Computation of Degree Constrained Rational Interpolants with Non-Strictly Positive Parametrizing Functions via Homotopy Continuation

Abstract

A numerically stable homotopy continuation method was first proposed by Enqvist for computing degree constrained rational covariance extensions. The approach was later adapted in the works of Nagamune, and Blomqvist and Nagamune, to the Nevanlinna-Pick interpolation problem and more general complexity constrained problems. However, the method has not been developed to the fullest extent as all the previous works limit the associated parametrizing function (in the form of a generalized pseudopolynomial) to be strictly positive definite on the unit circle, or equivalently, that all spectral zeros should lie inside the unit circle. The purpose of this paper is to show that the aforementioned restriction is not essential and that the method is equally applicable when some spectral zeros are on the unit circle. We show that even in this case, the modified functional of Enqvist has a stationary minimizer. Several numerical examples are provided herein to demonstrate the applicability of the method for computing degree constrained interpolants with spectral zeros on the unit circle, including solutions which may have poles on the unit circle.