How to generalize the fixed point iteration

In summary, the fixed point iteration method is a numerical technique used to approximate the root or solution of a given function. It involves repeatedly applying a fixed formula or rule to an initial guess until the resulting values converge to the desired root. Two main assumptions are made in this method: the given function must have a fixed point and it must be within the initial guess. The initial guess should be chosen carefully and trial and error can be used. The method can fail to converge if the initial guess is too far from the actual root or if the function does not have a fixed point. To improve convergence, a better initial guess, a different formula, and a stopping criterion can be used.
  • #1
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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
$$\Lambda(x,y)=h(x,y).$$
 
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  • #2
Yes, of course. Just think of (x, y) as a single two dimensional variable, z and solve g(z)= z.
 
  • #3
Excellent info can be found about these kinds of boards. Thanks folks.
 
  • #4
HallsofIvy,

What a simple solution!

Thanks
 
  • #5


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.
 

1. How does the fixed point iteration method work?

The fixed point iteration method is a numerical technique used to approximate the root or solution of a given function. It involves repeatedly applying a fixed formula or rule to an initial guess until the resulting values converge to the desired root.

2. What are the assumptions made in the fixed point iteration method?

Two main assumptions are made in the fixed point iteration method: first, the given function must have a fixed point, meaning that there exists a value where the function output is equal to its input; second, the fixed point must be within the initial guess and the resulting sequence of values must converge to the fixed point.

3. How do I choose the initial guess in fixed point iteration?

The initial guess should be chosen carefully as it can greatly affect the convergence of the method. It is recommended to choose an initial guess that is close to the actual root and within the interval where the fixed point exists. Trial and error can also be used to find a suitable initial guess.

4. Can the fixed point iteration method fail to converge?

Yes, there are cases where the fixed point iteration method may fail to converge. This can happen if the initial guess is too far from the actual root, or if the function does not have a fixed point, or if the fixed point is not within the interval of the initial guess. In such cases, alternative methods may be used to approximate the root.

5. How can I improve the convergence of the fixed point iteration method?

There are several ways to improve the convergence of the fixed point iteration method. One way is to choose a better initial guess. Another way is to use a different fixed point iteration formula or function. Additionally, using a stopping criterion, such as a maximum number of iterations or a tolerance for the difference between consecutive values, can help prevent the method from iterating indefinitely.

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