Toward high-performance polynomial system solvers based on triangular decompositions

Abstract

This thesis is devoted to the design and implementation of polynomial system solvers based on symbolic computation. Solving systems of non-linear, algebraic or differential equations, is a fundamental problem in mathematical sciences. It has been studied for centuries and still stimulates many research developments, in particular on the front of high-performance computing. Triangular decompositions are a highly promising technique with the potential to produce high-performance polynomial system solvers. This thesis makes several contributions to this effort. We propose asymptotically fast algorithms for the core operations on which triangular decompositions rely. Complexity results and comparative implementation show that these new algorithms provide substantial performance improvements. We present a fundamental software library for polynomial arithmetic in order to support the implementation of high-performance solvers based on triangular decompositions. We investigate strategies for the integration of this library in high-level programming environments where triangular decompositions are usually implemented. We obtain a high performance library combining highly optimized C routines and solving procedures written in the Maple computer algebra system. The experimental result shows that our approaches are very effective, since our code often outperforms pre-existing solvers in a significant manner.