Rational Hermite-Pad\'e Approximants vs Pad\'e Approximants. Numerical Results

TL;DR

This paper compares Hermite-Padé and Padé rational approximants, demonstrating through numerical examples and theoretical analysis that Hermite-Padé approximants have advantages in approximating certain functions, such as the frequency function of the Van der Pol equation.

Contribution

It provides numerical evidence and theoretical insights showing the superiority of Hermite-Padé approximants over Padé approximants for specific analytical functions.

Findings

Hermite-Padé approximants outperform Padé in numerical tests.

Theoretical analysis confirms the advantage for the Van der Pol frequency function.

Numerical examples illustrate practical benefits of Hermite-Padé methods.

Abstract

The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Pad\'e polynomials over the Pad\'e approximation procedure. The first part of the paper is devoted to some numerical examples in this direction. The second part will be devoted to some theoretical results. In particular, we demonstrate our ideas about the advantage of rational Hermite-Pad\'e approximants over Pad\'e approximants analyzing the analytical structure of the frequency function $\nu$ of the free Van der Pol equation.