Fixed Point Iteration Method Algorithm
Fixed point iteration method is open and simple method for finding real root of non-linear equation by successive approximation. It requires only one initial guess to start. Since it is open method its convergence is not guaranteed. This method is also known as Iterative Method
To find the root of nonlinear equation f(x)=0 by fixed point iteration method, we write given equation f(x)=0 in the form of x = g(x).
If x0 is initial guess then next approximated root in this method is obtaine by:
x1 = g(x1)
And similarly, next to next approximated root is obtained by using value of x1 i.e.
x2 = g(x2)
And the process is repeated until we get root within desired accuracy.
Note: While expressing f(x)=0 to x = g(x) we can have many different forms. For convergence, following criteraia must be satisfied.
Algorithm for Fixed Point Iteration Method
1. Start 2. Define function f(x) 3. Define function g(x) which is obtained from f(x)=0 such that x = g(x) and |g'(x) < 1| 4. Choose intial guess x0, Tolerable Error e and Maximum Iteration N 5. Initialize iteration counter: step = 1 6. Calculate x1 = g(x0) 7. Increment iteration counter: step = step + 1 8. If step > N then print "Not Convergent" and goto (12) otherwise goto (10) 9. Set x0 = x1 for next iteration 10. If |f(x1)| > e then goto step (6) otherwise goto step (11) 11. Display x1 as root. 12. Stop