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[Axiom-developer] Computational Mathematics

From: daly
Subject: [Axiom-developer] Computational Mathematics
Date: Sun, 24 Aug 2014 12:08:03 -0500


For those who don't know, I'm Tim Daly, Lead Developer on Axiom, an
open source computer algebra system

I saw slides from a presentation about the NIST Digital Repository of
Mathematical Formulae. I'm interested in helping.

There appears to be an effort to move the book into an online form. I
saw items in there about "a high degree of computer verification". 

There are years of effort on a Computer Algebra Test Suite (CATS)
from published tables and private communications. This effort includes
many thousands of test cases run in Axiom, all in machine readable
form. Axiom is mostly converted to "literate programs" in Latex and these
documents are also literate latex.

While the NIST DLMF is an amazing collection of formulas, it does not
address computational implementation issues such as the dependence of
integration algorithms on factoring algorithms.

The Axiom mailing list missives have been trying to raise interest in
a related area. We have been making computer algebra systems for the
last 40 years, accumulating implementations of algorithms in
computational mathematics. Unfortunately there is no "standard
reference" for these algorithms so it is highly likely that each CAS
will generate different answers.

Worse, as the computer algebra systems die, either from company deaths
(Symbolics/MACSYMA), purchase (Soft Warehouse/DERIVE), withdrawal
(NAG/Axiom), or loss of leader (Hearn/REDUCE) we lose valuable
implementations. The remaining systems (e.g. MATHEMATICA, MAPLE,
MATLAB, etc) will likely meet similar ends. 

This will leave a "black hole" in computational mathematics. Unlike
the 70s and 80s, there is no public financial support for building
large systems so these algorithmic implementations will simply
disappear. We must ensure that computational mathematics outlives
its founders.

We need to develop a standard set of algorithms along with
proofs, test cases, and implementation details. To this end I've
started collecting a bibliographic database of publications
by walking the bibliograpies of published papers. Currently it has
about 650 references, gaining about 50 per day. Also in-plan is 
providing links to published, freely available papers. 

>From this the plan is to collect the papers associated with an algorithm
and then show a series of ever more complex implementations, starting
with algorithms already implemented in Axiom.

Axiom has been instrumented to give detailed trace information so we can
build a tower of the algorithms required to do things like integration.

I believe it is important to get the computer algebra community
involved in the standard process. I'm willing to help where I can.

Tim Daly

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