References

The most important routine in this package is the computation of the 𝔇 matrices — or more specifically, of terms proportional to parts of the 𝔇 matrices. This mostly follows the treatment of Gumerov and Duraiswami [8] (with minor modifications to account for errors in their presentation, as described here). To seed the recursions they present, we also need to calculate the associated Legendre functions. This is now done using the "fully normalized column-wise recurrence formula" (fnCWF) given by Eqs. (12)—(14) of Xing et al. [10]. This improves significantly over the older implementation using the "modified forward row method" of Holmes and Featherstone [12], for which the results would fail to be finite starting at ℓ=22 for Float16, ℓ=183 for Float32, and ℓ=1474 for Float64. Another approach that was never precisely implemented in this package was due to Fukushima [13], who showed that using "X-numbers", wherein the exponent is stored as a separate integer, (implemented in this package) in the core of the recursion could increase the range to ℓ≈2³². Xing et al. showed that Fukushima's results exhibited increased error for certain angles, whereas their Eqs. (12)—(14) could be used directly to obtain results with greater accuracy for those certain angles, and comparable accuracy for other angles.

The other major functionality of this package is map2salm / salm2map, which decomposes function values on regular grids into mode weights (coefficients), and vice versa. The approach used here is taken from Reinecke and Seljebotn [3], with weights based on the method by Waldvogel [6]. However, this interface has been superseded by the SSHT object, which implements several approaches, including the Reinecke-Seljebotn-Waldvogel method, as well as the optimal-dimensionality method due to Elahi et al. [2], as well as a new unpublished optimal-dimensionality method I (Mike Boyle) created.

Bibliography