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.
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
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