The worst-case root-convergence factor of GMRES(1)

TL;DR

This paper analyzes the worst-case convergence factors of GMRES(1) for linear systems, revealing how symmetry properties of the matrix influence convergence behavior and providing theoretical and numerical validation.

Contribution

It derives the asymptotic and worst-case root-convergence factors of GMRES(1) for symmetric and skew-symmetric cases, offering new insights into its convergence properties.

Findings

Convergence depends on initial guess for symmetric matrices.

GMRES(1) converges unconditionally when I-A is skew-symmetric.

Theoretical results are validated by numerical experiments.

Abstract

In this work, we analyze the asymptotic convergence factor of minimal residual iteration (MRI) (or GMRES(1)) for solving linear systems $Ax=b$ based on vector-dependent nonlinear eigenvalue problems. The worst-case root-convergence factor is derived for linear systems with $A$ being symmetric or $I-A$ being skew-symmetric. When $A$ is symmetric, the asymptotic convergence factor highly depends on the initial guess. While $M=I-A$ is skew-symmetric, GMRES(1) converges unconditionally and the worst-case root-convergence factor relies solely on the spectral radius of $M$. We also derive the q-linear convergence factor, which is the same as the worst-case root-convergence factor. Numerical experiments are presented to validate our theoretical results.