Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Barel, Marc Van; Tisseur, Françoise

doi:10.1016/j.laa.2017.04.025

Cited by 12 publications

(18 citation statements)

References 18 publications

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“…PEPs arise in many applications, either directly or as approximations of other nonlinear eigenvalue problems [4,20,21,32,36,41]. However, the numerical algorithms currently available for the solution of PEPs are not completely satisfactory, despite of recent impressive advances in this area, both for small to medium size dense PEPs as well as for large-scale sparse PEPs [6,12,22,27,30,42,43,45].…”

Section: Introductionmentioning

confidence: 99%

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

Bueno

Dopico

Furtado

et al. 2018

Calcolo

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Section: Introductionmentioning

confidence: 99%

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

Bueno

Dopico

Furtado

et al. 2018

Calcolo

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“…Unfortunately, nowadays, such "ideal" algorithm exists only for degrees k = 1 (the QZ algorithm for generalized eigenvalue problems) and k = 2, in this case via linearizations and delicate scalings of P [15,16,37]. The recent work [34] shows that there is still some hope of finding an "ideal" algorithm for PEPs with degree k > 2.…”

Section: Properties Of Möbius Transformationsmentioning

confidence: 99%

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

2019

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Möbius transformations have been used in numerical algorithms for computing eigenvalues and invariant subspaces of structured generalized and polynomial eigenvalue problems (PEPs). These transformations convert problems with certain structures arising in applications into problems with other structures and whose eigenvalues and invariant subspaces are easily related to the ones of the original problem. Thus, an algorithm that is efficient and stable for some particular structure can be used for solving efficiently another type of structured problem via an adequate Möbius transformation. A key question in this context is whether these transformations may change significantly the conditioning of the problem and the backward errors of the computed solutions, since, in that case, their use might lead to unreliable results. We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducing the Möbius transformation is well conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the whole polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP is used.

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“…Note that tropical algebra is a semiring which recently has found many applications in numerical linear algebra. For example, tropical roots , the roots of tropical polynomials, are used to approximate eigenvalues of matrix polynomials 5‐9 . In References 10‐12, the authors used tropical algebra as a tool for an optimal diagonal scaling of a matrix and approximating incomplete LU and Cholesky factorizations preconditioners, respectively.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Tavakolipour

Shakeri

2020

Numerical Linear Algebra App

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Summary Let P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for n − 2 largest magnitude eigenvalues of P(t) we compute the asymptotics as n → ∞, in addition to t. When P(t) is also symmetric, these formulas allow us to compute the asymptotics of the 2‐norm condition number. The number of arithmetic operations involved, does not depend on n. We illustrate our results by some numerical tests.

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Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Cited by 12 publications

References 18 publications

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

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