Richardson, Neville

I am indebted to my students and co-workers whose many contributions made this chapter possible. And it is a pleasure to acknowledge helpful comments from Geoffrey Bond, Noel Cant, Renee Diehl, Hans Niemantsverdriet, Gianfranco Pacchioni, and Neville Richardson. 

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. 

In principle, we could determine h and cr immediately from the slope and y intercept of the linear part of Fig. 1. In practice, one must use more sophisticated procedures in order to extrapolate the ratios to their asymptotic limit, since the actual Ri do not become precisely linear until i is very large. We have available [9] the expansion coefficients for through order 49, although the higher-order values suffer some from roundoff error. Using Neville-Richardson extrapolation, we have been able to establish [9,10] that a = 1/2, so that (j) represents a square-root branch point. 

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