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Evaluation of Functions part 15

Figure 5.13.1 shows the discrepancies for the first five iterations of ratlsq when it is applied to find the m = k = 4 rational fit to the function f (x) = cos x/(1 + ex ) in the interval (0, π). One sees that after the first iteration, the results are virtually as good as the minimax solution. The iterations do not converge in the order that the figure suggests: In fact, it is the second iteration that is best (has smallest maximum deviation). The routine ratlsq accordingly returns the best of its iterations, not necessarily the last one;. | 208 Chapter5. Evaluation ofFunctions Figure 5.13.1 shows the discrepancies for the first five iterations of ratlsq when it is applied to find the m k 4 rational fit to the function f x cos x 1 ex in the interval 0 w . One sees that after the first iteration the results are virtually as good as the minimax solution. The iterations do not converge in the order that the figure suggests In fact it is the second iteration that is best has smallest maximum deviation . The routine ratlsq accordingly returns the best of its iterations not necessarily the last one there is no advantage in doing more than five iterations. CITED REFERENCES AND FURTHER READING Ralston A. and Wilf H.S. 1960 Mathematical Methods for Digital Computers New York Wiley Chapter 13. 1 5.14 Evaluation of Functions by Path Integration In computer programming the technique of choice is not necessarily the most efficient or elegant or fastest executing one. Instead it may be the one that is quick to implement general and easy to check. One sometimes needs only a few or a few thousand evaluations of a special function perhaps a complex valued function of a complex variable that has many different parameters or asymptotic regimes or both. Use of the usual tricks series continued fractions rational function approximations recurrence relations and so forth may result in a patchwork program with tests and branches to different formulas. While such a program may be highly efficient in execution it is often not the shortest way to the answer from a standing start. A different technique of considerable generality is direct integration of a function s defining differential equation - an ab initio integration for each desired function value along a path in the complex plane if necessary. While this may at first seem like swatting a fly with a golden brick it turns out that when you already have the brick and the fly is asleep right under it all you have to do is let it fall As a specific example let us consider the