On an Economic Arnoldi Method for $BML$-Matrices

Beckermann, Bernhard; Mertens, Clara; Vandebril, Raf

doi:10.1137/17m1114363

Cited by 2 publications

(3 citation statements)

References 19 publications

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“…(3.4) and (3.5)] introduced in [Gra93,JR94]. For further details we also refer to [BGF97,Sch08,BMV18]. We also recapitulate the isometric Arnoldi method in Algorithm 2.…”

Section: And Together Withmentioning

confidence: 99%

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Jawecki¹

2022

Preprint

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The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of the given matrix, and Gaussian quadrature for the Riemann-Stieltjes integral associated with this spectral distribution. Similar relations hold true for rational Krylov subspaces. In the present work, separation theorems are reviewed in the context of Krylov subspaces including rational Krylov subspaces with a single complex pole of higher multiplicity and some extended Krylov subspaces. For rational Gaussian quadrature related to some classes of rational Krylov subspaces with a single pole, the underlying separation theorems are newly introduced here.

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“…(3.4) and (3.5)] introduced in [Gra93,JR94]. For further details we also refer to [BGF97,Sch08,BMV18]. We also recapitulate the isometric Arnoldi method in Algorithm 2.…”

Section: And Together Withmentioning

confidence: 99%

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Jawecki¹

2022

Preprint

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“…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%

“…Liesen examined in more detail the relation between a matrix and a rational function of its adjoint [29]. An alternative manner to devise the short recurrences was proposed by Beckermann, Mertens, and Vandebril [7]. 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 and Reichel [8], this method was tuned later on and stabilized by Embree et al [22].…”

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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On an Economic Arnoldi Method for $BML$-Matrices

Cited by 2 publications

References 19 publications

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

When is a matrix unitary or Hermitian plus low rank?

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