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Fixed point method for nonlinear systems - complex roots

  1. Dec 4, 2015 #1
    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?
     

    Attached Files:

  2. jcsd
  3. Dec 9, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Dec 9, 2015 #3
    I kept working on the problem and already found an answer. I was solving it the wrong way, but I'm all good now! Thanks :)
     
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