Stan’s current method for solving systems of (nonlinear) algebraic equations is Powell’s dogleg method, as implemented in Eigen. There have been a few examples where Newton’s method is, up to an order of magnitude, faster. Examples include Precise but wrong results for a drug interaction model and Algebraic sovler problems (Laplace approximation in Stan), plus some other experiment I conducted myself.
This warrants creating a new solver. I’ll detail out the difference between the two solvers, along with recommendations on when to use which. Before opening an issue on GitHub, I however wanted to start a conversation here. The way I see it, Newton’s solver would have the same signature as Powell’s solver. We may however want to add other options, such as an adaptive step size.