Implementing Restarted GMRES for Large-Scale Matrix Computations in MATLAB

This article presents a comprehensive guide to implementing the restarted GMRES (Generalized Minimal Residual) algorithm in MATLAB for large-scale matrix computations. While GMRES efficiently handles large linear systems, complex matrices may pose computational challenges requiring optimization techniques. We demonstrate MATLAB programming strategies to enhance matrix operation efficiency and provide executable code examples for the restarted GMRES implementation. The discussion includes key algorithmic components such as: Arnoldi iteration for constructing orthogonal bases, Hessenberg matrix reduction, and restart mechanisms to control memory usage. We further analyze the algorithm's mathematical foundation, covering Krylov subspace methods and convergence properties, along with practical applications in scientific computing and engineering simulations. The implementation leverages MATLAB's built-in functions like gmres() while illustrating custom coding approaches for preconditioning and tolerance control to improve numerical stability. This technical exploration aims to deepen understanding of iterative methods and their effective deployment in computational mathematics.