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Re: [Axiom-math] Axiom's integration non-deterministic?
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From: |
William Sit |
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Subject: |
Re: [Axiom-math] Axiom's integration non-deterministic? |
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Date: |
Sun, 26 Mar 2006 06:16:21 -0500 |
Here is more detail, including loading and timing:
AXIOM Computer Algebra System
Version of Tuesday November 30, 2004 at 21:11:14
-----------------------------------------------------------------------------
Issue )copyright to view copyright notices.
Issue )summary for a summary of useful system commands.
Issue )quit to leave AXIOM and return to shell.
-----------------------------------------------------------------------------
(1) -> )set mess time on
(1) -> a:=integrate(sqrt(1+x^(-2/3)),x)
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/EXPR.o for domain
Expression
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/KERNEL.o for
domain Kernel
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/BOP.o for domain
BasicOperator
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/SCACHE.o for
package SortedCache
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/VOID.o for domain
Void
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/UPMP.o for package
UnivariatePolynomialMultiplicationPackage
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/AN.o for domain
AlgebraicNumber
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IAN.o for domain
InnerAlgebraicNumber
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ACFS-.o for domain
AlgebraicallyClosedFunctionSpace&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FS-.o for domain
FunctionSpace&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/COMMONOP.o for
package CommonOperators
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/KDAGG-.o for
domain KeyedDictionary&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/DIAGG-.o for
domain Dictionary&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/BOP1.o for package
BasicOperatorFunctions1
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ACF-.o for domain
AlgebraicallyClosedField&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ES-.o for domain
ExpressionSpace&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FRETRCT-.o for
domain FullyRetractableTo&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/EVALAB-.o for
domain Evalable&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/TRANFUN-.o for
domain TranscendentalFunctionCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/BASTYPE-.o for
domain BasicType&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IEVALAB-.o for
domain InnerEvalable&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/TRIGCAT-.o for
domain TrigonometricFunctionCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ATRIG-.o for
domain ArcTrigonometricFunctionCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/HYPCAT-.o for
domain HyperbolicFunctionCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ELEMFUN-.o for
domain ElementaryFunctionCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/RADCAT-.o for
domain RadicalCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/AF.o for package
AlgebraicFunction
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/POLYCATQ.o for
package PolynomialCategoryQuotientFunctions
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/POLYROOT.o for
package PolynomialRoots
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/PGCD.o for package
PolynomialGcdPackage
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FSAGG-.o for
domain FiniteSetAggregate&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FLASORT.o for
package FiniteLinearAggregateSort
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FSINT.o for
package FunctionSpaceIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/TRIGMNIP.o for
package TrigonometricManipulations
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTFACT.o for
package IntegerFactorizationPackage
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IROOT.o for
package IntegerRoots
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FACTFUNC.o for
package FactoredFunctions
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/EF.o for package
ElementaryFunction
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IDPOAMS.o for
domain IndexedDirectProductOrderedAbelianMonoidSup
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IDPOAM.o for
domain IndexedDirectProductOrderedAbelianMonoid
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/SIGNEF.o for
package ElementaryFunctionSign
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MULTSQFR.o for
package MultivariateSquareFree
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/UPSQFREE.o for
package UnivariatePolynomialSquareFree
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/HEUGCD.o for
package HeuGcd
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INMODGCD.o for
package InnerModularGcd
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/EMR.o for domain
EuclideanModularRing
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MDDFACT.o for
package ModularDistinctDegreeFactorizer
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MODRING.o for
domain ModularRing
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/TOOLSIGN.o for
package ToolsForSign
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/EFSTRUC.o for
package ElementaryFunctionStructurePackage
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/POLYLIFT.o for
package PolynomialCategoryLifting
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTTOOLS.o for
package IntegrationTools
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ALGMANIP.o for
package AlgebraicManipulations
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/LIST2MAP.o for
package ListToMap
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTEF.o for
package ElementaryIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FSPRMELT.o for
package FunctionSpacePrimitiveElement
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/PRIMELT.o for
package PrimitiveElement
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/PRS.o for package
PseudoRemainderSequence
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FACUTIL.o for
package FactoringUtilities
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTPAF.o for
package PureAlgebraicIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTG0.o for
package GenusZeroIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/UPCDEN.o for
package UnivariatePolynomialCommonDenominator
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/CDEN.o for package
CommonDenominator
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTRAT.o for
package RationalIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTTR.o for
package TranscendentalIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTHERTR.o for
package TranscendentalHermiteIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MONOTOOL.o for
package MonomialExtensionTools
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IR.o for domain
IntegrationResult
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IR2.o for package
IntegrationResultFunctions2
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IR2F.o for package
IntegrationResultToFunction
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/COMPLEX.o for
domain Complex
+---------+
3+-+2 3+-+ |3+-+2 3+-+2 3+-+ 2
(4x \|x + 3\|x + 7x)\|\|x + 1 + 6\|x + 9x\|x + 4x + 1
(1) ---------------------------------------------------------------
+---------+
3+-+2 |3+-+2 3+-+
(4\|x + 1)\|\|x + 1 + 3\|x + 4x
Type: Union(Expression Integer,...)
