{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\n*Numerical analysis* is the branch of mathematics that studies algorithms for computing approximate values of quantities defined by problems of continuous mathematics.\nIts objects are the following.\n\n1. A *problem*: a map $f$ from data $x$ to an exact answer $y = f(x)$, such as a root or an integral.\n2. A *numerical method*: a finite procedure, or a [[sequence|0G9EKM0E15]] of them, that computes an approximation $\\tilde y$ of $y$ from $x$ by arithmetic operations.\n3. An *error analysis*: a bound on the error $\\tilde y - y$, or a proof that the approximations have [[limit|05M5SWSZZD]] $y$, together with the cost of the method.\n\n## Remarks\n\nExample: the problem is to compute $\\sqrt{2}$, and the method is the iteration $x_{k+1} = (x_k + 2/x_k)/2$ from $x_0 = 1$.\nThe error analysis shows that $x_k \\to \\sqrt{2}$ and that the error is roughly squared at each step.\nNon-example: the symbolic simplification $\\int_0^1 2x \\, dx = 1$ by an antiderivative is exact computation, not an approximation, so it is not a numerical method in this sense.\nThe error of a method has two parts: the error of the method in exact arithmetic, and the error caused by rounding in a computer.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "B6E54YTR64",
    "kind": "definition",
    "msc": [
      "65-01",
      "65G50"
    ],
    "name": "Numerical analysis",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Numerical_analysis"
      }
    ],
    "tag": "J0NR3YRW07",
    "uses": [
      "0G9EKM0E15",
      "05M5SWSZZD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/J0NR3YRW07/",
  "links": [
    {
      "field": "defines",
      "target": "B6E54YTR64"
    },
    {
      "field": "uses",
      "target": "0G9EKM0E15"
    },
    {
      "field": "uses",
      "target": "05M5SWSZZD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/J0NR3YRW07.md",
  "msc": [
    "65-01",
    "65G50"
  ],
  "name": "Numerical analysis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J0N/J0NR3YRW07.md",
  "status": null,
  "tag": "J0NR3YRW07",
  "uri": "m2://J0NR3YRW07",
  "url": "https://tamnd.github.io/math2/wiki/numerical-analysis-definition/",
  "writing": {
    "tag": "J0NR3YRW07",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}