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Re: [Axiom-developer] Failure of Axiom? (was: Static versus Dynamically

From: William Sit
Subject: Re: [Axiom-developer] Failure of Axiom? (was: Static versus Dynamically typed) )
Date: Fri, 23 Sep 2005 08:11:57 -0400

Bill Page wrote:
> On September 22, 2005 11:41 PM M. Edward (Ed) Borasky wrote:
> > Bill Page worte:
> > > In a sense, Axiom is/was an experiment in the application of
> > > strongly typed programming languages in computer algebra and
> > > to be quite honest and blunt, for the most part the experiment
> > > seems to have failed. :(

Just to provide a better balance of these discussions, here are some quotations
taken from research papers involving Axiom published during 1999 to 2001. Note
that these authors had used other computer algebra systems for the SAME research
earlier. I am including three representatives from major research areas of
computer algebra:  abstract mathematics, applied computation, and comparison of
algorithms. For details, see the original articles.

(1) Ito calculus is a formalism to study continuous random
processes. Kendall [1] wrote:

"This article reports on progress in the implementation of Ito
calculus in the powerful and innovative computer algebra package
AXIOM, in the context of a decade of previous implementations and
applications.  It is shown how the elegant algebraic structure
underlying the expressive and effective formalism of \ito
calculus can be implemented directly in AXIOM using the package's
programmable facilities for 'strong typing' of computational
objects.  An application is given of the use of the implementation to
provide calculations for a new proof, based on stochastic
differentials, of the Mardia-Dryden distribution from statistical
shape theory."

And from the summary (the earlier implementations are in Reduce and

"In this article we have described how the innovative features
of AXIOM can be used to exploit the elegant formalism of the
\ito calculus, producing an implementation (Kendall 1998a) of
Itovsn3 substantially improving on its predecessors (Kendall
1991a, b, 1993b)."

(2) When algorithms for computing three-loop (Feynman) diagrams in Heavy
Quark Effective Theory (HQET) using integration-by-parts recurrence
relations are implemented in both Reduce and Axiom, there is substantial
increase in speed in Axiom.  Grozin reported [2]:

"I also re-implemented it in Axiom
[25].  All expressions involved are linear combinations of basis
integrals with coefficients which are rational functions of $d$.  It
is convenient to use Axiom Vector domain, which has all the necessary
operations.  This makes intermediate expressions shorter than in the
case when multivariate rational functions are used for entire
expressions, because, typically, not all basis integrals are
accompanied by every possible denominator.  The amount of GCD
calculations is thus reduced.  This improvement can be, in principle,
back-propagated to the REDUCE implementation by using matrices.
However, working with matrices in REDUCE is awkward, because there are
no local matrix variables, and no easy way for a function to return a
matrix.  I made no systematic benchmarking; one test set of 9375
recurrence relations took 2019 seconds in REDUCE and 350 seconds in

(3) In choosing a system to compare different algorithms for computations with
triangular sets of polynomial equations using Gr\"obner basis (GB), Aubry and
Maza \cite[p.~137]{Aubry} selected Axiom for the following reasons:

"We thought that the AXIOM computer algebra system (Jenks and Sutor,
1992; Broadbery et al., 1994), with its strongly typed and
object-oriented language, is convenient to satisfy our first
requirement.  We defined categories corresponding to the different
properties of triangular sets, packages and domains for the common
data structures and sub-routines.  Furthermore, AXIOM (version 1.2) is
connected with GB, the very powerful Gr\"obner engine developed by
Faug\`ere (1994).  This allowed us to run the non-trivial Gr\"obner
basis computations that are required in order to satisfy our other two

References for these quotations:
[1] Kendall WS, Symbolic Ito calculus in AXIOM: An ongoing story

[2] Grozin AG, Calculating three-loop diagrams in heavy quark effective theory
with integration-by-parts recurrence relations

[3] Aubry P, Maza MM, Triangular sets for solving polynomial systems: a
comparative implementation of four methods

You can make your own conclusion.


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