@hidden @: C-x * d @: @:
@:calc-embedded-duplicate@:}
address@hidden @: C-x * e @: @: 34 @:calc-embedded@:}
address@hidden @: C-x * f @:formula @:
@:calc-embedded-new-formula@:}
address@hidden @: C-x * g @: @: 35
@:calc-grab-region@:}
address@hidden @: C-x * i @: @: @:calc-info@:}
address@hidden @: C-x * j @: @:
@:calc-embedded-select@:}
address@hidden @: C-x * k @: @: @:calc-keypad@:}
address@hidden @: C-x * l @: @:
@:calc-load-everything@:}
address@hidden @: C-x * m @: @:
@:read-kbd-macro@:}
address@hidden @: C-x * n @: @: 4
@:calc-embedded-next@:}
address@hidden @: C-x * o @: @:
@:calc-other-window@:}
address@hidden @: C-x * p @: @: 4
@:calc-embedded-previous@:}
address@hidden @: C-x * q @:formula @: @:quick-calc@:}
address@hidden @: C-x * r @: @: 36
@:calc-grab-rectangle@:}
address@hidden @: C-x * s @: @:
@:calc-info-summary@:}
address@hidden @: C-x * t @: @: @:calc-tutorial@:}
address@hidden @: C-x * u @: @:
@:calc-embedded-update-formula@:}
address@hidden @: C-x * w @: @:
@:calc-embedded-word@:}
address@hidden @: C-x * x @: @: @:calc-quit@:}
address@hidden @: C-x * y @: @:1,28,49
@:calc-copy-to-buffer@:}
address@hidden @: C-x * z @: @:
@:calc-user-invocation@:}
address@hidden @: C-x * : @: @: 36
@:calc-grab-sum-down@:}
address@hidden @: C-x * _ @: @: 36
@:calc-grab-sum-across@:}
address@hidden @: C-x * ` @:editing @: 30
@:calc-embedded-edit@:}
address@hidden @: C-x * 0 @:(zero) @: @:calc-reset@:}
-
address@hidden
address@hidden @: 0-9 @:number @: @:@:number}
address@hidden @: . @:number @: @:@:0.number}
address@hidden @: _ @:number @: @:-@:number}
address@hidden @: e @:number @: @:@:1e number}
address@hidden @: # @:number @: @:@:address@hidden
address@hidden @: P @:(in number) @: @:+/-@:}
address@hidden @: M @:(in number) @: @:mod@:}
address@hidden @: @@ ' " @: (in number)@: @:@:HMS form}
address@hidden @: h m s @: (in number)@: @:@:HMS form}
-
address@hidden
address@hidden @: ' @:formula @: 37,46 @:@:formula}
address@hidden @: $ @:formula @: 37,46 @:$@:formula}
address@hidden @: " @:string @: 37,46 @:@:string}
-
address@hidden
address@hidden a b@: + @: @: 2 @:add@:(a,b) a+b}
address@hidden a b@: - @: @: 2 @:sub@:(a,b)
address@hidden
address@hidden a b@: * @: @: 2 @:mul@:(a,b) a b,
a*b}
address@hidden a b@: / @: @: 2 @:div@:(a,b) a/b}
address@hidden a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
address@hidden a b@: I ^ @: @: 2 @:nroot@:(a,b)
a^(1/b)}
address@hidden a b@: % @: @: 2 @:mod@:(a,b) a%b}
address@hidden a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
address@hidden a b@: : @: @: 2 @:fdiv@:(a,b)}
address@hidden a b@: | @: @: 2 @:vconcat@:(a,b)
a|b}
address@hidden a b@: I | @: @: @:vconcat@:(b,a)
b|a}
address@hidden a b@: H | @: @: 2 @:append@:(a,b)}
address@hidden a b@: I H | @: @: @:append@:(b,a)}
address@hidden a@: & @: @: 1 @:inv@:(a) 1/a}
address@hidden a@: ! @: @: 1 @:fact@:(a) a!}
address@hidden a@: = @: @: 1 @:evalv@:(a)}
address@hidden a@: M-% @: @: @:percent@:(a) a%}
-
address@hidden
address@hidden ... a@: @key{RET} @: @: 1 @:@:... a a}
address@hidden ... a@: @key{SPC} @: @: 1 @:@:... a a}
address@hidden a b@: @key{TAB} @: @: 3 @:@:... b a}
address@hidden a b c@: address@hidden @: @: 3 @:@:... b c
a}
address@hidden a b@: @key{LFD} @: @: 1 @:@:... a b a}
address@hidden ... a@: @key{DEL} @: @: 1 @:@:...}
address@hidden a b@: address@hidden @: @: 1 @:@:... b}
address@hidden @: address@hidden @: @: 4
@:calc-last-args@:}
address@hidden a@: ` @:editing @: 1,30 @:calc-edit@:}
-
address@hidden
address@hidden ... a@: C-d @: @: 1 @:@:...}
address@hidden @: C-k @: @: 27 @:calc-kill@:}
address@hidden @: C-w @: @: 27
@:calc-kill-region@:}
address@hidden @: C-y @: @: @:calc-yank@:}
address@hidden @: C-_ @: @: 4 @:calc-undo@:}
address@hidden @: M-k @: @: 27
@:calc-copy-as-kill@:}
address@hidden @: M-w @: @: 27
@:calc-copy-region-as-kill@:}
-
address@hidden
address@hidden @: [ @: @: @:@:[...}
address@hidden a b@: ] @: @: @:@:[a,b]}
address@hidden @: ( @: @: @:@:(...}
address@hidden(.. a b@: ) @: @: @:@:(a,b)}
address@hidden @: , @: @: @:@:vector or rect
complex}
address@hidden @: ; @: @: @:@:matrix or polar
complex}
