How to generalize the fixed point iteration

[Charles49]

If we want to solve $$f(x)=0$$ we can re-write the equation as
$$g(x)=x$$ and use the fixed point method, i.e, $$x_{n+1}=g(x_n)$$ starting with a guess $$x_0.$$ I was wondering if something similar can be done with
$$\Lambda(x,y)=h(x,y).$$

 

[HallsofIvy]

Yes, of course. Just think of (x, y) as a single two dimensional variable, z and solve g(z)= z.

 

[racefan4TX]

Excellent info can be found about these kinds of boards. Thanks folks.

 

[Charles49]

HallsofIvy,

What a simple solution!

Thanks

 

[blue_raver22]

The fixed point iteration method is a powerful tool for solving equations of the form f(x) = 0. It works by finding a function g(x) such that g(x) = x, and then iterating the function starting from an initial guess x0 until it converges to a solution.

In order to generalize the fixed point iteration method to solve equations of the form Λ(x,y) = h(x,y), we can follow a similar approach. First, we need to find a function g(x,y) such that g(x,y) = (x,y), where (x,y) is the solution to the equation Λ(x,y) = h(x,y). This function g(x,y) can be obtained by manipulating the original equation Λ(x,y) = h(x,y) in such a way that it becomes a fixed point problem.

Once we have the function g(x,y), we can use the same fixed point iteration method as before, with the only difference being that we will now have two variables x and y instead of just one. The iteration process will then be given by the following equations:

xn+1 = g(xn,yn)
yn+1 = h(xn,yn)

We can start with an initial guess (x0,y0) and iterate until we reach a solution (x*,y*), where Λ(x*,y*) = h(x*,y*).

In summary, the fixed point iteration method can be generalized to solve equations of the form Λ(x,y) = h(x,y) by finding a suitable function g(x,y) and using the same iteration process as before. This approach can be applied to a wide range of problems in science and engineering, making it a valuable tool for researchers and practitioners.