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 four-point 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).
2015 by the Meteorological Society of Japan