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)
xn = pi(xn-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


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