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Re: [Axiom-math] Curious behavior of Taylor series
From: |
Jay Belanger |
Subject: |
Re: [Axiom-math] Curious behavior of Taylor series |
Date: |
Tue, 22 Aug 2006 11:17:36 -0500 |
User-agent: |
Gnus/5.11 (Gnus v5.11) Emacs/22.0.50 (gnu/linux) |
Ralf Hemmecke <address@hidden> writes:
>>> (113) -> y := taylor x
>>> (113) x
>>> Type: UnivariateTaylorSeries(Expression
>>> Integer,x,0)
>>> (114) -> x*y
>>> (114) x x
>>> Type: UnivariateTaylorSeries(Expression
>>> Integer,x,0)
>>> (115) -> coefficient(%,1)
>>> (115) x
>>> Type: Expression
>>> Integer
...
>>>>> But Axiom coerced the two x to the same domain!!!
>>
>> Looking back, it didn't. One of the x's is in Expression Integer, and
>> the other is UTS.
>
> In some sense you are rigth, BUT, for that to claim you would have to
> know the internal representation of
> UnivariateTaylorSeries(Expression Integer,x,0).
> What if you are a poor Axiom user who doesn't have access to the source
> code? How could you justify your statement. All you see is ONE type and
> that is UnivariateTaylorSeries(Expression Integer,x,0) and nothing else.
I was looking at the Type: part of the output in (113) and (115).
>>> Well, but how can you tell this to Axiom?
>
>> Axiom could try to coerce x to be the same type as y, or y to be the
>> same type as x. The latter would lose structure, and should fail.
>> So x should be coerced to the same domain as y.
>
> As I said, Axiom does exactly that.
Did you? Well, I guess I need to be told several times before it
sinks in. Maybe if I learned to read, I could have saved up both
some typing. But thanks for your patience.
>>> But see, x is a symbol which should be coerce into a Taylor
>>> series. The interpreter has several choices. So assuming the ideal
>>> that the interpreter should have no mathematical knowledge itself,
>>> it can only take the available information from the library. But
>>> there are several available ways to go from x to UTS(Expression
>>> Integer,x,0). So how can the interpreter ever know that it does the
>>> wrong thing?
>>
>> The problem is that it didn't even try to go from x to UTS; x ends up
>> as an Expression Integer. I think it should have tried coercing x
>> before multiplying x and y.
>
> Let's abbreviate U := UnivariateTaylorSeries(Expression Integer,x,0) and
> E := Expression(Integer).
>
> That's what Axiom does. It tries to coerce to U, there is no such coerce
> function.
But x::U works just fine.
> Then maybe after checking some other coerce functions that
> fail, it finds a coercion to Expression(Integer), furthermore there is
> function *: (E, U) -> U. So Axiom does a minimal thing: it interprets
>
> x*y as (x::E)*y
>
> note that the coercion to U would go like
>
> (x::E::U) * y
So x::U is equivalent to x::E::U?
> and would then be more costly. You surely also consider that
> unreasonable to go the long way if there is a shorter path. No?
I'm just wondering why it doesn't try x::U right off.
I'm getting the (probably wrong) impression that it looks at x::U,
decides that that's too long, then goes for x::E. I would think that
deciding x::U is too long would be too long.
Jay
- Re: [Axiom-math] Curious behavior of Taylor series, (continued)
- Re: [Axiom-math] Curious behavior of Taylor series, Jay Belanger, 2006/08/20
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Jay Belanger, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Martin Rubey, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Igor Khavkine, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Jay Belanger, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/21
- Re: [Axiom-math] Curious behavior of Taylor series,
Jay Belanger <=
- Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/22
Re: [Axiom-math] Curious behavior of Taylor series, Igor Khavkine, 2006/08/21
Re: [Axiom-math] Curious behavior of Taylor series, William Sit, 2006/08/30
Re: [Axiom-math] Curious behavior of Taylor series, Ralf Hemmecke, 2006/08/20