Tutorial: Radial Basis Functions

In this tutorial we will show how to obtain the radial basis functions $g_{n\ell}(r)$, for given $k_{\rm max}$, $r_{\rm min}$, and $r_{\rm max}$.

Further, we assume that SphericalFourierBesselDecompositions.jl is already installed, and that you have a basic understanding of what the package is supposed to do.

First, load the package and create a shortcut

using SphericalFourierBesselDecompositions
const SFB = SphericalFourierBesselDecompositions

We will always assume that this shortcut has been created, as the package SFB does not export any symbols itself. Make the shortcut const can be important for performance.

To perform a SFB decomposition, we create a modes object amodes. That will construct the n,l,m-modes and basis functions.

kmax = 0.05
rmin = 500.0
rmax = 1500.0
amodes = SFB.AnlmModes(kmax, rmin, rmax)

Here, kmax is the maximum k-mode to be calculated, rmin and rmax are the radial boundaries.

To access the radial basis functions gnl(r), use

r = 555.5
rr = range(500.0, 1000.0, length=100)
gnl_r = amodes.basisfunctions.gnl[n,l+1](r)
gnlr = @. amodes.basisfunctions.gnl[n,l+1](rr)

Note that n ranges from 1 to amodes.nmax_l[l+1], and l ranges from 0 toamodes.lmax.

The associated wavenumbers can be accessed with

k = amodes.knl[n,l+1]

If the result is a NaN, then it is outside the parameters, i.e., k > kmax.