A numerical method for locating the zeros of an analytic function

Delves, L. M.; Lyness, J. N.

doi:10.1090/s0025-5718-1967-0228165-4

Cited by 286 publications

(132 citation statements)

References 10 publications

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“…Here, only some key references are mentioned without having the intention of being complete. Based on the pioneering work of Delves and Lyness [6], the author of this paper together with Peter Kravanja developed several methods to compute the zeros of a scalar analytic function t(z) (for a synthesis of these results, see [14]) reducing the problem to a generalized eigenvalue problem involving a Hankel matrix as well as a shifted Hankel matrix consisting of the moments of the analytic function t(z). Later on Tetsuya Sakurai joined us in our study and co-authored some papers [13,17,12].…”

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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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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“…The solution of eigenvalue problems (4) and (5) is based on the idea described in [7] and [8]. It is known from the theory of complex variables that the number of zeros ,.., 2 , 1 k n  It is suggested in [7] and [8] that using (7) one can construct a polynomial of degree k roots of which are the same as the roots of ) (z…”

Section: Numerical Proceduresmentioning

confidence: 99%

Calculation of Eigenvalues for Eddy Current Testing Problems

Koliskina

2015

Boundary Field Problems and Computer Simulation

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− Semi-analytical solutions of eddy current testing problems require several computational steps. One of the steps where numerical methods are needed is calculation of complex eigenvalues without good initial approximation for the roots. In the presented paper we describe three eddy current testing problems with cylindrical symmetry where a cylindrical inclusion in a conducting medium is of finite size. In all three cases eigenvalue problem reduces to transcendental equations containing Bessel functions in a complex plane. The algorithm of the solution of such problems is described in the paper. Results of numerical computation are presented.

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“…The radiation coupling coefficient can be calculated using the argument principle method (APM) [12], [13]. The latter method is a rigorous mathematical technique based on complex analysis [14] and is capable of finding the zeros (leaky-mode propagation constants ) of any analytic function (the waveguide dispersion relation) in the complex plane. Therefore, the radiation coupling coefficient in (1) can be easily determined.…”

Section: A Evanescent Coupling From Waveguide To Substrate-embedded mentioning

confidence: 99%

Substrate-embedded and flip-chip-bonded photodetector polymer-based optical interconnects: analysis, design, and performance

Glytsis

Jokerst

Villalaz

et al. 2003

J. Lightwave Technol.

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A numerical method for locating the zeros of an analytic function

Cited by 286 publications

References 10 publications

Designing rational filter functions for solving eigenvalue problems by contour integration

Designing rational filter functions for solving eigenvalue problems by contour integration

Calculation of Eigenvalues for Eddy Current Testing Problems

Substrate-embedded and flip-chip-bonded photodetector polymer-based optical interconnects: analysis, design, and performance

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