I am facing problems in making the initial guess while applying Newton-Raphson method to equations of more than one variable.Is ther some thumb rule for making the initial guess in the general case?To be specific let the system of equations be $$f_i(x,y,w,z) ,i=1,2,3,4$$,where all the four functions are algebraic functions.Thanks for any help in advance .

  • $\begingroup$ Can you provide an actual set of equations to solve? $\endgroup$ – Moo Aug 5 '17 at 5:36
  • $\begingroup$ As Moo commented, it would be good to know the equations. Let me play with them. $\endgroup$ – Claude Leibovici Aug 5 '17 at 9:18

Without knowing the functions, it could be hard to say.

One thing you can consider if $$\Phi(x,y,w,z)=\sum_{i=1}^4 f^2_i(x,y,w,z)$$ which should be zero at the solution.

So, and this is a thing I use to do, compute $\Phi(x,y,w,z)$ over a four dimension grid and select as a starting point the point corresponding to the minimum value you noticed.

The problem is that it does not ensure that this is the "good" starting point (in particular if multiple solutions do exist).


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