Given the n × n matrix polynomial $P ( x ) = \sum_{i = 0}^k 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, symplec- tic, etc., where the eigenvalues have special symmetries in the com- plex 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
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
BINI, DARIO ANDREA;NOFERINI, VANNI
2013-01-01
Abstract
Given the n × n matrix polynomial $P ( x ) = \sum_{i = 0}^k 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, symplec- tic, etc., where the eigenvalues have special symmetries in the com- plex 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 reportedI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
