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Fast computation of incomplete elliptic integral of first kind by half argument transformation. (English) Zbl 1201.65035
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, $R _{F}$.

65D20Computation of special functions, construction of tables
33E05Elliptic functions and integrals
33F05Numerical approximation and evaluation of special functions
NSWC; Mathematica
Full Text: DOI
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