An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices Part II

Abstract

this paper, we have presented an implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. Here, we show how the look-ahead Lanczos process --- combined with a quasi-minimal residual (QMR) approach --- can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. In particular, BCG iterates can be recovered stably from the QMR process. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problems and linear systems. Also, program listings of FORTRAN implementations of the look-ahead Lanczos algorithm and the QMR method are included. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra--eigenvalues; linear systems (direct and iterative methods); sparse and very large systems; G.4 [Mathematics of Computing]: Mathematical Software General Terms: Large sparse linear systems and eigenvalue problems, iterative methods Additional Key Words and Phrases: Non-Hermitian linear systems, quasi-minimal residual property, Lanczos method, biconjugate gradients The work of R. W. Freund and N. M. Nachtigal was supported in part by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA). Authors' addresses: R. W. Freund, RIACS, Mail Stop Ellis Street, NASA Ames Research Center, Moffett Field, CA 94035, and Institut fur Angewandte Mathematik, Universit at Wurzburg, D--W8700 Wurzburg, Federal Republic of Germany; N. ..