Rapid Application of the Spherical Harmonic Transform via Interpolative Decomposition Butterfly Factorization
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We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. While existing methods have total running times (including all precomputations) which grow at least as fast as O(N^3), where N is the degree of the transform, the computational complexity of our method is O( N^2 log^3(N) ). It is based on a new method for rapidly computing the Legendre Transform (LT) by hierarchically applying the interpolative decomposition butterfly factorization (IDBF). Numerical results are provided to demonstrate the effectiveness and numerical stability of the new framework.
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