[Maxima] Algebraic Risch
fateman at cs.berkeley.edu
Fri Oct 16 18:39:50 CDT 2009
David Simcha wrote:
> While Maxima has several practical advantages over, e.x., Axiom in terms
> of ease of use, etc., it seems that the symbolic integration
> capabilities are relatively weak. Apparently, the algebraic case of the
> Risch algorithm has not been implemented at all. Is this on the agenda,
> or will improvements to the implementation of the Risch algorithm likely
> not happen in the foreseeable future?
> Maxima mailing list
> Maxima at math.utexas.edu
Since no one seems to have answered this, it seems that it is not on
You are presumably welcome to contribute more to this capability.
I believe that the relatively low interest in the subject may have to do
with certain difficulties, but also the realization that the algebraic case
of the Risch "algorithm" has more to do with the theoretical decision
antiderivatives in closed form in terms of elementary functions. While
a historically interesting narrative from Liouville to Ritt to
Rosenlicht to Risch to
others (Moses, Wang, Davenport, Trager, Bronstein, Cherry, Llopis, and
others whose names I will remember
after I send off this note...), the actual interest in the applied math
is quite slim. For example, the information that an integral does not exist
in terms of elementary functions is perhaps charming, but not really useful.
And in the pure math community the information that an integral exists
be quite a separate question from whether it exists in terms of
Indeed, finding the explicit form tends to be treated as a freshman calculus
task solved by heuristics. If it were important, wouldn't the Risch
be taught to freshman? Or even mentioned?
You might think that implementing this program would mean that one could
do all those integrals in those tables of integrals ... but that
integration, which is not done by Risch (except when it is trivially a
of the fundamental theorem of integral calculus). And those are usually
not in the tables.
Furthermore, from an application standpoint,
definite integrals can often be done quite satisfactorily, numerically.
Not always of course.
Sometimes, using a computer algebra system we do a better job if we
functions can also be approximated symbolically (e.g. by Taylor series,
Pade approximations, or
Fourier series), which makes integrating them quite easy.
There may be other reasons for people not choosing to implement this
stuff, but that's my response,
off the top of my head.
But if someone wants to do this, check out the "parallel" Risch
algorithm first, and also
Bronstein's various documents.
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