An Algorithm for the Rapid Evaluation of Special Function Transforms

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. We introduce a fast algorithm for the numerical application to arbitrary vectors of several special function transforms. The algorithm requires O(n log(n)) operations to apply to an arbitrary vector any n×n matrix such that the rank of any p×q contiguous submatrix is bounded by a constant times pq/n. These rank bounds are proven here for the case of the Fourier-Bessel transform. Numerical experiments demonstrate a much wider applicability. The performance of the algorithm is illustrated via several numerical examples.

Fast wave computation via Fourier integral operators

“…Many of the algorithmic tools used in those papers are present in different forms or different contexts; let us for instance mention work on angular decompositions of the symbol of pseudodifferential operators [1], work on the butterfly algorithm and its applications in [21,23,38,37,35], work on fast "beamforming" methods for filtered backprojection for Radon and generalized Radon transforms (a problem similar to wave propagation) in [36,22,2], work on the plane-wave time-domain fast multipole method summarized in Chapters 18 and 19 of [9], and work on "phase-screen" methods in geophysics [11].…”

“…Combining these observations with the structure of the Butterfly algorithm [21,23], we have the following algorithm for applying the Fourier integral operators. Essentially, this computes {δ AB t } for all admissible pairs (A, B) efficiently in a recursive fashion.…”