72 research outputs found
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
Structured eigenvalue backward errors of matrix pencils and polynomials with palindromic structures
We derive formulas for the backward error of an approximate eigenvalue of a *-palindromic matrix polynomial with respect to *-palindromic perturbations. Such formulas are also obtained for complex T-palindromic pencils and quadratic polynomials. When the T-palindromic polynomial is real, then we derive the backward error of a real number considered as an approximate eigenvalue of the matrix polynomial with respect to real T-palindromic perturbations. In all cases the corresponding minimal structure preserving perturbations are obtained as well. The results are illustrated by numerical experiments. These show that there is a significant difference between the backward errors with respect to structure preserving and arbitrary perturbations in many cases
A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems
AbstractWe propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues on the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are presented. Our experience shows that our algorithm is efficient in comparison to the few existing approaches for small to medium size problems
The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis
An algorithm based on the Ehrlich-Aberth root-finding method is presented for
the computation of the eigenvalues of a T-palindromic matrix polynomial. A
structured linearization of the polynomial represented in the Dickson basis is
introduced in order to exploit the symmetry of the roots by halving the total
number of the required approximations. The rank structure properties of the
linearization allow the design of a fast and numerically robust implementation
of the root-finding iteration. Numerical experiments that confirm the
effectiveness and the robustness of the approach are provided.Comment: in press in Linear Algebra Appl. (2011
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