Structure preserving algorithms for perplectic eigenproblems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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“…Structure-preserving Jacobi algorithms for symmetric persymmetric and skew-symmetric persymmetric matrices have been recently developed in [90].…”

Section: Persymmetric Matricesmentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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“…In the context of a fixed scalar product or a restricted class of scalar products, the pursuit of spectral decompositions, Jordan canonical forms, and other condensed forms under structure preserving similarities or general similarities has been the focus of intense study. The literature on this subject is extensive: see for example [1], [2], [12], [15], [18], [36], [38], [40], [41], [44], [43] and the references therein.…”

Section: Also We Must Havementioning

confidence: 99%

Structured Factorizations in Scalar Product Spaces

Mackey¹,

Mackey²,

Tisseur³

2005

SIAM J. Matrix Anal. & Appl.

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Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products.A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes is given.Key words. automorphism group, Lie group, Lie algebra, Jordan algebra, bilinear form, sesquilinear form, scalar product, indefinite inner product, orthosymmetric, adjoint, factorization, symplectic, Hamiltonian, pseudo-orthogonal, polar decomposition, matrix sign function, matrix square root, generalized polar decomposition, eigenvalues, eigenvectors, singular values, structure preservation.AMS subject classifications. 15A18, 15A21, 15A23, 15A57, 15A631. Introduction. The factorization of a general matrix into a product of structured factors plays a key role in theoretical and computational linear algebra. In this work we address the following question: if we apply one of the standard factorizations to a matrix that is already structured, to what extent do the factors have additional structure related to that of the original matrix?Many applications generate structure and there are potential benefits to be gained by exploiting it when developing theory and deriving algorithms. For example, algorithms that preserve structure may have reduced storage requirements and operation counts, may be more accurate, and may also provide solutions that are more physically meaningful in the presence of rounding and truncation errors.The structured matrices we consider belong to the automorphism group G, the Lie algebra L, and the Jordan algebra J associated with a scalar product, that is, a nondegenerate bilinear or sesquilinear form on K n . These classes of matrices include linear structures such as complex symmetric, pseudo-symmetric, and Hamiltonian matrices, as well as nonlinear structures such as complex orthogonal, pseudo-orthogonal and symplectic matrices. Section 2 introduces concepts and notation needed for our unified treatment of structured factorizations in scalar product spaces. We introduce two important special types of scalar product, termed unitary and orthosymmetric, and describe several equivalent ways to characterize them. The proofs of these equivalences have been delegated to an appendix to avoid disrupting the flow of the p...

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“…When these algorithms preserve structure (see, for example, [2], [4], [13], and the literature cited therein) it is often appropriate to consider condition numbers that measure the sensitivity to structured perturbations. In this paper we investigate the effect of structure-preserving perturbations on linearly and nonlinearly structured eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

Structured Eigenvalue Condition Numbers

Karow

Kreßner²,

Tisseur

2006

SIAM J. Matrix Anal. & Appl.

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Abstract. This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.

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Structure preserving algorithms for perplectic eigenproblems

Cited by 25 publications

References 20 publications

Structured Eigenvalue Problems

Structured Eigenvalue Problems

Structured Factorizations in Scalar Product Spaces

Structured Eigenvalue Condition Numbers

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