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

摘要

The generalized Fermi-Dirac integral, F-k(eta, beta), is approximated by a group of polynomials of beta as F-k(eta, beta) approximate to Sigma(J)(j=0) g(j)beta F-j(k+j) where J = 1(1)10. Here F-k(eta) is the Fermi-Dirac integral of order k while gi are the numerical coefficients of the single and double precision minimax polynomial approximations of the generalization factor as root 1 + x/2 approximate to Sigma(J)(j=0) g(j)x(j) is not so beta large, an appropriate combination of these approximations computes Fk(r p) precisely when eta 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 F-k(eta, beta) of the orders, k = -1/2(1)5/2, when -infinity < < 8.92 and -infinity < 0.2113. Also, a degree 7 double precision polynomial approximation assures the 15 digit accuracy of F(tj, p) of the same orders when -infinity < s < 29.33 and 0 < beta < 3.999 x 10-3. Thanks to the piecewise,minimax rational approximations of Fk(n) (Fukushima, 2015 [181), the averaged CPU time of the new method is roughly the same as that of single evaluation of the integrand of Fk(n, /3). Since most of Ft(g) are commonly used in the approximation of F-k(eta, beta) of multiple contiguous orders, the simultaneous computation of F-k(eta, beta) 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 F-k(eta, beta). 2015 Elsevier Inc. All rights reserved.