Using covariance extension equation to solve the Nevanlinna-Pick interpolation with degree constraint

TL;DR

This paper tackles Nevanlinna-Pick interpolation with a degree constraint by reformulating the problem through the covariance extension equation (CEE) and solving it with a homotopy continuation method. Unlike traditional convex optimization, the continuation approach avoids reinitialization and remains robust when interpolant poles lie near the unit circle. Key contributions include a diffeomorphic path-tracking framework, a predictor-corrector scheme for stable convergence, and demonstrations on system identification, degree detection, and model reduction to validate robustness and efficiency. The work advances practical spectral estimation and identification tasks where degree constraints pose numerical challenges for existing methods.

Abstract

Nevanlinna-Pick interpolation problem has been widely studied in recent decades, however, the known algorithm is not simplistic and robust enough. This paper provide a new method to solve the Nevanlinna-Pick interpolation problem with degree constraint. It is based on the covariance extension equation proposed by Byrnes and Lindquist. A reformulation of the Nevanlinna-Pick interpolation problem is achieved and then solved by continuation method. This method need not calculate the initial value and a numerical example illustrates robustness and effciency of the proposed procedure

Figures (10)

• Figure 1: The locations of poles at $\nu=0$
• Figure 2: The locations of poles at $\nu=1$
• Figure 3: The trajectories of the poles
• Figure 4: The locations of poles at $\nu=0$
• Figure 5: The given spectral factor and its estimated ones

Theorems & Definitions (1)

• proof