Restarted Hessenberg method for solving shifted nonsymmetric linear systems

TL;DR

This paper introduces a restarted Hessenberg method for shifted nonsymmetric linear systems, demonstrating improved convergence and efficiency over existing methods through theoretical analysis and extensive numerical experiments.

Contribution

It proposes a new restarted Hessenberg method that accelerates convergence for shifted systems, outperforming existing methods like shifted FOM and GMRES in efficiency.

Findings

The new method converges faster in numerical experiments.

It reduces CPU time compared to existing shifted solvers.

The residuals remain collinear, ensuring stability.

Abstract

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.