Fast algorithms for spherical harmonic expansions, III

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.

[1]  Ming Gu,et al.  Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization , 1996, SIAM J. Sci. Comput..

[2]  P. Swarztrauber,et al.  SPHEREPACK 3.0: A Model Development Facility , 1999 .

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[5]  Mark Tygert,et al.  Fast Algorithms for Spherical Harmonic Expansions , 2006, SIAM J. Sci. Comput..

[6]  S. Goreinov,et al.  The maximum-volume concept in approximation by low-rank matrices , 2001 .

[7]  Mark A Ratner,et al.  A fast method for solving both the time-dependent Schrödinger equation in angular coordinates and its associated "m-mixing" problem. , 2009, The Journal of chemical physics.

[8]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[9]  Lexing Ying,et al.  Sparse Fourier Transform via Butterfly Algorithm , 2008, SIAM J. Sci. Comput..

[10]  Stanley C. Eisenstat,et al.  A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem , 1995, SIAM J. Matrix Anal. Appl..

[11]  P. Swarztrauber,et al.  Generalized Discrete Spherical Harmonic Transforms , 2000 .

[12]  Mark Tygert,et al.  Fast algorithms for spherical harmonic expansions, II , 2008, J. Comput. Phys..

[13]  Alfred V. Aho,et al.  Data Structures and Algorithms , 1983 .

[14]  Eugene E. Tyrtyshnikov,et al.  Incomplete Cross Approximation in the Mosaic-Skeleton Method , 2000, Computing.

[15]  Per-Gunnar Martinsson,et al.  On the Compression of Low Rank Matrices , 2005, SIAM J. Sci. Comput..

[16]  Laurent Demanet,et al.  A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators , 2008, Multiscale Model. Simul..

[17]  E. Michielssen,et al.  A multilevel matrix decomposition algorithm for analyzing scattering from large structures , 1996 .

[18]  Per-Gunnar Martinsson,et al.  On interpolation and integration in finite-dimensional spaces of bounded functions , 2005 .

[19]  Michael O'Neil,et al.  An algorithm for the rapid evaluation of special function transforms , 2010 .