Avoidance of numerical instabilities in computation of higher-order gravity

Abstract

HE calculation of higher-order gravity involves the solution of Laplace's equation in spherical coordinates. The solution requires the evaluation of associated Legendre functions. These functions can be computed explicitly or via recursion relationships. The use of the explicit relations necessitates excessive computer execution time or memory. On the other hand, uncritical use of standard recursion relationships, particularly on limited word-length machines, can lead to numerical instabilities near the poles. Alternative applications of recursion relationships are presented that avoid instabilities while retaining the advantages of minimal computer execution time and memory.