EFFICIENT PRECONDITIONING FOR SEQUENCES OF PARAMETRIC COMPLEX SYMMETRIC LINEAR SYSTEMS
Daniele Bertaccini · Cineca Institutional Research Information System (Tor Vergata University) · 2004
Abstract. Solution of sequences of complex symmetric linear systems of the form Ajxj = bj, j = 0,..., s, Aj = A + αjEj, A Hermitian, E0,..., Es complex diagonal matrices and α0,..., αs scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-andinvert and Jacobi–Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then Aj is complex symmetric. The case A Hermitian positive semidefinite, Re(αj) ≥ 0 and such that the diagonal entries of Ej, j = 0,..., s have nonnegative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A −1 are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches. Key words. Complex symmetric linear systems; preconditioning; parametric algebraic linear systems; incomplete factorizations; sparse approximate inverses. AMS subject classifications. 65F10, 65N22, 15A18. 1. Introduction. The