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  "body": "\n## Definition\n\nLet $D \\subseteq \\R^n$ and let $f \\colon D \\to \\R^n$ be a [[continuous function|0X3WGX95FE]]; the case $n = 1$ is the most common, and $\\C$ may replace $\\R$.\nA *zero* (or *root*) of $f$ is a point $r \\in D$ with $f(r) = 0$.\nA *root-finding algorithm* is a procedure that takes $f$ and starting data.\nIt produces a [[sequence|04CC6K5VDT]] $(x_k)$ in $D$, or a sequence of nested sets containing a zero.\nIt uses $f$ only through values of $f$ and possibly of its derivatives.\nIt is designed so that $(x_k)$ has a zero of $f$ as its [[limit|05M5SWSZZD]], or so that the sets shrink to a zero.\n\n## Remarks\n\nExample: the iteration $x_{k+1} = x_k - (x_k^2 - 2)/(2x_k)$ from $x_0 = 1$ gives $1, 1.5, 1.41667, 1.414216, \\ldots$, which tends to the zero $\\sqrt{2}$ of $f(x) = x^2 - 2$.\nNon-example: the iteration $x_{k+1} = 2/x_k$ from $x_0 = 1$ oscillates between $1$ and $2$, so it does not tend to a zero of $x^2 - 2$, although $\\sqrt{2}$ is its fixed point.\nAn algorithm is *globally convergent* on a class of functions if it converges from every admissible start, and *locally convergent* if it converges from starts close enough to a zero.\nIts speed is measured by its order of convergence, see [[rate of convergence|YWZQC5E54B]].\n",
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