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In [[numerical analysis]], the '''Bulirsch–Stoer algorithm''' is a method for the [[numerical ordinary differential equations|numerical solution of ordinary differential equations]] which combines three powerful ideas, [[Richardson extrapolation]], the use of [[rational function extrapolation]] in Richardson-type applications, and the [[modified midpoint method]], to obtain numerical solutions to [[ordinary differential equation|ordinary differential equations]] (ODEs) with high accuracy and comparatively little computational effort. It is named after [[Roland Bulirsch]] and [[Josef Stoer]]. It is sometimes called the '''Gragg–Bulirsch–Stoer (GBS) algorithm''' because of the importance of a result about the error function of the modified midpoint method, due to [[William B. Gragg]].
==Underlying
The idea of Richardson extrapolation is to consider a numerical calculation whose accuracy depends on the used stepsize ''h'' as an (unknown) [[analytic function]] of the stepsize ''h'', performing the numerical calculation with various values of <math>h</math>, fitting a (chosen) analytic function to the resulting points, and then evaluating the fitting function for ''h'' = 0, thus trying to approximate the result of the calculation with infinitely fine steps.
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