Solving the Generalized Eigenvalue Problem for Rational Toeplitz Matrices
Dario A. Bini, Fabio Di Benedetto · SIAM Journal on Matrix Analysis and Applications · 1990
The generalized eigenvalue problem ${\bf Ax} = \lambda {\bf Bx}$ for Toeplitz matrices generated by a rational function is considered from a computational point of view. Three different functions having the same zeros as the polynomial $p(\lambda ) = \det ({\bf A} - \lambda {\bf B})$ are introduced and fast methods for the evaluation at a point of these functions and their derivatives are presented. Such methods, which can be used for the numerical computation of the eigenvalues as zeros of $p(\lambda )$, generalize some results holding in the particular case of banded Toeplitz matrices.