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A continuation method for a weakly elliptic two-parameter eigenvalue problem

IMA JOURNAL OF NUMERICAL ANALYSIS 2001年第1期21卷 199-216页

作者： Plestenjak, B Univ Ljubljana IMFM TCS SI-1000 Ljubljana Slovenia

We show that the continuation method can be used to solve a weakly elliptic two-parameter eigenvalue problem. We generalize the continuation method for a nonsymmetric eigenvalue problem Ax = lambdax by T. Y. Li, Z. Ze... 详细信息

关键词： weakly elliptic two-parameter eigenvalue problem continuation method homotopy bifurcation

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On linearizations of the quadratic two-parameter eigenvalue problem

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LINEAR ALGEBRA AND ITS APPLICATIONS 2012年第8期436卷 2725-2743页

作者： Hochstenbach, Michiel E. Muhic, Andrej Plestenjak, Bor Inst Math Phys & Mech SI-1000 Ljubljana Slovenia TU Eindhoven Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands Univ Ljubljana Dept Math SI-1000 Ljubljana Slovenia

We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparameter eigenvalue problem. two of these transformations are generalizations of the well-known linearization of the quadratic eigenvalue problem and linearize the QMEP as a singular two-parameter eigenvalue problem. The third replaces all nonlinear terms by new variables and adds new equations for their relations. The QMEP is thus transformed into a nonsingular five-parameter eigenvalue problem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparameter eigenvalue problems. We also consider several special cases of the QMEP, where some matrix coefficients are zero. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Quadratic two-parameter eigenvalue problem Linearization two-parameter eigenvalue problem

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Asymptotic Behaviour of the Non-real Pair-eigenvalues of a two parameter eigenvalue problem

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COMPLEX ANALYSIS AND OPERATOR THEORY 2023年第4期17卷 1-18页

作者： Ozturk, Hasen Mekki Ordu Univ Fac Arts & Sci Dept Math Ordu Turkiye

This article focuses on a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, Levitin and ozturk showed that the real pair-eigenvalues of a two-parameter eigenvalue problem lie in a union of rectangular regions;however, there has been little written about the non-real pair-eigenvalues. This research deals mainly with the non-real pair-eigenvalues. By using formal asymptotic analysis, we prove that as the norm of an off-diagonal operator diverges to infinity there exists a family of non-real pair-eigenvalues, and each component of the pair-eigenvalues lies approximately on a circle in its corresponding complex plane. Afterwards, we establish a Gershgorin-type result for the localisation of the spectrum of a two-parameter eigenvalue problem, which is a more general enclosure result for the pair-eigenvalues, derived from an enclosure result of Feingold and Varga.

关键词： two-parameter eigenvalue problem Complex eigenvalues Block-operator matrices Multiparameter spectral problems Non-self-adjoint problems Asymptotic expansion

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NONLINEARIZING two-parameter eigenvalue problemS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2021年第2期42卷 775-799页

作者： Ringh, Emil Jarlebring, Elias KTH Royal Inst Technol SeRC Swedish E Sci Res Ctr Dept Math Lindstedtsvagen 25 SE-10044 Stockholm Sweden

We investigate a technique to transform a linear two-parameter eigenvalue problem into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other two-parameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigensolver is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.

关键词： two-parameter eigenvalue problem nonlinear eigenvalue problem multiparameter eigenvalue problem iterative algorithms implicit function theorem

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Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

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LINEAR ALGEBRA AND ITS APPLICATIONS 2009年第3-4期431卷 369-380页

作者： Jarlebring, Elias Hochstenbach, Michiel E. Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium Eindhoven Univ Technol Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Delay-differential equations two-parameter eigenvalue problem Multiparameter eigenvalue problem Critical delays Robustness Stability Asymptotic stability Companion form Quadratic eigenvalue problem Polynomial eigenvalue problem Quadratic two-parameter eigenvalue problem Polynomial two-parameter eigenvalue problem

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Solving two-parameter eigenvalue problems using an alternating method

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LINEAR ALGEBRA AND ITS APPLICATIONS 2022年 643卷 137-160页

作者： Eisenmann, Henrik Nakatsukasa, Yuji Max Planck Inst Math Sci D-04103 Leipzig Germany Univ Oxford Math Inst Oxford OX2 6GG England

