Overcoming Instability in Evaluation of Generalized Hypergeometric Integrals in the Case of Crowding of Singular Points

摘要

A method for evaluating integral (1.1) with an arbitrarily high accuracy in the case of an arbitrary number of singular points has been developed. The method has linear complexity. It is based on the construction of linear recurrent equations for the coefficients of the solution expansion in different neighborhoods and on the numerical matching of the solutions at the points where the neighborhoods overlap. When the singular points are close to each other, one faces the cancellation phenomenon resulting in great loss of accuracy. The use of symbolic transformations in the computer algebra system Maple 9 makes it possible to overcome numerical instability and ensures effectiveness of the algorithm in the case of high-precision computations. The suggested method can also be used in many other numerically instable problems.