Iteration Method in Numerical Analysis

Iteration Method
Let the given equation be f(x) = 0 and the value of x to be determined. By using the Iteration method you can find the roots of the equation. To find the root of the equation first we have to write equation like below
x = pi(x)
Let x=x₀ be an initial approximation of the required root α then the first approximation x₁ is given by x₁ = pi(x0).

Similarly for second, thrid and so on. approximation
x₂ = pi(x₁)
x₃ = pi(x₂)
x₄ = pi(x₃)
x_n = pi(x_n-1)

Iteration Method Example:
Find the real root of the equation x³ + x ² = 1 by iteration method.
Solution:
We can rewrite the above equation by
x³ + x ² - 1 = 0;
Let f(x) = x³ + x ² - 1
f(0) = -1 (positive)
f(1) = 1 (negative)
Hence the root value lie between 0 to 1

x³ + x ² - 1 = 0
x² (x + 1) = 1
x² = 1/ (x + 1)
x = 1/ √(x + 1)
pi(x) = 1/ √(x + 1)

Let the initial approximation be x₀ = 0.5

x₁ = pi(x₀) = 1/√1+ 0.5 = 0.81649

x₂ = pi(x₁) = 1/√1+ 0.81649 = 0.74196

x₃ = pi(x₂) = 1/√1+ 0.74196 = 0.75767

x₄ = pi(x₃) = 1/√1+ 0.75767 = 0.75427

x₅ = pi(x₄) = 1/√1+ 0.75427 = 0.75500

x₆ = pi(x₅) = 1/√1+ 0.75500 = 0.75485

x₇ = pi(x₆) = 1/√1+ 0.75485 = 0.75488

Since the difference between x₆ and x₇ are very small, so the root is 0.75488.

Iteration method Practice problem:
1. Solve by iteration method 2x - logx - 7 = 0
2. Find the root of the equation x log x = 1.2 by iteration method
3. Compute the real root of 3x - cosx - 1 = 0 by iteration method
4. Find the root of the equation sin x = 1 + x3 between ( -2,-1) to 3 decimal places by Iteration method