The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order

AbstractThe QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the Hermitian quasiseparable structure (and thus generalizing the classical tridiagonal scheme) is presented. The algorithm is based on an earlier work [Y. Eidelman and I. Gohberg, A modification of the Dewilde–van der Veen method for inversion of finite structured matrices, Linear Algebra Appl. 343–344 (2002) 419–450], and it applies to the general case of Hermitian quasiseparable matrices of an arbitrary order.The algorithm operates on generators (i.e., a linear set of parameters defining the quasiseparable matrix), and the storage and the cost of one iteration are only linear. The results of some numerical experiments are presented.An application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere