A new second order Taylor-like theorem with an optimized reduced remainder

J. Chaskalovic; F. Assous

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arXiv · 2304.08136

A new second order Taylor-like theorem with an optimized reduced remainder

J. Chaskalovic·F. Assous

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Abstract

In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function $f$ defined on the interval $[a,b]$, this formula is derived by introducing a linear combination of $f'$ computed at $n+1$ equally spaced points in $[a,b]$, together with $f''(a)$ and $f''(b)$. We then consider two classical applications of this Taylor-like expansion: the interpolation error and the numerical quadrature formula. We show that using this approach improves both the Lagrange $P_2$ - interpolation error estimate and the error bound of the Simpson rule in numerical integration.

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J. Chaskalovic, F. Assous. 2023-04-17. A new second order Taylor-like theorem with an optimized reduced remainder. https://arxiv.org/abs/2304.08136

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