Newton's method for a system of equations

[defunc]

Under what circumstances will Newtons method for a system of nonlinear equations converge? Are there any criteria at all which guarantees convergence?

Regards

[arildno]

Interesting question!

I googled a bit, and found a lecture note that might be of interest:
http://www.math.ntnu.no/emner/TMA4122/2006h/notat-src/nr-systems-a4.pdf

(This is from the Technical University of Trondheim (NTNU), which has overall good standards, also internationally)
NOTE OF CAUTION:

I haven't as yet read it through; so I cannot say whether it is mathematically acceptable.

[slider142]

Kantorovitch's theorem (http://planetmath.org/encyclopedia/KantorovitchsTheorem.html) will tell you that Newton's method applied to a vector-valued multivariable function converges if started from an input which satisfies some inequalities. There are several versions for stronger types of convergence.

[whs]

Remember, Newtons method is basically the same as fixed-point iteration.

So, I'm assuming you know what needs to happen for convergence of a single variable in the fixed point case. Simply extend that to the multivariable case.

i.e, There exists a 'K' between 0 < K < 1, where you will start adding absolute values of partial derivatives, which will end up being less than K. If you have a text book, hopefully it will either mention it in the main text, or as an exercise.

It's kind of important. If this is for an introductory numerical analysis class, it might not be covered in an introductory text.

[defunc]

Thank you all for your replies!