DGMRES method augmented with eigenvectors for computing the Drazin-inverse solution of singular linear systems

TL;DR

This paper enhances the DGMRES method for solving singular linear systems by augmenting it with eigenvectors, improving convergence and overcoming stagnation issues, supported by theoretical derivation and numerical examples.

Contribution

It introduces an augmented DGMRES method with eigenvectors, improving convergence for singular systems, extending prior work on eigenvector augmentation in iterative methods.

Findings

Enhanced convergence with eigenvector augmentation

Overcomes stagnation in DGMRES

Numerical examples demonstrate advantages

Abstract

The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R.Morgan in [R.Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J.Matrix Anal.Appl. 16 (1995)1154-1171]. We derive the implementation of this method and present some numerical examples to show the advantages of this method.