We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparametereigenvalueproblem. two of these transformations a...
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We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparametereigenvalueproblem. two of these transformations are generalizations of the well-known linearization of the quadraticeigenvalueproblem and linearize the QMEP as a singular two-parametereigenvalueproblem. 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-parametereigenvalueproblem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparametereigenvalueproblems. We also consider several special cases of the QMEP, where some matrix coefficients are zero. (C) 2011 Elsevier Inc. All rights reserved.
We propose a homotopy method to solve the quadratic two-parameter eigenvalue problems, which arise frequently in the analysis of the asymptotic stability of the delay differential equation. Our method does not require...
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We propose a homotopy method to solve the quadratic two-parameter eigenvalue problems, which arise frequently in the analysis of the asymptotic stability of the delay differential equation. Our method does not require to form coupled generalized eigenvalueproblems with Kronecker product type coefficient matrices and thus can avoid the increasing of the computational cost and memory storage. Numerical results and the applications in the delay differential equations are presented to illustrate the effectiveness and efficiency of our method. It appears that our method tends to be more effective than the existing methods in terms of speed, accuracy and memory storage as the problem size grows.
Generalized eigenvalueproblems 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 ...
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Generalized eigenvalueproblems 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.
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 produ...
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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 eigenvalueproblem: the polynomial two-parametereigenvalueproblem, 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 eigenvalueproblem opens a door to efficient computation of DDE stability. (C) 2009 Elsevier Inc. All rights reserved.
Given a quadratictwo-parameter matrix polynomial Q(?,?mu), we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. The purpose for constructing this vector space is t...
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Given a quadratictwo-parameter matrix polynomial Q(?,?mu), we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. The purpose for constructing this vector space is that potential linearizations of Q(?,?mu) lie in it. Then, we identify a set of linearizations and describe their constructions. Finally, we determine a class of linearizations for a quadratic two-parameter eigenvalue problem.
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