Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

Guettel, Stefan; Polizzi, Eric; Tang, Ping; Viaud, G

doi:10.48550/arxiv.1407.8078

Cited by 2 publications

(7 citation statements)

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“…Although quite effective, the Gauss-quadrature approach along a circular contour that was proposed in the original FEAST article [29], is clearly not the only possible choice for optimizing the convegence ratio. Three other options have already been considered for the Hermitian problem including [16]: (i) the Trapezoidal rule; (ii) different contour shapes beside a circle such as a finely tuned flat ellipse; (iii) a new approximation of the spectral projector based on a Zolotarev approximant to the sign function which, after transformations, provides complex poles on the unit circle [42,16]. Both Gauss and Zolotarev are well-suited choices for the Hermitian problem since they favor an accentuation of the decay of |ρ a | at the boundaries of the interval along the real axis.…”

Section: Discussion On Convergencementioning

confidence: 99%

“…A detailed numerical analysis on FEAST was completed recently in [38], placing the algorithm on a more solid theoretical foundation. In particular, a relatively small number of quadrature nodes (using Gauss, Trapezoidal or Zolotarev [16] rules) on a circular contour suffices to produce a rapid decay of the function ρ a from ≈ 1 within the search contour to ≈ 0 outside. In comparison with more standard polynomial filtering [35,32], the rational filter (1.4) can lead to a very fast convergence of the subspace iteration procedure.…”

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confidence: 99%

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Feast Eigensolver for Non-Hermitian Problems

Kestyn¹,

Polizzi²,

Tang³

2016

SIAM J. Sci. Comput.

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Abstract. A detailed new upgrade of the FEAST eigensolver targeting non-Hermitian eigenvalue problems is presented and thoroughly discussed. It aims at broadening the class of eigenproblems that can be addressed within the framework of the FEAST algorithm. The algorithm is ideally suited for computing selected interior eigenvalues and their associated right/left bi-orthogonal eigenvectors, located within a subset of the complex plane. It combines subspace iteration with efficient contour integration techniques that approximate the left and right spectral projectors. We discuss the various algorithmic choices that have been made to improve the stability and usability of the new nonHermitian eigensolver. The latter retains the convergence property and multi-level parallelism of Hermitian FEAST, making it a valuable new software tool for the scientific community.Key words. non-Hermitian eigenproblem, FEAST, spectral projectors, contour integration, right/left eigenvectors, bi-orthogonal vectors AMS subject classifications. 65F15, 15A18 34L16 65Y05 35P991. Introduction. The generalized eigenvalue problem AX = BXΛ with A and B square matrices and Λ diagonal, is a central topic in numerical linear algebra and arises from a broad and diverse set of disciplines in mathematics, science and engineering (the problem is said "standard" if B ≡ I or "generalized" otherwise). Solving the interior eigenvalue problem consists of determining nontrivial solutions {λ i , x i } (i.e. eigenpairs with x i = Xe i and λ i = Λ i,i ) located anywhere inside the spectrum. Most common numerical applications lead to symmetric eigenvalue problems where A is real symmetric or complex Hermitian, B is symmetric or Hermitian positive definite (hpd), and all the obtained eigenvalues λ i are real. Non-symmetric and non-Hermitian eigenvalue problem (including the case where A is complex symmetric) can also be encountered in a variety of situations resulting in complex values for λ i . In this case x i is called the right eigenvector associated with λ i , while one can also define a left eigenvector. Although many software packages are available for symmetric (or Hermitian) matrices (see e.g. [17,23,33,18,25,5,36,20]), relatively few algorithms and software can handle the non-Hermitian problem [22,4,23,3,13]. The FEAST eigensolver [29, 10], proven to be a robust and efficient tool for computing the partial eigenspectrum of Hermitian system matrices [38], can also be generalized and applied to arbitrary non-Hermitian systems [21,41,37].FEAST is a subspace iteration method that uses the Rayleigh-Ritz projection and an approximate spectral projector as a filter [38]. Given a Hermitian generalized eigenvalue problem AX = BXΛ of size n, the algorithm in Figure 1 outlines the main steps of a generic Rayleigh-Ritz subspace iteration procedure for computing m eigenpairs. At convergence, the algorithm yields the B-orthonormal eigensubspace Y m ≡ X m = {x 1 , x 2 , . . . , x m } n×m and associated eigenvalues Λ Q m ≡ Λ m . Taking ρ(B −1 A) = B −1 A, yields the...

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Section: Discussion On Convergencementioning

confidence: 99%

mentioning

confidence: 99%

Feast Eigensolver for Non-Hermitian Problems

Kestyn¹,

Polizzi²,

Tang³

2016

SIAM J. Sci. Comput.

Self Cite

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“…, LM, there exists a unique vector s i ∈ K M (C, V ) such that P LM s i = x i . In the diagonalizable case, for the error analysis of the block SS-RR method and the FEAST eigensolver, the following inequality was given in [15] and [9,23] for M = 1:…”

Section: Error Bounds Of the Block Ss-rr Methods And The Feast Eigens...mentioning

confidence: 99%

“…Therefore, the FEAST eigensolver computes the eigenvalues located in Ω and their corresponding eigenvectors. For numerical integration, the FEAST eigensolver uses the Gauß-Legendre quadrature or the Zolotarev quadrature; see [9,17].…”

Section: The Feast Eigensolvermentioning

confidence: 99%

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Relationships among contour integral-based methods for solving generalized eigenvalue problems

Imakura

Sakurai

2016

Japan J. Indust. Appl. Math.

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Recently, contour integral-based methods have been actively studied for solving interior eigenvalue problems that find all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we reconsider the algorithms of the five typical contour integral-based eigensolvers from the viewpoint of projection methods, and then map the relationships among these methods. From the analysis, we conclude that all contour integral-based eigensolvers can be regarded as projection methods and can be categorized based on their subspace used, the type of projection and the problem to which they are applied implicitly.

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Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

Cited by 2 publications

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Feast Eigensolver for Non-Hermitian Problems

Feast Eigensolver for Non-Hermitian Problems

Relationships among contour integral-based methods for solving generalized eigenvalue problems

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