Comparison of computational methods of associated legendre functions

Abstract

Comparisons are made between two computational methods for associated Legendre functions across truncation wave numbers between 39 and 10239. One of the two methods uses the fourpoint recurrence in double precision (the Fourier method) and the other uses the three-point recurrence in the extended arithmetic (the X-number method). Both methods avoid the shortcomings of the traditional method using the three-point recurrence in double precision and generate values accurate enough to enable stable Legendre transforms at large truncation wave numbers ( > 1700). The errors for the Fourier method are found to be much smaller than those for the X-number method and have little latitudinal dependencies. The errors for the Fourier method, however, are found to grow rapidly with large degrees n > 2048. Two alternatives are proposed to calculate the scaling factor of the Fourier coefficients of the associated Legendre functions accurately with errors in O(√n ).