Numerical stability of branched continued fraction expansions of Lauricella–Saran’s hypergeometric function F K ratios

Abstract The paper considers the numerical stability of the backward recurrence algorithm for computing approximants of branched continued fraction expansions for the Lauricella–Saran’s hypergeometric function F K ratios. For the first time, estimates of the relative errors of approximants computations are obtained, showing the dependence of the error of the approximant on the magnitude of the rounding errors of its elements and the values of the coefficients of the branched continued fractions. Also, for the above-mentioned approximants, sufficient conditions for the sets of numerical stability are established for the first time. Numerical experiments are conducted comparing the efficiency of the backward recurrence algorithm with the forward recurrence algorithm and the Lentz algorithm. It is shown that the backward recurrence algorithm provides high accuracy of computations even for a high order of approximants. Numerical experiments allowed us to evaluate the practical effectiveness of the proposed theoretical results.