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Mathematics of Computation

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Precise and fast computation of the general complete elliptic integral of the second kind


Author: Toshio Fukushima
Journal: Math. Comp. 80 (2011), 1725-1743
MSC (2010): Primary 33E05
DOI: https://doi.org/10.1090/S0025-5718-2011-02455-5
Published electronically: May 27, 2011
Previous version: Originally posted February 1, 2011
Current version: Corrects copyright year
MathSciNet review: 2785476
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Abstract | References | Similar Articles | Additional Information

Abstract: We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind $ B(m)$ and $ D(m)$ by using their Taylor series expansions, the definition of Jacobi's nome, and Legendre's relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch's cel2 and 16 times faster than Carlson's $ R_F$ and $ R_D$.


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Additional Information

Toshio Fukushima
Affiliation: National Astronomical Observatory of Japan, 2-21-1, Ohsawa, Mitaka, Tokyo 181-8588, Japan
Email: Toshio.Fukushima@nao.ac.jp

DOI: https://doi.org/10.1090/S0025-5718-2011-02455-5
Keywords: Elliptic integral
Received by editor(s): February 10, 2010
Received by editor(s) in revised form: April 22, 2010
Published electronically: May 27, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.