Restarted Hessenberg method for solving shifted nonsymmetric linear systems
For researchers solving shifted linear systems, this method offers a more efficient alternative to existing restarted shifted solvers, though the improvement is incremental.
The authors propose a restarted Hessenberg method for solving shifted nonsymmetric linear systems, demonstrating that it achieves faster convergence in terms of number of restarts and lower CPU time compared to restarted FOM and other shifted solvers, as validated by numerical experiments on time fractional differential equations.
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, Méthode de Hessenberg Généralisée et Applications, Ph.D. Thesis, Université 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.