{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $s \\ge 1$ be an integer.\nAn *$s$-stage Butcher tableau* is an array\n$$\\begin{array}{c|c} c \u0026 A \\\\ \\hline \u0026 b^{T} \\end{array} \\;=\\; \\begin{array}{c|ccc} c_1 \u0026 a_{11} \u0026 \\cdots \u0026 a_{1s} \\\\ \\vdots \u0026 \\vdots \u0026 \u0026 \\vdots \\\\ c_s \u0026 a_{s1} \u0026 \\cdots \u0026 a_{ss} \\\\ \\hline \u0026 b_1 \u0026 \\cdots \u0026 b_s \\end{array}$$\nwith real entries: a vector $c \\in \\R^s$, a matrix $A = (a_{ij}) \\in \\R^{s \\times s}$ and a vector $b \\in \\R^s$.\nIt specifies the $s$-stage [[Runge–Kutta method|7DBPJKYC6C]] for $y' = f(t, y)$ with step $h$:\n$$K_i = f\\Big(t_n + c_i h, \\; y_n + h \\sum_{j=1}^{s} a_{ij} K_j\\Big), \\quad i = 1, \\ldots, s, \\qquad y_{n+1} = y_n + h \\sum_{i=1}^{s} b_i K_i.$$\n\n## Remarks\n\nThe method is explicit when $a_{ij} = 0$ for $j \\ge i$, so the tableau is strictly lower triangular and blank entries are read as $0$.\nUsually $c_i = \\sum_j a_{ij}$ for each $i$.\nAn embedded pair adds a second row $\\hat b^{T}$ of weights that gives a method of another order from the same stages.\nExample: the classical fourth-order method has $c = (0, \\tfrac12, \\tfrac12, 1)^T$, $a_{21} = a_{32} = \\tfrac12$, $a_{43} = 1$, all other $a_{ij} = 0$, and $b = (\\tfrac16, \\tfrac13, \\tfrac13, \\tfrac16)^T$.\nCheck of two order conditions: $\\sum b_i = 1$ and $\\sum b_i c_i = \\frac16 + \\frac16 + \\frac16 = \\frac12$.\nNon-example: an array with $c \\in \\R^2$, $A \\in \\R^{2 \\times 2}$ and $b \\in \\R^3$ is not a Butcher tableau, since the number of weights must equal the number of stages.\n\n## Explanations\n\n### L1 human\n\nA Butcher tableau is a small table of numbers that encodes a Runge–Kutta method completely.\nThe left column says when in the step each stage is evaluated.\nThe square block says how earlier stages enter it, and the bottom row says how to average the stages.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "RH8BD7PW5F",
    "kind": "definition",
    "msc": [
      "65L06"
    ],
    "name": "Butcher tableau",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods"
      }
    ],
    "tag": "65H4W6TYKF",
    "uses": [
      "7DBPJKYC6C"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/65H4W6TYKF/",
  "links": [
    {
      "field": "defines",
      "target": "RH8BD7PW5F"
    },
    {
      "field": "uses",
      "target": "7DBPJKYC6C"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/65H4W6TYKF.md",
  "msc": [
    "65L06"
  ],
  "name": "Butcher tableau",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/65H/65H4W6TYKF.md",
  "status": null,
  "tag": "65H4W6TYKF",
  "uri": "m2://65H4W6TYKF",
  "url": "https://tamnd.github.io/math2/wiki/butcher-tableau-definition/",
  "writing": {
    "tag": "65H4W6TYKF",
    "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
      }
    ]
  }
}