# Best way to numerically solve a nonlinear system $f(x)=0$, $f:\mathbb{R}^n\to\mathbb{R}^m$

I am aware that that there are a lot of methods known to solving a nonlinear system $$\mathbf{f}(\mathbf{x})=0$$, if $$\mathbf{f}:\mathbb{R}^n\to\mathbb{R}^n$$, assuming the Jacobian is non-singular. However, I am wondering what happens if $$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^n$$, with $$m\neq n$$. Of course, there can be infinitely solutions, or no solutions at all, but what are good numerical ways for solving such a system? I am aware that we can alter the Newton-Rhapson iteration by replacing $$\mathbf{J}_f(\mathbf{x^k})^{-1}$$, by its Moore-Penrose inverse. However, the following paper by Levin and Israel suggests generalizing this to arbitrary $$\{2\}$$-inverses. I am wondering what the numerical advantage of this higher generalization is, compared to the Moore-Penrose inverse. I don't see why it would reduce the computation time, since it computes and SVD of the Jaobian $$\mathbf{J}_f(\mathbf{x}^k)$$ anyway, so why would one not directly obtain the Moore-Penrose inverse, but construct the $$\{2\}$$-inverse $$\Sigma^{(2)}$$.

EDIT: I might think the construction of $$\Sigma^{(2)}$$ is beneficial if the singular values are small, preventing the emergence of enormous in the iteration. If the singular values are not very small, the $$\Sigma^{(2)}$$ matrix will be the Moore-Penrose inverse.

A NEWTON METHOD FOR SYSTEMS OF m EQUATIONS IN n VARIABLES

Another paper by the same authors suggests an inverse-free method for solving the system, by Newton's directional method. What advantage could this method have over the one described above, using the pseudo-inverses.

AN INVERSE-FREE DIRECTIONAL NEWTON METHOD FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

As you might understand, I am getting a little bit lost in the different methods, and I am wondering if anyone could give me a good overview in which situation, which method can best be applied. Thanks in advance!

• Nonlinear is extremely broad! If the equations are linear or polynomial, you have whole fields of study devoted to them — linear algebra and algebraic geometry, respectively. Nov 27 '20 at 17:18

I am not familiar with the first method involving $$\Sigma^{(2)}$$. It vaguely resembles Jacobi preconditioning on your singular values. Hopefully another answer can supplement this one.