Newton Raphson method initial guess

Question

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 .

Answer 1

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).