# Fixed point method for nonlinear systems - complex roots

1. Dec 4, 2015

### RicardoMP

1. The problem statement, all variables and given/known data
I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point method with the iterative function G (that I've also uploaded). I must also justify the convergence or divergence of its iterations.

2. Relevant equations
The F and G functions are in the .png files that I've uploaded, where x=(x1,x2).

3. The attempt at a solution
Before I tried to verify the convergence criteria for the fixed point method, I tried to find the roots and proceeded to rewrite G in the form x=G(x). And so I arrived to the expressions: $x_1=\pm \sqrt{-x_2^2}$ and $x_2=\pm \sqrt[4]{\frac{1-x_1^2}{4}}$.
Obviously faced with complex roots, before applying the fixed point method, I went on and tried to verify the convergence of its iterations, by studying the max norm of the Jacobian matrix of the rewritten G function. In the end, I got the matrix that I've uploaded (systemJG.png). Given a certain real interval to $x_1$ and $x_2$, I have to compare the absolute value of a complex number and a real number. Anyway, I think I'm not going in the right path and I'm overlooking something and complicating the whole problem, starting with the fact I've only considered the positive roots of the $x_1$ and $x_2$ in the matrix... Any tips in how to tackle this problem or any similar example I can study?

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2. Dec 9, 2015