Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey, D. Steven; Mackey, Niloufer; Tisseur, Françoise

doi:10.1007/978-3-319-15260-8_12

Cited by 22 publications

(29 citation statements)

References 70 publications

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“…For several other structured classes of matrix polynomials of odd grade there are indeed structure preserving companion linearizations [16,38,39] (see also [15,Sect. 7] and [41], and the references therein), but there are no stratification results available in the literature for these structures. Since in this paper the stratification results for skew-symmetric pencils and polynomials previously developed in [19,22] have played a key role, we see again that the techniques of this paper cannot be directly used to get similar results for other structured matrix polynomials.…”

Section: Discussionmentioning

confidence: 99%

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

Dmytryshyn

Dopico

2018

Linear Algebra and its Applications

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We show that the set of m × m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m × m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d = 1).

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Section: Discussionmentioning

confidence: 99%

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

Dmytryshyn

Dopico

2018

Linear Algebra and its Applications

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“…. , k, we have that the orthogonal complement, computed with the Gram-Schmidt process, is ψ ⊥ (θ) = ψ(θ) − Ψ k (θ)h. Using the Observation 4 we obtain directly (23). We express ψ(θ) as (16) and, the columns of Ψ k as…”

Section: Orthogonalizationmentioning

confidence: 99%

On restarting the tensor infinite Arnoldi method

Mele

Jarlebring

2017

Bit Numer Math

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An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov-Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by

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“…For instance, it was shown that constructing pencils in these spaces corresponding to a given ansatz vector is very simple and almost all the resulting pencils are linearizations of P(λ ) from which the eigenvalues and corresponding eigenvectors can be easily recovered. In further work [18,20,12], it was shown that if P(λ ) has some special structure like, Hermitian, symmetric, ⋆-alternating and ⋆-palindromic, (see [22] for definitions), then there exist subspaces of L 1 (P) and L 2 (P) with the property that almost every pencil of the subspace is a structure preserving linearization of P(λ ) from which both finite and infinite eigenvalues of P(λ ) and corresponding eigenvectors can be easily recovered.…”

Section: Introductionmentioning

confidence: 99%

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

Das

Bora

2019

ELA

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The seminal work [21] introduced vector spaces of matrix pencils, with the property that almost all the pencils in the spaces are strong linearizations of a given square regular matrix polynomial. This work was subsequently extended to include the case of square singular matrix polynomials in [5]. We extend this work to non-square matrix polynomials by proposing similar vector spaces of rectangular matrix pencils that are equal to the ones in [21] when the polynomial is square. Moreover, the properties of these vector spaces are similar to those in [5] for the singular case. In particular, the complete eigenvalue problem associated with the matrix polynomial can be solved by using almost every matrix pencil from these spaces. Further, almost every pencil in these spaces can be 'trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily solve the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in [7]. Further, the global backward error analysis in [10] applied to these linearizations, shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

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Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Cited by 22 publications

References 70 publications

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

On restarting the tensor infinite Arnoldi method

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

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