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Evaluation of Functions part 3

into equation (5.1.11), and then setting z = 1. Sometimes you will want to compute a function from a series representation even when the computation is not efficient. | 5.2 Evaluation ofContinued Fractions 169 into equation 5.1.11 and then setting z 1. Sometimes you will want to compute a function from a series representation even when the computation is not efficient. For example you may be using the values obtained to fit the function to an approximating form that you will use subsequently cf. 5.8 . If you are summing very large numbers of slowly convergent terms pay attention to roundoff errors In floating-point representation it is more accurate to sum a list of numbers in the order starting with the smallest one rather than starting with the largest one. It is even better to group terms pairwise then in pairs of pairs etc. so that all additions involve operands of comparable magnitude. CITED REFERENCES AND FURTHER READING Goodwin E.T. ed. 1961 Modern Computing Methods 2nd ed. New York Philosophical Library Chapter 13 van Wijngaarden s transformations . 1 Dahlquist G. and Bjorck A. 1974 Numerical Methods Englewood Cliffs NJ Prentice-Hall Chapter 3. Abramowitz M. and Stegun I.A. 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York 3.6. Mathews J. and Walker R.L. 1970 Mathematical Methods of Physics 2nd ed. Reading MA W.A. Benjamin Addison-Wesley 2.3. 2 5.2 Evaluation of Continued Fractions Continued fractions are often powerful ways of evaluating functions that occur in scientific applications. A continued fraction looks like this f x bo ai bi a2 5.2.1 b2 a3 3 a4 a5 4 Printers prefer to write this as f x bo ai bi ag as a4 a5 bg bs b4 b- 5.2.2 In either 5.2.1 or 5.2.2 the as and Vs can themselves be functions of x usually linear or quadratic monomials at worst i.e. constants times x or times x2 . For example the continued fraction representation of the tangent function is x tan x ---- 1 2 x2 2 x2 2 x2 Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 3 5 7 5.2.3 Continued .