This is a compact version of the paper arXiv 1805.09550.
First we define the NFW PDF for any input scale radius $q=R/R_{vir}$
$$ d(q,c) = \frac{c^2}{(c q + 1)^2 \left(\frac{1}{c + 1} + \ln(c + 1) - 1\right)} $$
We define the un-normalised integral up to a normalised radius $q$ as
$$ p'(q,c) = \ln(1 + c q)-\frac{c q}{1 + c q}. $$
We can define $p$, the correctly normalised probability (where $p(q=1,c)=1$ with a domain [0,1]) as
$$ p(q,c) = \frac{p'(q,c)}{p'(1,c)} \Rightarrow p'(q,c)=p(q,c) p'(1,c) $$
Using the un-normalised variant $p'$ we can state that
$$ p'+1 = \ln(1 + c q) - \frac{c q}{1 + c q} + \frac{1 + c q}{1 + c q} = \ln(1 + c q) + \frac{1}{1 + c q}. $$
Taking exponents and setting equal to $y$ we get
$$ y = e^{p'+1} = (1 + c q) e^{1/(1 + c q)}. $$
We can the define $x$ such that
$$ x = 1 + c q, \ y = x e^{1/x}. $$
Here we can use the Lambert W function to solve for $x$, since it generically solves for relations like y = x e^{x} (where $x = W_0(y)$)). The exponent has a $1/x$ term, which modifies that solution to the slightly less elegant
$$ x = -\frac{1}{W_0(-1/y)} = -\frac{1}{W_0(-1/e^{(p'+1)})}, \ \text{sub for } q = \frac{x - 1}{c}, \ q = -\frac{1}{c}\left(\frac{1}{W_0(-1/e^{(p'+1)})}+1\right). $$
Given the above, this opens up an analytic route for generating exact random samples of the NFW for any $c$ (where we must be careful to scale such that $p'(q,c)=p(q,c) p'(1,c)$) via,
$$ r([0,1]; c) = q(p=U[0,1]; c). $$
library(NFWdist, quietly=TRUE)
Both the PDF (dnfw) integrated up to x, and CDF at q (pnfw) should be the same (0.373, 0.562, 0.644, 0.712):
for(con in c(1,5,10,20)){ print(integrate(dnfw, lower=0, upper=0.5, con=con)$value) print(pnfw(0.5, con=con)) }
The qnfw should invert the pnfw, returning the input vector (1:9)/10:
for(con in c(1,5,10,20)){ print(qnfw(p=pnfw(q=(1:9)/10,con=con), con=con)) }
The sampling from rnfw should recreate the expected PDF from dnfw:
for(con in c(1,5,10,20)){ par(mar=c(4.1,4.1,1.1,1.1)) plot(density(rnfw(1e6,con=con), bw=0.01)) lines(seq(0,1,len=1e3), dnfw(seq(0,1,len=1e3),con=con),col='red') legend('topright',legend=paste('con =',con)) }
Happy sampling!
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