A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions

Abstract

A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order 0 as well as evaluating Schlömilch and Fourier-Bessel expansions in O(N (log N ) 2 / loglog N ) operations. The algorithm is based on an asymptotic expansion for Bessel functions of large arguments, the fast Fourier transform, and the Neumann addition formula. All the algorithmic parameters are selected from error bounds to achieve a near-optimal computational cost for any accuracy goal. Numerical results demonstrate the efficiency of the resulting algorithm. 1 2. Existing methods. We now give a brief description of some existing methods for evaluating (1.2) that roughly fall into four categories: (1) Direct summation; (2) Evaluation via an integral representation; (3) Evaluation using asymptotic expansions, and; (4) Butterfly schemes. For other approaches see [5].2.1. Direct summation. One of the simplest ideas is to evaluate J ν (r k ω n ) for 1 ≤ k, n ≤ N and then naively compute the sums in (1.2). By using the Taylor series expansion in (3.3) for small z, the asymptotic expansion in (3.1) for large z, and Miller's algorithm [18, Sec. 3.6(vi)] (backward recursion with a three-term recurrence) otherwise, each evaluation of J ν (z) costs O(1) operations [2]. Therefore, 2 One could now compute J ν,M ( r ω ⊺ )c by first computing the vector J ASY ν,M ( r ω ⊺ )c and then correcting it by R ν,M ( r ω ⊺ )c. The matrix-vector product J ASY ν,M ( r ω ⊺ )c 9

Interpolative Butterfly Factorization

“…Another key problem in modern large-scale computation is the parallel scalability of fast algorithms. Although there have been various fast algorithms like the (nonuniform) fast Fourier transform (FFT) [1,14,15], the FFT in polar and spherical coordinates [19,29,31,33], the parallel scalability of some of these traditional algorithms might be still limited in high performance computing. This motivates much effort to improve their parallel scalability [2,26,30,37].…”

Interpolative Decomposition Butterfly Factorization

“…Other similar examples when ν = 0 can be found in [43] and they can be also evaluated by IDBF with the same operation counts. Figure 12 summarizes the results of this example for different problem sizes N with different parameter pairs ( , k).…”

“…Example 2. Next, we provide an example of a special function transform, the evaluation of Schlömilch expansions [43] at g k = k−1 N for 1 ≤ k ≤ N :…”

Fast Steerable Principal Component Analysis

“…We note that the Fourier-Bessel expansion coefficients can be computed in O(nL 2 log L) operations using algorithms for rapid evaluation of special functions [23] or a fast analysisbased Fourier-Bessel expansion [24]. However, such "fast" algorithms may only lead to a marginal improvement for two reasons.…”