Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library. Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver. Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices. Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++