Let $f(x)$ be a density function on $\mathbb R$; I want to find numerically the $\alpha$ quantile of the associated distribution, i.e. I want to find $c$ such that
$$\int_{-\infty}^c f(x)dx = \alpha$$
$f(x)$ is reasonably fast to compute, but if I apply the usual root-finding algorithms (say, newton-raphson) I would need to compute the integral of $f(x)$ at every iteration, which is impractical.
Do you have any references as of how solve the problem numerically? Is there a root-finding algorithm that doesn't need the evaluation of the function, but only of its derivative (in this case $f(x)$) ?