Coding up Newton's method for a mapping from R^2 to R -- the Jacobian wouldn't be invertible

I'm trying to code up in Matlab a multivariable Newton's method, for a mapping from R^2 to R, but the Jacobian would be a 2x1 matrix, not square, so it wouldn't be invertible.

Does this mean that Newton's method can't be used for root-finding, when mappings are from R^2 to R?

Would I then need to implement a derivative-free method instead, or is there a workaround?

Thanks,

Newton's method can refer either to a method for solving $$f(x)=0$$ where $$f: R^{n} \rightarrow R^{n}$$, or to a method for minimizing/maximizing a function $$g: R^{n} \rightarrow R$$ by solving the system of equations $$\nabla g(x)=0$$.
Your function $$h$$ maps $$R^{2}$$ to $$R$$ and you want to find a zero of the function. This is typically done by minimizing
$$\min h(x)^{2}$$
Newton's method for minimizing a function can be applied to minimizing $$h(x)^{2}$$.
• It might be that h(x) takes on positive and negative values. If you minimize $h(x)$ you'll find the most negative value of $h(x)$, which isn't what you want. Sep 22 '20 at 22:01