Computing GMRES for Large Matrices

GMRES (Generalized Minimal Residual Method) is an efficient iterative algorithm for solving large sparse linear systems, particularly suited for non-symmetric matrices. The algorithm progressively approximates solutions through Krylov subspace projections and possesses inherent capabilities to handle ill-conditioned problems automatically.

In MATLAB, GMRES implementation typically follows two approaches: Direct utilization of the built-in gmres function, which features optimized restart logic where users simply specify tolerance, maximum iterations, and restart parameters. Manual implementation of core GMRES workflow including Arnoldi process for orthogonal basis construction and least-squares problem solving - this approach offers greater flexibility but requires careful handling of numerical stability issues through proper Householder transformations or modified Gram-Schmidt orthogonalization.

The restart mechanism is crucial in GMRES, periodically resetting Krylov subspace dimensions to prevent excessive memory consumption. MATLAB's gmres function controls restart steps via input parameters, while manual implementations must ensure proper residual vector transfer after restarts using residual norm preservation techniques.

In practical applications, combining preconditioning techniques (like ILU decomposition) significantly accelerates convergence. For ill-conditioned matrices, consider adjusting GMRES reorthogonalization strategies or switching to flexible FGMRES variants that allow variable preconditioners across iterations. Key implementation considerations include monitoring residual norms, handling breakdown conditions in Arnoldi process, and optimizing matrix-vector products for sparse storage formats.