A numerical method for nonlinear eigenvalue problems using contour integrals

Asakura, Junko; Sakurai, Tetsuya; Tadano, Hiroshi; Ikegami, Tsutomu; Kimura, Kuniko

doi:10.14495/jsiaml.1.52

Cited by 163 publications

(201 citation statements)

References 5 publications

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“…The function m approximates that jump discontinuity by a sequence of poles. 10 In the upper-left position of the tables, for example, after dividing by −iz −1 , we note that…”

Section: Rational Approximationmentioning

confidence: 99%

“…An obvious choice for some applications is the trapezoidal rule in a circle, though FEAST itself is based on Gauss quadrature. Nonlinear analogues have been considered [10,22,188], and spectral projectors are analyzed from other angles in [17]. We expect that algorithms of this kind will be applied to many problems in the years ahead.…”

Section: Functions and Eigenvalues Of Matrices And Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

470

362

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Abstract.It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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“…The function m approximates that jump discontinuity by a sequence of poles. 10 In the upper-left position of the tables, for example, after dividing by −iz −1 , we note that…”

Section: Rational Approximationmentioning

confidence: 99%

Section: Functions and Eigenvalues Of Matrices And Operatorsmentioning

confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

470

362

View full text Add to dashboard Cite

show abstract

“…In [23], we designed a variant of Beyn's algorithm [5] showing that this eigenvalue problem can be solved via contour integration of the resolvent function T (z) 1 . Consider the following contour integrals, called moments, based on the resolvent function applied to a rectangular matrixV :…”

Section: Algorithm Based On Approximate Contour Integrationmentioning

confidence: 99%

“…Later on Tetsuya Sakurai joined us in our study and co-authored some papers [13,17,12]. 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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“…Finding the y-values such that the resultant matrix is singular is equivalent to a polynomial eigenvalue problem, and many techniques exist such as methods based on contour integrals [2,6], Newton-type methods and inverse iteration methods (see the review [34]), the Ehrlich-Aberth method [10], and the standard approach of solving via linearization [19]. In our implementation we use linearization, which replaces a polynomial eigenvalue problem with a generalized eigenvalue problem with the same eigenvalues and Jordan structure [19].…”

Section: Resultant Methodsmentioning

confidence: 99%

Computing the common zeros of two bivariate functions via Bézout resultants

2014

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Abstract. The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented Matlab for computing with bivariate functions.

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A numerical method for nonlinear eigenvalue problems using contour integrals

Cited by 163 publications

References 5 publications

The Exponentially Convergent Trapezoidal Rule

The Exponentially Convergent Trapezoidal Rule

Designing rational filter functions for solving eigenvalue problems by contour integration

Computing the common zeros of two bivariate functions via Bézout resultants

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