Numerical Solution of Matrix Polynomial Equations by Newton's Method

Kratz, Werner; Stickel, Eberhard

doi:10.1093/imanum/7.3.355

Cited by 34 publications

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“…i − C The general approach for solving (3.4) is to solve the n 2 × n 2 linear system derived by vec function and Kronecker product [11,15] such as…”

Section: Convergence Of Newton's Methodsmentioning

confidence: 99%

Newton's Method for Solving a Quadratic Matrix Equation With Special Coefficient Matrices

Seo

Kim³

2013

Honam Mathematical Journal

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Abstract.We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton's method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Fréchet derivative at the elementwise minimal positive solvent is nonsingular. Although the Fréchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.

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“…i − C The general approach for solving (3.4) is to solve the n 2 × n 2 linear system derived by vec function and Kronecker product [11,15] such as…”

Section: Convergence Of Newton's Methodsmentioning

confidence: 99%

Newton's Method for Solving a Quadratic Matrix Equation With Special Coefficient Matrices

Seo

Kim³

2013

Honam Mathematical Journal

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“…The work in [19] uses interval arithmetic to compute an interval matrix containing the exact solution to the quadratic matrix equation. For the case of the general matrix solvent problem, we can also cite [9], [30] and [23]. On the other hand, the question of designing symbolic algorithms for computing solvents remains relatively unexplored.…”

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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“…Kratz and Stickel [26] proved that Newton's method could also be applied to solve this general case. Seo and Kim [38] studied the relaxed Newton's method for finding the minimal nonnegative solution of the MPE (1.1) and they also proved that the relaxed Newton's method could work more efficiently than the general Newton's method.…”

Section: Introductionmentioning

confidence: 99%

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

2018

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We consider a matrix polynomial equation (MPE) A n X n + A n−1 X n−1 + · · · + A 0 = 0, where A n , A n−1 , . . . , A 0 ∈ R m×m are the coefficient matrices, and X ∈ R m×m is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.

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Numerical Solution of Matrix Polynomial Equations by Newton's Method

Cited by 34 publications

References 0 publications

Newton's Method for Solving a Quadratic Matrix Equation With Special Coefficient Matrices

Newton's Method for Solving a Quadratic Matrix Equation With Special Coefficient Matrices

A contour integral approach to the computation of invariant pairs

Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

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