Incomplete factorization-based preconditionings for solving the Helmholtz equation

Preconditioning techniques based on incomplete factorization of matrices are investigated, to solve highly inde nite complex-symmetric linear systems. A novel preconditioning is introduced. The real part of the matrix is made positive de nite, or less inde nite, by adding properly de ned perturbations to the diagonal entries, while the imaginary part is unaltered. The resulting preconditioning matrix, which is obtained by applying standard methods to the perturbed complex matrix, turns out to perform signi cantly better than classical incomplete factorization schemes. For realistic values of the GMRES restart parameter, spectacular reduction of iteration counts is observed. A theoretical spectral analysis is provided, in which the spectrum of the preconditioner applied to inde nite matrix is related to the spectrum of the same preconditioner applied to a Stieltjes matrix extracted from the inde nite matrix. Results of numerical experiments are reported, which display the e ciency of the new preconditioning.