Closure to 'Efficient Algorithm for Computing Einstein Integrals' by Junke Guo and Pierre Y. Julien

摘要

We would like to thank all the discussers for their constructive comments and for providing alternative approximations of the Einstein integrals. It is commonly agreed that these two integrals remain elusive to exact solutions. However, a lot of progress has been made in developing fast and accurate approximations. Abad and Garcia emphasize the need for simpler algorithms and propose a polynomial approximation. Srivastava provides keen insight into the convergence of the series and offers improvements. Roland et al. present an error analysis as well as a valuable comparison of the computational time of various algorithms. Besides polynomial approximations and different series expansions, other e-mail communications by A. R. Kacimov pointed to the possible use of hypergeometric series as well as software packages like MatLab, Maple, and Mathematica. Four main issues are raised in the discussions and they are addressed in the following sequence: (1) convergence; (2) algorithm efficiency; (3) accuracy; and (4) computational time. This closure also includes a comparative analysis of the different algorithms and an application example on the Missouri River.