Numerical Computation of Ultra-High-Degree Legendre Function

The computations of an ultra-high degree associated Legendre functions and its first derivative up to degree and order of 10800 are reported. Not only the magnitude of orders for the ultra-high degree calculation is presented but the numerical stability and accuracy of the computed values are described in detail. The accuracy on the order of $10^{-25}\;and\;10^{-15}$ was obtained for the values of Legendre function and the first derivatives of Legendre functions, respectively. The computable highest degree and order of Legendre function in terms of latitudes and the linear relationship between the magnitude of the function with respect to degrees and orders is found. It is expected that the computed Legendre functions contribute in many geodetic and geophysical applications for simulations as well as theoretical verifications.