Efficient numerical method for the discrete-time symmetric matrix polynomial equation

A numerical procedure is proposed to solve a matrix polynomial equation frequently encountered in discrete-time control and signal processing. The algorithm is based on a simple rewriting of the original equation in terms of a reduced Sylvester matrix. In contrast to previously published methods, it does not make use of elementary polynomial operations. Moreover, and most notably, it is numerically reliable. Basic examples borrowed from control and signal processing literature are aimed at illustrating the simplicity and efficiency of this new numerical method.