Iteration functions for Fixed Point method

In summary, the Fixed Point method is an iterative method used to find the roots of a given function by repeatedly applying a function to an initial guess until the resulting value converges to a fixed point. It is important to use iteration in this method as it allows for approaching the root of a function. The convergence of an iteration function can be determined by examining its behavior as the number of iterations increases. Some limitations of the Fixed Point method include potential non-convergence, convergence to a different root or non-root value, and slow convergence rate compared to other methods.
  • #1
carlosbgois
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Hi there. I need to find some iteration functions for [itex]x - 2\frac{sin(x)}{cos(x)}=0[/itex], as [itex]g(x)=2\frac{sin(x)}{cos(x)}[/itex] does not converge. I can't find any others, maybe I didn't quite undertood how they're built. Any help will be appreciated

Thanks
 
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  • #2
You could check [itex]g(x)=arcsin(\frac{x \cos(x)}{2})[/itex] and [itex]g(x)=arccos(\frac{2 \sin(x)}{x})[/itex] and g(x)=arctan(x/2)? That is just your formula, with different ways to isolate an x at one side.
 

1. What is the Fixed Point method in mathematics?

The Fixed Point method is an iterative method used to find the roots of a given function. It involves repeatedly applying a function to an initial guess until the resulting value converges to a fixed point, which is the root of the original function.

2. How does the Fixed Point method work?

The Fixed Point method works by starting with an initial guess and repeatedly applying a function to it until the resulting value converges to a fixed point. This fixed point is the root of the original function and can be found by rearranging the function into the form x = g(x) and finding the value of x that satisfies this equation.

3. What is the importance of iteration in the Fixed Point method?

Iteration is crucial in the Fixed Point method as it allows us to approach the root of a function by repeatedly applying the function to an initial guess. The more iterations we perform, the closer we get to the actual root of the function.

4. How do you determine the convergence of an iteration function?

The convergence of an iteration function can be determined by examining the behavior of the function as the number of iterations increases. If the values of the function approach a fixed point, the function is said to converge. However, if the values oscillate or diverge, the function is said to not converge.

5. What are some limitations of the Fixed Point method?

The Fixed Point method may not always converge to a root, especially if the initial guess is far from the actual root. Additionally, the method may converge to a different root or even a non-root value if the function is not well-behaved. Furthermore, the convergence rate of the method may be slow compared to other root-finding methods.

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