Taylor Polynomials in High Arithmetic Precision as Universal Approximators

TL;DR

This paper demonstrates that high-precision Taylor polynomial approximations serve as universal tools for various computational tasks, offering high accuracy, predictable errors, and elimination of common issues like Runge phenomenon.

Contribution

It introduces a unified high-precision Taylor polynomial approach for diverse approximation tasks, with rigorous error analysis and predictable convergence properties.

Findings

Errors are precisely predictable across tasks.

Runge phenomenon is eliminated with high-precision methods.

Extrapolation extent can be anticipated a-priori.

Abstract

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is demonstrated, by utilizing partial sums of Taylor series in high arithmetic precision. In particular, the proposed method is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method is based on the utilization of Taylor polynomials, by exploiting some hundreds of computer digits, resulting in highly accurate calculations. Interestingly, some well-known problems were found to reason by calculations accuracy, and not methodological inefficiencies, as supposed. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated and the extrapolation extent may a-priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provide a rigor estimation of the prediction ability. The approximation errors have comprehensively been analyzed, for a variety of calculation digits and test problems.