Fast computation of elliptic integrals and their generalizations

A method is developed for the fast and highly accurate computation of complete and incomplete elliptic integrals of the first, second, and third kinds, as well as their generalizations—integrals of the hypergeometric type. The method is based on the reduction of this problem to solving a second-order linear differential equation with polynomial coefficients. Using the expansions of the solution to this equation about its regular and singular points, linear recursions are obtained for the coefficients of these expansions, and the stability of their solutions is examined. The use of the Padé approximation resulted in a significant improvement of the convergence of these expansions. In the case when the singular points are close to each other, which causes strong numerical instability, symbolic transformations for calculating the coefficients in these expansions are used. This radically improves the stability of the method. The execution time of the algorithm is shown to be a linear function of the required accuracy.