Computing the Nearest Rank-Deficient Matrix Polynomial

Giesbrecht, Mark; Haraldson, Joseph; Labahn, George

doi:10.1145/3087604.3087648

Cited by 7 publications

(8 citation statements)

References 31 publications

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“…The Lagrange multipliers will not be unique in this particular scenario and the rate of convergence may degrade if Newton's method is used. In the instance of r = 1 then we present the following result (Giesbrecht et al, 2017).…”

Section: The Jacobianmentioning

confidence: 85%

“…In the case of finding the nearest singular matrix pencil this problem was solved by the present authors in Giesbrecht et al (2017). Previous to that this problem was posed for linear matrix pencils in Byers and Nichols (1993) and followed up in Byers et al (1998).…”

Section: Previous Researchmentioning

confidence: 99%

“…If we do not allow perturbations to zero-coefficients, that is, A 0 [1, 1] and A 0 [2, 3] may not be perturbed, then after five iterations of plain Newton's method (see (Giesbrecht et al, 2017)) we compute Guglielmi et al (2017) report an upper-bound on the distance to singularity allowing complex perturbations, that is ∆A ∈ C[t] n×n of ∆ C A F ≈ 0.1357 in Example 2.10. In Example 2.12, Guglielmi et al (2017) report an upper-bound on the distance to singularity allowing real perturbations, ∆ R A F ≈ 0.1366.…”

Section: Nearest Affinely Structured Examples Imentioning

confidence: 99%

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Computing lower rank approximations of matrix polynomials

Giesbrecht

Haraldson

Labahn

2020

Journal of Symbolic Computation

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Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest matrix polynomial that is algebraically singular with a prescribed lower bound on the dimension given in a previous paper by the authors. In this paper we prove that such lower rank matrices at minimal distance always exist, satisfy regularity conditions, and are all isolated and surrounded by a basin of attraction of non-minimal solutions. In addition, we present an iterative algorithm which, on given input sufficiently close to a rank-at-most matrix, produces that matrix. The algorithm is efficient and is proven to converge quadratically given a sufficiently good starting point. An implementation demonstrates the effectiveness and numerical robustness of our algorithm in practice.

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Section: The Jacobianmentioning

confidence: 85%

Section: Previous Researchmentioning

confidence: 99%

Section: Nearest Affinely Structured Examples Imentioning

confidence: 99%

See 1 more Smart Citation

Computing lower rank approximations of matrix polynomials

Giesbrecht

Haraldson

Labahn

2020

Journal of Symbolic Computation

Self Cite

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“…The results of this paper can be applied to a number of problems in numerical linear algebra. One example is solving various distance problems for matrix polynomials if the corresponding problems are solved for matrix pencils, e.g., finding a singular matrix polynomial nearby a given matrix polynomial [4,18,20]. Another application lies in the stratification theory [8,14]: constructing an explicit perturbation of a matrix polynomial when a perturbation of its linearization is known.…”

Section: Introductionmentioning

confidence: 99%

Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

Dmytryshyn

2021

Bit Numer Math

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A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

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“…The results of this paper can be applied to a number of problems in numerical linear algebra. One example is solving various distance problems for matrix polynomials if the corresponding problems are solved for matrix pencils, e.g., finding a singular matrix polynomials nearby a given matrix polynomial [4,18,19]. Another application lies in the stratification theory [8,14]: constructing an explicit perturbation of a matrix polynomial when a perturbation of its linearization is known.…”

Section: Introductionmentioning

confidence: 99%

Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

Dmytryshyn¹

2020

Preprint

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A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we present an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

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Computing the Nearest Rank-Deficient Matrix Polynomial

Cited by 7 publications

References 31 publications

Computing lower rank approximations of matrix polynomials

Computing lower rank approximations of matrix polynomials

Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

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