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Zbl 1201.65035
Fukushima, Toshio
Fast computation of incomplete elliptic integral of first kind by half argument transformation. (English)
[J] Numer. Math. 116, No. 4, 687-719 (2010). ISSN 0029-599X; ISSN 0945-3245/e

Summary: We develop a new method to calculate the incomplete elliptic integral of the first kind, ${F(\varphi|m)}$, by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of ${\varphi}$ by repeated usage of the formulas while fixing $m$. The method is sufficiently precise in the sense that the maximum relative error is 3--5 machine epsilons at most. Thanks to the simplicity of the half argument formulas, the new procedure is significantly faster than the existing procedures. For example, it runs 20--60\% faster than Bulirsch' function, el1, and 1.9--2.2 times faster than the method using Carlson's function, $R _{F}$.

MSC 2000:: *65D20 Computation of special functions
33E05 Elliptic functions and integrals
33F05 Numerical approximation of special functions

Keywords: algorithm; numerical examples; incomplete elliptic integral; half argument formulas; Jacobian elliptic functions; Bulirsch' function; Carlson's function

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