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61
QMR: a QuasiMinimal Residual Method for NonHermitian Linear Systems
, 1991
"... ... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from t ..."
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Cited by 337 (26 self)
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... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead 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.
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
, 1994
"... This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied ..."
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Cited by 171 (5 self)
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This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied
An Implementation of the LookAhead Lanczos Algorithm for NonHermitian Matrices Part I
, 1991
"... ..."
Iterative Solution of Linear Systems
 Acta Numerica
, 1992
"... 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 nonHermitian matrices. We also review GMRES ..."
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Cited by 101 (8 self)
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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 nonHermitian matrices. We also review GMRES and related methods, as well as CGlike 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 Lanczosbased iterative methods for general nonHermitian 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 lookahead techniques. Moreover, we describe the quasiminimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the lookahead Lanczos process to obtain quasioptimal approximate solutions. Next, a survey of transposefree Lanczosbased 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 parameterdependent algorithms that, unlike CGtype schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameterdependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal
A restarted GMRES method augmented with eigenvectors
 SIAM J. Matrix Anal. Appl
, 1995
"... Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that appr ..."
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Cited by 81 (11 self)
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Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that approximate eigenvectors corresponding to a few of the smallest eigenvalues be formed and added to the subspace for GMRES. The convergence can be much faster, and the minimum residual property is retained. Key words. GMRES, conjugate gradient, Krylov subspaces, iterative methods, nonsymmetric systems AMS subject classifications. 65F15, 15A18
Krylov subspace methods on supercomputers
 SIAM J. SCI. STAT. COMPUT
, 1989
"... This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three dimensional model ..."
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Cited by 68 (4 self)
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This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel / vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. We describe in detail a few approaches consisting of implementing efficient forward and backward triangular solutions. Then we discuss polynomial preconditioning as an alternative to standard incomplete factorization techniques. Another efficient approach is to reorder the equations so as improve the structure of the matrix to achieve better parallelism or vectorization. We give an overview of these ideas and others and attempt to comment on their effectiveness or potential for different types of architectures.
Row Projection Methods For Large Nonsymmetric Linear Systems
 SIAM J. Scientific and Statistical Computing
, 1992
"... . Three conjugate gradient accelerated row projection (RP) methods for nonsymmetric linear systems are presented and their properties described. One method is based on Kaczmarz's method and has an iteration matrix that is the product of orthogonal projectors; another is based on Cimmino's ..."
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Cited by 42 (5 self)
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. Three conjugate gradient accelerated row projection (RP) methods for nonsymmetric linear systems are presented and their properties described. One method is based on Kaczmarz's method and has an iteration matrix that is the product of orthogonal projectors; another is based on Cimmino's method and has an iteration matrix that is the sum of orthogonal projectors. A new RP method which requires fewer matrixvector operations, explicitly reduces the problem size, is error reducing in the 2norm, and consistently produces better solutions than other RP algorithms is also introduced. Using comparisons with the method of conjugate gradient applied to the normal equations, the properties of RP methods are explained. A row partitioning approach is described which yields parallel implementations suitable for a wide range of computer architectures, requires only a few vectors of extra storage, and allows computing the necessary projections with small errors. Numerical testing verifies the robu...
Shiftinvert and Cayley transforms for the detection of eigenvalues with largest real part of nonsymmetric matrices
, 1993
"... This manuscript is concerned with the determination of the rightmost eigenvalues of large sparse real nonsymmetric matrices. Specifically, the use of subspace iteration preconditioned by the Cayley transform and/or shiftinvert is discussed. The convergence properties of subspace iteration are used ..."
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Cited by 32 (6 self)
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This manuscript is concerned with the determination of the rightmost eigenvalues of large sparse real nonsymmetric matrices. Specifically, the use of subspace iteration preconditioned by the Cayley transform and/or shiftinvert is discussed. The convergence properties of subspace iteration are used to construct a strategy to validate the rightmost eigenvalue, which is computed by an iterative method. The motivation behind this paper is that rational preconditioners are very reliable in general but they can miss rightmost eigenvalues with large imaginary part. Numerical examples are given to illustrate the theory. Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, 3001 Heverlee, Belgium, Karl.Meerbergen@cs.kuleuven.ac.be y University of Bath, School of Mathematical Sciences, Claverton Down, Bath, BA2 7AY, United Kingdom, A.Spence@maths.bath.ac.uk z Katholieke Universiteit Leuven, Department of Computer Science, Dirk.Roose@cs.kuleuven.ac.be D...
Geometric Aspects in the Theory of Krylov Subspace Methods
 Acta Numerica
, 1999
"... The recent development of Krylov subspace methods for the solution of operator equations has shown that two basic construction principles, the orthogonal residual (OR) and minimal residual (MR) approaches, underlie the most commonly used algorithms. It is shown that these can both be formulated ..."
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Cited by 29 (2 self)
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The recent development of Krylov subspace methods for the solution of operator equations has shown that two basic construction principles, the orthogonal residual (OR) and minimal residual (MR) approaches, underlie the most commonly used algorithms. It is shown that these can both be formulated as techniques for solving an approximation problem on a sequence of nested subspaces of a Hilbert space, a problem not necessarily related to an operator equation. Most of the familiar Krylov subspace algorithms result when these subspaces form a Krylov sequence. The wellknown relations among the iterates and residuals of OR/MR pairs are shown to hold also in this rather general setting. We further show that a common error analysis for these methods involving the canonical angles between subspaces allows many of the recently developed error bounds to be derived in a simple manner. An application of this analysis to compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge qsuperlinearly when applied to such operator equations.
An Arnoldi code for computing selected eigenvalues of sparse real unsymmetric matrices
, 1995
"... Arnoldi methods can be more effective than subspace iteration methods for computing the dominant eigenvalues of a large sparse real unsymmetric matrix. A code, EB12, for the sparse unsymmetric eigenvalue problem based on a subspace iteration algorithm, optionally combined with Chebychev acceleration ..."
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Cited by 28 (4 self)
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Arnoldi methods can be more effective than subspace iteration methods for computing the dominant eigenvalues of a large sparse real unsymmetric matrix. A code, EB12, for the sparse unsymmetric eigenvalue problem based on a subspace iteration algorithm, optionally combined with Chebychev acceleration, has recently been described by Duff and Scott (1993) and is included in the Harwell Subroutine Library (Anon 1993). In this paper we consider variants of the method of Arnoldi and discuss the design and development of a code to implement these methods. The new code, which is called EB13, offers the user the choice of a basic Arnoldi algorithm, an Arnoldi algorithm with Chebychev acceleration, and a Chebychev preconditioned Arnoldi algorithm. Each method is available in blocked and unblocked form. The code may be used to compute either the rightmost eigenvalues, the eigenvalues of largest absolute value, or the eigenvalues of largest imaginary part. The performance of each option in the EB...