address@hidden @: .. @: @: @:@:interval}
-
address@hidden
address@hidden @: ~ @: @: @:calc-num-prefix@:}
address@hidden @: < @: @: 4
@:calc-scroll-left@:}
address@hidden @: > @: @: 4
@:calc-scroll-right@:}
address@hidden @: @{ @: @: 4
@:calc-scroll-down@:}
address@hidden @: @} @: @: 4 @:calc-scroll-up@:}
address@hidden @: ? @: @: @:calc-help@:}
-
address@hidden
address@hidden a@: n @: @: 1 @:neg@:(a)
@minus{}a}
address@hidden @: o @: @: 4 @:calc-realign@:}
address@hidden @: p @:precision @: 31 @:calc-precision@:}
address@hidden @: q @: @: @:calc-quit@:}
address@hidden @: w @: @: @:calc-why@:}
address@hidden @: x @:command @: @:M-x
calc-@:command}
address@hidden a@: y @: @:1,28,49
@:calc-copy-to-buffer@:}
-
address@hidden
address@hidden a@: A @: @: 1 @:abs@:(a)}
address@hidden a b@: B @: @: 2 @:log@:(a,b)}
address@hidden a b@: I B @: @: 2 @:alog@:(a,b) b^a}
address@hidden a@: C @: @: 1 @:cos@:(a)}
address@hidden a@: I C @: @: 1 @:arccos@:(a)}
address@hidden a@: H C @: @: 1 @:cosh@:(a)}
address@hidden a@: I H C @: @: 1 @:arccosh@:(a)}
address@hidden @: D @: @: 4 @:calc-redo@:}
address@hidden a@: E @: @: 1 @:exp@:(a)}
address@hidden a@: H E @: @: 1 @:exp10@:(a) 10.^a}
address@hidden a@: F @: @: 1,11 @:floor@:(a,d)}
address@hidden a@: I F @: @: 1,11 @:ceil@:(a,d)}
address@hidden a@: H F @: @: 1,11 @:ffloor@:(a,d)}
address@hidden a@: I H F @: @: 1,11 @:fceil@:(a,d)}
address@hidden a@: G @: @: 1 @:arg@:(a)}
address@hidden @: H @:command @: 32 @:@:Hyperbolic}
address@hidden @: I @:command @: 32 @:@:Inverse}
address@hidden a@: J @: @: 1 @:conj@:(a)}
address@hidden @: K @:command @: 32 @:@:Keep-args}
address@hidden a@: L @: @: 1 @:ln@:(a)}
address@hidden a@: H L @: @: 1 @:log10@:(a)}
address@hidden @: M @: @:
@:calc-more-recursion-depth@:}
address@hidden @: I M @: @:
@:calc-less-recursion-depth@:}
address@hidden a@: N @: @: 5 @:evalvn@:(a)}
address@hidden @: P @: @: @:@:pi}
address@hidden @: I P @: @: @:@:gamma}
address@hidden @: H P @: @: @:@:e}
address@hidden @: I H P @: @: @:@:phi}
address@hidden a@: Q @: @: 1 @:sqrt@:(a)}
address@hidden a@: I Q @: @: 1 @:sqr@:(a) a^2}
address@hidden a@: R @: @: 1,11 @:round@:(a,d)}
address@hidden a@: I R @: @: 1,11 @:trunc@:(a,d)}
address@hidden a@: H R @: @: 1,11 @:fround@:(a,d)}
address@hidden a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
address@hidden a@: S @: @: 1 @:sin@:(a)}
address@hidden a@: I S @: @: 1 @:arcsin@:(a)}
address@hidden a@: H S @: @: 1 @:sinh@:(a)}
address@hidden a@: I H S @: @: 1 @:arcsinh@:(a)}
address@hidden a@: T @: @: 1 @:tan@:(a)}
address@hidden a@: I T @: @: 1 @:arctan@:(a)}
address@hidden a@: H T @: @: 1 @:tanh@:(a)}
address@hidden a@: I H T @: @: 1 @:arctanh@:(a)}
address@hidden @: U @: @: 4 @:calc-undo@:}
address@hidden @: X @: @: 4
@:calc-call-last-kbd-macro@:}
-
address@hidden
address@hidden a b@: a = @: @: 2 @:eq@:(a,b) a=b}
address@hidden a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
address@hidden a b@: a < @: @: 2 @:lt@:(a,b) a<b}
address@hidden a b@: a > @: @: 2 @:gt@:(a,b) a>b}
address@hidden a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
address@hidden a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
address@hidden a b@: a @{ @: @: 2 @:in@:(a,b)}
address@hidden a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
address@hidden a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
address@hidden a@: a ! @: @: 1,45 @:lnot@:(a) !a}
address@hidden a b c@: a : @: @: 45 @:if@:(a,b,c)
a?b:c}
address@hidden a@: a . @: @: 1 @:rmeq@:(a)}
address@hidden a@: a " @: @: 7,8
@:calc-expand-formula@:}
-
address@hidden
address@hidden a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
address@hidden a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
address@hidden a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
address@hidden a b@: a _ @: @: 2 @:subscr@:(a,b)
a_b}
-
address@hidden
address@hidden a b@: a \ @: @: 2 @:pdiv@:(a,b)}
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@:calc-embedded-preserve-modes@:}
address@hidden @: m f @: @: 12 @:calc-frac-mode@:}
address@hidden @: m g @: @: 52 @:calc-get-modes@:}
address@hidden @: m h @: @: @:calc-hms-mode@:}
