NUMERICAL APPROXIMATION OF THE FERMI-DIRAC INTEGRAL OF ORDER 1/2 BY MEANS OF COMPOSITE GAUSS-LEGENDRE QUADRATURE

摘要

We approximate the Fermi-Dirac integral ∫_0~∞√t/e~(t-x)+1 dt, by means of composite ten-point Gauss-Legendre quadrature, for values of x in the range x e [-100, 100]. For any x, the integral is approximated by composite quadrature on the interval 0 < t < 150, which is subdivided into a number of subintervals. We achieve a relative error of no more than 10~(-3) with 10 subintervals, and an error of no more than 10~(-14) with 89 subintervals. On our computational platform, the real-time duration of the computation is faster than two milliseconds for any x ∈ [-100, 100].