Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
On the parallel Risch Algorithm (II)
ACM Transactions on Mathematical Software (TOMS)
User-based integration software
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
| Hi-index | 0.00 |
In 1976 Risch [1] proposed a scheme for finding the integrals of forms built up out of transcendental functions that viewed general functions as rational forms in a suitable differential field and represented the polynomial parts of those forms in a distributed rather than recursive way. By using a data representation where all variables were (more or less) equally important this new method seemed to side-step some of the complications that had appeared in his previous scheme [2] where various side-constraints had to be propagated between the levels present in a tower of separate extensions of differential fields, otherwise seen as levels in recursive datastructures.An initial implementation of the method was prepared in the context of the SCRATCHPAD/1 algebra system and demonstrated at the 1976 SYMSAC meeting at Yorktown Heights, a subsequent version for Reduce [3][5] came after that, and made it possible to try the method on a large range of integrals. These practical studies showed up some problems with the method and its implementation.The presentation given here re-expresses the 1976 Risch method in terms of rewrite rules, and thus exposes the major problem it suffers from as a manifestation of the fact that in certain circumstances the set of rewrites generated is not confluent. This difficulty is then attacked using a critical-pair/completion (CPC) approach. For very many integrands it is then easy to see that the initial set of rewrites used in the early implementations [1] and [3] do not need any extension, and this fact explains the high level of competence of the programs involved despite their shaky theoretical foundations. For a further large collection of problems even a simple CPC scheme converges rapidly; when the techniques presented here are applied to the REDUCE integration test suite in all applicable cases a short computation succeeds in completing the set of rewrites and hence gives a secure basis for testing for integrability.This paper describes the implementation of the CPC process and discusses current limitations to and possible future extended applications of it.