Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

Ferreira, Carla; Parlett, Beresford Ν.; Dopico, Froilán M.

doi:10.1007/s00211-012-0470-z

Cited by 10 publications

(5 citation statements)

References 14 publications

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“…In this paper, we extend the development of structured condition numbers with respect to perturbations of the parameters to the fundamental case of the solutions of linear systems of equations with low-rank structured coefficient matrices, and we will prove that, also in this case, the solutions may be much better conditioned with respect to perturbations of the parameters than with respect to perturbations of the entries of the matrix, but that the opposite can not happen. This work is influenced by the recent references [5,11], which deal with the sensitivity of eigenvalues of some low-rank structured matrices, but also by the classical reference [13], in which the use of differential calculus for developing condition numbers was introduced.…”

Section: Introductionmentioning

confidence: 99%

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

Dopico

Pomés

2016

Numer Algor

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Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n) parameters, where n × n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters Partially supported by Ministerio de Economía y Competitividad of Spain through grants describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1, 1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.

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Section: Introductionmentioning

confidence: 99%

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

Dopico

Pomés

2016

Numer Algor

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“…We have given a rather thorough account of all the cases when a unique real tridiagonal matrix can make an approximate eigentriple ( λ, x, y * ) exact. Our results suggest two related avenues for future work extending the results in [3]. The first is to exploit the apparent locality of the formula in (4.3) for the off-diagonal entries in the (T, S) formulation to make well chosen perturbations to just a few entries of the computed eigenvector in order to reduce the backward error even further.…”

Section: Discussionmentioning

confidence: 82%

The Inverse Eigenvector Problem for Real Tridiagonal Matrices

Parlett¹,

Dopico²,

Ferreira³

2016

SIAM J. Matrix Anal. & Appl.

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A little known property of a pair of eigenvectors (column and row) of a real tridiagonal matrix is presented. With its help we can define necessary and sufficient conditions for the unique real tridiagonal matrix for which an approximate pair of complex eigenvectors are exact. Similarly we can designate the unique real tridiagonal matrix for which two approximate real eigenvectors, with different real eigenvalues, are also exact. We close with an illustration that these unique "backward error" matrices are sensitive to small rounding errors in certain partial sums which play a key role in determining the matrices.

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“…Compare with y * ∆T = λy * to see that y * = x T ∆. See [13,31]. The so-called twisted factorizations generalize the lower and upper bidiagonal factorizations.…”

Section: Shift Strategymentioning

confidence: 99%

“…In the unsymmetric case even the absolute condition numbers can rise to ∞ and little is known about relative errors. In [13] several relative condition numbers with respect to eigenvalues were derived. Some of them use bidiagonal factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries.…”

Section: Relative Eigenvalue Condition Numbersmentioning

confidence: 99%

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

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Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

Cited by 10 publications

References 14 publications

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

The Inverse Eigenvector Problem for Real Tridiagonal Matrices

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

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