Hankel Transformations using method of Ogata 2005
This is a very simple module designed to implement the Hankel transformation following the method of Ogata 2005.
It is a fast and accurate way of integrating functions of the form f(x)J(x), where f is an arbitrary slowly decreasing function, and J(x) is an arbitrary Bessel function of the first kind.
Either clone the repository at github.com/steven-murray/hankel and use python setup.py install, or simply install using pip install hankel.
The only dependencies are numpy and scipy.
This implementation is set up to allow more efficient calculation of multiple functions f(x). To do this, the format is class-based, with the main object taking as arguments the order of the bessel function, and the number and size of the integration steps. For example, to integrate the function J_0(x) (ie. f(x) = 1, cf. Ogata’s paper) one would do the following:
from hankel import HankelTransform f = lambda x: 1 #Define the input function f(x) h = HankelTransform(nu=0,N=120,h=0.03) #Create the HankelTransform instance h.transform(f) #Should give [1.0000000000003544, -9.8381428368537518e-15]
The correct answer is 1, so we have done quite well. The second element of the returned result is an estimate of the error (it is the last term in the summation). Here we used 120 steps of size 0.03. Difference applications will need to tune these parameters to get best results. In the above example, one may modify the function f and recall h.transform(f) without re-instantiating. This avoids unnecessary recalculation. Note that currently only integer orders are implemented!
Also included in the module is a subclass called SphericalHankelTransform. This is dedicated to integrating functions of the form f(x)j(x), where j(x) is a spherical Bessel function of arbitrary order. It is called in exactly the same way. Note that currently, only zeroth order is implemented for this class. An example:
from hankel import SphericalHankelTransform f = lambda x: x/(x**3+1) #Define the input function f(x) h = SphericalHankelTransform(nu=0,N=500,h=0.005) #Create the HankelTransform instance h.transform(f) #Should give [0.61092293340214776, -1.4163951324130801e-14]
Mathematica gives the answer as 0.610913. Note that the zeroth order spherical bessel function is just sin(x)/x.
In terms of the implementation, the main limitation is that not any arbitrary order is supported (only integer orders for the standard case, and only zeroth order for the spherical case). This is due to a limitation in scipy, but may be addressed at some point.
Another implementation-specific limitation is that the method is not perfectly efficient in all cases. Care has been taken to make it efficient in the general sense. However, for specific orders and functions, simplifications may be made which reduce the number of trigonometric functions evaluated. For instance, for an order 0 spherical transform, the weights are analytically always identically 1.
In terms of limitations of the method, they are very dependent on the form of the function chosen. Notably, functions which tend to infinity at x=0 will be poorly approximated in this method, and will be highly dependent on the step-size parameter, as the information at low-x will be lost between 0 and the first step. As an example consider the simple function f(x) = 1 with a zeroth order spherical bessel function. This tends to 1 at x=0, rather than 0:
f = lambda x: 1 h = SphericalHankelTransform(0,120,0.03) h.transform(f) #[1.5461236955707951, -3.5905712375161296e-16]
The true answer is pi/2, which is a difference of about 3%. Modifying the step size and number of steps to gain accuracy we find:
h = SphericalHankelTransform(0,10000,0.0001) h.transform(f) #[1.5706713512229455, -0.00010492204442285768]
This has much better than percent accuracy, but uses almost 100 times the amount of steps. The key here is the reduction of h to “get inside” the low-x information. This limitation is amplified for cases where the function really does tend to infinity at x=0, rather than a finite positive number, such as f(x) = 1/x.
Based on the algorithm provided in
H. Ogata, A Numerical Integration Formula Based on the Bessel Functions, Publications of the Research Institute for Mathematical Sciences, vol. 41, no. 4, pp. 949-970, 2005.
Also draws inspiration from
Fast Edge-corrected Measurement of the Two-Point Correlation Function and the Power Spectrum Szapudi, Istvan; Pan, Jun; Prunet, Simon; Budavari, Tamas (2005) The Astrophysical Journal vol. 631 (1)
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