Recursive computation of finite difference of associated Legendre functions

The existing methods to compute the definite integral of associated Legendre function (ALF) with respect to the argument suffer from a loss of significant figures independently of the latitude. This is caused by the subtraction of similar quantities in the additional term of their recurrence formulas, especially the finite difference of their values between two endpoints of the integration interval. In order to resolve the problem, we develop a recursive algorithm to compute their finite difference. Also, we modify the algorithm to evaluate their definite integrals assuming that their values at one endpoint are known. We numerically confirm a significant increase in computing precision of the integral by the new method. When the interval is one arc minute, for example, the gain amounts to 2–4 digits for the degree of harmonics in the range 2 ≤ n ≤ 2,048. This improvement in precision is achieved at a negligible increase in CPU time, say less than 5%.