Bulirsch-Stoer method

When the quadrature of eq 2 cannot be performed analytically the integration should be carried out numerically by robust routines such as the Runge-Kutta, Adams-Moulton predictor-corrector or Bulirsch-Stoer methods with step size and error control [53, 55, 56], These routines can also be found in computer codings at Netlib and in standard books on computer codes [53]. 

The extrapolation is performed using a rational function rather than a polynomial (Bulirsch-Stoer method rather than Neville). 

BDF methods. Use either J or tol or omit both. Bulirsch-Stoer method (for smoothly varying systems). RadauS method. Use any combination of J, M, and tol. Fourth-order Runge-Kutta method with fixed step. Fourth-order Runge-Kutta method with an adaptive step. Bulirsch-Stoer method (for stiff systems). 

The quasiclassical trajectory method was used to study this system, and the variable step size modified Bulirsch-Stoer algorithm was specially developed for recombination problems such as this one. Comparisons were made with the fourth order Adams-Bashforth-Moulton predictor-corrector algorithm, and the modified Bulirsch-Stoer method was always more efficient, with the relative efficiency of the Bulirsch-Stoer method increasing as the desired accuracy increased. We measure the accuracy by computing the rms relative difference between the initial coordinates and momenta and their back-integrated values. For example, for a rms relative difference of 0.01, the ratio of the CPU times for the two methods was 1.6, for a rms relative difference of 0.001 it was 2.0, and for a rms relative difference of 10 it was 3.3. Another advantage of the variable step size method is that the errors in individual trajectories are more similar, e.g. a test run of ten trajectories yielded rms errors which differed by a factor of 53 when using the modified Bulirsch-Stoer... 

Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Richardson extrapolation and the Bulirsch-Stoer method. In Numerical recipes in FORTRAN the art of scientific computing. Cambridge University Press, Cambridge, 2nd edn., pp 718-725... 

It is useliil to multiply the estimation by a safety coefficient (i.e., 5). If extrapolation techniques are adopted, it is possible to estimate the error by calculating the difference between the predictions obtained with the Bulirsch-Stoer algorithm (or Neville, alias the Romberg method) using different support points, and hence different T... 

The algorithm usually adopted to prepare the points for the extrapolation, obtained by varying h, is a variant of the central point method proposed by Gragg (Stoer and BuUrsch, 1983), while the extrapolation for -> 0 is performed either with the Richardson method applied to polynomials or the Bulirsch-Stoer (Stoer and BuUrsch, 1983) method applied to rational functions. 

Once the value of is calculated for different Hn, an extrapolation for A = 0 is performed. For the extrapolation, we can use a polynomial approximation with the Neville algorithm, which is equivalent to the Richardson extrapolation for this specific case, or better still, a rational function using the Bulirsch-Stoer algorithm. The convergence of the method can be assessed by comparing the two different values of the extrapolation. 

If an inverse interpolation method is used and the Bulirsch-Stoer algorithm in particular is applied to rational functions, the differences between the predictions of different rational functions can be calculated in order to check whether they tend to zero as well as to estimate the error arising in using the differences between the various previsions. 

In spite of the modifications introduced by Bulirsch and Stoer, the method derived from the extended trapezoid rule suffers from several shortcomings that make it less efficient than other methods described later, when implemented in a general integration program. 

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