Package to compute integrals involving Bessels functions such as
or Hankel transforms
The integrals are performed using the FFTLog algorithm (Hamilton 2000).
Given an array logarithmically spaced r and a function f evaluated over this array, we
want to evaluate the Hankel transform for the multipole values contained in the array Ell.
For instance, let us consider the following Hankel pair
Since we know the analytical transform, we can perform a check
- Instantiate an object
HankelPlan
HankelTest = FFTLog.HankelPlan(x = k)- Perform some precomputations
prepare_Hankel!(HankelTest, Ell)- Compute the Hankel transform
Fy = evaluate_Hankel(HankelTest, Pk)- If needed, the array
y(the counterpart of the arrayr) can be obtained with
y = get_y(HankelTest)Now, let us compare the numerical and the analytical transforms
We can also plot the relative difference
Quite good, isn't it?
| Step | Status | Comment |
|---|---|---|
| Integrals with a Bessel function | ✅ | Implemented, needs some polishing |
| Hankel Transforms | ✅ | Implemented, needs some polishing |
| Multithreading | ✔️ | Implemented |
| Integrals with a Bessel derivative | ✔️ | Implemented |
| Automatic Differentiation | 🚧 | WIP |
| GPU compatibility | 🚧 | WIP |
| Integrals with two Bessel functions | 🚧 | WIP |
| Python wrapper | 🚧 | WIP |
- Marco Bonici, PostoDoctoral Researcher at INAF-IASF Milano