Romberg Algorithm

Quantitative estimation of /q (and of the following ones) is performed using the Romberg algorithm in Mathcad software with accuracy parameter Tol=10. Iq values are listed in Table 8.1. 

In actual calculations four values of the external field have been applied Ej = — 0.002, — 0.001, 0.001, and 0.002 au.< > The numerical stability of the result was checked by the Richardson extrapolation procedure coupled with the Romberg algorithm using the above four values of E. In this way the values obtained for a can be improved, and one can also detect numerical instabilities which in most cases are caused by insufficiently accurate calculated values of iif (E ). 

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... 

To have the same number of elements with the Romberg method, we need 33 function calculations (extremes included). If we use the Bulirsch series, the number of points is, again, 13, but the more precise integral is obtained with step /z/8, with respect toh/Vl using this algorithm. 

This assumes that the panel width is changed by a factor of 2 in the two calculations. This is easily implemented as previously discussed with the trapezoidal algorithm as all the previously summed function values can be reused as the panel size is reduced at each iterative step by a factor of 2. The use of Richardson s extrapolation when applied to integration is known in the literature as Romberg integration. 

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