Inverse cdf of a mixture of normal distributions

General
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1

If I have a mixture distribution with two normal components like so

wN(\mu_1, \sigma^2_1) + (1 - w)N(\mu_2, \sigma^2_2)

it is relatively simple to calculate the pdf and the cdf of this distribution. I know that the cdf is

F(x) = wF_1(x) + (1-w)F_2(x)

I would like to calculate the inverse cdf F^{-1}(x). I’m able to do this in R using the uniroot or the optim function, but how would I go about doing this in stan?

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2

The CDF of the mixture from two normal distributions is:
image
Then use this in the Newton Solver to calculate its inverse.