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  "body": "\n## Definition\n\nLet $y \\in \\R^n$ be a quantity defined by a mathematical problem, such as a zero of a function or the value of an integral.\nA *validated* (or *verified*) computation of $y$ is an algorithm, run in floating-point arithmetic, with the following property.\nIt either reports failure or outputs a set $Y \\subseteq \\R^n$, usually a box of [[intervals|QPKKTE1ZY0]], such that $y \\in Y$ is guaranteed.\nThe guarantee must account for every [[round-off error|CPN6SD3BTV]] and every [[truncation error|4AKCQERM89]] of the computation.\n*Validated numerics* is the study and use of such algorithms.\n\n## Remarks\n\nExample: the interval Newton test.\nLet $f(x) = x^2 - 2$ and $X = [1.4, 1.5]$; with midpoint $m = 1.45$ and the enclosure $[2.8, 3.0]$ of $f'$ on $X$, the interval Newton step is $N(X) = m - f(m)/[2.8, 3.0] = 1.45 - 0.1025/[2.8, 3.0] \\subseteq [1.4133, 1.4159]$.\nSince $N(X) \\subseteq X$ and $0 \\notin [2.8, 3.0]$, the interval Newton theorem shows that $f$ has exactly one zero in $X$, and it lies in $N(X)$.\nDone with outward rounding, this proves $1.4133 \\le \\sqrt{2} \\le 1.4159$.\nNon-example: running Newton's method in floating point until two iterates agree to $15$ digits gives an excellent approximation, but no guarantee, so it is not a validated computation.\n",
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