FOXBOX

Abstract

The dreaded phenomenon of expression swell in symbolic computation can be palliated by adopting implicit representations for symbolic objects, such as straight-line programs or so-called black box representations. In the latter, each expression is a symbolic object, more specifically, a computer program with a set of statically initialized data, which takes as input a value for each variable and then produces the value of the symbolic object it represents at the specified point. In this thesis we introduce F sc OXB sc OX, a software package that puts in practice the black box representation of symbolic objects and provides algorithms for performing the symbolic calculus with such representations. Improved versions of the algorithms found in Kaltofen and Trager (Journal of Symbolic Computing, vol. 9, nr. 3, p. 311 (1990)) and Kaltofen and Diaz (International Symposium on Symbolic and Algebraic Computation '95,) p. 232) are discussed. Also we describe an interpolation scheme based on a Zippel's algorithm (Journal of Symbolic Computing, vol. 9, nr. 3, p. 375 (1990)) that optimizes the number of required black box evaluations. The design of F sc OXB sc OX is intended to demonstrate how plug-in software components can be employed for generally used symbolic systems. Our implementation incorporates data types parameterized by arbitrary coefficient domains and generic algorithms. By providing a mechanism for interfacing to general purpose computer algebra systems, we broaden F sc OXB scOX's applicability. Furthermore we provide a distribution mechanism that allows for parallel and distributed execution of F sc OXB sc OX programs independent of the underlying parallel architecture. Finally, we present the results of several challenge problems which exercise our F sc OXB sc OX library and represent the first symbolic solutions of such problems.