# 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} }

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

### Fast Discrete Finite Hankel Transform for Equations in a Thin Annulus

- MathematicsComputational 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…

### Computing with Functions in Spherical and Polar Geometries II. The Disk

- Computer ScienceSIAM 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.

### Computing with Functions in Spherical and Polar Geometries I. The Sphere

- Computer ScienceSIAM 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.

### Interpolative Butterfly Factorization

- Computer ScienceSIAM 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…

### Interpolative Decomposition Butterfly Factorization

- Computer ScienceSIAM 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.

### Sine-Cosine Wavelets Approach in Numerical Evaluation of Hankel Transform for Seismology

- Computer Science
- 2016

### INTERPOLATIVE DECOMPOSITION BUTTERFLY FACTORIZATION\ast

- Computer Science
- 2020

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.

### Reducing dimensionality to model 2D rotating and standing waves in a delayed nonlinear optical system with thin annulus aperture

- MathematicsNonlinear Analysis: Real World Applications
- 2018

### Fast Steerable Principal Component Analysis

- Computer ScienceIEEE 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.

### An efficient procedure for custom beam-profile convolution in polar coordinates: testing, benchmarking and application to biophotonics

- 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

### A Fast, Simple, and Stable Chebyshev-Legendre Transform Using an Asymptotic Formula

- Computer Science, MathematicsSIAM 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…

### Dual algorithms for fast calculation of the Fourier-Bessel transform

- Mathematics
- 1981

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…

### A Fast Algorithm for the Evaluation of Legendre Expansions

- Computer ScienceSIAM 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.

### The fast Hankel transform as a tool in the solution of the time dependent Schrödinger equation

- Physics
- 1985

### Approximation for bessel-functions and their application in the computation of hankel-transforms

- Mathematics
- 1982

### A Hankel transform approach to tomographic image reconstruction.

- MathematicsIEEE 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.