New scheme for numerical computation of high degree Legendre polynomials

Spherical harmonics are widely used to represent the gravitational field of the Earth and other celestial bodies. These harmonics include Legendre’s polynomials. However, the numerical computation of these polynomials is a difficult issue, especially to a high degree. In the literature, several algebraic or numerical methods have been proposed to compute Legendre polynomials with acceptable precision within reasonable time, avoiding instabilities due to underflow or overflow problems. These methods have their advantages and disadvantages. For example, cannot be used for parallel computations. In this contribution, we are developing a new scheme for the numerical computation of Legendre polynomials. The problem of underflow/overflow is managed by using trigonometric identities and a technique based on successive ratios. The performance of the proposed scheme is demonstrated by several numerical experiments. In addition, the results are compared with other methods in terms of stability, precision, and speed. Finally, we suggest means for the generalization of the proposed algorithm in the numerical computation of the related Legendre functions for high degrees and orders.