{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1FMBCFJ7K1"
    },
    {
      "field": "uses",
      "target": "7R0F4956PX"
    },
    {
      "field": "uses",
      "target": "TVSVBH4TT4"
    },
    {
      "field": "specializes",
      "target": "FH0538XCZR"
    },
    {
      "field": "specializes",
      "target": "MNXC26G3BQ"
    },
    {
      "field": "about",
      "target": "TVSVBH4TT4"
    },
    {
      "field": "defines",
      "target": "W9ARPZGPM6"
    }
  ],
  "body": "\n## Summary\n\nThe Newton–Cotes formulas are the [[quadrature rules|ADSVMCY77A]] obtained by integrating the interpolating polynomial of the integrand at equally spaced nodes.\nThe closed formulas include the endpoints; the trapezoidal rule and Simpson's rule are the first two.\nThey are simple and exact for polynomials of the interpolation degree.\nHigh order formulas have negative weights and are unstable, so in practice low order formulas are applied on many small subintervals.\n\n## Explanations\n\n### L1 human\n\nSample the function at evenly spaced points, draw the polynomial curve through the samples, and take the area under that curve as the estimate.\nWith two points the curve is a line and the estimate is a trapezoid.\nWith three points the curve is a parabola, and the result is Simpson's rule with weights proportional to $1, 4, 1$.\nUsing many points at once does not work well, because polynomials through many equally spaced points can swing wildly.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65D32",
      "41A55"
    ],
    "name": "Newton–Cotes formulas",
    "same_as": {
      "wikidata": "Q944241",
      "wikipedia": "Newton–Cotes formulas"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas"
      }
    ],
    "specializes": [
      "ADSVMCY77A"
    ],
    "tag": "BQPQAY1FDJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/BQPQAY1FDJ/",
  "links": [
    {
      "field": "specializes",
      "target": "ADSVMCY77A"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BQPQAY1FDJ.md",
  "msc": [
    "65D32",
    "41A55"
  ],
  "name": "Newton–Cotes formulas",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BQP/BQPQAY1FDJ.md",
  "status": null,
  "tag": "BQPQAY1FDJ",
  "uri": "m2://BQPQAY1FDJ",
  "url": "https://tamnd.github.io/math2/wiki/newton-cotes-formulas/",
  "writing": {
    "tag": "BQPQAY1FDJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}