Parallel stochastic estimation method of eigenvalue distribution

Futamura, Yasunori; Tadano, Hiroshi; Sakurai, Tetsuya

doi:10.14495/jsiaml.2.127

Cited by 39 publications

(40 citation statements)

References 11 publications

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“…Now, consider the pencil H 1 −λH 0 with H 0 and H 1 defined as in (15). Because of (16), and of the fact that λ 0 , .…”

Section: Proofmentioning

confidence: 99%

“…. , λ m all the eigenvalues of P (λ) in the interior of Γ and the matrices H 0 and H 1 defined as in (15). For k = 0, 1, .…”

Section: Computing Invariant Pairs Via Moment Pencilsmentioning

confidence: 99%

“…Even an approximate estimate can be useful to choose Γ and k consistently. An answer to this problem is provided in [14] and [15]. In particular, Theorem 2 in [14] points out that the number m of eigenvalues of P (λ) that are inside Γ is given by:…”

Section: (Y T ) Is a Simple Invariant Pair For P (λ)mentioning

confidence: 99%

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A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and the backward error for invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

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“…Now, consider the pencil H 1 −λH 0 with H 0 and H 1 defined as in (15). Because of (16), and of the fact that λ 0 , .…”

Section: Proofmentioning

confidence: 99%

“…. , λ m all the eigenvalues of P (λ) in the interior of Γ and the matrices H 0 and H 1 defined as in (15). For k = 0, 1, .…”

Section: Computing Invariant Pairs Via Moment Pencilsmentioning

confidence: 99%

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A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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“…This method was introduced and demonstrated in condensed matter physics [7]. For the estimation of the level density, we count the number of the eigenvalues in a specified energy region, Γ.…”

Section: Framework Of the Stochastic Estimation Of The Eigenvalue Countmentioning

confidence: 99%

“…the shell model Monte Carlo (SMMC) [3], the moment-based methods [4,5], and the Lanczos-based method [6]. In this proceedings, we review stochastic estimation of the eigenvalue distribution based on shifted Krylov-subspace method [7] and its application to nuclear shell model calculations [8]. We also discuss its validity and feasibility.…”

Section: Introductionmentioning

confidence: 99%

Stochastic estimation of level density in nuclear shell-model calculations

Shimizu

Utsuno

Futamura

et al. 2016

EPJ Web of Conferences

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Abstract. An estimation method of the nuclear level density stochastically based on nuclear shell-model calculations is introduced. In order to count the number of the eigenvalues of the shell-model Hamiltonian matrix, we perform the contour integral of the matrix element of a resolvent. The shifted block Krylov subspace method enables us its efficient computation. Utilizing this method, the contamination of center-of-mass motion is clearly removed.

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Efficient estimation of eigenvalue counts in an interval

Napoli

Polizzi

Saad

2016

Numerical Linear Algebra App

112

142

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Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications, and it is a prerequisite of eigensolvers based on a divide-andconquer paradigm. Often, an exact count is not necessary, and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure. We also discuss how the latter method is particularly wellsuited for the FEAST eigensolver.ESTIMATING EIGENVALUE COUNTS 675 count OEa; b in OEa; b. While this method yields an exact count, it requires two complete LDL T factorizations, and this can be quite expensive for realistic eigenproblems.This paper discusses two alternative methods that provide only an estimate for OEa; b , but which are relatively inexpensive. Both methods work by estimating the trace of the spectral projector P associated with the eigenvalues inside the interval OEa; b. This spectral projector is expanded in two different ways, and its trace is computed by resorting to stochastic trace estimators, for example, [10,11]. The first method utilizes filtering techniques based on Chebyshev polynomials. The resulting projector is expanded as a polynomial function of A. In the second method, the projector is constructed by integrating the resolvent of the eigenproblem along a contour in the complex plane enclosing the interval OEa; b. In this case, the projector is approximated by a rational function of A.For each of the aforementioned methods, we present various implementations depending on the nature of the eigenproblem (generalized versus standard) and cost considerations. Thus, in the polynomial expansion case, we propose a barrier-type filter when dealing with a standard eigenproblem and two high/low pass filters in the case of generalized eigenproblems. In the rational expansion case, we have the choice of using an LU factorization or a Krylov subspace method to solve linear systems. The optimal implementation of each method used for the eigenvalue count depends on the situation at hand and involves compromises between cost and accuracy. While it is not the aim of this paper to explore detailed analysis of these techniques, we will discuss various possibilities and provide illustrative examples.The polynomial and rational expansion methods are motivated by two distinct approaches recently suggested in the context of electronic structure calculations: (i) spectrum slicing and (ii) Cauchy integral eigen-projection. In the spectrum slicing techniques [1], the eigenpairs are computed by dividing the spectrum in many small subintervals, called 'slices' or 'windows'. For each window, a barrier function is approximated by Chebyshev-Jackson polynomials in order to select only the portion of the spectrum in the slice. In this method, it is important to determine an approximate c...

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Parallel stochastic estimation method of eigenvalue distribution

Cited by 39 publications

References 11 publications

A contour integral approach to the computation of invariant pairs

A contour integral approach to the computation of invariant pairs

Stochastic estimation of level density in nuclear shell-model calculations

Efficient estimation of eigenvalue counts in an interval

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