The Pad\'e matrix pencil method with spurious pole information assimilation

TL;DR

This paper introduces a new Padé approximant method that effectively eliminates spurious poles caused by noise or numerical issues, improving the reliability of complex function approximations, especially near branch points.

Contribution

The method uniquely assimilates information from eliminated spurious poles to produce reduced order Padé approximants, enhancing accuracy and robustness over existing approaches.

Findings

Effective elimination of spurious poles due to noise and precision limitations.

Comparable performance to Robust Padé Approximation (RPA) method.

Improved reliability in approximating functions with branch points.

Abstract

We present a novel method for calculating Pad\'e approximants that is capable of eliminating spurious poles placed at the point of development and of identifying and eliminating spurious poles created by precision limitations and/or noisy coefficients. Information contained in in the eliminated poles is assimilated producing a reduced order Pad\'e approximant (PA). While the [m+k/m] conformation produced by the algorithm is flexible, the m value of the rational approximant produced by the algorithm reported here is determined by the number of spurious poles eliminated. Spurious poles due to coefficient noise/precision limitations are identified using an evidence-based filter parameter applied to the singular values of a matrix comprised of the series coefficients. The rational function poles are found directly by solving a generalized eigenvalue problem defined by a matrix pencil. Spurious poles place at the point of development, responsible in some algorithms for degeneracy, are identified by their magnitudes. Residues are found by solving an overdetermined linear matrix equation. The method is compared with the so-called Robust Pad\'e Approximation (RPA) method and shown to be competitive on the problems studied. By eliminating spurious poles, particularly in functions with branch points, such as those encountered solving the power-flow problem, solution of these complex-valued problems is made more reliable.