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  "body": "\n## Definition\n\nLet $X$ be a set, let $f \\colon X \\to X$ be a map, and let $x_0 \\in X$.\nThe *fixed-point iteration* of $f$ starting at $x_0$ is the [[sequence|0QE8STGJH2]] $(x_n)_{n \\ge 0}$ defined by\n$$x_{n+1} = f(x_n) \\quad \\text{for } n \\ge 0.$$\nA *fixed point* of $f$ is a point $x^* \\in X$ with $f(x^*) = x^*$.\n\n## Remarks\n\nIf $X$ is a metric space, $f$ is [[continuous|0P27W9RW0G]], and the iteration [[converges|06JYPJ4WHH]] to $x^*$, then $x^*$ is a fixed point, since $f(x^*) = \\lim f(x_n) = \\lim x_{n+1} = x^*$.\nExample: for $f(x) = x/2 + 1/x$ on $(0, \\infty)$ and $x_0 = 1$, the iterates are $1, 1.5, 1.41\\overline{6}, \\ldots$, and they converge to $\\sqrt{2}$, the fixed point, since $x = x/2 + 1/x$ means $x^2 = 2$.\nNon-example of convergence: for $f(x) = 2x$ on $\\R$ and $x_0 = 1$, the iterates $2^n$ diverge, although $0$ is a fixed point.\nThe [[Banach fixed point theorem|0VK6VEH345]] says the iteration converges to the unique fixed point when $f$ is a [[contraction|0QN69YA357]] of a nonempty complete metric space.\nIf $q \u003c 1$ is the contraction constant, the error after $n$ steps is at most $q^n d(x_1, x_0) / (1 - q)$.\nThe sequence is also written $x_n = f^n(x_0)$, the $n$-th iterate of $f$ applied to $x_0$.\n",
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