Aspectos teóricos e implementación del método iterativo GMRES para la resolución de sistemas lineales

Abstract

[en] In this work we study different theoretical aspects of the GMRES algorithm (Generalized Minimal RESidual), including the use of Krylov subspaces, general results on convergence and the adapted version with restarted GMRES(m). GMRES is an iterative method that approximates the solution of a linear system $Ax = b$ by looking for the solution that minimizes the residue within the Krylov subspace. This method can be applied to general linear systems, because it does not require an specific structure of the matrix $A$ of the system. It is especially suitable for system of high dimension when it is not possible to solve it by direct methods and when the classical iterative methods like Jacobi, Gauss-Seidel or SOR do not converge or do not converge in a reasonable amount of time. We complement the work with several remarks and pseudo-code schemes useful for the implementation (in C language) that we have done of the method. We use our own implementation of GMRES in several examples, improving the implementation until we reach the final version (included in the Appendix). In the memory we present some examples to illustrate some aspects of the convergence of GMRES for different spectra of $A$.