Precise and fast computation of generalized Fermi–Dirac integral by parameter polynomial approximation

The generalized Fermi–Dirac integral, Fk(η, β), is approximated by a group of polynomials of β   as Fk(η,β)≈∑j=0JgjβjFk+j(η) where J=1(1)10J=1(1)10. Here Fk(η) is the Fermi-Dirac integral of order k while gj   are the numerical coefficients of the single and double precision minimax polynomial approximations of the generalization factor as 1+x/2≈∑j=0Jgjxj. If β is not so large, an appropriate combination of these approximations computes Fk(η, β) precisely when η is too small to apply the optimally-truncated Sommerfeld expansion (Fukushima, 2014 [15]). For example, a degree 8 single precision polynomial approximation guarantees the 24 bit accuracy of Fk(η, β  ) of the orders, k=−1/2(1)5/2,k=−1/2(1)5/2, when −∞<η≤8.92−∞<η≤8.92 and β ≤ 0.2113. Also, a degree 7 double precision polynomial approximation assures the 15 digit accuracy of Fk(η, β  ) of the same orders when −∞<η≤29.33−∞<η≤29.33 and 0≤β≤3.999×10−30≤β≤3.999×10−3. Thanks to the piecewise minimax rational approximations of Fk(η) (Fukushima, 2015 [18]), the averaged CPU time of the new method is roughly the same as that of single evaluation of the integrand of Fk(η, β). Since most of Fk(η) are commonly used in the approximation of Fk(η, β) of multiple contiguous orders, the simultaneous computation of Fk(η, β) of these orders is further accelerated by the factor 2–4. As a result, the new method runs 70–450 times faster than the direct numerical integration in practical applications requiring Fk(η, β).