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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(ϕ|m), by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of ϕ 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:
65D20Computation of special functions, construction of tables
33E05Elliptic functions and integrals
33F05Numerical approximation and evaluation of special functions
Software:
Mathematica
References:
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