Synthesis of Algebraic and Numerical Algorithms

本项目研究了有效的代数和数值算法的综合及其在实践和理论上的改进。这包括多项式寻根的实用数值算法的综合研究，这是代数计算的中心主题。这项研究应该包括许多技术上的创新。该项目涵盖的另一个基本主题是结构化矩阵的计算，它在科学和工程计算以及信号处理的许多领域都有广泛的应用。该项目还包括多项式寻根和结构化矩阵的各种扩展和应用，特别是多项式gcd的近似，多项式插值和多点计算，以及多项式方程组的求解。这项工作也适用于符号确定和模块化约简，这是代数和几何计算的核心主题，以及解码BCH纠错码，特别是在实践中最重要的里德-所罗门码。

This project investigates the synthesis of effective algebraic and numerical algorithms and on their practical and theoretical improvement. This includes the comprehensive study of practical numerical algorithms for polynomial rootfinding, which is a central subject of algebraic computing. The study should include many technical novelties. Another fundamental subject covered by the project and having ubiquitous applications to many areas of scientific and engineering computing and signal processing is the computations with structured matrices. The project also includes various extensions and applications of polynomial rootfinding and structured matrices, in particular to approximating polynomial gcd, to polynomial interpolation and multipoint evaluation, and to solving polynomial systems of equations. The work also applies to sign determination and modular reduction, which is a central subject of algebraic and geometric computations, and on decoding the BCH error-correcting codes, in particular Reed-Solomon codes most important in practice.