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

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.