I am looking into methods of approximating zeros of functions, in particular for systems of nonlinear equations.
I have discovered that Newton's Method can easily be applied to an m-by-m system of equations, but convergence seems to be difficult to prove...
For m=1 systems, the bisection method is guaranteed to converge, but I haven't found any sources outlining how to achieve guaranteed convergence for systems of equations.
Are there any methods with guaranteed convergence for systems of equations? If not, can anybody recommend sources outlining how to analyse whether or not Newton's method will converge?
Thank you all for your time.