Computing the Zeros of Analytic Functions

Kravanja, Peter; Barel, Marc Van

doi:10.1007/bfb0103927

Cited by 79 publications

(78 citation statements)

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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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“…Second, this mapping reduces the zeros finding problem for the exterior of Σ to the same problem for the exterior of a bounded set Φ ⊂ C. Speaking more precisely, the conditions guarantying that all roots of P (z) lie outside Φ or equivalently Ker B(z) ⊂ D Q \ Ω Q would be necessary and sufficient to prove that existence and uniquenesses of solution (10). The majority of results related to such conditions for polynomials are devoted to the situation when a circle is considered in place of Φ (to review existent results in that field, see [18,19] as well as [26,28]). That is why we firstly encircle Φ and then use readily available zero-free conditions for that circle.…”

Section: Zeros Of B(z) and Equivalent Problem For Polynomialmentioning

confidence: 99%

Existence of the solution to a nonlocal-in-time evolutional problem

Макаров

Sytnyk

Vasylyk

2014

NAMC

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Abstract. This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal problem to its classical initial value analogue, uses more advanced analysis. That is a validation of the correctness in definition of the general solution representation via the Dunford-Cauchy formula. Such approach allows us to reduce the given existence problem to the problem of locating zeros of a certain entire function. It results in the necessary and sufficient conditions for the existence of a generalized (mild) solution to the given nonlocal problem. Aside of that we also present new sufficient conditions which in the majority of cases generalize existing results.

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“…Using the fact that g is analytic over the complex plane (in β), low order modes are numerically calculated with a technique described by Kravanja and Van Barel [5] and already used by the authors for the modeling of dissipative silencers [6].…”

Section: S-matrix Of the Acoustic Systemmentioning

confidence: 99%

An efficient FEM-BEM procedure for the multi-frequency acoustical analysis of lined ducts containing passive components

Maréchal

Perrey‐Debain

Ville

et al. 2010

Boundary Elements and Other Mesh Reduction Methods XXXII

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This paper deals with strategies for computing efficiently the propagation of sound waves in ducts with acoustic lining at its walls. Although efficient, these treatments seem to have reached their limit and there is still a need for considering other passive techniques to reduce further the sound radiation at the duct exit. In most cases of practical interest, these added acoustics components can be modeled as acoustic cavities, which are connected to the duct and can be either purely reactive or dissipative. The assessment of the efficiency of such a system requires a precise knowledge of the acoustic field in the duct. Although standard Finite Element (FE) software could, in principle, be used for this purpose, a full FE model would be extremely demanding, particularly in the mid-frequency range and this can have a negative impact when, for instance, some efficient optimizations are needed. In the present work, we present a new numerical procedure that judiciously exploits the benefit of the FEM and the BEM approach. Firstly, a set of FE eigenmodes are computed in the cavity to produce a numerical impedance matrix connecting the pressure and the acoustic velocity on the duct wall interface. Then an integral representation for the acoustic pressure in the main duct is used. The presence of acoustic liners on the walls of the duct is taken into account via an appropriate modal decomposition of the Green's function. Typical applications involving Helmholtz resonators and Herschel-Quincke tubes are presented. We show that our algorithm allows a very fast and accurate computation of the scattering matrix of such a system with a numerical complexity that grows very mildly with the frequency.

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Computing the Zeros of Analytic Functions

Cited by 79 publications

References 0 publications

Designing rational filter functions for solving eigenvalue problems by contour integration

Designing rational filter functions for solving eigenvalue problems by contour integration

Existence of the solution to a nonlocal-in-time evolutional problem

An efficient FEM-BEM procedure for the multi-frequency acoustical analysis of lined ducts containing passive components

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