First and foremost it is important to realize there is a
computational history before Axiom. What I think is important
from this history is that several programs have been developed to find
approximate results also known as non-exact or numerical results for
problems. I will call them approximating programs as a set.
These include Fortran, C, etc..
Now for where I think I differ with you Bill. Axiom is a
CAS. As such it has built in capabilities and certain tools that
are different than an approximating programs. It has the ability
to help us solve a problem exactly. This makes Axiom as well as
all other CAS's vastly different than approximating programs.
Should Axiom be an all in one computer program for solving math
problems? I believe this is asking too much for any
program. Should an NR program be converted to SPAD? As such this
is in my opinion very low if non-existent on the agenda for
The one way gpl'd programs have an advantage on proprietary software is
that they can use existing gpl'd programs without worrying about
licenses etc.. For instance consider all the work done in Fortran
like quadpack etc.. Axiom if asked for a numerical result could
if linking was considered could use the existing libraries to compute
practically any function.
In fact this is just the strategy used by the proprietary Axiom.
To link it to NAG was what was done. There are plenty of high
quality libraries that can be used for numerical approximations in
Fortran that are gpl: lapack is but one of them.
Is this better than Mathematica? Well I guess the analogy may be
something like is it better to sleep with many partners than with just
one. By this I mean, problems of using other programs to
calculate numerical values may include trusting the approximating
program that you are using and the library that it uses to find a
numerical value. That is if many sources are used then we have to
trust each one is not faulty. Mathematica or Maple probably rely
on there own programs for approximate results. However there are
trusted libraries to approximately solve math problems that are gpl'd
and that people are working right now on, like GSL.
If I am not seeing the full picture please enlighten me.
Otherwise I do not understand why any approximation for a function need
be incorporated into SPAD.