address@hidden @: m i @: @: 12,13
@:calc-infinite-mode@:}
address@hidden @: m m @: @: @:calc-save-modes@:}
address@hidden @: m p @: @: 12 @:calc-polar-mode@:}
address@hidden @: m r @: @:
@:calc-radians-mode@:}
address@hidden @: m s @: @: 12
@:calc-symbolic-mode@:}
address@hidden @: m t @: @: 12
@:calc-total-algebraic-mode@:}
address@hidden @: m v @: @: 12,13
@:calc-matrix-mode@:}
address@hidden @: m w @: @: 13 @:calc-working@:}
address@hidden @: m x @: @:
@:calc-always-load-extensions@:}
-
address@hidden
address@hidden @: m A @: @: 12
@:calc-alg-simplify-mode@:}
address@hidden @: m B @: @: 12
@:calc-bin-simplify-mode@:}
address@hidden @: m C @: @: 12
@:calc-auto-recompute@:}
address@hidden @: m D @: @:
@:calc-default-simplify-mode@:}
address@hidden @: m E @: @: 12
@:calc-ext-simplify-mode@:}
address@hidden @: m F @:filename @: 13
@:calc-settings-file-name@:}
address@hidden @: m N @: @: 12
@:calc-num-simplify-mode@:}
address@hidden @: m O @: @: 12
@:calc-no-simplify-mode@:}
address@hidden @: m R @: @: 12,13
@:calc-mode-record-mode@:}
address@hidden @: m S @: @: 12
@:calc-shift-prefix@:}
address@hidden @: m U @: @: 12
@:calc-units-simplify-mode@:}
-
address@hidden
address@hidden @: s c @:var1, var2 @: 29
@:calc-copy-variable@:}
address@hidden @: s d @:var, decl @:
@:calc-declare-variable@:}
address@hidden @: s e @:var, editing @: 29,30
@:calc-edit-variable@:}
address@hidden @: s i @:buffer @:
@:calc-insert-variables@:}
address@hidden @: s k @:const, var @: 29
@:calc-copy-special-constant@:}
address@hidden a b@: s l @:var @: 29 @:@:a (letting
var=b)}
address@hidden a ...@: s m @:op, var @: 22,29 @:calc-store-map@:}
address@hidden @: s n @:var @: 29,47 @:calc-store-neg@:
(v/-1)}
address@hidden @: s p @:var @: 29
@:calc-permanent-variable@:}
address@hidden @: s r @:var @: 29 @:@:v (recalled
value)}
address@hidden @: r 0-9 @: @:
@:calc-recall-quick@:}
address@hidden a@: s s @:var @: 28,29 @:calc-store@:}
address@hidden a@: s 0-9 @: @:
@:calc-store-quick@:}
address@hidden a@: s t @:var @: 29 @:calc-store-into@:}
address@hidden a@: t 0-9 @: @:
@:calc-store-into-quick@:}
address@hidden @: s u @:var @: 29 @:calc-unstore@:}
address@hidden a@: s x @:var @: 29
@:calc-store-exchange@:}
-
address@hidden
address@hidden @: s A @:editing @: 30
@:calc-edit-AlgSimpRules@:}
address@hidden @: s D @:editing @: 30 @:calc-edit-Decls@:}
address@hidden @: s E @:editing @: 30
@:calc-edit-EvalRules@:}
address@hidden @: s F @:editing @: 30
@:calc-edit-FitRules@:}
address@hidden @: s G @:editing @: 30
@:calc-edit-GenCount@:}
address@hidden @: s H @:editing @: 30
@:calc-edit-Holidays@:}
address@hidden @: s I @:editing @: 30
@:calc-edit-IntegLimit@:}
address@hidden @: s L @:editing @: 30
@:calc-edit-LineStyles@:}
address@hidden @: s P @:editing @: 30
@:calc-edit-PointStyles@:}
address@hidden @: s R @:editing @: 30
@:calc-edit-PlotRejects@:}
address@hidden @: s T @:editing @: 30
@:calc-edit-TimeZone@:}
address@hidden @: s U @:editing @: 30 @:calc-edit-Units@:}
address@hidden @: s X @:editing @: 30
@:calc-edit-ExtSimpRules@:}
-
address@hidden
address@hidden a@: s + @:var @: 29,47 @:calc-store-plus@:
(v+a)}
address@hidden a@: s - @:var @: 29,47
@:calc-store-minus@: (v-a)}
address@hidden a@: s * @:var @: 29,47
@:calc-store-times@: (v*a)}
address@hidden a@: s / @:var @: 29,47 @:calc-store-div@:
(v/a)}
address@hidden a@: s ^ @:var @: 29,47
@:calc-store-power@: (v^a)}
address@hidden a@: s | @:var @: 29,47
@:calc-store-concat@: (v|a)}
address@hidden @: s & @:var @: 29,47 @:calc-store-inv@:
(v^-1)}
address@hidden @: s [ @:var @: 29,47 @:calc-store-decr@:
(v-1)}
address@hidden @: s ] @:var @: 29,47 @:calc-store-incr@:
(v-(-1))}
address@hidden a b@: s : @: @: 2 @:assign@:(a,b) a
@tfn{:=} b}
address@hidden a@: s = @: @: 1 @:evalto@:(a,b) a
@tfn{=>}}
-
address@hidden
address@hidden @: t [ @: @: 4
@:calc-trail-first@:}
address@hidden @: t ] @: @: 4 @:calc-trail-last@:}
address@hidden @: t < @: @: 4
@:calc-trail-scroll-left@:}
address@hidden @: t > @: @: 4
@:calc-trail-scroll-right@:}
address@hidden @: t . @: @: 12
@:calc-full-trail-vectors@:}
-
address@hidden
address@hidden @: t b @: @: 4
