numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

, the Bulirsch–Stoer algorithm is a method for the numerical solution of ordinary differential equations which combines three powerful ideas:

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 A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, we ...

, the use of

rational function extrapolation Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...

in Richardson-type applications, and the modified midpoint method, to obtain numerical solutions to

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s (ODEs) with high accuracy and comparatively little computational effort. It is named after

Roland Bulirsch Roland Zdeněk Bulirsch (10 November 1932 – 21 September 2022) was a German mathematician specialising in numerical analysis. He studied and taught at the Technical University of Munich, and taught internationally as visiting professor. He was ...

and

Josef Stoer Josef Stoer (born 21 June 1934) is a German mathematician specializing in numerical analysis and professor emeritus of the Institut für Mathematik of Universität Würzburg. He was born in Meschede, and earned his Ph.D. in 1961 at Johannes Gute ...

. 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 ideas

The idea of Richardson extrapolation is to consider a numerical calculation whose accuracy depends on the used stepsize ''h'' as an (unknown)

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

of the stepsize ''h'', performing the numerical calculation with various values of ''h'', 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. Bulirsch and Stoer recognized that using rational functions as fitting functions for Richardson extrapolation in numerical integration is superior to using polynomial 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 the fourth-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 size ''H'', consisting of ''n'' substeps of size ''h'' = ''H''/''n'' each, and expressed as a power series in ''h'', contains only even powers of ''h''. 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 interval ''H'' with increasing numbers of substeps are combined. , in their discussion of the method, say that rational extrapolation in this case is nearly never an improvement over polynomial interpolation . 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 .

References

* . * . * * .

External links

ODEX.F
implementation of the Bulirsch–Stoer algorithm by Ernst Hairer and Gerhard Wanner (for other routines and license conditions, see thei

page).
BOOST library
implementation in C++.

implementation in Java. {{DEFAULTSORT:Bulirsch-Stoer algorithm Numerical integration (quadrature)