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GMRES with deflated restarting
 SIAM J. Sci. Comput
"... Abstract. A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon’s thick restarting approach. It has the efficiency of implicit restarting, but is simpler and does not have the same numerical concerns ..."
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Cited by 60 (8 self)
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Abstract. A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon’s thick restarting approach. It has the efficiency of implicit restarting, but is simpler and does not have the same numerical concerns. The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES. Also, it is demonstrated that using harmonic Ritz vectors is important, because then the whole subspace is a Krylov subspace that contains certain important smaller subspaces.
A restarted Krylov subspace method for the evaluation of matrix functions
 SIAM J. Numer. Anal
"... Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the recipro ..."
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Cited by 58 (8 self)
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Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions. We further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments.
Numerical methods for large eigenvalue problems
, 2002
"... Over the past decade considerable progress has been made towards the numerical solution of largescale eigenvalue problems, particularly for nonsymmetric matrices. Krylov methods and variants of subspace iteration have been improved to the point that problems of the order of several million variable ..."
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Cited by 25 (1 self)
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Over the past decade considerable progress has been made towards the numerical solution of largescale eigenvalue problems, particularly for nonsymmetric matrices. Krylov methods and variants of subspace iteration have been improved to the point that problems of the order of several million variables can be solved. The methods and software that have led to these advances are surveyed.
Deflated iterative methods for linear equations with multiple righthand sides
, 2004
"... Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems ar ..."
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Cited by 20 (7 self)
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Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is given showing significant improvement for a problem from quantum chromodynamics. The second and subsequent righthand sides are solved much quicker than without the deflation. This new approach is relatively simple to implement and is very efficient compared to other deflation methods.
Deflated GMRES for systems with multiple shifts and multiple righthand sides
, 2007
"... Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including latti ..."
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Cited by 15 (1 self)
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Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and nonHermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRESDR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple righthand sides with multiple shifts for each righthand side. We develop a method that efficiently solves the multiple righthand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.
Anasazi software for the numerical solution of largescale eigenvalue problems
 ACM TOMS
"... Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of largescale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furt ..."
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Cited by 13 (0 self)
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Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of largescale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furthermore, Anasazi provides implementations for some of the most recent eigensolver methods. The purpose of our paper is to describe the design and development of the Anasazi framework. A performance comparison of Anasazi and the popular FORTRAN 77 code ARPACK is given.
The QR Algorithm Revisited
 SIAM REVIEW
, 2008
"... The QR algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices. Most discussions of the QR algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used. This paper outlines a pedagogical p ..."
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Cited by 12 (0 self)
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The QR algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices. Most discussions of the QR algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used. This paper outlines a pedagogical path that leads directly to the implicit multishift QR algorithms that are used in practice, bypassing the basic QR algorithm completely.
DEFLATED RESTARTING FOR MATRIX FUNCTIONS ∗
"... Abstract. We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted ap ..."
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Cited by 11 (0 self)
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Abstract. We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course of the underlying restarted Arnoldi process by extracting Ritz vectors and using those closest to the spectral region of interest as exact shifts. The approximation is constructed with the help of a generalization of Krylov decompositions to linearly dependent vectors. A description of the restarted process as a successive interpolation scheme at Ritz values is given in which the exact shifts are replaced with improved approximations of eigenvalues in each restart cycle. Numerical experiments demonstrate the efficacy of the approach.
Block algorithms for reordering standard and generalized Schur forms
 ACM Transactions on Mathematical Software
, 2006
"... Abstract. Block algorithms for reordering a selected set of eigenvalues in a standard or generalized Schur form are proposed. Efficiency is achieved by delaying orthogonal transformations and (optionally) making use of level 3 BLAS operations. Numerical experiments demonstrate that existing algorith ..."
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Cited by 10 (5 self)
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Abstract. Block algorithms for reordering a selected set of eigenvalues in a standard or generalized Schur form are proposed. Efficiency is achieved by delaying orthogonal transformations and (optionally) making use of level 3 BLAS operations. Numerical experiments demonstrate that existing algorithms, as currently implemented in LAPACK, are outperformed by up to a factor of four. Key words. Schur form, reordering, invariant subspace, deflating subspace. AMS subject classifications. 65F15, 65Y20. 1. Introduction. Applying
Multiple explicitly restarted Arnoldi method for solving large eigenproblems
 SIAM Journal on scientific computing SJSC, Volume 27, Number
, 2005
"... In this paper we propose a new approach for calculating some eigenpairs of large sparse nonHermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is particularly well suited for environments that combine different parallel programming paradigms. This technique is bas ..."
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Cited by 8 (5 self)
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In this paper we propose a new approach for calculating some eigenpairs of large sparse nonHermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is particularly well suited for environments that combine different parallel programming paradigms. This technique is based on a multiple use of Explicitly Restarted Arnoldi method and improves its convergence. This technique is implemented and tested on a distributed enviromnment consisting of two interconnected parallel machines. MERAM technique is compared to Explicitly Restarted Arnoldi (ERAM) method, and one can notice that the accelaration of convergence is improved effectively. In some cases, more than one twofold improvement can be seen in MERAM results. We also implemented MERAM on a cluster of workstations. According to our experiments, MERAM converges better than Explicitly Restarted Block Arnoldi method and, for some matrices, more quickly than PARPACK package which implements Implicitly Restarted Arnoldi Method. Key words. Large eigenproblem, Arnoldi method, explicit restarting, parallel programming, asynchronous communication, heterogeneous environment.