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Abstract

Global modeling of gravity and magnetic field typically involves the use of spherical harmonics. The expression of potential includes the corresponding Legendre functions. However, the numerical computation of these functions is difficult at higher degrees and calls for considerable attention. While reviewing the literature, it has been noted that standard textbooks on spherical harmonics provide several algebraic or numerical methods for computing associated Legendre functions with acceptable precision within a reasonable time, avoiding instabilities due to underflow or overflow problems. These methods have their advantages and disadvantages. In this contribution, we are developing a new scheme for the numerical computation of associated Legendre functions, at least not in the way they have been traditionally used in geodesy and geophysics. The problem of underflow/overflow is treated by the use of trigonometric identities and a technique based on successive relationships distinguishes four different cases. The performance and limitations of the proposed scheme are demonstrated through multiple numerical experiments. Finally, the results are compared with other methods as regards stability, precision, and speed.