{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $N \\ge 1$ and let $x_k = \\cos(k\\pi/N)$ for $k = 0, 1, \\dots, N$.\nFor a continuous function $f$ on $[-1, 1]$, let $p_N$ be the interpolating polynomial of degree at most $N$ with $p_N(x_k) = f(x_k)$ for all $k$.\nThe $(N + 1)$-point *Clenshaw–Curtis quadrature* rule is the [[quadrature rule|ADSVMCY77A]]\n$$\\int_{-1}^{1} f(x)\\,dx \\approx \\int_{-1}^{1} p_N(x)\\,dx = \\sum_{k=0}^{N} w_k f(x_k),$$\nwhere $w_k = \\int_{-1}^{1} \\ell_k(x)\\,dx$ and $\\ell_k$ is the Lagrange basis polynomial of the node $x_k$.\n\n## Remarks\n\nThe nodes are the extreme points of the [[Chebyshev polynomial|M6EAR1JHPP]] $T_N$ on $[-1, 1]$.\nWriting $p_N(\\cos\\theta) = \\sum_{j=0}^{N} c_j \\cos(j\\theta)$ gives $\\int_{-1}^{1} p_N = \\sum_j c_j \\int_0^\\pi \\cos(j\\theta)\\sin\\theta\\,d\\theta$, and $\\int_0^\\pi \\cos(j\\theta)\\sin\\theta\\,d\\theta$ is $2/(1 - j^2)$ for even $j$ and $0$ for odd $j$.\nThe rule is exact for polynomials of degree at most $N$, and of degree at most $N + 1$ when $N$ is even, by symmetry.\n\nExample: $N = 2$.\nThe nodes are $1, 0, -1$, and the interpolatory weights are $\\frac13, \\frac43, \\frac13$.\nThis is Simpson's rule; for $f = x^2$ it gives $\\frac13 + 0 + \\frac13 = \\frac23 = \\int_{-1}^{1} x^2\\,dx$.\n\nNon-example: the interpolatory rule at the equally spaced points $-1, -\\frac13, \\frac13, 1$ is a Newton–Cotes rule, not a Clenshaw–Curtis rule, because its nodes are not of the form $\\cos(k\\pi/3)$, which are $1, \\frac12, -\\frac12, -1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VEB7J3C74F",
    "kind": "definition",
    "msc": [
      "65D32",
      "41A55"
    ],
    "name": "Clenshaw–Curtis quadrature",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Clenshaw–Curtis_quadrature"
      }
    ],
    "tag": "YVDWN0KVG4",
    "uses": [
      "ADSVMCY77A",
      "M6EAR1JHPP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YVDWN0KVG4/",
  "links": [
    {
      "field": "defines",
      "target": "VEB7J3C74F"
    },
    {
      "field": "uses",
      "target": "ADSVMCY77A"
    },
    {
      "field": "uses",
      "target": "M6EAR1JHPP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YVDWN0KVG4.md",
  "msc": [
    "65D32",
    "41A55"
  ],
  "name": "Clenshaw–Curtis quadrature",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YVD/YVDWN0KVG4.md",
  "status": null,
  "tag": "YVDWN0KVG4",
  "uri": "m2://YVDWN0KVG4",
  "url": "https://tamnd.github.io/math2/wiki/clenshaw-curtis-quadrature-definition/",
  "writing": {
    "tag": "YVDWN0KVG4",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}