Fixed Point Method to find the root of the non-linear equation

Fixed Point Method [Linearly Convergent]

1.      Assign an initial value to x₀ and accuracy level E.

2.      Reform given equation as x = g(x)

3.      Compute x₁ = g(x₀).

4.      Check the required accuracy level of x₁.

If absolute value of (x₁ – x₀) / x₁ < E then

Write x₁ as the root of the equation and go to 4.

Otherwise

            Set x₀ = x₁ and

            Go to step 2

5.      Stop

The equation x_i+1 = g (x_i) where I = 0, 1, 2 … is called the fixed point iteration formula.

Example: Find the square root of 5 using the equation x² - 5 = 0 by using fixed point iteration method.

Solⁿ: Here f(x) = x² – 5

Step 1: Assigning Initial value of x₀ = 1 and E = 0.01

Step 2: Reforming the equation x² - 5 = 0

            Or x² = 5

            Or x = 5/x

            Or 2x = 5/x + x [Adding x in both sides]

             x = [5/x + x] / 2

Step 3: Iteration I:

Computing x₁ = g(x₀).

                        x₁=[5/x₀ + x₀] / 2 = 3  [x_{0 = 1}]

Step 4: Checking accuracy level,          

            Absolute value of (x₁ – x₀) / x₁ < E

            (3 - 1) / 3 = 0.6667

            0.6667 < 0.01 which is false

            So, set x₀ = x₁ and go to step 3

            Iteration II:

            Computing x₁ = g(x₀).

                        x₁=[5/x₀ + x₀] / 2 = …

Repeat the iterations until error criteria satisfied.

If error criteria is satisfied then,

Write improved estimation x₁ as the root of the equation and stop.