A Block QMR Method for Computing Multiple Simultaneous Solutions to Complex Symmetric Systems

Abstract

The solution of complex symmetric indefinite systems of equations where multiple solutions are required is considered. The quasi-minimum residual (QMR) method, ideally suited for these matrices, is generalized using the block Lanczos algorithm to solve multiple solutions simultaneously. This modification alone is shown, through numerical examples involving large sparse matrices from finite element discretization of Maxwell’s equations, to accelerate the convergence by a factor almost as great as the number of simultaneous solutions. A natural convergence criterion for this method is presented that is shown to be as effective as, and easier to compute than, the usual equation residual. Finally, a numerical comparison of the classical incomplete Cholesky and a variant of the ILU(T) preconditioners is given showing superior performance by the latter.