@:calc-trail-backward@:}
address@hidden @: t d @: @: 12,50
@:calc-trail-display@:}
address@hidden @: t f @: @: 4
@:calc-trail-forward@:}
address@hidden @: t h @: @: @:calc-trail-here@:}
address@hidden @: t i @: @: @:calc-trail-in@:}
address@hidden @: t k @: @: 4 @:calc-trail-kill@:}
address@hidden @: t m @:string @:
@:calc-trail-marker@:}
address@hidden @: t n @: @: 4 @:calc-trail-next@:}
address@hidden @: t o @: @: @:calc-trail-out@:}
address@hidden @: t p @: @: 4
@:calc-trail-previous@:}
address@hidden @: t r @:string @:
@:calc-trail-isearch-backward@:}
address@hidden @: t s @:string @:
@:calc-trail-isearch-forward@:}
address@hidden @: t y @: @: 4 @:calc-trail-yank@:}
-
address@hidden
address@hidden d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)}
address@hidden oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
address@hidden d@: t D @: @: 15 @:date@:(d)}
address@hidden d@: t I @: @: 4 @:incmonth@:(d,n)}
address@hidden d@: t J @: @: 16 @:julian@:(d,z)}
address@hidden d@: t M @: @: 17 @:newmonth@:(d,n)}
address@hidden @: t N @: @: 16 @:now@:(z)}
address@hidden d@: t P @:1 @: 31 @:year@:(d)}
address@hidden d@: t P @:2 @: 31 @:month@:(d)}
address@hidden d@: t P @:3 @: 31 @:day@:(d)}
address@hidden d@: t P @:4 @: 31 @:hour@:(d)}
address@hidden d@: t P @:5 @: 31 @:minute@:(d)}
address@hidden d@: t P @:6 @: 31 @:second@:(d)}
address@hidden d@: t P @:7 @: 31 @:weekday@:(d)}
address@hidden d@: t P @:8 @: 31 @:yearday@:(d)}
address@hidden d@: t P @:9 @: 31 @:time@:(d)}
address@hidden d@: t U @: @: 16 @:unixtime@:(d,z)}
address@hidden d@: t W @: @: 17 @:newweek@:(d,w)}
address@hidden d@: t Y @: @: 17 @:newyear@:(d,n)}
-
address@hidden
address@hidden a b@: t + @: @: 2 @:badd@:(a,b)}
address@hidden a b@: t - @: @: 2 @:bsub@:(a,b)}
-
address@hidden
address@hidden @: u a @: @: 12
@:calc-autorange-units@:}
address@hidden a@: u b @: @: @:calc-base-units@:}
address@hidden a@: u c @:units @: 18
@:calc-convert-units@:}
address@hidden defn@: u d @:unit, descr @:
@:calc-define-unit@:}
address@hidden @: u e @: @:
@:calc-explain-units@:}
address@hidden @: u g @:unit @:
@:calc-get-unit-definition@:}
address@hidden @: u p @: @:
@:calc-permanent-units@:}
address@hidden a@: u r @: @:
@:calc-remove-units@:}
address@hidden a@: u s @: @: @:usimplify@:(a)}
address@hidden a@: u t @:units @: 18
@:calc-convert-temperature@:}
address@hidden @: u u @:unit @:
@:calc-undefine-unit@:}
address@hidden @: u v @: @:
@:calc-enter-units-table@:}
address@hidden a@: u x @: @:
@:calc-extract-units@:}
address@hidden a@: u 0-9 @: @:
@:calc-quick-units@:}
-
address@hidden
address@hidden v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
address@hidden v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
address@hidden v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
address@hidden v@: u G @: @: 19 @:vgmean@:(v)}
address@hidden a b@: H u G @: @: 2 @:agmean@:(a,b)}
address@hidden v@: u M @: @: 19 @:vmean@:(v)}
address@hidden v@: I u M @: @: 19 @:vmeane@:(v)}
address@hidden v@: H u M @: @: 19 @:vmedian@:(v)}
address@hidden v@: I H u M @: @: 19 @:vhmean@:(v)}
address@hidden v@: u N @: @: 19 @:vmin@:(v)}
address@hidden v@: u S @: @: 19 @:vsdev@:(v)}
address@hidden v@: I u S @: @: 19 @:vpsdev@:(v)}
address@hidden v@: H u S @: @: 19 @:vvar@:(v)}
address@hidden v@: I H u S @: @: 19 @:vpvar@:(v)}
address@hidden @: u V @: @:
@:calc-view-units-table@:}
address@hidden v@: u X @: @: 19 @:vmax@:(v)}
-
address@hidden
address@hidden v@: u + @: @: 19 @:vsum@:(v)}
address@hidden v@: u * @: @: 19 @:vprod@:(v)}
address@hidden v@: u # @: @: 19 @:vcount@:(v)}
-
address@hidden
address@hidden @: V ( @: @: 50
@:calc-vector-parens@:}
address@hidden @: V @{ @: @: 50
@:calc-vector-braces@:}
address@hidden @: V [ @: @: 50
@:calc-vector-brackets@:}
address@hidden @: V ] @:ROCP @: 50
@:calc-matrix-brackets@:}
address@hidden @: V , @: @: 50
@:calc-vector-commas@:}
address@hidden @: V < @: @: 50
@:calc-matrix-left-justify@:}
address@hidden @: V = @: @: 50
@:calc-matrix-center-justify@:}
address@hidden @: V > @: @: 50
@:calc-matrix-right-justify@:}
address@hidden @: V / @: @: 12,50
@:calc-break-vectors@:}
