广义极小残差法中基于Arnoldi过程的多项式预处理方法

A Polynomial Preprocessing Method Based on the Arnoldi Process in the Generalized Minimal Residual Method

摘要: 本文探讨了在求解大规模稀疏线性方程组时，多项式预处理技术在GMRES方法中的应用，提高了其计算效率和计算精度。我们分析了多项式预处理如何增加用于形成近似解的多项式的阶数。同时为了简化多项式预处理的过程，我们提出了基于Arnoldi过程的多项式预处理方法，通过直接利用Arnoldi基向量和递归系数来构造多项式$p\left(A\right)b$ ，从而有效避免了对多项式系数的直接计算。通过数值算例验证了这种方法简单且高效，为多项式预处理在GMRES中的应用提供了新的视角。

Abstract: In this paper, the application of polynomial preprocessing technology in the GMRES method is discussed when solving large-scale sparse linear equations, which improves its computational efficiency and computational accuracy. We analyze how polynomial preprocessing increases the order of the polynomial used to form an approximate solution. At the same time, in order to simplify the process of polynomial preprocessing, we propose a polynomial preprocessing method based on the Arnoldi process, which directly uses the Arnoldi basis vector and recursive coefficients to construct the polynomial $p\left(A\right)b$ , which effectively avoids the direct calculation of the polynomial coefficients. Numerical examples verify that this method is simple and efficient, which provides a new perspective for the application of polynomial preprocessing technology in the GMRES method.