We improve the precision and computation speed of the fully-normalized associated Legendre functions (fnALFs) for ultra-high degrees and orders of spherical harmonic transforms. We take advantage of their numerical behaviour of and propose two new methods for solving an underflow/overflow problem in their calculation. We specifically discuss the application of the two methods in the fixed-order increasing-degree recursion computation technique. The first method uses successive ratios of fnALFs and the second method, called the Midway method, starts iteration from tiny initial values, which are still in the range of the IEEE double-precision environment, rather than from sectorial fnALFs. The underflow/overflow problem in the successive ratio method is handled by using a logarithm-based method and the extended range arithmetic. We validate both methods using numerical tests and compare their results with the X-number method in terms of precision, stability, and speed. The results show that the relative precision of the proposed methods is better than 10⁻⁹ for the maximum degree of 100000, compared to results derived by the high precision Wolfram’s Mathematica software. Average CPU times required for evaluation of fnALFs over different latitudes demonstrate that the two proposed methods are faster by about 10–30% and 20–90% with respect to the X-number method for the maximum degree in the range of 50–65000.

我们针对超高阶数和阶数的球谐变换提高了完全归一化的关联勒让德函数 (fnALF) 的精度和计算速度。我们利用它们的数值行为并提出了两种新方法来解决计算中的下溢/溢出问题。我们具体讨论了这两种方法在定阶递增递归计算技术中的应用。第一种方法使用 fnALF 的连续比率，第二种方法称为 Midway 方法，从微小的初始值开始迭代，这些初始值仍在 IEEE 双精度环境的范围内，而不是从扇形 fnALF。通过使用基于对数的方法和扩展范围算法来处理连续比率方法中的下溢/溢出问题。我们使用数值测试验证这两种方法，并在精度、稳定性和速度方面将其结果与 X 数方法进行比较。结果表明，所提方法的相对精度优于10⁻⁹表示最大度数 100000，与高精度 Wolfram 的 Mathematica 软件得出的结果相比。评估不同纬度上的 fnALF 所需的平均 CPU 时间表明，对于最大度数为 50-65000 的 X 数方法，所提出的两种方法的速度分别提高了约 10-30% 和 20-90%。