address@hidden @: V . @: @: 12,50
@:calc-full-vectors@:}
-
address@hidden
address@hidden s t@: V ^ @: @: 2 @:vint@:(s,t)}
address@hidden s t@: V - @: @: 2 @:vdiff@:(s,t)}
address@hidden s@: V ~ @: @: 1 @:vcompl@:(s)}
address@hidden s@: V # @: @: 1 @:vcard@:(s)}
address@hidden s@: V : @: @: 1 @:vspan@:(s)}
address@hidden s@: V + @: @: 1 @:rdup@:(s)}
-
address@hidden
address@hidden m@: V & @: @: 1 @:inv@:(m) 1/m}
-
address@hidden
address@hidden v@: v a @:n @: @:arrange@:(v,n)}
address@hidden a@: v b @:n @: @:cvec@:(a,n)}
address@hidden v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
address@hidden v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
address@hidden m@: v c @:0 @: 31 @:getdiag@:(m)}
address@hidden v@: v d @: @: 25 @:diag@:(v,n)}
address@hidden v m@: v e @: @: 2 @:vexp@:(v,m)}
address@hidden v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
address@hidden v a@: v f @: @: 26 @:find@:(v,a,n)}
address@hidden v@: v h @: @: 1 @:head@:(v)}
address@hidden v@: I v h @: @: 1 @:tail@:(v)}
address@hidden v@: H v h @: @: 1 @:rhead@:(v)}
address@hidden v@: I H v h @: @: 1 @:rtail@:(v)}
address@hidden @: v i @:n @: 31 @:idn@:(1,n)}
address@hidden @: v i @:0 @: 31 @:idn@:(1)}
address@hidden h t@: v k @: @: 2 @:cons@:(h,t)}
address@hidden h t@: H v k @: @: 2 @:rcons@:(h,t)}
address@hidden v@: v l @: @: 1 @:vlen@:(v)}
address@hidden v@: H v l @: @: 1 @:mdims@:(v)}
address@hidden v m@: v m @: @: 2 @:vmask@:(v,m)}
address@hidden v@: v n @: @: 1 @:rnorm@:(v)}
address@hidden a b c@: v p @: @: 24 @:calc-pack@:}
address@hidden v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
address@hidden v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
address@hidden m@: v r @:0 @: 31 @:getdiag@:(m)}
address@hidden v i j@: v s @: @: @:subvec@:(v,i,j)}
address@hidden v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
address@hidden m@: v t @: @: 1 @:trn@:(m)}
address@hidden v@: v u @: @: 24 @:calc-unpack@:}
address@hidden v@: v v @: @: 1 @:rev@:(v)}
address@hidden @: v x @:n @: 31 @:index@:(n)}
address@hidden n s i@: C-u v x @: @: @:index@:(n,s,i)}
-
address@hidden
address@hidden v@: V A @:op @: 22 @:apply@:(op,v)}
address@hidden v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
address@hidden m@: V D @: @: 1 @:det@:(m)}
address@hidden s@: V E @: @: 1 @:venum@:(s)}
address@hidden s@: V F @: @: 1 @:vfloor@:(s)}
address@hidden v@: V G @: @: @:grade@:(v)}
address@hidden v@: I V G @: @: @:rgrade@:(v)}
address@hidden v@: V H @:n @: 31 @:histogram@:(v,n)}
address@hidden v w@: H V H @:n @: 31
@:histogram@:(v,w,n)}
address@hidden v1 v2@: V I @:mop aop @: 22
@:inner@:(mop,aop,v1,v2)}
address@hidden m@: V J @: @: 1 @:ctrn@:(m)}
address@hidden m@: V L @: @: 1 @:lud@:(m)}
address@hidden v@: V M @:op @: 22,23 @:map@:(op,v)}
address@hidden v@: V N @: @: 1 @:cnorm@:(v)}
address@hidden v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
address@hidden v@: V R @:op @: 22,23 @:reduce@:(op,v)}
address@hidden v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
address@hidden a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
address@hidden a@: I H V R @:op @: 22 @:fixp@:(op,a)}
address@hidden v@: V S @: @: @:sort@:(v)}
address@hidden v@: I V S @: @: @:rsort@:(v)}
address@hidden m@: V T @: @: 1 @:tr@:(m)}
address@hidden v@: V U @:op @: 22 @:accum@:(op,v)}
address@hidden v@: I V U @:op @: 22 @:raccum@:(op,v)}
address@hidden a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
address@hidden a@: I H V U @:op @: 22 @:afixp@:(op,a)}
address@hidden s t@: V V @: @: 2 @:vunion@:(s,t)}
address@hidden s t@: V X @: @: 2 @:vxor@:(s,t)}
-
address@hidden
address@hidden @: Y @: @: @:@:user commands}
-
address@hidden
address@hidden @: z @: @: @:@:user commands}
-
address@hidden
address@hidden c@: Z [ @: @: 45 @:calc-kbd-if@:}
address@hidden c@: Z | @: @: 45
@:calc-kbd-else-if@:}
address@hidden @: Z : @: @: @:calc-kbd-else@:}
address@hidden @: Z ] @: @: @:calc-kbd-end-if@:}
-
address@hidden
address@hidden @: Z @{ @: @: 4 @:calc-kbd-loop@:}
address@hidden c@: Z / @: @: 45 @:calc-kbd-break@:}
address@hidden @: Z @} @: @:
@:calc-kbd-end-loop@:}
address@hidden n@: Z < @: @: @:calc-kbd-repeat@:}
