A faster multipole Legendre-Chebyshev transform

TL;DR

This paper introduces a fast, efficient algorithm based on the fast multipole method for transforming Legendre and Chebyshev coefficients, significantly improving speed and simplicity over previous methods.

Contribution

It presents a novel modal Galerkin approach with Chebyshev basis functions and a faster spreading technique, reducing computation time by about 20% and initialization time by fivefold.

Findings

Approximately 20% faster execution due to fewer floating point operations

Initialization algorithm is roughly 5 times faster for large arrays

Transforms 1,000,000 coefficients in about 20 milliseconds on a MacBook Pro M3

Abstract

This paper describes a fast algorithm for transforming Legendre coefficients into Chebyshev coefficients, and vice versa. The algorithm is based on the fast multipole method and is similar to the approach described by Alpert and Rokhlin [SIAM J. Sci. Comput., 12 (1991)]. The main difference is that we utilise a modal Galerkin approach with Chebyshev basis functions instead of a nodal approach with a Lagrange basis. Part of the algorithm is a novel method that facilitates faster spreading of intermediate results through neighbouring levels of hierarchical matrices. This enhancement leads to a method that is approximately 20% faster to execute, due to less floating point operations. We also describe an efficient initialization algorithm that for the Lagrange basis is roughly 5 times faster than the original method for large input arrays. The described method has both a planning and execution stage that asymptotically require O(N) flops. The algorithm is simple enough that it can be implemented in 100 lines of vectorized Python code. Moreover, its efficiency is such that a single-threaded C implementation can transform 1000,000 coefficients in approximately 20 milliseconds on a new MacBook Pro M3, representing about 3 times the execution time of a well-planned (single-threaded) type 2 discrete cosine transform from FFTW (www.fftw.org). Planning for the 1000,000 coefficients requires approximately 50 milliseconds.