The Schur Algorithm for Matrix-Valued Meromorphic Functions

Abstract

This article extends the classical Schur algorithm to matrix-valued functions that are bounded on the unit circle and have a finite number of Smith–McMillan poles inside the unit disc. With each such function this article associates two infinite sequences: one is the well-known sequence of reflection coefficients (all less than one in magnitude), whereas the other is a sequence of signs. Under certain assumptions, the number of negative signs equals the number of poles within the unit disc. This article shows how to solve tangential interpolation problems using the algorithm and gives a simple proof for the connection between the number of poles inside the unit disc of each solution to the inertia of a certain Pick matrix. Also described is a numerically efficient procedure for carrying out the algorithm that involves only scalar operations.