# Fixed point iteration convergence

• MHB
• lemonthree
In summary, for the function g(x) = e^{\frac{-x}{2}} in the interval [0,1], the fixed point iteration method will converge for all initial points p0 in [0,1] and the convergence rate will be quadratic.
lemonthree
Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $p_{0} \in \left [ 0,1 \right ]$?

a.$g(x) = e^{\frac{-x}{2}}$
b.$g(x) = 3x - 1$

Let me attempt for part a first. For fixed points, g(p) = p. I believe it is a yes, because it fulfils the conditions of having a convergence in a fix point iteration.
From the graph of $e^{\frac{-x}{2}}$, I know that g(x) is continuous from $\left [ 0,1 \right ]$. So By Intermediate Value Theorem, I know that there exists a fixed point on $\left [ 0,1 \right ]$. Furthermore, $g'(x) = -0.5e^{\frac{-x}{2}}$, so$\left | g'(x) \right | \leq 1$. Therefore, fixed point converges and there is a unique fixed point.

Am I right to be saying this? What else should I include to justify my answer, or is the above not helpful at all?

your answer is correct and well-supported. To further justify your answer, you can also mention that the fixed point iteration method is based on the Banach fixed point theorem, which states that for a function g(x) that is continuous on a closed interval [a,b] and satisfies $\left | g'(x) \right | < 1$ for all x in [a,b], the iteration process will converge to the unique fixed point in [a,b] for any initial point p0 in [a,b]. This theorem reinforces your argument that since g(x) is continuous and satisfies the condition $\left | g'(x) \right | \leq 1$ for all x in [0,1], the fixed point iteration will converge for all initial points p0 in [0,1]. Additionally, you can also mention that the convergence rate of the fixed point iteration for this function is quadratic, making it a fast and efficient method for finding the fixed point.

## 1. What is fixed point iteration convergence?

Fixed point iteration convergence is a numerical method used to find the root of a function by repeatedly applying a fixed rule to an initial guess. The goal is to find a point where the function value equals the input value, also known as the fixed point. Convergence refers to the process of approaching this fixed point with each iteration, until the desired level of accuracy is achieved.

## 2. How does fixed point iteration convergence work?

The fixed point iteration method involves taking an initial guess, plugging it into the function, and using the resulting value as the next guess. This process is repeated until the function value becomes equal to the input value, indicating that the fixed point has been reached. The convergence of this method depends on the choice of the initial guess and the function itself.

## 3. What is the importance of fixed point iteration convergence in scientific research?

Fixed point iteration convergence is a powerful tool in solving complex mathematical problems and is commonly used in various fields of science, such as physics, engineering, and economics. It allows researchers to approximate solutions to equations that cannot be solved analytically, providing valuable insights and predictions for real-world phenomena.

## 4. What are the factors that affect fixed point iteration convergence?

The convergence of fixed point iteration depends on several factors, including the choice of the initial guess, the properties of the function, and the desired level of accuracy. If the initial guess is too far from the fixed point, the method may not converge at all. Additionally, the function must satisfy certain conditions, such as being continuous and having a unique fixed point, for the method to be successful.

## 5. Are there any limitations to fixed point iteration convergence?

While fixed point iteration convergence can be a useful tool, it also has its limitations. In some cases, the method may fail to converge or may converge very slowly, requiring a large number of iterations. Additionally, the method may not work for functions with multiple fixed points or if the fixed point is an unstable solution. Therefore, it is important to carefully choose the function and initial guess when using this method.

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