{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "AE08JXN7CA"
    }
  ],
  "body": "\n## Summary\n\nBirkhoff interpolation asks for a polynomial whose derivatives of chosen orders take chosen values at chosen nodes.\nUnlike Hermite interpolation, it does not require that all lower derivatives are also prescribed.\nIt contains [[polynomial interpolation|SS1YMS2NPC]] and [[Hermite interpolation|9Z50M2WPTP]] as special cases.\nUnlike these, it does not always have a unique solution, even when the number of conditions equals the number of coefficients.\nThe main question of the theory is which patterns of conditions, encoded in an incidence matrix, are poised, that is, solvable uniquely for all data.\n\n## Explanations\n\n### L1 human\n\nSuppose you know the value of a quadratic at $-1$ and at $1$ and its slope at $0$.\nThat is three conditions for three coefficients, so one might expect exactly one answer.\nBut for $p(x) = c_0 + c_1 x + c_2 x^2$ the slope at $0$ is $c_1$, and the two values only fix $c_0 + c_2$ and $c_1$.\nSo the slope is either forced or contradicted, and $c_0$ and $c_2$ are not fixed separately.\nBirkhoff interpolation studies when such gaps in the conditions are harmless.\nThe [[definition|AE08JXN7CA]] sets up the incidence matrix.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "41A05",
      "65D05"
    ],
    "name": "Birkhoff interpolation",
    "same_as": {
      "wikidata": "Q3800523",
      "wikipedia": "Birkhoff interpolation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Birkhoff_interpolation"
      }
    ],
    "tag": "FT1YKVBFC9"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/FT1YKVBFC9/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/FT1YKVBFC9.md",
  "msc": [
    "41A05",
    "65D05"
  ],
  "name": "Birkhoff interpolation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FT1/FT1YKVBFC9.md",
  "status": null,
  "tag": "FT1YKVBFC9",
  "uri": "m2://FT1YKVBFC9",
  "url": "https://tamnd.github.io/math2/wiki/birkhoff-interpolation/",
  "writing": {
    "tag": "FT1YKVBFC9",
    "kind": "concept",
    "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
      }
    ]
  }
}