Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

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

doi:10.1137/140980090

Cited by 77 publications

(143 citation statements)

References 34 publications

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“…To compute sign(A)x efficiently, they find the lowest degrees of P and Q such that the computed result has acceptable error (which they set to ≈ 10 −2 ). A recent work of Güttel, Polizzi, Tang and Viaud [27] computes partial eigenpairs of largesparse matrices using Zolotarev functions: they apply Z 2r+1 (A; ) to a vector (or a set of vectors) to obtain a "filter function" that gives a subspace rich in the eigenspace corresponding to the eigenvalues of A in a specified interval, and repeatedly apply the filter function to improve the accuracy. Druskin, Güttel, and Knizhnerman [18] use Zolotarev's function for the inverse square root function.…”

Section: Qr(x)mentioning

confidence: 99%

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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Abstract. The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divideand-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant of the QDWH iteration for the polar decomposition. The key idea of this algorithm comes from approximation theory: we use the best rational approximant for the scalar sign function due to Zolotarev in 1877. The algorithm exploits the extraordinary property enjoyed by the sign function that a high-degree Zolotarev function (best rational approximant) can be obtained by appropriately composing low-degree Zolotarev functions. This lets the algorithm converge in just two iterations in double-precision arithmetic, with the whopping rate of convergence seventeen. The resulting algorithms for the symmetric eigendecompositions and the SVD have higher arithmetic costs than the QDWH-based algorithms, but are better-suited for parallel computing and exhibit excellent numerical backward stability.

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Section: Qr(x)mentioning

confidence: 99%

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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“…To solve eigenvalue problems, Sakurai and his co-authors applied the idea of the generalized eigenvalue problem involving the Hankel and shifted Hankel matrix using moments based on the resolvent function [18,10,15,9,19,25,1,2]. Eric Polizzi and co-authors also used contour integrals based on the resolvent function resulting in the FEAST algorithm [16,22,7].…”

Section: Introductionmentioning

confidence: 99%

Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an effective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.Keywords : linear, polynomial, nonlinear eigenvalue problems; contour integration; filter function; rational approximation; nonlinear least squares; resolvent. MSC : Primary : 65-F15, Secondary : 47-J10, 30-E05. Designing rational filter functions for solving eigenvalue problems by contour integrationMarc Van Barel ⇤ January 20, 2015Abstract Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an e↵ective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.

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“…Polizzi's FEAST algorithm can be summarized as follows (see also Krämer et al [10]; Tang and Polizzi [11]; Laux [12]; Güttel et al [13]):…”

Section: Introductionmentioning

confidence: 99%

Nonlinear eigenvalue problems and contour integrals

Barel

Kravanja

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper Beyn's algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a Hankel tensor. A numerical example shows that the choice of the filter function is very important, particularly with respect to where it is positioned in the complex plane.

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Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

Cited by 77 publications

References 34 publications

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Designing rational filter functions for solving eigenvalue problems by contour integration

Nonlinear eigenvalue problems and contour integrals

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