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

Computer Science/Numerical Methods/ Iteration method ( with it’s 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 x0x_0x0​ is selected, and the next approximation is obtained using the formula:

xn+1=g(xn)x_{n+1} = g(x_n)xn+1​=g(xn​)

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

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

2. Convergence Condition

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

∣g′(α)∣<1| g'(\alpha) | < 1∣g′(α)∣<1

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

Explanation of Convergence Condition

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

3. Example of Iteration Method

Find the root of x3+x−1=0x^3 + x – 1 = 0x3+x−1=0 using iteration method.

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

x=1−x3x = 1 – x^3x=1−x3

Step 2: Select an initial guess

Let x0=0.5x_0 = 0.5x0​=0.5.

Step 3: Apply the iteration formula

Using xn+1=1−xn3x_{n+1} = 1 – x_n^3xn+1​=1−xn3​,

x1=1−(0.5)3=1−0.125=0.875x_1 = 1 – (0.5)^3 = 1 – 0.125 = 0.875x1​=1−(0.5)3=1−0.125=0.875 x2=1−(0.875)3=1−0.6699=0.3301x_2 = 1 – (0.875)^3 = 1 – 0.6699 = 0.3301x2​=1−(0.875)3=1−0.6699=0.3301 x3=1−(0.3301)3=1−0.0359=0.9641x_3 = 1 – (0.3301)^3 = 1 – 0.0359 = 0.9641x3​=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).

Numerical Methods for Solving Systems of Nonlinear …

MCA-08 / BCA-12: Computer Oriented Numerical Methods

B.Tech 4th Semester MATHEMATICS- …

Numerical Analysis