Rayleigh quotient iteration and simplified Jacobi–Davidson method with preconditioned iterative solves

Freitag, Melina A.; Spence, A.

doi:10.1016/j.laa.2007.11.013

Cited by 27 publications

(58 citation statements)

References 11 publications

(19 reference statements)

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“…In this section, we refine the equivalence results given in [11], [13]. We show that y kþ1 ¼ η k ðx þ s k Þ always holds if the same Krylov subspace method satisying the Galerkin condition (4.3) is applied to (4.4a) and…”

Section: ð3:14þsupporting

confidence: 62%

“…Equivalence of the inner solves of IRQI and single-vector JD. In this section, we refine the equivalence results of inner solves of IRQI and the single-vector JD method shown in [11], [13].…”

Section: ð3:14þmentioning

confidence: 85%

“…This result is extended in [15] to a two-sided RQI and a two-sided JD for nonHermitian problems, where the shifted linear system is solved iteratively by unpreconditioned BiCG method. The equivalence result is further extended in [11], [13] to preconditioned solves for both standard and generalized non-Hermitian problems. Assuming that the tuned preconditioner Q RQ is defined as in (3.9c), Freitag and Spence [11] and Freitag, Spence, and Vainikko [13] show that y kþ1 ¼ η k ðx þ s k Þ if the same Krylov subspace method satisfying the Galerkin condition (4.3) is applied to the preconditioned systems…”

Section: ð3:14þmentioning

confidence: 97%

“…with some vector w ðiÞ ; see, e.g., [16]. For one-sided RQI for computing only the right eigenpair, a common choice is w ðiÞ ¼ Bx ðiÞ (see [13], [14]), and therefore ðiÞ ⊥ Bx ðiÞ in the non-Hermitian case. From here through the end of the paper, we focus on the behavior of the inner solve -i.e., the solution of step 2 of Algorithm 2.1-in a given outer iteration, and we drop the superscript (i) denoting the number of outer iterations, unless otherwise stated.…”

Section: Preliminariesmentioning

confidence: 99%

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Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method

Szyld¹,

Xue²

2011

SIAM J. Matrix Anal. & Appl.

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Abstract. We study inexact Rayleigh quotient iteration (IRQI) for computing a simple interior eigenpair of the generalized eigenvalue problem Av ¼ λBv, providing new insights into a special type of preconditioners with "tuning" for the efficient iterative solution of the shifted linear systems that arise in this algorithm. We first give a new convergence analysis of IRQI, showing that locally cubic and quadratic convergence can be achieved for Hermitian and non-Hermitian problems, respectively, if the shifted linear systems are solved by a generic Krylov subspace method with a tuned preconditioner to a reasonably small fixed tolerance. We then refine the study by Freitag and Spence [Linear Algebra Appl., 428 (2008), pp. 2049-2060 on the equivalence of the inner solves of IRQI and single-vector Jacobi-Davidson method where a full orthogonalization method with a tuned preconditioner is used as the inner solver. We also provide some new perspectives on the tuning strategy, showing that tuning is essentially needed only in the first inner iteration in the non-Hermitian case. Based on this observation, we propose a flexible GMRES algorithm with a special configuration in the first inner step, and show that this method is as efficient as GMRES with the tuned preconditioner. 1. Introduction. Rayleigh quotient iteration (RQI) is a classical algorithm for computing a simple eigenpair of a matrix A or a matrix pencil ðA; BÞ. This algorithm has been extensively studied for more than fifty years, and is shown to exhibit locally cubic and quadratic convergence rates, respectively, for Hermitian and non-Hermitian problems; see [21], [29], and the references therein. In recent years, there has been considerable interest in inexact eigenvalue algorithms, including inexact Rayleigh quotient iteration (IRQI), with inner-outer iterations for computing eigenvalues of matrices around some specified shift, especially those lying in the interior of the spectrum to which regular Krylov subspace methods fail to provide rapid approximations. In each outer iteration, a shift-invert matrix-vector product is computed inexactly through the iterative solution (inner iteration) of the corresponding linear system. Inexact eigenvalue algorithms are of significant use if the matrices are too large for exact shift-invert matrix-vector products to be affordable, or if the matrices cannot be formed explicitly. In this paper, we provide some new insights into a special type of preconditioners with "tuning" for the efficient iterative solution (inner solves) of the shifted linear system of equations that arises in IRQI for computing a simple interior eigenpair of the generalized eigenvalue problem Av ¼ λBv.The original motivation of tuning the preconditioner is to resolve the difficulties arising in the preconditioned inner solves for inexact inverse iteration and IRQI. Specifically, a good preconditioner in the usual setting of solving linear systems generally

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Section: ð3:14þsupporting

confidence: 62%

Section: ð3:14þmentioning

confidence: 85%

Section: ð3:14þmentioning

confidence: 97%

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method

Szyld¹,

Xue²

2011

SIAM J. Matrix Anal. & Appl.

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“…Inexact implementation of the Rayleigh quotient iteration has been considered by various authors [5,13,19,21,22]. Other variable shifts were considered by Spence et al [11,12]. For non-Hermitian A, Rayleigh quotient methods using σ k = w * k Au k /w * k u k , where w k is the current approximation to the left eigenvector, can also be used, although this nearly doubles the amount of calculation required for each iteration.…”

Section: C240mentioning

confidence: 99%

Acceleration of inexact inverse iteration for eigenvalue problems

Andrew¹,

Foo²,

Tan³

et al. 2008

ANZIAMJ

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Many methods have been used to improve the efficiency of iterative numerical algorithms. Combining different methods is not always possible because the performance of acceleration methods usually depends critically on the precise form of the error in successive iterates, and this form often changes when other acceleration methods are used. Inexact implementation methods have proved particularly effective in increasing the efficiency of iterations involving sparse matrices. This article investigates the extent to which the efficiency of inexact inverse iteration and the inexact Rayleigh quotient algorithm, for the numerical computation of eigenvalues and eigenvectors of sparse matrices, may be further increased by the use of the scalar epsilon algorithm, a classical extrapolation technique. Some encouraging numerical results are presented and some pointers are given for future research.

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Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration

Freitag

Kürschner

2014

Numer. Linear Algebra Appl.

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SUMMARYConvergence results are provided for inexact two‐sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non‐Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two‐sided methods is considered, and the successful tuning strategy for preconditioners is extended to two‐sided methods, creating a novel way of preconditioning two‐sided algorithms. Furthermore, it is shown that inexact two‐sided Rayleigh quotient iteration and the inexact two‐sided Jacobi‐Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov–Galerkin–Krylov method is used and when this specific tuning strategy is applied. Copyright © 2014 John Wiley & Sons, Ltd.

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Rayleigh quotient iteration and simplified Jacobi–Davidson method with preconditioned iterative solves

Cited by 27 publications

References 11 publications

Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method

Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method

Acceleration of inexact inverse iteration for eigenvalue problems

Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration

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