We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue prob-lems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems, possibly after performing an affine transformation. This includes a class of Helmholtz equations when separation of variables is applied. We provide a convergence proof for extremal eigenvalues and empirical evidence along with a local convergence proof for other eigenvalues. (c) 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： Alternating optimization Helmholtz equation two-parameter eigenvalue problem

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A two-Dimensional Bisection Method for Solving two-parameter eigenvalue problems

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SIAM Journal on Matrix Analysis and Applications 1992年第4期13卷 1085-1093页

作者： Xingzhi Ji

It has long been known that separation of variables can be applied to the Helmholtz equation in 11 three-dimensional coordinate systems. As a result, a multiparameter eigenvalue problem is formed. In this paper, the well-known bisection method for ordinary eigenvalue problems is generalized to a special class of discrete two-parameter eigenvalue problems. The range of the real roots of the problem is discussed and some numerical results are given.

关键词： 65F15 65L10 65N25 two-parameter eigenvalue problem two-dimensional bisection range of real roots

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SOLVING SINGULAR GENERALIZED eigenvalue problemS BY A RANK-COMPLETING PERTURBATION

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2019年第3期40卷 1022-1046页

作者： Hochstenbach, Michiel E. Mehl, Christian Plestenjak, Bor TU Eindhoven Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands Tech Univ Berlin Inst Math Sekretariat MA 4-5 D-10623 Berlin Germany Univ Ljubljana Dept Math Ljubljana 1000 Slovenia

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, with respect to both accuracy and efficiency. The existing package Guptri is very elegant but may be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.

关键词： singular pencil singular generalized eigenvalue problem rank-completing perturbation Guptri differential algebraic equations two-parameter eigenvalue problem model updating double eigenvalues quadratic two-parameter eigenvalue problem

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Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

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JOURNAL OF COMPUTATIONAL PHYSICS 2015年 298卷 585-601页

作者： Plestenjak, Bor Gheorghiu, Calin I. Hochstenbach, Michiel E. Univ Ljubljana IMFM Jadranska 19 SI-1000 Ljubljana Slovenia Univ Ljubljana Dept Math SI-1000 Ljubljana Slovenia Romanian Acad T Popoviciu Inst Numer Anal Cluj Napoca 400110 Romania Eindhoven Univ Technol Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrodinger equation leads to a multiparameter eigenvalue problem (MEP);important cases include Mathieu's system, Lame's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Helmholtz equation Schrodinger equation Separation of variables Mathieu's system Lame's system Spectral methods Chebyshev collocation Laguerre collocation Multiparameter eigenvalue problem two-parameter eigenvalue problem Three-parameter eigenvalue problem Sylvester equation Bartels-Stewart method Subspace methods

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COMPUTING THE SIGNED DISTANCE BETWEEN OVERLAPPING ELLIPSOIDS

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SIAM JOURNAL ON OPTIMIZATION 2015年第4期25卷 2359-2384页

作者： Iwata, Satoru Nakatsukasa, Yuji Takeda, Akiko Univ Tokyo Grad Sch Informat Sci & Technol Dept Math Informat Tokyo 1138656 Japan

Computing the signed distance between two ellipsoids is a convex optimization problem when the two ellipsoids have no intersection, but it becomes nonconvex when the ellipsoids overlap. Efficient algorithms for convex optimization problems are thus not guaranteed to find the correct signed distance between overlapping ellipsoids. In this paper, we first show that computing the signed distance is equivalent to minimizing the norm along the boundary of the Minkowski difference. We then derive an algorithm with running time O(n(6)), where n is the dimension of the ellipsoids, that obtains a global minimizer on the boundary of the Minkowski difference and hence provides the exact signed distance. The algorithm first finds all the points that satisfy the Karush-Kuhn- Tucker (KKT) conditions, and then identifies a relevant KKT point with the smallest signed distance. The primary difficulty in computing the KKT points is that they are the solutions of two bivariate rational equations, whose poles are not known explicitly. Our key step is to convert the rational equations into polynomial equations, which we do by constructing certain bivariate matrix pencils whose zeros of determinants are the zeros of the rational functions. This reduces the problem to a two-parameter quadratic eigenvalue problem, which can be solved via a single-parameter linear eigenvalue problem of larger (squared) size, for which reliable algorithms are available. Thus we provide the first algorithm for computing the signed distance between overlapping ellipsoids with polynomial complexity.

关键词： ellipsoids signed distance nonconvex optimization KKT conditions Minkowski difference two-parameter eigenvalue problem Bezoutian

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