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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.

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  • $\begingroup$ 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. $\endgroup$
    – user251257
    Jul 26 '15 at 15:51
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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.

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  • $\begingroup$ to minimize $\Phi$, you usually apply some kind of Gauss-Newton type methods. There is no free lunch :) $\endgroup$
    – user251257
    Jul 26 '15 at 15:42
  • $\begingroup$ @user251257. Yes but it is much more stable and there are plenty of minimization methods which are totally available. $\endgroup$ Jul 26 '15 at 15:47
  • $\begingroup$ I would not say more stable. but one can introduce globalization techniques in a natural way. it may still perform poorly, if you are far away from optimum. $\endgroup$
    – user251257
    Jul 26 '15 at 15:49
  • $\begingroup$ I agree with you. This could be the start of a very interesting discussion. May I confess that this is the way I always solve systems of nonlinear equations. $\endgroup$ Jul 26 '15 at 16:01
  • $\begingroup$ well, the Newton step and the gauss Newton step are the same, if the Jacobian is non singular. so you aren't doing something wrong. however the condition for the gauss Newton system is worse. that's all. $\endgroup$
    – user251257
    Jul 26 '15 at 16:06

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