address@hidden @: Z > @: @:
@:calc-kbd-end-repeat@:}
address@hidden n m@: Z ( @: @: @:calc-kbd-for@:}
address@hidden s@: Z ) @: @:
@:calc-kbd-end-for@:}
-
address@hidden
address@hidden @: Z C-g @: @: @:@:cancel if/loop
command}
-
address@hidden
address@hidden @: Z ` @: @: @:calc-kbd-push@:}
address@hidden @: Z ' @: @: @:calc-kbd-pop@:}
address@hidden @: Z # @: @: @:calc-kbd-query@:}
-
address@hidden
address@hidden comp@: Z C @:func, args @: 50
@:calc-user-define-composition@:}
address@hidden @: Z D @:key, command @:
@:calc-user-define@:}
address@hidden @: Z E @:key, editing @: 30
@:calc-user-define-edit@:}
address@hidden defn@: Z F @:k, c, f, a, n@: 28
@:calc-user-define-formula@:}
address@hidden @: Z G @:key @:
@:calc-get-user-defn@:}
address@hidden @: Z I @: @:
@:calc-user-define-invocation@:}
address@hidden @: Z K @:key, command @:
@:calc-user-define-kbd-macro@:}
address@hidden @: Z P @:key @:
@:calc-user-define-permanent@:}
address@hidden @: Z S @: @: 30
@:calc-edit-user-syntax@:}
address@hidden @: Z T @: @: 12 @:calc-timing@:}
address@hidden @: Z U @:key @:
@:calc-user-undefine@:}
-
address@hidden format
-
address@hidden
-NOTES
-
address@hidden
address@hidden 1
address@hidden
-Positive prefix arguments apply to @expr{n} stack entries.
-Negative prefix arguments apply to the @expr{-n}th stack entry.
-A prefix of zero applies to the entire stack. (For @key{LFD} and
address@hidden@key{DEL}}, the meaning of the sign is reversed.)
-
address@hidden 2
address@hidden
-Positive prefix arguments apply to @expr{n} stack entries.
-Negative prefix arguments apply to the top stack entry
-and the next @expr{-n} stack entries.
-
address@hidden 3
address@hidden
-Positive prefix arguments rotate top @expr{n} stack entries by one.
-Negative prefix arguments rotate the entire stack by @expr{-n}.
-A prefix of zero reverses the entire stack.
-
address@hidden 4
address@hidden
-Prefix argument specifies a repeat count or distance.
-
address@hidden 5
address@hidden
-Positive prefix arguments specify a precision @expr{p}.
-Negative prefix arguments reduce the current precision by @expr{-p}.
-
address@hidden 6
address@hidden
-A prefix argument is interpreted as an additional step-size parameter.
-A plain @kbd{C-u} prefix means to prompt for the step size.
-
address@hidden 7
address@hidden
-A prefix argument specifies simplification level and depth.
-1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}.
-
address@hidden 8
address@hidden
-A negative prefix operates only on the top level of the input formula.
-
address@hidden 9
address@hidden
-Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
-Negative prefix arguments specify a word size of @expr{w} bits, signed.
-
address@hidden 10
address@hidden
-Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
-cannot be specified in the keyboard version of this command.
-
address@hidden 11
address@hidden
-From the keyboard, @expr{d} is omitted and defaults to zero.
-
address@hidden 12
address@hidden
-Mode is toggled; a positive prefix always sets the mode, and a negative
-prefix always clears the mode.
-
address@hidden 13
address@hidden
-Some prefix argument values provide special variations of the mode.
-
address@hidden 14
address@hidden
-A prefix argument, if any, is used for @expr{m} instead of taking
address@hidden from the stack. @expr{M} may take any of these values:
address@hidden
address@hidden@tableindent10pt
address@hidden iftex
address@hidden @asis
address@hidden Integer
-Random integer in the interval @expr{[0 .. m)}.
address@hidden Float
-Random floating-point number in the interval @expr{[0 .. m)}.
address@hidden 0.0
-Gaussian with mean 1 and standard deviation 0.
address@hidden Error form
-Gaussian with specified mean and standard deviation.
address@hidden Interval
-Random integer or floating-point number in that interval.
address@hidden Vector
-Random element from the vector.
address@hidden table
address@hidden
-}
address@hidden iftex
-
address@hidden 15
address@hidden
-A prefix argument from 1 to 6 specifies number of date components
-to remove from the stack. @xref{Date Conversions}.
