Solving polynomial eigenvalue problems by means of the Ehrlich–Aberth method

Given the n×n matrix polynomial P(x)=∑i=0kPixi, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial detP(x), is treated in polynomial form rather than in matrix form by means of the Ehrlich–Aberth iteration. The main...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 439; no. 4; pp. 1130 - 1149
Main Authors: Bini, Dario A., Noferini, Vanni
Format: Journal Article
Language:English
Published: Elsevier Inc 15.08.2013
Subjects:
Ehrlich–Aberth method
Polynomial eigenvalue problem
Root-finding algorithm
Structured polynomials
65F15
Ehrlich–Aberth method
Root-finding algorithm
Polynomial eigenvalue problem
Structured polynomials
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:Given the n×n matrix polynomial P(x)=∑i=0kPixi, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial detP(x), 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 polynomials is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2013.02.024