# Fixed-Point Iteration for Nonlinear System of Equations

1. Dec 20, 2006

### irony of truth

Hello:

I am solving for the fixed point of this nonlinear system:
x^2 - x + 2y^2 + yz - 10 = 0
5x - 6y + z = 0
-x^2 - y^2 + z = 0

Somehow, I got stuck with my function for g, g(x) = x. I ran this in a program applying the Newton's method and I got its solution easily. However, I find it difficult using fixed-point method.

My function g should look something like this: My first expression shall be "equated to x", but the equal sign should not appear. My second expression shall be "equated to y" but no equal sign should appear (and so forth). I have tried several forms of g but the iteration would simply diverge from the fixed point.

For example,
x = x^2 + 2y^2 + yz - 10; others: (x^2 + x+ 2y^2 + yz - 10)/2
y = (5x + y + z) /7; others: (5x + z)/6; (5x + 2y + z) /8
z = x^2 + y^2

So, my function would be:
(x^2 + 2y^2 + yz - 10)
g = ((5x + y + z) /7 )
( x^2 + y^2 )
(but for any initial guess, the fixed point can't be found (as it diverges)).

Is this possible to converge to a fixed point?

Any help will be appreciated.

2. Dec 27, 2006

if you put your system in the form

$$F(x,y,z)=0$$ i think the condition for convergence (no t pretty sure) is:

$$|GraF|<1$$ gra=gradient of the function....

3. Dec 27, 2006

### irony of truth

$$F(x,y,z)=0$$