Numerical Solution of a Quadratic Matrix Equation

Davis, George J.

doi:10.1137/0902014

Cited by 61 publications

(47 citation statements)

References 9 publications

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“…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]. The work in [19] uses interval arithmetic to compute an interval matrix containing the exact solution to the quadratic matrix equation.…”

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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“…When (7) has a solution X, the 2n eigenvalues of Q(λ) can be found by finding the eigenvalues of the matrix X and the matrix pencil λA + AX + B. This solvent approach has been explored in [4] and [5], and more recently in [9] and [11]. Suppose that X (1) and X (2) are two solvents of Q(X) and the spectra of these two solvents are disjoint.…”

Section: Solving Hyperbolic Qepsmentioning

confidence: 99%

Algorithms for hyperbolic quadratic eigenvalue problems

Guo

Lancaster

2005

Math. Comp.

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Abstract. We consider the quadratic eigenvalue problem (or the QEP) (λ 2 A + λB + C)x = 0, where A, B, and C are Hermitian with A positive definite. The QEP is called hyperbolic if (x * Bx) 2 > 4(x * Ax)(x * Cx) for all nonzero x ∈ C n . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if B is positive definite and C is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

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“…When A = A T , B = −B T , C = C T in the quadratic eigenvalue problem (2), it has a Hamiltonian eigenstructure, that is, the eigenvalues are symmetric with respect to both axes [9,11]. Motivation for finding skew-symmetric solvent of the quadratic matrix equation (1) comes from the quadratic eigenvalue problem (2), because any skew-symmetric matrix has a pair of purely imaginary eigenvalues.…”

Section: Introductionmentioning

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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Numerical Solution of a Quadratic Matrix Equation

Cited by 61 publications

References 9 publications

A contour integral approach to the computation of invariant pairs

A contour integral approach to the computation of invariant pairs

Algorithms for hyperbolic quadratic eigenvalue problems

Finding the Skew-Symmetric Solvent to a Quadratic Matrix Equation

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