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Canonical Wiener–Hopf and spectral factorization of large‐degree matrix polynomials
Author(s) -
Böttcher Albrecht,
Halwass Martin
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410389
Subject(s) - mathematics , polynomial matrix , quartic function , matrix decomposition , matrix (chemical analysis) , square matrix , factorization , polynomial , matrix polynomial , mathematical analysis , pure mathematics , symmetric matrix , eigenvalues and eigenvectors , algorithm , physics , materials science , quantum mechanics , composite material
Abstract We present a novel Newton method for canonical Wiener–Hopf and spectral factorization of matrix polynomials. The initial vector results from solving a block Toeplitz‐like system, and the Jacobi matrix governing the Newton iteration has nice structural and numerical properties. The local quadratic convergence of the method is proved and was tested numerically. For scalar polynomials of degree 20000, a superfast version of the method implemented on a laptop typically reqired about half a minute to produce an initial vector and then performed the Newton iteration within one second. In the matrix case, the method worked reproachless on a laptop with 8 Gigabyte RAM if the degree of the polynomial times the squared matrix dimension did not exceed 20000. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)