Time: 0.02 (IN) + 0.17 (EV) + 0.42 (OT) + 0.12 (GC) = 0.72 sec
(2) ->
(2) -> b:=integrate(sqrt(1+x^(-2/3)),x)
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/CHVAR.o for
package ChangeOfVariable
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ALGFF.o for domain
AlgebraicFunctionField
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MATRIX.o for
domain Matrix
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IIARRAY2.o for
domain InnerIndexedTwoDimensionalArray
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MATCAT-.o for
domain MatrixCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ARR2CAT-.o for
domain TwoDimensionalArrayCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTALG.o for
package AlgebraicIntegrate
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/SAE.o for domain
SimpleAlgebraicExtension
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/INTHERAL.o for
package AlgebraicHermiteIntegration
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FFINTBAS.o for
package FunctionFieldIntegralBasis
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MONOGEN-.o for
domain MonogenicAlgebra&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MATLIN.o for
package MatrixLinearAlgebraFunctions
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MHROWRED.o for
package ModularHermitianRowReduction
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/TRIMAT.o for
package TriangularMatrixOperations
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IBATOOL.o for
package IntegralBasisTools
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/VECTCAT-.o for
domain VectorCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FRAMALG-.o for
domain FramedAlgebra&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FFCAT-.o for
domain FunctionFieldCategory&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MATCAT2.o for
package MatrixCategoryFunctions2
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/IMATLIN.o for
package InnerMatrixLinearAlgebraFunctions
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FINRALG-.o for
domain FiniteRankAlgebra&
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/MCDEN.o for
package MatrixCommonDenominator
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/LIST2.o for
package ListFunctions2
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/FLAGG2.o for
package FiniteLinearAggregateFunctions2
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/ICDEN.o for
package InnerCommonDenominator
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/VECTOR2.o for
package VectorFunctions2
Loading j:/OpenAxiom/axiom014/mnt/windows/algebra/LSMP.o for package
LinearSystemMatrixPackage
+---------+
3+-+2 |3+-+2
(2) (\|x + 1)\|\|x + 1
Type: Union(Expression Integer,...)
Time: 0.02 (IN) + 6.18 (EV) + 0.12 (OT) + 0.78 (GC) = 7.10 sec
(3) ->
(3) -> c:=integrate(sqrt(1+x^(-2/3)),x)
+---------+
3+-+2 |3+-+2
(3) (\|x + 1)\|\|x + 1
Type: Union(Expression Integer,...)
Time: 6.40 (EV) + 0.45 (GC) = 6.85 sec
(4) -> a-b
(4) 0
Type: Expression Integer
Time: 0 sec
(5) -> interpret(a::InputForm) - interpret (b::InputForm)
(5) 0
Type: Expression Integer
Time: 0.03 (EV) + 0.03 (OT) = 0.07 sec
(6) -> t:Boolean:=(a::EXPR INT)=(b::EXPR INT)
(6) false
Type: Boolean
Time: 0 sec
(7) -> d:=integrate(sqrt(1+y^(-2/3)),y)
+---------+
3+-+2 |3+-+2
(7) (\|y + 1)\|\|y + 1
Type: Union(Expression Integer,...)
Time: 0.02 (IN) + 6.98 (EV) + 0.48 (GC) = 7.48 sec
I am no longer sure why (3) took almost as long as (2) (in fact longer EV time),
and why (7) took even longer EV time. However, the longer time compared to (1)
is explained by the extra domains that are loaded. So the system behavior (or at
least the integration algorithm) is affected by what domains are already loaded
(that explains why (7) is comparable to (2) and (3), not (1)). (6) shows that
equality in EXPR INT is not equivalent to zero testing of the difference. Note
that the domain CHVAR was the first loaded in (2), which enabled simplification
of radical expressions. So to answer Igor's question, we way try tracing what
caused CHVAR to be loaded.
William