Geodesic
Algorithms for computing complete elliptic integrals and some related functions
Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 2, pp. 21-32

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We give new algorithms for computing the complete elliptic integrals of the first and second kinds and some related functions. The algorithms are constructed on the base of rapidly converging power series; the fixed sign of the series guarantees their good conditionality (stability with respect to rounding errors). The algorithms turned out to be flexible and easily adaptable to any specific demands of computing practice.
Keywords: complete elliptic integrals of the first and second kinds, toroidal functions, conical functions. 
@article{SJIM_2012_15_2_a2,
     author = {V. N. Belykh},
     title = {Algorithms for computing complete elliptic integrals and some related functions},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {21--32},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a2/}
}
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V. N. Belykh. Algorithms for computing complete elliptic integrals and some related functions. Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 2, pp. 21-32. http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a2/