Martin, On January 3, 2006 6:49 AM you wrote: >... > "... Axiom's integrator gives you the answer when an answer > exists. If one does not, it provides a proof that there is > no answer." > > on page 2 is simply wrong. (Apart from containing a typo).
Rather than just state that there is a typo: please, please state exactly the error and the correction. Or better yet include a 'diff -au' patch to correct the error. > Even if read as > > ... Axiom's integrator gives you the answer when an answer > in terms of elementary functions exists. If one does not, > it provides a proof that there is no answer. > > it is not true. The Risch algorithm is *not* completely > implemented in Axiom, unfortunately. If, we can finish the implementation of this algorithm in Axiom, then would your re-worded statement above be correct? Of course we are really talking about "anti-derivatives", not integration as such. Manuel Bronstein has written extensively about this: http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications So what "elementary functions" are being considered by Axiom's algorithm? E.g. addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationals, subtraction, and trigonometric expressions such as suggested here: Eric W. Weisstein. "Elementary Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ElementaryFunction.html Or should we be talking about "closed forms" as in the reference: http://arxiv.org/abs/math.NT/9805045?front Date: Fri, 8 May 1998 22:49:15 GMT (12kb) What is a closed-form number? Authors: Timothy Y. Chow Comments: 11 pages; submitted to Amer. Math. Monthly If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described. -------- Or some other definition such as: http://en.wikipedia.org/wiki/List_of_functions#Elementary_functions > It might be more complete than the ones in MMA and Maple, I doubt > that it's better than MuPAD, but boasting that it is complete is > unwise, I'm afraid. Since it was obviously the intention of the author of Axiom's integration algorithm that it completely implement the Risch algorithm and in spite of: http://mathworld.wolfram.com/RischAlgorithm.html "The case of algebraic extensions is quite complicated and is therefore not completely implemented in any computer algebra system." http://en.wikipedia.org/wiki/Risch_algorithm "The Risch decision procedure is not formally an algorithm because it requires an oracle that decides whether a constant expression is zero, a problem shown by Daniel Richardson to be undecidable. Transforming the Risch decision procedure into an algorithm that can be executed by a computer is a complex task that requires the use of heuristics and many refinements." ------- What are the prospects for actually completing this programme in Axiom rather than "correcting" the documentation? So far we have documented the following cases: http://wiki.axiom-developer.org/198IntegrateSinX2XIsNotHandled http://wiki.axiom-developer.org/199IntegrateExpX2ExpXXX and the Risch algorithm is mentioned in the following places in the current Axiom source: http://wiki.axiom-developer.org/axiom--test--1/src/algebra/EfstrucSpad http://wiki.axiom-developer.org/axiom--test--1/src/algebra/IntafSpad http://wiki.axiom-developer.org/axiom--test--1/src/algebra/IntefSpad http://wiki.axiom-developer.org/axiom--test--1/src/algebra/IntrfSpad And we also have Manuel Bronstein's work on http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html pmint - The Poor Man's Integrator pmint is a very short (95 lines) Maple program for integrating transcendental elementary or special functions. It is based on recent improvements to a powerful heuristic called parallel integration. While it is not meant to be as complete as the large commercial integrators based on the Risch algorithm, its very small size makes it easy to port to any computer algebra system or library capable of factoring multivariate polynomials. Because it is applicable to special functions (such as Airy, Bessel, Whittaker, Lambert), it is able to compute integrals not handled by the existing integrators. pmint is not meant as a replacement for existing integrators, but either as an extension, or as a cheap and powerful alternative for any computer algebra project. ------- It would be wonderful to have this available in Axiom. Regards, Bill Page. _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
