{
  "id": "2304.08136",
  "version": "v1",
  "published": "2023-04-17T10:40:42.000Z",
  "updated": "2023-04-17T10:40:42.000Z",
  "title": "A new second order Taylor-like theorem with an optimized reduced remainder",
  "authors": [
    "J. Chaskalovic",
    "F. Assous"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-17T10:40:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D30",
      "65N15",
      "65N30",
      "65N75",
      "41A05"
    ],
    "keywords": [
      "second order taylor-like theorem",
      "optimized reduced remainder",
      "interpolation error estimate",
      "linear combination",
      "numerical quadrature formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}