# A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions

@article{Townsend2015AFA,
title={A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions},
author={Alex Townsend},
journal={SIAM J. Numer. Anal.},
year={2015},
volume={53},
pages={1897-1917}
}
• Alex Townsend
• Published 7 January 2015
• Computer Science
• SIAM J. Numer. Anal.
A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ 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…
12 Citations

## Figures and Tables from this paper

• Mathematics
Computational Mathematics and Modeling
• 2020
An algorithm is proposed for a fast discrete finite Hankel transform of a function in a thin annulus. The transform arises in a natural way in the Neumann boundary-value problem for the Poisson
• Computer Science
SIAM J. Sci. Comput.
• 2017
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method to solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop.
• Computer Science
SIAM J. Sci. Comput.
• 2016
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method to solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop.
• Computer Science
SIAM J. Sci. Comput.
• 2017
This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic
• Computer Science
SIAM J. Sci. Comput.
• 2020
This paper introduces a "kernel-independent" interpolative decomposition butterfly factorization as a data-sparse approximation for matrices that satisfy a complementary low-rank property and is a general framework for nearly optimal fast matvec useful in a wide range of applications.
This paper introduces a kernel-independent"" interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary lowrank property and its construction algorithms.
• Computer Science
IEEE Transactions on Computational Imaging
• 2016
An algorithm that efficiently and accurately performs principal component analysis (PCA) for a large set of 2-D images, and, for each image, the set of its uniform rotations in the plane and their reflections is introduced.
• Mathematics
• 2018
We discuss efficient algorithms for the accurate forward and reverse evaluation of the discrete Fourier–Bessel transform as numerical tools to assist in the 2D polar convolution of two radially

## References

SHOWING 1-10 OF 32 REFERENCES

• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 2014
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree $N$ polynomial in $\mathcal{O}(N (\log N)^2/\log \log N)$ operations is
This paper presents a new procedure for the fast calculation of the Fourier-Bessel transform. The computation is performed in a "dual" mode and involves two matched algorithms, the first of which
• Computer Science
SIAM J. Sci. Comput.
• 1991
An algorithm is presented for the rapid calculation of the values and coefficients of finite Legendre series and admits far-reaching generalizations and is currently being applied to several other problems.
• Mathematics
IEEE transactions on medical imaging
• 1988
The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm.