Evaluation of Methods for Spherical Harmonic Synthesis of the Gravitational Potential and its Gradients

This work is concerned with the comparison of four of the best-known methods for the computation of the gravitational potential and its gradients: the traditional formulation in terms of Associated Legendre Functions in spherical coordinates; the non-singular method of Pines; the algorithm developed by Cunningham and extended by Metris and collaborators; and a variant of the first method based on the Clenshaw summation formula. Extensive numerical tests in double and quadruple floating point precision have been performed in order to assess and compare the efficiency and precision of these algorithms. Results show that when properly optimized the algorithm of Clenshaw is the most efficient, closely followed by the traditional Legendre formulation. All four methods are characterized by a high level of precision, although care should be taken when approaching the geographic poles due to the singularities which affect the methods of Legendre and Clenshaw. The methods of Cunningham Metris and Pines are both characterized by some loss of relative precision at the equator, which is inherent in the choice of the coordinate system.