Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices

Embree, Mark; Sifuentes, Josef; Soodhalter, Kirk M.; Szyld, Daniel B.; Xue, Fei

doi:10.1137/110851006

Cited by 5 publications

(17 citation statements)

References 29 publications

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“…For larger values of k, the effect is more pronounced. In [18], the authors report a similar behavior for PGMRES in the context of skew-hermitian updates. Table 4 shows the effect of the rank when the update changes not only the elements but also the sparse pattern of the matrix.…”

Section: Numerical Experimentsmentioning

confidence: 62%

“…The matrix N can be approximated as a product of two matrices by applying a singular value decomposition [21], [23], or with probabilistic algorithms for constructing matrix decompositions [25]. In [18], the authors propose a preconditioner based on splitting a nearly Hermitian matrix A into its Hermitian and skew-Hermitian parts. Finally, in overdetermined least squares problems that involve the addition of a set of new equations, the normal equations can be formulated as a rank-k update of the normal equations for the initial matrix [24].…”

Section: Introductionmentioning

confidence: 99%

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Low-rank updates of balanced incomplete factorization preconditioners

2016

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Let Ax = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + P Q T where P , Q ∈ R n×k are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system Bx = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.

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Section: Numerical Experimentsmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Low-rank updates of balanced incomplete factorization preconditioners

2016

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“…We focus especially on the orthogonality of the obtained Arnoldi vectors. The orthogonality in the forthcoming figures is measured by a method described originally by Paige [6,15]. Given V * k V k − I = U k + U * k with U k strictly upper triangular, we define S k = (I + U k ) −1 U k .…”

Section: Numerical Examplesmentioning

confidence: 99%

On an Economic Arnoldi Method for $BML$-Matrices

Beckermann¹,

Mertens²,

Vandebril³

2018

SIAM J. Matrix Anal. & Appl.

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Matrices whose adjoint is a low rank perturbation of a rational function of the matrix naturally arise when trying to extend the well known Faber-Manteuffel theorem [7,8], which provides necessary and sufficient conditions for the existence of a short Arnoldi recurrence. We show that an orthonormal Krylov basis for this class of matrices can be generated by a short recurrence relation based on GMRES residual vectors. These residual vectors are computed by means of an updating formula. Furthermore, the underlying Hessenberg matrix has an accompanying low rank structure, which we will investigate closely.

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“…Liesen examined in more detail the relation between a matrix and a rational function of its adjoint . An alternative manner to devise the short recurrences was proposed by Beckermann et al An algorithm to exploit the short recurrences to develop efficient solvers for Hermitian plus low‐rank case is the progressiver GMRES method, proposed by Beckermann et al, this method was tuned later on and stabilized by Embree et al…”

Section: Introductionmentioning

confidence: 99%

“…Liesen examined in more detail the relation between a matrix and a rational function of its adjoint. 31 An alternative manner to devise the short recurrences was proposed by Beckermann et al 32 An algorithm to exploit the short recurrences to develop efficient solvers for Hermitian plus low-rank case is the progressiver GMRES method, proposed by Beckermann et al, 33 this method was tuned later on and stabilized by Embree et al 34 In this article, we characterize unitary and Hermitian plus low-rank matrices by examining their singular-and eigenvalues. We prove that a matrix having at most k singular values less than 1 and at most k greater than 1 is unitary plus rank k. Similarly, by examining the eigenvalues of the skew-Hermitian part of a matrix we show that if at most k of these eigenvalues are greater than 0 and at most k are smaller than zero, the matrix is Hermitian plus rank k. These characterizations enable us to determine the closest unitary or Hermitian plus rank k matrices in the spectral and Frobenius norms by setting some well-chosen singular-or eigenvalues to 1 or 0.…”

Section: Introductionmentioning

confidence: 99%

When is a matrix unitary or Hermitian plus low rank?

Corso

Poloni

Robol

et al. 2019

Numerical Linear Algebra App

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Summary Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing complexity once the matrix has been reduced, for instance, to tridiagonal or Hessenberg form. Recently, fast and reliable eigensolvers dealing with low‐rank perturbations of unitary and Hermitian matrices have been proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, for example, the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of a Hermitian or unitary matrix plus a low‐rank perturbation. In this paper, we give necessary and sufficient conditions characterizing the class of Hermitian or unitary plus low‐rank matrices. The number of singular values deviating from 1 determines the rank of a perturbation to bring a matrix to unitary form. A similar condition holds for Hermitian matrices; the eigenvalues of the skew‐Hermitian part differing from 0 dictate the rank of the perturbation. We prove that these relations are linked via the Cayley transform. Then, based on these conditions, we identify the closest Hermitian or unitary plus rank k matrix to a given matrix A, in Frobenius and spectral norm, and give a formula for their distance from A. Finally, we present a practical iteration to detect the low‐rank perturbation. Numerical tests prove that this straightforward algorithm is effective.

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Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices

Cited by 5 publications

References 29 publications

Low-rank updates of balanced incomplete factorization preconditioners

Low-rank updates of balanced incomplete factorization preconditioners

On an Economic Arnoldi Method for $BML$-Matrices

When is a matrix unitary or Hermitian plus low rank?

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