oldk1331 wrote:
>
> Thanks for your informative replies, Waldek Hebisch.
> I have a question related about Risch algorithm and its
> implementation in FriCAS.
>
> I asked my math professor about Risch algorithm before,
> he never heard of it and he seemed to not care about it,
> because "numerical methods are enough, nobody cares
> about symbolic results". I want to know your perspective
> of the future of Risch algorithm and its implementation in
> FriCAS.
Well, if you want numbers use numerical methods. However,
I care about formulas. And Risch algorithm is a powerful
method to produce new formulas.
Let me add a philosophical remark: thery behind Risch algorithm
deals with fundamental properties of symbolicaly defined
functions. In analogy to number theory it could be called
function theory, only most mathematicians hearing "function
theory" would imagine quite a different thing. Interestingly,
nonconstant functions in a sense are better behaved than numbers.
And results of such study seem to be more useful for computations
than results of number theory.
Concerning future of Risch algorithm, there is still theoretical
work to do. First, finding more efficient variant for elementary
functions. Second, extending algorithm to larger classes of
functions. Some generalizations are already implemented, for example
several parts writen by Bronstein deal with so called monomial
extensions. I added code to handle undefined functions, like:
(1) -> f := operator 'f
(1) f
Type: BasicOperator
(2) -> integrate(f(x)*D(f(x), x), x)
2
f(x)
(2) -----
2
Type: Union(Expression(Integer),...)
I wrote about method to find interal in terms of exponential integrals
and incomplete Gamma functions (with rational first argument). However
we would like to allow more Liouvilian functions as integrals,
in particular polylogaritms.
To put it differently, there is a long way to go. Most attention
will probably go into handling special functions (both as part
of integrand and as possible integrals). Certainly we want
to "complete" implementation for elementary functions and possibly
for larger classes. For algebraic functions probably most
important is work on other parts of FriCAS: we could greatly
improve speed for such integrals using better algorithms for
gcd, determinants, resultants, etc... For definite integrals
we need to improve limits. So beside Risch algorithm we need
to work on other parts to get a smootly working whole.
--
Waldek Hebisch
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