Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

Question

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure out an automatic way to have a good initial guess for the iterative methods and for high dimensional problems. For one dimensional problems, I know that we can use the bisection method to have a good guess for the newton method which is not so practical for high dimensional problems. So I am wondering if there are other methods for the multi-dimensional case.

Answer 1

Suppose that you have $n$ equations for $n$ unknowns of the form $$f_i=f_i(x_1,x_2,\cdots,x_n)=0$$ I think that the fastest way to solve is to minimize function $$\Phi(x_1,x_2,\cdots,x_n)=\sum_{i=1}^n f_i^2$$ and not use Newton-Raphson which, as you wrote, can be slow.

If the solution is unique, no problem.

Generating good starting values is too much problem dependent to get an answer. May be, you could provide an example.