-
address@hidden 16
address@hidden
-A prefix argument specifies a time zone; @kbd{C-u} says to take the
-time zone number or name from the top of the stack. @xref{Time Zones}.
-
address@hidden 17
address@hidden
-A prefix argument specifies a day number (0-6, 0-31, or 0-366).
-
address@hidden 18
address@hidden
-If the input has no units, you will be prompted for both the old and
-the new units.
-
address@hidden 19
address@hidden
-With a prefix argument, collect that many stack entries to form the
-input data set. Each entry may be a single value or a vector of values.
-
address@hidden 20
address@hidden
-With a prefix argument of 1, take a single
address@hidden @address@hidden
address@hidden @address@hidden
-matrix from the stack instead of two separate data vectors.
-
address@hidden 21
address@hidden
-The row or column number @expr{n} may be given as a numeric prefix
-argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
-from the top of the stack. If @expr{n} is a vector or interval,
-a subvector/submatrix of the input is created.
-
address@hidden 22
address@hidden
-The @expr{op} prompt can be answered with the key sequence for the
-desired function, or with @kbd{x} or @kbd{z} followed by a function name,
-or with @kbd{$} to take a formula from the top of the stack, or with
address@hidden'} and a typed formula. In the last two cases, the formula may
-be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
-may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
-last argument of the created function), or otherwise you will be
-prompted for an argument list. The number of vectors popped from the
-stack by @kbd{V M} depends on the number of arguments of the function.
-
address@hidden 23
address@hidden
-One of the mapping direction keys @kbd{_} (horizontal, i.e., map
-by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
-reduce down), or @kbd{=} (map or reduce by rows) may be used before
-entering @expr{op}; these modify the function name by adding the letter
address@hidden for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
-or @code{d} for ``down.''
-
address@hidden 24
address@hidden
-The prefix argument specifies a packing mode. A nonnegative mode
-is the number of items (for @kbd{v p}) or the number of levels
-(for @kbd{v u}). A negative mode is as described below. With no
-prefix argument, the mode is taken from the top of the stack and
-may be an integer or a vector of integers.
address@hidden
address@hidden@tableindent-20pt
address@hidden iftex
address@hidden @cite
address@hidden -1
-(@var{2}) Rectangular complex number.
address@hidden -2
-(@var{2}) Polar complex number.
address@hidden -3
-(@var{3}) HMS form.
address@hidden -4
-(@var{2}) Error form.
address@hidden -5
-(@var{2}) Modulo form.
address@hidden -6
-(@var{2}) Closed interval.
address@hidden -7
-(@var{2}) Closed .. open interval.
address@hidden -8
-(@var{2}) Open .. closed interval.
address@hidden -9
-(@var{2}) Open interval.
address@hidden -10
-(@var{2}) Fraction.
address@hidden -11
-(@var{2}) Float with integer mantissa.
address@hidden -12
-(@var{2}) Float with mantissa in @expr{[1 .. 10)}.
address@hidden -13
-(@var{1}) Date form (using date numbers).
address@hidden -14
-(@var{3}) Date form (using year, month, day).
address@hidden -15
-(@var{6}) Date form (using year, month, day, hour, minute, second).
address@hidden table
address@hidden
-}
address@hidden iftex
-
address@hidden 25
address@hidden
-A prefix argument specifies the size @expr{n} of the matrix. With no
-prefix argument, @expr{n} is omitted and the size is inferred from
-the input vector.
-
address@hidden 26
address@hidden
-The prefix argument specifies the starting position @expr{n} (default 1).
-
address@hidden 27
address@hidden
-Cursor position within stack buffer affects this command.
-
address@hidden 28
address@hidden
-Arguments are not actually removed from the stack by this command.
-
address@hidden 29
address@hidden
-Variable name may be a single digit or a full name.
-
address@hidden 30
address@hidden
-Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
address@hidden, or in some cases @key{RET}) to finish the edit, or kill the
-buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents
evaluation
-of the result of the edit.
-
address@hidden 31
address@hidden
-The number prompted for can also be provided as a prefix argument.
-
address@hidden 32
address@hidden
-Press this key a second time to cancel the prefix.
-
address@hidden 33
address@hidden
-With a negative prefix, deactivate all formulas. With a positive
-prefix, deactivate and then reactivate from scratch.
-
address@hidden 34
address@hidden
-Default is to scan for nearest formula delimiter symbols. With a
-prefix of zero, formula is delimited by mark and point. With a
-non-zero prefix, formula is delimited by scanning forward or
-backward by that many lines.
-
address@hidden 35
address@hidden
-Parse the region between point and mark as a vector. A nonzero prefix
-parses @var{n} lines before or after point as a vector. A zero prefix
-parses the current line as a vector. A @kbd{C-u} prefix parses the
-region between point and mark as a single formula.
-
address@hidden 36
address@hidden
-Parse the rectangle defined by point and mark as a matrix. A positive
-prefix @var{n} divides the rectangle into columns of width @var{n}.
