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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f$ be a continuous function on $[a, b]$.\nFor $n \\ge 0$ let $h_n = (b - a)/2^n$, and let $R(n, 0)$ be the composite [[trapezoidal rule|FH0538XCZR]] with $2^n$ subintervals:\n$$R(n, 0) = h_n \\Big( \\tfrac12 f(a) + \\sum_{k=1}^{2^n - 1} f(a + k h_n) + \\tfrac12 f(b) \\Big).$$\nFor $1 \\le m \\le n$ define\n$$R(n, m) = R(n, m - 1) + \\frac{R(n, m - 1) - R(n - 1, m - 1)}{4^m - 1}.$$\n*Romberg's method* approximates $\\int_a^b f(x)\\,dx$ by the entries of the triangular table $R(n, m)$, usually by the diagonal entry $R(n, n)$.\n\n## Remarks\n\nEach column is a [[Richardson extrapolation|WMTZG71K2Y]] of the previous one, which removes the term $h^{2m}$ from the error expansion.\nIf $f$ is smooth enough, the error of $R(n, m)$ is $O(h_n^{2m + 2})$ as $n \\to \\infty$.\nThe values $R(n, 0)$ can be computed recursively: $R(n, 0) = \\frac12 R(n - 1, 0) + h_n \\sum_{k=1}^{2^{n-1}} f(a + (2k - 1) h_n)$, so no function value is computed twice.\n\nExample: $f(x) = x^2$ on $[0, 1]$.\nThen $R(0, 0) = \\frac12 (0 + 1) = \\frac12$ and $R(1, 0) = \\frac12 \\big( \\frac12 \\cdot 0 + \\frac14 + \\frac12 \\cdot 1 \\big) = \\frac38$.\nSo $R(1, 1) = \\frac38 + \\frac{3/8 - 1/2}{3} = \\frac13$, which is the exact integral, as Simpson's rule is exact for quadratics.\n\nNon-example: replacing $4^m - 1$ by $2^m - 1$ gives a different table that does not cancel the even powers of $h$, so it is not Romberg's method.\n",
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