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Topic: A bug in Reduce package 'algint'?
Replies: 9   Last Post: Feb 1, 2013 6:02 PM

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clicliclic@freenet.de

Posts: 980
Registered: 4/26/08
Re: A bug in Reduce package 'algint'?
Posted: Feb 1, 2013 6:02 PM
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clicliclic@freenet.de schrieb:
>
> Axel Vogt schrieb:

> >
> > I have not checked, how Maple 'finds' the solution, but it does
> > not have implemented the 'catalog' of rules you have worked out
> > and at hand now. Unfortunately.

>
> I suspect you may be referring to a paper discussed last autumn,
> something about umpteen theorems on a few pages. As Maple appears to
> be comparatively weak on pattern matching, it would probably be better
> here to implement this kind of reduction by setting up and solving the
> systems of linear equations anew for each problem (by the Extended
> Euclidean Algorithm if you like), and generalizing to arbitrary order
> at the same time. [...]
>


Actually, the situation is not as bad as we thought: Maple seems to
employ "Hermite reduction" in order to raise exponents in elliptic
integrands at least. The procedure appears to be detailed in: Labahn &
Mutrie, "Reduction of Elliptic Integrals to Legendre Normal Form"
(University of Waterloo Tech. Report 97-21, Department of Computer
Science, 1997).

In a 1999 paper entitled "Symbolic integration of elliptic integrals",
B.C. Carlson summarizes this report as follows:

"In a recent report Labahn and Mutrie (1997) chose Legendre's canonical
forms but made improvements on the classical method. They used Hermite
reduction (Hermite, 1912, pp. 35-54; Hermite, 1917, pp. 120-126,
201-205) (Geddes et al., 1992) to arrive at integrands without multiple
poles, and the terms with simple poles are decomposed by an implicit
full partial-fraction expansion to avoid working in algebraic extensions
as far as possible. To reduce to Legendre's canonical forms they finally
chose one of 21 transformations according to the zeros of s^2 and the
limits of integration. The reductions that they used take special care
to minimize the radical extensions required to express their answer."

The Hermite references are to the collected "Oeuvres" vol. III and IV,
and s^2 denotes the square of the elliptic radicand. I haven't seen the
Labahn-Mutrie report myself.

Martin.



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