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  "body": "\n## Summary\n\nRomberg's method computes a definite integral by applying [[Richardson extrapolation|WMTZG71K2Y]] again and again to the [[trapezoidal rule|FH0538XCZR]] with step sizes $h, h/2, h/4, \\dots$.\nIt works because the error of the trapezoidal rule for a smooth function has an expansion in even powers of $h$.\nThis expansion comes from the [[Euler–Maclaurin formula|9P839VB8S7]].\nEach extrapolation step removes one more term of this expansion.\nThe results are arranged in a triangular table whose diagonal converges fast for smooth integrands.\n\n## Explanations\n\n### L1 human\n\nThe trapezoidal rule is crude, but its error shrinks in a very regular way when the step is halved.\nSince the leading error is about four times smaller after halving, a suitable combination of two results cancels it.\nRepeating this trick removes the next error term, then the next one.\nFor example, combining the trapezoidal values for $1$ and $2$ subintervals in this way gives exactly Simpson's rule.\n",
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