[EMAIL PROTECTED] wrote:
Andrew Coppin cites me and asks:
I find that statement interesting. I have never come across *any*
other package that can perform _symbolic_ mathematics.
(Sure, there are packages that can perform specific operations -
solving certain kinds of equations, transforming matricies,
rearranging formulas. But I have never seen any other package where
you can just do *anything*.)
You must be joking, but OK, I am naive, and I will answer as it were
serious. Really, haven't heard about Maple???
http://www.maplesoft.com/
Last I heard, Maple is simply another fast number-chrunking engine.
Its limited library is integrated within Matlab Symbolic Toolbox, and if
you *have* Maple and Matlab, the latter can use the full force of the
former.
Now Matlab I have heard of. (And I hate it with a passion.)
Maple is commercial as Mathematica, but there are perfectly
usable free packages as well, for example Axiom and Maxima (Macsyma
rekindled).
http://wiki.axiom-developer.org/FrontPage
http://maxima.sourceforge.net/
Now this actually looks vaguely promising. (Rather confused at this
point as to what the relationship between Axiom, Maxima, Reduce and Sage
is...)
The are more free stuff, GAP, MaCaulay,... In general, check
http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
I did indeed check that list a while back. I looked at a whole heap of
systems listed, but... half of them didn't seem to exist any more, and
the other half looked very clunky, had strange syntax, and only handled
a few specific constructs.
But yes - I have tried to implement that pattern matching engine a
couple of times in Pascal. (Remember Pascal?) Getting it to work for
a few test cases is easy. Getting it to *properly* handle
associativity and commutivity is really nontrivial. (I mean *really*
nontrivial! Or perhaps I am an inferior programmer - one of the two!)
The mathematically sensitive matching/substitution is a hard task even if
you have at your disposal a reasonably full unifier. Forget Pascal, take
Prolog, which will save several days/weeks of the implementation of basic
stuff. Even then, it will be quite tedious to write a package able
say, to
reduce rational formulae, to reduce a polynomial modulo an ideal, to
implement the basic trig identities, find a reasonable common form for
expressions containing complex exponentials AND trigonometrics, etc.
LOL! I did try to learn Prolog once... I never did understand how it was
able to make impossible leaps of deduction like that. And yet, in other,
similar cases, it couldn't figure out the result.
Of course, now I know that the answer is a magic trick known as
unification (although I still don't know much about that topic). Not
really sure how a unifyer would help you manipulate symbolic expressions...
If I get time, I might have another go at implementing the pattern
matching algorithm in Haskell. Of course, even if it works, that's still
a pretty small problem compared to finding a set of transformation rules
that cover all the cases you care about, is reasonably efficient,
produces the right answers...etc...etc...
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