Richardson extrapolation

Combining two approximations computed at different step sizes to cancel their shared leading error term, producing a new approximation accurate to a higher order.

Richardson extrapolation combines an approximation $A(h)$ known to have error of the pure form $Ch^p + O(h^{p+1})$ with $A(h/2)$ , via $\dfrac{2^pA(h/2)-A(h)}{2^p-1}$ , to cancel the $Ch^p$ term exactly and leave an error of order $h^{p+1}$ or smaller. Applied to the central-difference formula ( $p=2$ ), this is $\dfrac{4D(h/2)-D(h)}{3}$ .

Examples

Romberg integration is Richardson extrapolation applied over and over to the composite trapezoidal rule, each pass buying two more orders of accuracy.