A fast and accurate DE-formula algorithm to evaluate Ambartsumian-Chandrarasekhar $H$-function for isotropic scattering

Abstract

In the present work, a compact, fast, and yet accurate algorithm is developed to calculate the numerical values of the Ambartsumian-Chandrasekhar’s \(H\)-function for isotropic scattering and its moments on the basis of the double exponential (DE) formula of Takahashi and Mori (RIMS, Kyoto Univ., 9:721, 1974). The main improvement made in the new method is an elimination of the iterative procedure for automatic adjustment of the step-size of integrations carried out with the DE-formula. Instead, a set of optimal values for the upper limit of integration \(T_{\text{max}}\) and the number of division points \(N_{\text{T}}\) to specify the step-size of the quadrature is predetermined for calculations of the \(H\)-function with a 15-digit accuracy, and also another for the evaluations of the moments with an accuracy of 14-digits or better. FORTRAN90 subroutines HFISCA for the \(H\)-function and HFMOMENT for the moments of arbitrary degrees are subsequently constructed (their source codes and a driver together with a sample set of output are shown in Appendix of this paper). Tables of sample calculations of the \(H\)-function and its moments of degree −1 through 6 carried out by these programs are also presented. The routines HFISCA and HFMOMENT should prove useful not only in astrophysical applications but also in other disciplines of science such as the electron transports in condensed matter and remote-sensing data analyses. A request for a copy of the Fortran 90 source code of the program can be made by writing to kawabata@rs.kagu.tus.ac.jp.