On the Adaptive Quadrature of Fermi-Dirac Functions and their Derivatives
Keywords:Fermi-Dirac Integrals, Partial Derivatives, QUADPACK Adaptive Quadrature, Optimised Break Points
In this paper, using the Python SciPy module “quad”, a fast auto-adaptive quadrature solver based on the pre-compiled QUADPACK Fortran package, computational research is undertaken to accurately integrate the generalised Fermi-Dirac function and all its partial derivatives up to the third order. The numerical results obtained with quad method when combined with optimised break points achieve an excellent accuracy comparable to that obtained by other publications using fixed-order quadratures.
 R.B. Larson, and P.R. Demarque, An application of Henyey's approach to the integration of the equations of stellar structure, The Astrophysical Journal, 140, 524, 1964.
 D.D. Clayton, Principles of stellar evolution and nucleosynthesis, McGraw-Hill, 1968
 J.P. Cox and R.T. Giuli, 1968, Principles of stellar structure, Gordon & Breach, 1968.
 L.D. Cloutman, Numerical evaluation of FDIs, Ap. J. Supp., 7, 677-699, 1989.
 F.X.Timmes, http:// cococubed.asu.edu/ code_pages/ fermi_dirac.shtml
 F.X.Timmes, http://cococubed.asu.edu/code_pages/eos.shtml[
 Z. Gong etal., “Generalized FD Functions and Derivatives: Properties and Evaluation”, arXiv:astro-ph/0102329v1.20Feb.2001.
 https:// github.com /scipy/scipy/ tree/master/scipy/ integrate/ quadpack
 J.D. Cook, https://www.johndcook.com/blog/2012/ 03/20/scipy-integration/