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?



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 $h(x)^{2}$.

  • $\begingroup$ What's the benefit of minimizing h^2 instead of h? A more informative Hessian matrix, e.g. either with all nonnegative eigenvalues or nonpositive eigenvalues, perhaps? $\endgroup$ – user37077 Sep 22 '20 at 21:55
  • $\begingroup$ 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. $\endgroup$ – Brian Borchers Sep 22 '20 at 22:01

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