Characterization of worst-case GMRES

TL;DR

This paper investigates the worst-case residual bounds for GMRES, characterizing initial vectors that attain these bounds, and explores the relationship between worst-case and ideal GMRES approximations, including complex extensions.

Contribution

It provides a comprehensive analysis of worst-case GMRES residuals, characterizes vectors attaining these bounds, and compares worst-case and ideal approximations, including complex polynomial considerations.

Findings

Worst-case GMRES behavior is the same for A and A^T.

Initial vectors attaining worst-case residuals satisfy a 'cross equality'.

The relation between worst-case and ideal GMRES can be sharply characterized.

Abstract

Given a matrix $A$ and iteration step $k$, we study a best possible attainable upper bound on the GMRES residual norm that does not depend on the initial vector $b$. This quantity is called the worst-case GMRES approximation. We show that the worst case behavior of GMRES for the matrices $A$ and $A^T$ is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we show that such vectors satisfy a certain "cross equality", and we characterize them as right singular vectors of the corresponding GMRES residual matrix. We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is sharp at all iteration steps $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems in the context of worst-case and ideal GMRES for a real matrix change, when one considers complex (rather than real) polynomials and initial vectors in these problems.