Richardson extrapolation

In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value ${\displaystyle A^{\ast }=\lim _{h\to 0}A(h)}$. In essence, given the value of ${\displaystyle A(h)}$ for several values of ${\displaystyle h}$, we can estimate ${\displaystyle A^{\ast }}$ by extrapolating the estimates to ${\displaystyle h=0}$. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century,^[1][2] though the idea was already known to Christiaan Huygens in his calculation of ${\displaystyle \pi }$.^[3] In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."^[4]

Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations.