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

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.

• unfortunately, there is no universal techniques to provide a good initial guess. to be sure, you need to implement globalization techniques, like line search or trust region. Jul 26 '15 at 15:51

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.
• to minimize $\Phi$, you usually apply some kind of Gauss-Newton type methods. There is no free lunch :) Jul 26 '15 at 15:42