Matrix methods for Pad\'e approximation: numerical calculation of poles, zeros and residues

TL;DR

This paper presents matrix-based methods for Padé approximation of Z-transforms, focusing on numerical stability and error analysis for calculating poles, zeros, and residues efficiently.

Contribution

It introduces new formulas linking Padé approximation components to a tridiagonal matrix, enhancing numerical stability and error computation methods.

Findings

Formulas for poles, zeros, residues in terms of matrix J

Numerical stability tests and comparisons

Methods for computing forward and backward errors

Abstract

A representation of the Pad\'e approximation of the $Z$-transform of a signal as a resolvent of a tridiagonal matrix $J$ is given. Several formulas for the poles, zeros and residues of the Pad\'e approximation in terms of the matrix $J$ are proposed. Their numerical stability is tested and compared. Methods for computing forward and backward errors are presented.