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  "body": "\n## Definition\n\nLet $A^*$ be a number and let $A(h)$ be approximations of $A^*$, defined for small $h \u003e 0$.\nAssume that there are a constant $c$ and exponents $0 \u003c k \u003c k'$ such that\n$$A(h) = A^* + c\\, h^k + O(h^{k'}) \\quad (h \\to 0),$$\nin [[big O notation|0RYTGBJAGN]].\nFor a fixed ratio $t \u003e 1$, the *Richardson extrapolation* of $A$ is\n$$R(h) = \\frac{t^k A(h/t) - A(h)}{t^k - 1}.$$\nThen $R(h) = A^* + O(h^{k'})$.\n\n## Remarks\n\nThe last claim is a direct computation: in $t^k A(h/t) - A(h)$ the terms $t^k c (h/t)^k$ and $c h^k$ cancel, and $t^k A^* - A^* = (t^k - 1)A^*$.\nExample: for a function $f$ with five continuous derivatives, the central difference $A(h) = \\frac{f(x+h) - f(x-h)}{2h}$ satisfies $A(h) = f'(x) + \\frac{f'''(x)}{6} h^2 + O(h^4)$.\nWith $k = 2$ and $t = 2$, the extrapolation $\\frac{4A(h/2) - A(h)}{3}$ approximates the [[derivative|0F1WR9C83K]] $f'(x)$ with error $O(h^4)$.\nNon-example: if the true error is $A(h) = A^* + h$ but one uses $k = 2$, then $R(h) = A^* + \\frac{h}{3}$, which is still of first order; the exponent $k$ must be the true leading exponent.\nThe procedure can be repeated with the next exponent of the error expansion, which gives a triangular table of extrapolated values.\n\n## Explanations\n\n### L1 human\n\nCompute with two step sizes, and use the known shape of the error to remove its main part.\n",
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