-A zero or @kbd{C-u} prefix parses each line as one formula. A negative
-prefix suppresses special treatment of bracketed portions of a line.
-
address@hidden 37
address@hidden
-A numeric prefix causes the current language mode to be ignored.
-
address@hidden 38
address@hidden
-Responding to a prompt with a blank line answers that and all
-later prompts by popping additional stack entries.
-
address@hidden 39
address@hidden
-Answer for @expr{v} may also be of the form @expr{v = v_0} or
address@hidden - v_0}.
-
address@hidden 40
address@hidden
-With a positive prefix argument, stack contains many @expr{y}'s and one
-common @expr{x}. With a zero prefix, stack contains a vector of
address@hidden and a common @expr{x}. With a negative prefix, stack
-contains many @expr{[x,y]} vectors. (For 3D plots, substitute
address@hidden for @expr{y} and @expr{x,y} for @expr{x}.)
-
address@hidden 41
address@hidden
-With any prefix argument, all curves in the graph are deleted.
-
address@hidden 42
address@hidden
-With a positive prefix, refines an existing plot with more data points.
-With a negative prefix, forces recomputation of the plot data.
-
address@hidden 43
address@hidden
-With any prefix argument, set the default value instead of the
-value for this graph.
-
address@hidden 44
address@hidden
-With a negative prefix argument, set the value for the printer.
-
address@hidden 45
address@hidden
-Condition is considered ``true'' if it is a nonzero real or complex
-number, or a formula whose value is known to be nonzero; it is ``false''
-otherwise.
-
address@hidden 46
address@hidden
-Several formulas separated by commas are pushed as multiple stack
-entries. Trailing @kbd{)}, @kbd{]}, @address@hidden, @kbd{>}, and @kbd{"}
-delimiters may be omitted. The notation @kbd{$$$} refers to the value
-in stack level three, and causes the formula to replace the top three
-stack levels. The notation @kbd{$3} refers to stack level three without
-causing that value to be removed from the stack. Use @key{LFD} in place
-of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
-to evaluate variables.
-
address@hidden 47
address@hidden
-The variable is replaced by the formula shown on the right. The
-Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
-assigns
address@hidden @math{x \coloneq a-x}.
address@hidden @expr{x := a-x}.
-
address@hidden 48
address@hidden
-Press @kbd{?} repeatedly to see how to choose a model. Answer the
-variables prompt with @expr{iv} or @expr{iv;pv} to specify
-independent and parameter variables. A positive prefix argument
-takes @address@hidden vectors from the stack; a zero prefix takes a matrix
-and a vector from the stack.
-
address@hidden 49
address@hidden
-With a plain @kbd{C-u} prefix, replace the current region of the
-destination buffer with the yanked text instead of inserting.
-
address@hidden 50
address@hidden
-All stack entries are reformatted; the @kbd{H} prefix inhibits this.
-The @kbd{I} prefix sets the mode temporarily, redraws the top stack
-entry, then restores the original setting of the mode.
-
address@hidden 51
address@hidden
-A negative prefix sets the default 3D resolution instead of the
-default 2D resolution.
-
address@hidden 52
address@hidden
-This grabs a vector of the form address@hidden, @var{wsize}, @var{ssize},
address@hidden, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
address@hidden, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
-grabs the @var{n}th mode value only.
address@hidden enumerate
-
address@hidden
-(Space is provided below for you to keep your own written notes.)
address@hidden
address@hidden
address@hidden iftex
-
-
address@hidden [end-summary]
-
address@hidden Key Index, Command Index, Summary, Top
address@hidden Index of Key Sequences
-
address@hidden ky
-
address@hidden Command Index, Function Index, Key Index, Top
address@hidden Index of Calculator Commands
-
-Since all Calculator commands begin with the prefix @samp{calc-}, the
address@hidden key has been provided as a variant of @kbd{M-x} which
automatically
-types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
address@hidden calc-last-args}.
-
address@hidden pg
-
address@hidden Function Index, Concept Index, Command Index, Top
address@hidden Index of Algebraic Functions
-
-This is a list of built-in functions and operators usable in algebraic
-expressions. Their full Lisp names are derived by adding the prefix
address@hidden, as in @code{calcFunc-sqrt}.
address@hidden
-All functions except those noted with ``*'' have corresponding
-Calc keystrokes and can also be found in the Calc Summary.
address@hidden iftex
-
address@hidden tp
-
address@hidden Concept Index, Variable Index, Function Index, Top
address@hidden Concept Index
-
address@hidden cp
-
address@hidden Variable Index, Lisp Function Index, Concept Index, Top
address@hidden Index of Variables
-
-The variables in this list that do not contain dashes are accessible
-as Calc variables. Add a @samp{var-} prefix to get the name of the
-corresponding Lisp variable.
-
-The remaining variables are Lisp variables suitable for @code{setq}ing
-in your Calc init file or @file{.emacs} file.
-
address@hidden vr
-
address@hidden Lisp Function Index, , Variable Index, Top
address@hidden Index of Lisp Math Functions
-
-The following functions are meant to be used with @code{defmath}, not
address@hidden definitions. For names that do not start with @samp{calc-},
-the corresponding full Lisp name is derived by adding a prefix of
address@hidden
-
address@hidden fn
-
address@hidden
-
-
address@hidden
- arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0
address@hidden ignore
- [Emacs-diffs] Changes to calc.texi,
Glenn Morris <=