Richardson Extrapolation

Updated 15 Sep 2026

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What it is
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  a technique that combines two central-difference estimates at different step sizes ( $h$ and $h/2$ ) to cancel out the leading truncation-error term, improving accuracy from $O(h^2)$ to $O(h^4)$
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Formula
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  $D_{extrap} = \dfrac{4D(h/2) - D(h)}{3}$
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Worked example: $f(x)=e^x$ , estimating $f'(0)=1.0$
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  $D(0.2) = 1.006680$ , $D(0.1) = 1.001668$ (error roughly quartered when $h$ is halved, consistent with $O(h^2)$ )
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  $D_{extrap} = 0.9999973$ — error drops to about $2.7\times10^{-6}$ , over 600× more accurate than either individual estimate
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Related
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  Numerical Differentiation

Linked references 2