Fast algorithms for spherical harmonic expansions, III

TL;DR

This paper introduces a butterfly scheme-based algorithm to accelerate spherical harmonic transforms, emphasizing efficient precomputations through depth-first traversal, demonstrated with numerical examples.

Contribution

It presents a novel butterfly scheme approach for faster spherical harmonic transforms with an optimized precomputation strategy using depth-first traversal.

Findings

Significant speedup in spherical harmonic transform computations.

Precomputation management improves efficiency.

Numerical examples validate the algorithm's effectiveness.

Abstract

We accelerate the computation of spherical harmonic transforms, using what is known as the butterfly scheme. This provides a convenient alternative to the approach taken in the second paper from this series on "Fast algorithms for spherical harmonic expansions." The requisite precomputations become manageable when organized as a "depth-first traversal" of the program's control-flow graph, rather than as the perhaps more natural "breadth-first traversal" that processes one-by-one each level of the multilevel procedure. We illustrate the results via several numerical examples.