Algorithms for Solvents of Matrix Polynomials

Dennis, John E.; Traub, Joseph F.; Weber, R.P.

doi:10.1137/0715034

Cited by 50 publications

(33 citation statements)

References 5 publications

(1 reference statement)

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“…The relation between the Riccati and the quadratic matrix equation is highlighted in [7], whereas a study on the existence of solvents can be found in [13]. Several works address the problem of computing a numerical approximation for the solution of the quadratic matrix equation: an approach to compute, when possible, the dominant solvent is proposed in [12]. Newton's method and some variations are also used to approximate solvents numerically: see for example [11], [22], [21], [27].…”

Section: P (S) := J=0mentioning

confidence: 99%

A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and the backward error for invariant pairs and solvents. We then adapt the Sakurai-Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

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Section: P (S) := J=0mentioning

confidence: 99%

A contour integral approach to the computation of invariant pairs

Barkatou

Boito

Ugalde

2017

Theoretical Computer Science

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“…which is dealt by Dennis, Traub and Weber [4]. It has an infinite number of solvents which have a form:…”

Section: Numerical Experimentsmentioning

confidence: 99%

Finding the Skew-Symmetric Solvent to a Quadratic Matrix Equation

Han¹,

Kim²

2012

East Asian mathematical journal

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Abstract. In this paper we consider the quadratic matrix equation which can be defined bywhere X is a n × n unknown real matrix; A, B and C are n × n given matrices with real elements. Newton's method is considered to find the skew-symmetric solvent of the nonlinear matrix equations Q(X). We also show that the method converges the skew-symmetric solvent even if the Fréchet derivative is singular. Finally, we give some numerical examples.

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“…The problem of reducing an algebraic Riccati equation to (1.2) has been analyzed in [4]. Numerical methods for solving (1.2) has been considered including two linearly convergent algorithms for computing a dominant solvent [8,10]. Davis [6,7] used Newton's method for solving (1.2).…”

Section: B Hashemi and M Dehghanmentioning

confidence: 99%

Efficient computation of enclosures for the exact solvents of a quadratic matrix equation

Hashemi

Dehghan

2010

ELA

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Abstract. None of the usual floating point numerical techniques available for solving the quadratic matrix equation AX 2 + BX + C = 0 with square matrices A, B, C and X, can provide an exact solution; they can just obtain approximations to an exact solution. We use interval arithmetic to compute an interval matrix which contains an exact solution to this quadratic matrix equation, where we aim at obtaining narrow intervals for each entry. We propose a residual version of a modified Krawczyk operator which has a cubic computational complexity, provided that A is nonsingular and X and X + A −1 B are diagonalizable. For the case that A is singular or nearly singular, but B is nonsingular we provide an enclosure method analogous to a functional iteration method. Numerical examples have also been given.

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Algorithms for Solvents of Matrix Polynomials

Abstract: Abstract. In an earlier paper we developed the algebraic theory of matrix polynomials. Here we introduce two algorithms for computing "dominant" solvents. Global convergence of the algorithms under certain conditions is established.

Cited by 50 publications

References 5 publications

A contour integral approach to the computation of invariant pairs

A contour integral approach to the computation of invariant pairs

Finding the Skew-Symmetric Solvent to a Quadratic Matrix Equation

Efficient computation of enclosures for the exact solvents of a quadratic matrix equation

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