Let g R rightarrow R be such a function that g elementof C1

Let g: R rightarrow R_+ be such a function that g elementof C^1 (R) and for all x elementof R. - 1

Solution

Fixed point :  A point, say, s is called a fixed point if it satisfies the equation x = g(x).

Fixed point Iteration : The transcendental equation f(x) = 0 can be converted algebraically into the form x = g(x) and then using the iterative scheme with the recursive relation

x_n+1= g(x_n),           i = 0, 1, 2, . . .,

with some initial guess x₀  is called the fixed point iterative scheme.

Algorithm - Fixed Point Iteration Scheme

Given an equation f(x) = 0  
Convert f(x) = 0 into the form x = g(x)  
Let the initial guess be x₀
Do  
       x_n+1= g(x_n)
while (none of the convergence criterion C1 or C2 is met)

C1. Fixing apriority the total number of iterations N .

C2. By testing the condition | x_n+1 - g(x_n) | (where i is the iteration number) less than some tolerance limit, say epsilon, fixed apriority.

Algorithm - Fixed Point Iteration Scheme Given an equation f(x) = 0   Convert f(x) = 0 into the form x = g(x)   Let the initial guess be x0 Do          xn+1= g(xn) while (none of the convergence criterion C1 or C2 is met)