Stability of Structured Hamiltonian Eigensolvers

Tisseur, Françoise

doi:10.1137/s0895479800368007

Cited by 20 publications

(14 citation statements)

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“…It is expected that the recent analysis by Tisseur [25] of a related family of algorithms can also be applied to this work to show that these methods are not only backward stable, but in fact strongly backward stable.…”

Section: Discussionmentioning

confidence: 96%

“…A noteworthy advantage of these methods is that the rich eigenstructure of the initial matrix is not obscured by rounding errors during the computation. Such algorithms also exhibit greater numerical stability, and are likely to be strongly backward stable [25]. Storage requirements are appreciably lowered by working with a truncated form of the matrix.…”

Section: Introductionmentioning

confidence: 99%

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Structure preserving algorithms for perplectic eigenproblems

Mackey¹,

Mackey²,

Dunlavy³

2005

ELA

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Structured real canonical forms for matrices in R n×n that are symmetric or skew-symmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthogonal bases for the invariant subspaces of the associated matrix. In addition to preserving structure, these methods are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.

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Section: Discussionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Structure preserving algorithms for perplectic eigenproblems

Mackey¹,

Mackey²,

Dunlavy³

2005

ELA

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“…Thus (3.2) is not a formula we would evaluate routinely in the course of solving a problem. Nevertheless, it is useful as a tool when testing the stability of newly developed structure-preserving algorithms, as shown in [27], or to illustrate instability of wellknown algorithms.…”

Section: Nowmentioning

confidence: 99%

“…This factorization is discussed in [7, Cor. 4.5(ii)] and [27].We make frequent use of the following lemmas. …”

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confidence: 99%

A Chart of Backward Errors for Singly and Doubly Structured Eigenvalue Problems

Tisseur¹

2003

SIAM J. Matrix Anal. & Appl.

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Abstract. We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a factor √ 2 of the unstructured backward error. This paper collects, unifies, and extends existing work on this subject.Key words. eigenvalue, eigenvector, symmetric matrix, Hermitian matrix, skew-symmetric matrix, skew-Hermitian matrix, symplectic matrix, conjugate symplectic matrix, Hamiltonian matrix, backward error, condition number AMS subject classifications. 65F15, 65F20, 65H10, 65L15, 65L20, 15A18, 15A57 PII. S089547980139995X1. Introduction. Byers, and Mehrmann [8] present a chart of numerical methods for structured eigenvalue problems for which the matrices have more than one of the properties defined as follows: −InIn 0 ], I n being the n×n identity matrix. Structured eigenvalue problems occur in numerous applications and we refer to [8] for a list of them and pointers to the relevant literature. In this paper we present a chart of computable backward errors for approximate eigenpairs and condition numbers for simple eigenvalues of matrices having one or two of these special structures.The importance of condition numbers for characterizing the sensitivity of solutions to problems and backward errors for assessing the stability and quality of numerical algorithms is widely appreciated. A backward error of an approximate eigenpair (x, λ) of a matrix A is a measure of the smallest perturbation E such that (A + E)x = λx. This backward error has two main uses. First, it can be used to determine if A natural definition of the normwise backward error of an approximate eigenpairwhere α is a positive parameter that allows freedom in how the perturbations are measured and · denotes any vector norm and the corresponding subordinate matrix norm. Deif [9] derived the explicit expression for the 2-norm (also valid for any subordinate norm and the Frobenius norm),showing that the normwise backward error is a scaled residual. Also of interest is the backward error of a set of approximate eigenpairs (For a measure of the backward error we use the natural generalization of (1.1),for which an explicit expression is available for any unitarily invariant norm if rank(The measure η is not entirely appropriate for our structured eigenvalue problems, as it does not respect any structure in A. Similar remarks can be made about condition numbers. Standard condition numbers are derived without requiring that perturbations preserve structure. As a consequence, standard condition numbers usually exceed the actual condition number for an eigenvalue problem subject to structured perturbation. In the last few years, efforts have been concentrated on deriving new structure-preserving algorithms for the solution of structured eigenvalue problems for both the dense case [21], to cite just a ...

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“…The solution of quadratic eigenvalue problems is typically done via a linearization procedure, where the quadratic problem is embedded into a double size linear generalized eigenvalue problem. Apart from the doubling of the dimension there are other disadvantages to this linearization procedure, like the increase of the condition number of the problem, i.e., the linearized system is sometimes much more sensitive to perturbations in the data than the original problem, see [33]. On the other hand there are no efficient methods known that work directly with the quadratic eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%