When is Newton's Method guaranteed to converge to a good solution (non-linear system)? My knowledge of Newton's Method is partial. I am trying to understand what guarantees this method can give on the solution of systems of non-linear equations.
Specifically, I have a set of non-linear equations that are easily twice differentiable. What additional conditions do I need to fulfill in order to guarantee that Newton's Method finds a good solution? How important is the starting point? if it is important, how can I guarantee that I find a good starting point?
 A: The items below should help you to look up further details of Newton's Method for system of nonlinear equations.
Advantages:


*

*Q-quadratically convergent from good starting guesses if the Jacobian $J(x_*)$ is nonsingular

*Exact solution in one iteration for an affine $F$ (exact at each iteration for any finite component functions of $F$)


Disadvantages:


*

*Not globally convergent for many problems

*Requires $J(x_k)$ at each iteration

*Each iteration requires the solution of a system of linear equations that may be singular or ill-conditioned


References:


*

*See Section 3

*This has a worked example
Notes


*

*For Newton's method, you would choose a tolerance and use some vector norm to test that the result is good enough. 

*If you choose a bad starting value, all bets are off. 

*You might also want to look into quasi-Newton methods. 

*For good starting values, you want to look into the Steepest Descent method, which is used to find accurate starting approximations for the Newton-based techniques.

*As an aside, you probably also want to look at and understand "Constrained" versus "Unconstrained" methods.

