Matrix Algorithms, Volume II: Eigensystems

Stewart, G. W.; Mahajan, Aditya

doi:10.1115/1.1523352

Cited by 266 publications

(479 citation statements)

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“…4.3] derives an extension of Saad's result (Theorem 1) to approximate eigenspaces, presenting bounds in the spectral norm applicable in the context of self-adjoint linear operators in a Hilbert space. Stewart [21,22] proves an analogous result applicable to non-Hermitian matrices.…”

Section: Bounds For Approximate Eigenspacementioning

confidence: 83%

Accuracy of singular vectors obtained by projection-based SVD methods

Nakatsukasa

2017

Bit Numer Math

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The standard approach to computing an approximate SVD of a large-scale matrix is to project it onto lower-dimensional trial subspaces from both sides, compute the SVD of the small projected matrix, and project it back to the original space. This results in a low-rank approximate SVD to the original matrix, and we can then obtain approximate left and right singular subspaces by extracting subsets from the approximate SVD. In this work we assess the quality of the extraction process in terms of the accuracy of the approximate singular subspaces, measured by the angle between the exact and extracted subspaces (relative to the angle between the exact and trial subspaces). The main message is that the extracted approximate subspaces are optimal usually to within a modest constant.

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Section: Bounds For Approximate Eigenspacementioning

confidence: 83%

Accuracy of singular vectors obtained by projection-based SVD methods

Nakatsukasa

2017

Bit Numer Math

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“…During the process, B-orthogonality of the columns of X k is explicitly enforced. It can be shown that X k eventually spans a subspace that contains the required eigenvectors [4]. This procedure requires the solution of p linear systems of equations with coefficient matrix A at each step k (one system per each column of X k ).…”

Section: Trace Minimization Eigensolvermentioning

confidence: 99%

“…The trace of a square matrix, tr(·), is defined as the sum of its diagonal elements, and it can be shown to be equal to the sum of its eigenvalues [4].…”

Section: Trace Minimization Eigensolvermentioning

confidence: 99%

“…Many different methods have been proposed for solving the above problem, including subspace iteration, Krylov projection methods such as Lanczos or Krylov-Schur, and Davidson-type methods such as Jacobi-Davidson. Details of these methods can be found in [1][2][3][4]. Subspace iteration and Krylov methods perform best when computing largest eigenvalues, but usually fail to compute the smallest or interior eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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A Parallel Implementation of the Trace Minimization Eigensolver

Romero

Román

2008

High Performance Computing for Computational Science - VECPAR 2008

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Abstract. In this paper we describe a parallel implementation of the trace minimization method for symmetric generalized eigenvalue problems proposed by Sameh and Wisniewski. The implementation includes several techniques proposed in a later article of Sameh, such as multishifting, preconditioning and adaptive inner solver termination, which accelerate the method and make it much more robust. A Davidson-type variant of the algorithm has been also considered. The different methods are analyzed in terms of sequential and parallel efficiency.Topics. Numerical algorithms for CS&E, parallel and distributed computing.

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“…Consequently, (u c , α c ) can be located by monitoring the rightmost eigenvalue of (1.2) along a path of stable steady states. Commonly used iterative eigenvalue solvers such as Arnoldi's method and its variants (see Stewart (2001)) work well when a small set of eigenvalues of (1.2) near a given point σ ∈ C (called the "shift") are sought. Thus, a good estimate for the rightmost eigenvalue of (1.2) would be an ideal choice for σ.…”

Section: Introductionmentioning

confidence: 99%

Efficient iterative algorithms for linear stability analysis of incompressible flows

Elman¹,

Rostami

2015

IMA J Numer Anal

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Linear stability analysis of a dynamical system entails finding the rightmost eigenvalue for a series of eigenvalue problems. For large-scale systems, it is known that conventional iterative eigenvalue solvers are not reliable for computing this eigenvalue. A more robust method recently developed in and Meerbergen & Spence (2010), Lyapunov inverse iteraiton, involves solving large-scale Lyapunov equations, which in turn requires the solution of large, sparse linear systems analogous to those arising from solving the underlying partial differential equations. This study explores the efficient implementation of Lyapunov inverse iteration when it is used for linear stability analysis of incompressible flows. Efficiencies are obtained from effective solution strategies for the Lyapunov equations and for the underlying partial differential equations. Existing solution strategies are tested and compared, and a modified version of a Lyapunov solver is proposed that achieves significant savings in computational cost.

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Matrix Algorithms, Volume II: Eigensystems

Cited by 266 publications

References 0 publications

Accuracy of singular vectors obtained by projection-based SVD methods

Accuracy of singular vectors obtained by projection-based SVD methods

A Parallel Implementation of the Trace Minimization Eigensolver

Efficient iterative algorithms for linear stability analysis of incompressible flows

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