基于SOR预条件的全局GMRES方法求解Sylvester矩阵方程

SOR Preconditioned Global GMRES Method for Solving Sylvester Matrix Equations

摘要: Sylvester矩阵方程在控制理论、图像处理及微分方程数值解等领域应用广泛。对于大规模问题，直接求解方法因计算量和存储需求巨大而不再适用。全局GMRES (Generalized Minimal Residual Method)方法是求解此类大规模方程的有效迭代方法。为提升其收敛速度，本文引入了基于SOR (Successive Over-Relaxation)迭代的预条件技术，构建了SOR预条件全局GMRES方法。数值实验表明，与传统的全局GMRES方法相比，新方法能显著减少迭代步数与计算时间，验证了其有效性。

Abstract: The Sylvester matrix equation is widely used in fields such as control theory, image processing, and numerical solutions to differential equations. For large-scale problems, direct solution methods become impractical due to their high computational cost and storage requirements. The global Generalized Minimal Residual Method (GMRES) is an effective iterative approach for solving such large-scale equations. To enhance its convergence speed, this paper introduces a preconditioning technique based on the Successive Over-Relaxation (SOR) iteration, establishing the SOR-preconditioned global GMRES method. Numerical experiments demonstrate that, compared to the traditional global GMRES method, the proposed approach significantly reduces the number of iteration steps and computational time, confirming its effectiveness.