Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

Mehrmann, Volker; Voß, Heinrich

doi:10.1002/gamm.201490007

Cited by 222 publications

(205 citation statements)

References 91 publications

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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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Section: Resultant Methodsmentioning

confidence: 99%

Computing the common zeros of two bivariate functions via Bézout resultants

2014

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“…In many applications, the study of spectral properties of matrix polynomials is a central topic [8,21,22,24,34,38]. A common technique to find the eigenvalues of a matrix polynomial is converting to a linear problem using the following method.…”

Section: Linearizations Of Matrix Polynomialsmentioning

confidence: 99%

Duality of matrix pencils, Wong chains and linearizations

Noferini

Poloni

2015

Linear Algebra and its Applications

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We consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils. One is dual pencils, a pair of pencils with the same regular part and related singular structures. They were introduced by V. Kublanovskaya in the 1980s. The other is Wong chains, families of subspaces, associated with (possibly singular) matrix pencils, that generalize Jordan chains. They were introduced by K.T. Wong in the 1970s. Together, dual pencils and Wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications, especially in the context of linearizations of matrix polynomials.We first give a self-contained introduction to these two concepts, using modern language and extending them to a more general form; we describe the relation between them and show how they act on the Kronecker form of a pencil and on spectral and singular structures (eigenvalues, eigenvectors and minimal bases). Then we present several new applications of these results to more recent topics in matrix pencil theory, including: constraints on the minimal indices of singular Hamiltonian and symplectic pencils, new sufficient conditions under which pencils in L 1 , L 2 linearization spaces are strong linearizations, a new perspective on Fiedler pencils, and a link between the Möller-Stetter theorem and some linearizations of matrix polynomials.

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“…The stability analysis of delay-differential equations leads to an RCR-palindromic QEP [28], while a variant of RCR-alternating structure (without conjugation) arises in linear response theory from quantum chemistry [66]. Further details on these and other applications can be found in [52,69,79], chapter What about chapter 2 and 6, too? 3 of this Festschrift, and the references therein.…”

Section: Definition 8 (Adjoint Of Matrix Polynomials)mentioning

confidence: 99%

“…Such problems arise directly from applications, from finite element discretizations of continuous models, or as approximations to more general nonlinear eigenproblems, as detailed in the survey articles [69,79]. The trend towards extreme designs, such as high speed trains, optoelectronic devices, micro-electromechanical systems, and "superjumbo" jets such as the Airbus 380, presents a challenge for the computation of the resonant frequencies of these structures as these extreme designs often lead to eigenproblems with poor conditioning.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey

Tisseur

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Matrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems. One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial by a matrix pencil with the same spectrum, and then computes with the pencil. Many matrix polynomials arising from applications have additional algebraic structure, leading to symmetries in the spectrum that are important for any computational method to respect. Thus it is useful to employ a structured linearization for a matrix polynomial with structure. This essay surveys the progress over the last decade in our understanding of linearizations and their construction, both with and without structure, and the impact this has had on numerical practice.

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Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods

Cited by 222 publications

References 91 publications

Computing the common zeros of two bivariate functions via Bézout resultants

Computing the common zeros of two bivariate functions via Bézout resultants

Duality of matrix pencils, Wong chains and linearizations

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

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