How to check curvature of a vector valued function

Question

In terms of numerical optimization, the newton-rapson method requires a pos. definite Hessian $\nabla^2f$ respectively pos. curvature for computing the next step $p_k$ by solving $$\nabla^2 f p_k = -\nabla f$$

If I'm dealing with a vector valued function, e.g. a system of equations $F$, the newton step $p_k$ gets computed by $\nabla F p_k = - F$ without any calculation of the hessian. If in the latter case the newton-rapson diverges from my current point, would there be a possibility to check about the curvature of $F$?

Answer 1

The condition for finding a root that corresponds to positive definiteness of the Hessian in optimization, is that the function grows strictly monotonically.

But this is not a useful condition. That is because even in optimization, positive definiteness of the Hessian does not actually guarantee convergence of the unmodified Newton method. What is important is to focus on the remedy to the problem. The remedy is to use a line search procedure. If you understand how to write a line search procedure for optimization (e.g., based on the Wolfe and Goldstein conditions), then you will also know how to write a line search procedure for root finding.