The application of the Ehrlich‐Aberth method to structured polynomial eigenvalue problems

Noferini, Vanni

doi:10.1002/pamm.201110439

Cited by 2 publications

(2 citation statements)

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“…Finally, we have proposed and tested a structured version of the algorithm that is able to catch certain structures in the spectrum. New numerical experiments on the matter confirm and extend the results of [13,36] about the efficiency and the accuracy of the structured EAI.…”

Section: Conclusion and Lines Of Future Researchsupporting

confidence: 72%

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

Бини

Noferini

2013

Linear Algebra and its Applications

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Given the n × n matrix polynomial P (x) = k i=0 P i x i , we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial det P (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 polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.

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Section: Conclusion and Lines Of Future Researchsupporting

confidence: 72%

“…Discussions and analyses of the behaviour of different structured versions of the EAI applied to structured PEPs have already appeared in [13] for palindromic polynomials and in [36] for even/odd polynomials.…”

Section: Structured Casementioning

confidence: 99%