Approximation for Bessel functions and their application in the computation of Hankel transforms

Piessens, Robert; Branders, Mieke

doi:10.1016/0898-1221(82)90012-8

Cited by 10 publications

(4 citation statements)

References 8 publications

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“…The two basic computational steps in evaluating ) (ρ F are to evaluate the projection ) (x p by equation (4) and to evaluate the one-dimensional Fourier transform by equation (5). Assuming…”

Section: Projection Methods For Fhtmentioning

confidence: 99%

Fast phase retrieval algrithm for rotational symmetric DOE designing

Wang

Zhang

et al. 2003

SPIE Proceedings

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For the design of DOE with rotational symmetric phase distribution, the Fourier Transform (FT) and Inverse Fourier Transform (IFT) take the form of the first order of Hankel transform. The speed of numerical calculation for Hankel transform becomes the bottleneck of phase retrieval algorithms. A new fast Hankel transform method is applied to DOE designing process, in which Hankel transform is achieved by one time's of projection integration and one time's of Fast Fourier Transform (FFT). Utilizing the merits of FFT, speed of Hankel transform is greatly improved. Therefore, rotational symmetric DOE can be designed with fast speed and good designed result can be obtained more easily. Some examples of DOE designing with the help of FHT are given.

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Section: Projection Methods For Fhtmentioning

confidence: 99%

Fast phase retrieval algrithm for rotational symmetric DOE designing

Wang

Zhang

et al. 2003

SPIE Proceedings

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“…(B16) with Λ = 0.5401m IR = 0.10802 corresponding to m IR = 0.2 has been used. Because of the Bessel function the integral is rapidly varying and special methods can be developed for accurate evaluation [33]. In Mathematica NIntegrate with Method → "ExtrapolatingOscillatory" can be applied.…”

Section: Appendix B: Wilson Line Correlatorsmentioning

confidence: 99%

“…The expectation value of a single Wilson line thus effectively vanishes. varying and special methods can be developed for accurate evaluation [33]. In Mathematica NIntegrate with Method → "ExtrapolatingOscillatory" can be applied.…”

Section: (B11)mentioning

confidence: 99%

Gluon radiation from a classical point particle. II. Dense gluon fields

Kajantie¹,

McLerran²,

Paatelainen³

2020

Phys. Rev. D

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The goal of this paper is to extend the results of Ref. [1], where formulae were derived for gluonic radiation for a high energy nucleus colliding with a classical colored particle. In Ref.[1], we computed the amplitudes for radiation in the fragmentation region of the particle for a dilute gluonic field. In this paper, we compute the radiation by solving the fluctuation equations of the dense background field in a specific gauge which makes it simple to solve the asymptotic radiation from an initial condition immediately after the passage of the nucleus. We identify and compute two components of gluon radiation, a bulk component which extends to the central region and bremsstrahlung, which may give rise to an experimentally observable intensity peak in the target fragmentation region.

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“…A fast, but perhaps numerically unstable (see Figure 4.1), algorithm then follows by exploiting DCTs and DSTs. Existing work has approximated J ν (z) using asymptotic expansions with rational coefficients [20], the approximation 2/(πz) cos(z − (2ν + 1)π/4) [4], and for half-integer ν the asymptotic expansion in (3.1) [24]. However, none of these approaches have rigorously determined the regime in which the employed asymptotic expansion is an accurate approximation and instead involve dubiously chosen algorithmic parameters.…”

Section: Evaluation Via An Integral Representationmentioning

confidence: 99%

A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions

Townsend¹

2015

SIAM J. Numer. Anal.

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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

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Approximation for Bessel functions and their application in the computation of Hankel transforms

Cited by 10 publications

References 8 publications

Fast phase retrieval algrithm for rotational symmetric DOE designing

Fast phase retrieval algrithm for rotational symmetric DOE designing

Gluon radiation from a classical point particle. II. Dense gluon fields

A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions

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