GMRES, pseudospectra, and Crouzeix's conjecture for shifted and scaled Ginibre matrices

TL;DR

This paper analyzes the GMRES algorithm's behavior on shifted and scaled Ginibre matrices, providing exact asymptotics for residual errors and proving a restricted form of Crouzeix's conjecture related to pseudospectra.

Contribution

It offers new insights into GMRES performance on random matrices and establishes a restricted version of Crouzeix's conjecture for Ginibre matrices.

Findings

Exact large-N behavior of GMRES residuals for independent right-hand sides.

Analysis of pseudospectra and numerical range of Ginibre matrices.

Proof of a restricted Crouzeix's conjecture for these matrices.

Abstract

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted $N\times N$ matrix whose entries are independent complex Gaussians. When the right hand side of this linear system is independent of this random matrix, the $N\to\infty$ behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix's conjecture.