Computer Science/Numerical Methods/ Iteration method ( with it’s convergence condition).

Ali Seron Jason

6 days ago

https://www.gyanodhan.com/video/7A2.%20Computer%20Science/Numerical%20Methods/673.%20Iteration%20method%20%20%28%20with%20it%27s%20convergence%20condition%29.mp4

Contents

0.1 Iteration Method in Numerical Methods
0.2 1. General Form of Iteration Method

1 2. Convergence Condition

Iteration Method in Numerical Methods

The Iteration Method is a technique used to find the root of an equation $f (x) = 0$ . It is based on approximating the root through successive iterations, refining the value until it reaches the desired accuracy.

1. General Form of Iteration Method

The given equation $f (x) = 0$ is rewritten in the form:

$x = g (x)$

where $g (x)$ is a function derived from $f (x)$ .

An initial guess $x_0$ is selected, and the next approximation is obtained using the formula:

$x_{n+1} = g(x_n)$

This process continues until the absolute difference between successive approximations is less than a predefined tolerance $ϵ\epsilon$ , i.e.,

$∣xn+1−xn∣<ϵ|x_{n+1} – x_n| < \epsilon$

2. Convergence Condition

For the iteration method to converge to a root $α\alpha$ , the function $g (x)$ must satisfy:

$g'(\alpha) | < 1$

where $g′(α)g'(\alpha)$ is the derivative of $g (x)$ evaluated at the root $α\alpha$ .

Explanation of Convergence Condition

If $∣ g^{'} (x) ∣ < 1$ in the neighborhood of the root, the successive approximations get closer to the root.
If $∣ g^{'} (x) ∣ > 1$ , the method diverges, meaning it moves away from the actual root.

3. Example of Iteration Method

Find the root of $x^3 + x – 1 = 0$ using iteration method.

Step 1: Rewrite in the form $x = g (x)$

$x = 1 – x^3$

Step 2: Select an initial guess

Let $x_0 = 0.5$ .

Step 3: Apply the iteration formula

Using $x_{n+1} = 1 – x_n^3$ ,

$x_1 = 1 – (0.5)^3 = 1 – 0.125 = 0.875$ $x_2 = 1 – (0.875)^3 = 1 – 0.6699 = 0.3301$ $x_3 = 1 – (0.3301)^3 = 1 – 0.0359 = 0.9641$

This process continues until convergence.

4. Advantages and Disadvantages

Advantages:

Simple and easy to implement.
Does not require complex calculations like derivatives.

Disadvantages:

May not always converge.
Requires a good initial guess.

If you need more examples or explanations, let me know!

Computer Science/Numerical Methods/ Iteration method ( with it’s convergence condition).

Iteration Method in Numerical Methods

1. General Form of Iteration Method

2. Convergence Condition

Explanation of Convergence Condition

3. Example of Iteration Method

Find the root of $x^3 + x – 1 = 0$ using iteration method.

Step 1: Rewrite in the form $x = g (x)$

Step 2: Select an initial guess

Step 3: Apply the iteration formula

4. Advantages and Disadvantages