Fixed-Point Iteration for Nonlinear System of Equations

In summary, A person is asking for help with finding the fixed point of a nonlinear system using the fixed-point method. They have tried various forms of the function g, but have not been able to find a fixed point that converges. They are wondering if it is possible to converge to a fixed point and are seeking assistance.
  • #1
irony of truth
90
0
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.
 
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  • #2
if you put your system in the form

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

[tex] |GraF|<1 [/tex] gra=gradient of the function...
 
  • #3
Hi Karlisbad:

[tex] F(x,y,z)=0 [/tex]

I believe that this will be effective for the Newton's Method, which I got one of its solutions using a scilab program. It's pretty difficult for me to use the fixed point iteration, but I am hoping that somehow, I can get that fixed point.
 

What is fixed-point iteration?

Fixed-point iteration is a numerical method used to solve a system of nonlinear equations. It involves repeatedly updating an initial guess for the solution until it converges to the correct solution.

How does fixed-point iteration work?

Fixed-point iteration works by taking an initial guess for the solution and using that to calculate a new estimate for the solution. This process is repeated until the estimates converge to the correct solution.

What are the advantages of using fixed-point iteration?

Fixed-point iteration is relatively easy to implement and can be applied to a wide range of nonlinear systems. It also has the advantage of being able to handle multiple variables and equations simultaneously.

Are there any limitations to fixed-point iteration?

Fixed-point iteration may not always converge to the correct solution, especially if the initial guess is too far from the true solution. It also requires that the equations in the system have a unique fixed point, which may not always be the case.

How do I know if fixed-point iteration has converged?

Fixed-point iteration has converged when the difference between consecutive estimates for the solution is smaller than a predetermined tolerance. This tolerance is usually set by the user and can be adjusted to achieve the desired level of accuracy.

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