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

2023/04/17 by J. Chaskalovic, Chaskalovic, J., F. Assous +1
Computer Science · Decision Sciences · #41A05 #65D30 #65N15 #65N30 #65N75 #Digital Filter Design and Implementation #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2304.08136

openalex publication_date 2023/04/17 · openalex created_date 2023/04/21 · openalex updated_date 2026/07/28

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 P₂ - interpolation error estimate and the error bound of the Simpson rule in numerical integration.

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