Taylor Polynomials in High Arithmetic Precision as Universal Approximators

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.

The Lagrangian remainder of Taylor's series, distinguishes $\mathcal{O}(f(x))$ time complexities to polynomials or not

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The purpose of this letter is to investigate the time complexity consequences of the truncated Taylor series, known as Taylor Polynomials [1][2][3]. In particular, it is demonstrated that the examination of the P = NP equality, is associated with the determination of whether the n th derivative of a particular solution is bounded or not.

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