I want to find roots of a $\mathbb{R}^n \to \mathbb{R}$-function. The problem is that I can only find algorithms for $\mathbb{R}^n \to \mathbb{R}^n$- and $\mathbb{R} \to \mathbb{R}$-functions. I know very little about the roots, apart from that they are a $n-1$ dimensional submanifold of $\mathbb{R}^n$. I could also probably make some guesses of the whereabouts of a few.
I will accept an answer that either (a) provides a reference for the statement that there are no root-finding methods for $\mathbb{R}^n \to \mathbb{R}$-functions of the kind I am looking for or (b) that provides a list of a few of the most popular ones or (c) a way I can adapt an existing method.
EDIT: Ideally I'd like to find as large parts of the manifold of zeros as possible, if there is some way to make that extension, please put it in the answer! Would be super helpful.
Not for finding "roots", but for local/global minima/maxima there exist plenty of algorithms to find minima/maxima of functions. The most algorithms rely on the gradient descent method, which can be described as follows $$ x_{n+1} := x_n - \gamma \nabla f(x_n) $$ where $\gamma>0$ is some (maybe adaptively updated) constant. Beware though: if your function has more local minima and a unique global minimum, it might happen that this finds just a local minimum. There are several upgrades to this algorithm to overcome this (stochastic gradient method, genetic algorithm etc.), but the choice highly depends on the problems itself.