QMR: A Quasi-Minimal Residual method for non-Hermitian linear systems

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. A novel BCG like approach is presented called the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

It is shown 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. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation is presented of a look-ahead version of the Lanczos algorithm that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead

Recent advances in Lanczos-based iterative methods for nonsymmetric linear systems

In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been proposed. This paper gives a survey of some of these recent developments

Iterative Solution of Linear Systems

this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to non-Hermitian matrices. We also review GMRES and related methods, as well as CG-like algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczos-based iterative methods for general non-Hermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using look-ahead techniques. Moreover, we describe the quasi-minimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the look-ahead Lanczos process to obtain quasi-optimal approximate solutions. Next, a survey of transposefree Lanczos-based methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameter-dependent algorithms that, unlike CG-type schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameter-dependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtiga

Ornl/tm-12873

First principles methods are used to aid the metallurgist in the investigation and design of new materials. However, these methods suffer from an O(N 3 ) scaling which restricts the problem sizes that can be addressed. By incorporating the QMR method into a first principles algorithm we are able to attain O(N) scaling for the problem sizes of interest. This advancement will provide researchers with the necessary tools to treat large systems. - v - 1. Introduction The design of new technologically advanced materials is of extreme importance to the industrial and economical competitiveness of the U.S. For example, the design of a high temperature ductile intermetallic alloy could save the power generation and aerospace industries billions of dollars. The modern design of new materials makes use of local-density approximation (LDA) based, first principles electronic structure calculations. These methods are parameter-free methods, requiring no experimental input. They are predictive ..