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We describe a method for the numerical evaluation of normalized versions ofthe associated Legendre functions $P_\nu^{-\mu}$ and $Q_\nu^{-\mu}$ of degrees$0 \leq \nu \leq 1,000,000$ and orders $-\nu \leq \mu \leq \nu$ on the interval$(-1,1)$. Our algorithm, which runs in time independent of $\nu$ and $\mu$, isbased on the fact that while the associated Legendre functions themselves areextremely expensive to represent via polynomial expansions, the logarithms ofcertain solutions of the differential equation defining them are not. Weexploit this by numerically precomputing the logarithms of carefully chosensolutions of the associated Legendre differential equation and representingthem via piecewise trivariate Chebyshev expansions. These precomputedexpansions, which allow for the rapid evaluation of the associated Legendrefunctions over a large swath of parameter domain mentioned above, aresupplemented with asymptotic and series expansions in order to cover itentirely. The results of numerical experiments demonstrating the efficacy ofour approach are presented, and our code for evaluating the associated Legendrefunctions is publicly available.

An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order