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, though the idea was already known to Christiaan Huygens in his calculation of ${\displaystyle \pi }$. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."

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.