دانلود رایگان مقاله ارزیابی عددی بسیار دقیق انتگرال تعمیم یافته فرمی - دیراک

Abstract

We indicate a new and a very accurate algorithm for the evaluation of the Generalized Fermi–Dirac Integral with a relative error less than 10−20. The method involves Double Exponential, Trapezoidal and Gauss–Legendre quadratures. For the residue correction of the Gauss–Legendre scheme, a simple and precise continued fraction algorithm is used.

7. Conclusions

We have indicated two new algorithms for the evaluation of the GFDI. The first one provides the values of the GFDI with near double precision accuracy. Compared to its earlier variant, the computational effort is halved and this recipe is suitable for routine evaluation. The second algorithm provides the values of the GFDI with a relative error less than 10−20. As far as we know, this is the most accurate algorithm at present that can be used for reference purposes. Compared to the methods indicated in the literature, the present one cuts down the function evaluation roughly by a factor 10 which is a significant reduction. These methods are backed by rigorous error estimates [15,26,27,34]. In particular, the effect of the branch point of the integrand of the GFDI is dealt with thoroughly. These methods can be easily extended to the generalized Boltzmann integrals [13]. The Matlab and the Fortran routines that evaluate these integrals can be obtained from the first author upon request.