Generation and Verification of Algorithms for Symbolic-Numeric Processing

This paper presents a new method in which the algorithms suitable for implementation in numerical code are designed by human and coded in language of a CAS. The algorithms are tested by comparing their results on representative input data sets with results of the symbolic algorithms included in the CAS. The comparision is done in precise arithmetics and at the algebraic level in the CAS. The numerical implementation is then automatically generated from the same source, provided that the computer algebra system in question contains a facility to convert both mathematical functions and its own control language into a code in programming language suitable for the numerical applications. The method uses a CAS for advanced debugging of symbolic-numeric algorithms and allows also comparisions of the algorithms in different floating-point arithmetics. By such approach one can have strong confidence that the numerical code is correct. So here the knowledge from the computer algebra system is used to verify the correctness of proposed algorithms. In the work reported here, the computer algebra system REDUCE (Hearn, 1993) with the standard code generation package GENTRAN (Gates, 1986) is used to develop codes in FORTRAN. Generally we deal here with the development of a particular symbolic processing algorithm which is usually used as a part of a large numerical code. Typically the algorithm deals only with a special domain of formulas. Many papers e.g. (Steinberg, 1985, Wang, 1986, Dewar and Richardson, 1990, Cook, 1992, Kant, 1993) were dealing with code generation of numerical algorithms. Some work has also been reported on program transformation techniques e.g. (Zippel, 1992) and automatic differentiation of numerical codes (Rostaing, Dalmas and Galligo, 1993), however we are ..