Precise and fast computation of Fermi–Dirac integral of integer and half integer order by piecewise minimax rational approximation

Abstract

Piecewise quadruple precision approximations of the Fermi–Dirac integral of integer order, 0(1)10, and half integer order, -9/2(1)21/2, are developed by combining (i) the optimally truncated Sommerfeld expansion, (ii) the piecewise truncated Chebyshev series expansion, and/or (iii) the reflection formula. They are used in constructing the double precision piecewise minimax rational approximations of the integral of the same orders. The relative errors of the new minimax approximations, which are all due to rounding off, are 3–13 machine epsilons at most and less than 4 machine epsilons typically while their CPU times are only 0.44–1.1 times that of the exponential function when -5⩽η⩽45. As a result, the new approximations run 16–31 times faster than the piecewise Chebyshev polynomial approximations (Macleod, 1998) for the physically important orders, -1/2(1)5/2, and 330–720 and 50–93 times faster than the combined series expansions (Goano, 1995) for the half integer orders, -1/2(1)21/2, and the integer orders, 1(1)9, respectively. A file of the Fortran codes of the obtained approximations and their test program and sample output is named xfdh.txt and located at: https://www.researchgate.net/profile/Toshio_Fukushima/.