Risch Algorithm
Jemfinch02
Jeremy
Carlos Bazzarella
Indefinite Integral Algorithm. In another words; An Algorithm to
find the closed form solution of an integral in finite terms.
For a good description of the Algorithm get a hold of the following book
"Algorithms for Computer Algebra
Keith O. Geddes
Stephen R. Czapor
George Labahn
Kluwer Academic Publishers
ISBN 0-7923-9259-0
Carlos.
Andreas Eder
> What is the Risch Algorithm? Answers, Websites, anything is welcome. THanks
If you look for a good book an symbolic integration, I would have a
look in:
Manuel Bronstein
'Symbolic Integration I : transcendental functions'
Spinger-Verlag Berlin Heidelberg New York 1997
ISBN 3-540-60521-5
There, you will also find the original Risch papers in the
bibliography.
Andreas
--
Wherever I lay my .emacs, there's my $HOME
Bill Bauldry
>Jeremy
The Risch algorithm is a (set of) method(s) for determining an
indefinite integral and can answer the question of whether one exists or
not. If you have some background in Abstract Algebra, then take a look
at Geddes et al, "Algorithms for Computer Algebra". The book has a very
nice intro to the algorithm and a good bibliography. Contrary to
advertising claims, no current CAS implements the full algorithm.
Regards,
Bill
===================================
Wm C Bauldry, PhD
Professor and Chairperson
Department of Mathematical Sciences
Appalachian State University
Boone, NC 28608
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Jemfinch02
Andreas Jung
: In article <19990312013001...@ng-cf1.aol.com>, Jemfinch02 <jemfi...@aol.com> wrote:
: > Why doesn't any CAS implement the full algorithm?
: Axiom claims to, I believe.
But it's only a claim. For at least the following integrals,
the 1996 version of Axiom failed to find out if a symbolic
integral exists or not, but printed an error message instead:
integrate(asin(exp(x)), x)
integrate(acos(exp(x)), x)
integrate(asin(log(x)), x)
integrate(acos(log(x)), x)
integrate(log(1+sqrt(1-q^2*sin(x)^2)), x)
Perhaps, someone can feed these examples into an actual version
of Axiom and tell us if things have changed.
Greetings,
Andreas Jung.
--
Andreas Gisbert Jung DL9AAI Tel:0381/498-3364 Fax:0381/498-3366
Theoretische Informatik mailto:aj...@informatik.uni-rostock.de
Universitaet Rostock http://www.informatik.uni-rostock.de/~ajung/
PGP fingerprint = 8A 0B 05 CA EE AB 7B 01 D9 07 6A D0 84 38 BB 82
Richard J. Fateman
Andreas Jung <aj...@informatik.uni-rostock.de> wrote:
>G. A. Edgar (ed...@math.ohio-state.edu.nospam) wrote:
>: In article <19990312013001...@ng-cf1.aol.com>, Jemfinch02 <jemfi...@aol.com> wrote:
>
>: > Why doesn't any CAS implement the full algorithm?
>
>: Axiom claims to, I believe.
>
>But it's only a claim. For at least the following integrals,
There is a fallacy in claiming the Risch "algorithm" is an
algorithm at all: it depends, for solution of subproblems,
on heuristics to tell if expressions are equivalent to zero.
It was shown in 1968 or so by Daniel Richardson, that the
zero equivalence problem over a class of expressions including
the Risch target class, is recursively unsolvable. You
can always tell if a polynomial is identically zero. But you cannot
tell by a uniform algorithm if a rational expression in 1 variable
including exp sin sqrt, composition, pi, +-/, constants is zero.
[I think I got that about right..].
But that is not the problem with most implementations of the
Risch algorithm. They mostly have not been programmed completely,
because, even assuming you can solve the zero-equivalence problem,
the procedure is hard to program. Why aren't people pushing
to do this more?...
Another reason is, it is not nearly as useful as you might think,
because it returns algebraic antiderivatives whose validity may
be on a set of measure zero. It may also, in the vast majority
of problems, simply say, after an impressive pause, nope. can't do it.
Also, most people are interested in definite, not INdefinite integrals,
at least once they've finished with Freshman calculus.
And in cases where approximate solutions are easily obtained
for the corresponding definite integral, the Risch algorithm
may grind on for a while and then say there is no closed form.
The theory of algebraic integration, and the history of this
problem is interesting, but the connection with computer algebra
systems aimed at solving applied mathematics problems in
complex analysis is not nearly as central as one might think.
My opinion, anyway.
--
Richard J. Fateman
fat...@cs.berkeley.edu http://http.cs.berkeley.edu/~fateman/