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