GMRES Method for Solving Large Linear Systems

This document provides a comprehensive overview of the GMRES (Generalized Minimal Residual) method for solving large-scale linear systems. As an iterative solver, GMRES is particularly effective for handling sparse matrix systems commonly encountered in scientific computing. The core algorithm operates by constructing an orthogonal basis through Arnoldi iteration and minimizing the residual norm in Krylov subspaces. Key implementation aspects include: utilizing Householder reflections for improved numerical stability, managing restart strategies to control memory usage, and employing preconditioning techniques to accelerate convergence. We present detailed algorithmic workflows with pseudocode examples, highlighting critical functions like orthogonalization procedures and residual norm calculations. The document includes practical MATLAB/Python code snippets demonstrating how to implement the restart mechanism and handle Hessenberg matrix formations. Furthermore, we analyze GMRES's convergence behavior compared to other iterative methods (like Conjugate Gradient and BiCGSTAB), discussing its advantages in handling nonsymmetric matrices and limitations regarding memory requirements. Through comparative performance benchmarks and application case studies, readers will gain practical insights for effectively deploying GMRES in real-world computational problems involving large sparse systems.