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There are four recursive algorithms used in the computation of the fully normalized associated Legendre functions (FNALFs): the standard forward column algorithm, the standard forward row algorithm, the recursive algorithm between every other degree, and the Belikov algorithm. These algorithms were evaluated in terms of their first relative numerical accuracy, second relative numerical accuracy, and computation speed and efficiency. The results show that when the degree  n  reaches 3000, both the recursive algorithm between every other degree and the Belikov algorithm are applicable for | cos  θ  | ∈[0, 1], with the latter better second relative numerical accuracy than the former at a slower computation speed. In terms of | cos  θ  | ∈[0, 1], the standard forward column algorithm, the recursive algorithm between every other degree, and the Belikov algorithm are applicable within degree  n  of 1900, and the standard forward column algorithm has the highest computation speed. The standard forward column algorithm is applicable for | cos  θ  | ∈[0, 1] within degree  n  of 1900. This algorithm’s range of applicability decreases as the degree increases beyond 1900; however, it remains applicable within a minute range when | cos  θ  | is approximately equal to 1. The standard forward row algorithm has the smallest range of applicability: it is only applicable within degree  n  of 100 for | cos  θ  | ∈[0, 1], and its range of applicability decreases rapidly when the degree is greater than 100. The results of this research are expected to be useful to researchers in choosing the best algorithms for use in the computation of the FNALFs.

Evaluating Applicability of Four Recursive Algorithms for Computation of the Fully Normalized Associated Legendre Functions