Precise and Fast Computation of Elliptic Integrals and Elliptic Functions

Summarized is the recent progress of the new methods to compute Legendre's complete and incomplete elliptic integrals of all three kinds and Jacobian elliptic functions. Also reviewed are the entirely new methods to (i) compute the inverse functions of complete elliptic integrals, (ii) invert a general incomplete elliptic integral numerically, and (iii) evaluate the partial derivatives of the elliptic integrals and functions recursively. In order to avoid the information loss against small parameter and/or characteristic, newly introduced are the associate complete and incomplete elliptic integrals. The main techniques used are (i) the piecewise approximation for single variable functions, and (ii) a systematic utilization of the half and double argument transformations and the truncated Maclaurin series expansions for the others. The new methods are of the errors of 5 ulps at most without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) slightly faster than Bulirsch's procedure for the incomplete elliptic integral of the first kind, (ii) 1.5 times faster than Bulirsch's procedure for Jacobian elliptic functions, (iii) 2.5 times faster than Cody's and Bulirsch's procedures for the complete elliptic integrals, and (iv) 3.5 times faster than Carlson's procedures for the incomplete elliptic integrals of the second and third kind. Their Fortran programs are available at https://www.researchgate.net/profile/Toshio_Fukushima/.