A New First Order Taylor-like Theorem With An Optimized Reduced Remainder

TL;DR

This paper introduces a novel first order Taylor-like theorem with a significantly reduced remainder, achieved through an optimal linear combination of derivatives at equally spaced points, enhancing approximation accuracy.

Contribution

The paper presents a new Taylor-like formula with minimized remainder using optimal weights, improving approximation accuracy over classical Taylor's theorem.

Findings

Reduced remainder compared to classical Taylor's formula

Optimal weights minimize the approximation error

Enhanced accuracy in interpolation and quadrature estimates

Abstract

This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error estimate and also the trapezoidal quadrature error, in order to assess the gain of accuracy we obtain using this new Taylor-like formula.