Convergence of the Newton-Raphson method applied to a nonlinear system

Question

I'm working on a nonlinear system of 2 equations and try to solve it with the Newton-Raphson method. I have a function $f(x,y)$. In order to use the method, I have to find a starting vector $(x,y)$. But the method will only converge for specific starting vectors.

Is there a formula that gives the convergence of the Newton-Raphson method on a system of equations ?

Thank you all.

Answer 1

As long as you pick even a somewhat reasonable guess, the method should find the closest root. If the equations were linear, and you never encounter a situation where you divide by zero, you won't even have to use a reasonable guess.

$$ {\displaystyle x_{n+1}=x_{n}-m{\frac {f(x_{n})}{f'(x_{n})}}.} $$

Answer 2

No, in general there is no easy way to determine whether or not Newton-Raphson will converge for a given starting value, other than to try it. The region of convergence can be very complicated, and may have a fractal boundary.