A Simple and Accurate Method for the Calculation of Generalized Fermi Functions

Abstract

We present a simple method for the calculation of generalized Fermi functions. It is based on a straight integration using quadratures, with an adequate interval subdivision and variable choice. The result is a fast and accurate algorithm, which is compared with other existing methods in accuracy, versatility, and speed.

Generalized Fermi–Dirac functions and derivatives: properties and evaluation

“…The positions of the break points are optimized such that the combined errors from all the pieces contributing to the generalized FD functions are minimized and equally distributed among the pieces. Fortunately, the choice of break points turns out rather uncritical for each individual derivative [see also [14]]. …”

“…For the generalized FD functions we compare the results of [14] (the method adopted in the present paper) with those of [13] evaluated in double precision. We find that they agree with each other to 14-digit accuracy for η up to 1000, and to 10-digit accuracy for η up to 10000.…”

“…We propose an accurate and relatively fast numerical method to evaluate the derivatives of the generalized FD functions, following the algorithm introduced by [14]. Limitations of direct numerical differentiation are also discussed.…”

“…The numerical method to calculate derivatives of the generalized FD functions up to the third order is introduced in Sect. 3, based on the scheme of [14]. At the end we check the efficiency and validity of some popularly used analytic approximation formulae in several extreme cases in Sect.…”

“…New recursion relations of the generalized Fermi-Dirac functions have been found. An effective numerical method to evaluate the derivatives of the generalized Fermi-Dirac functions up to third order with respect to both degeneracy and temperature is then proposed, following Aparicio [14]. A Fortran program based on this method, together with a sample test case, is provided.…”