QMR-based projection techniques for the solution of non-Hermitian systems with multiple right-hand sides (Q2780551)

The authors present two new projection approaches, based on quasi-minimal residual (QMR) and block QMR algorithms for solving systems of linear equations with non-Hermitian coefficient matrices and multiple right-hand sides all of which are available simultaneously. An outline of the single-seed projection approach and its block variant is reported. The theory of the single-seed case helps to explain the convergence behavior under certain conditions. As a result of the underlying unsymmetric Lanczos process: (i) the basic vectors do not need to be stored; (ii) the subspace dimension does not need to be predetermined.NEWLINENEWLINENEWLINEThe authors show that the approximate solutions and residuals to the non-seed systems are cheaply computed and they are available at every stage of the algorithm because they are updated with short-term recurrences. Some implementation details of both algorithms are discussed and advantages of the block-seed approach in cache-based serial and parallel computers are noted. NEWLINENEWLINENEWLINEThe computational savings of the suggested two algorithms over using the classical QMR to solve each system individually are illustrated in two examples: (1) for finding the scattered electric fields caused when plane waves at various angles impinge on a horizontal air-soil interface and (2) for the scattered electric field from a buried object when the source of the incident field is a point source.