Innumerical analysis, theBulirsch–Stoer algorithm is a method for thenumerical solution of ordinary differential equations which combines three powerful ideas:Richardson extrapolation, the use ofrational function extrapolation in Richardson-type applications, and the modified midpoint method,[1] to obtain numerical solutions toordinary differential equations (ODEs) with high accuracy and comparatively little computational effort. It is named afterRoland Bulirsch andJosef Stoer. It is sometimes called theGragg–Bulirsch–Stoer (GBS) algorithm because of the importance of a result about the error function of the modified midpoint method, due toWilliam B. Gragg.
The idea of Richardson extrapolation is to consider a numerical calculation whose accuracy depends on the used stepsizeh as an (unknown)analytic function of the stepsizeh, performing the numerical calculation with various values ofh, fitting a (chosen) analytic function to the resulting points, and then evaluating the fitting function forh = 0, thus trying to approximate the result of the calculation with infinitely fine steps.
Bulirsch and Stoer recognized that usingrational functions as fitting functions for Richardson extrapolation in numerical integration is superior to usingpolynomial functions because rational functions are able to approximate functions with poles rather well (compared to polynomial functions), given that there are enough higher-power terms in the denominator to account for nearby poles. While a polynomial interpolation or extrapolation only yields good results if the nearest pole is rather far outside a circle around the known data points in the complex plane, rational function interpolation or extrapolation can have remarkable accuracy even in the presence of nearby poles.
The modified midpoint method by itself is a second-order method, and therefore generally inferior to fourth-order methods like thefourth-order Runge–Kutta method. However, it has the advantage of requiring only one derivative evaluation per substep (asymptotically for a large number of substeps), and, in addition, as discovered by Gragg, the error of a modified midpoint step of sizeH, consisting ofn substeps of sizeh =H/n each, and expressed as a power series inh, contains only even powers ofh. This makes the modified midpoint method extremely useful to the Bulirsch–Stoer method as the accuracy increases two orders at a time when the results of separate attempts to cross the intervalH with increasing numbers of substeps are combined.
Hairer, Nørsett & Wanner (1993, p. 228), in their discussion of the method, say that rational extrapolation in this case is nearly never an improvement over polynomial interpolation (Deuflhard 1983). Furthermore, the modified midpoint method is a modification of the regular midpoint method to make it more stable, but because of the extrapolation this does not really matter (Shampine & Baca 1983).