Fixed Point Iteration / Repeated Substitution Method

This is most easiest of all method. The logic is very simple. Given an equation, take an initial guess and and find the functional value for that guess, in the subsequent iteration the result obtained in last iteration will be new guess. Continue this process until get the required accuracy. This means that absoulte differebce between two subsequent functional values in two iterations is too small to be neglected.

But all functions don’t converge. If for given function

|f′(x)|< 1

Then this method will converge to actual solution or in other cases, it will diverge.

Implementation in C

#include< stdio.h>
#include< stdlib.h>
#include< math.h>

#define g(x) sqrt(x+6)
#define gd(a0) (0.5/sqrt(a0+6))
#define f(x) (x*x-x-6)
#define MAX_COUNT 5000
int main()
{
int count=0;
float x0=0,x1=0,error=0;
char iffound=0;
printf("Please enter the initial value: ");
scanf("%f",&x0);
do{
        x1=g(x0);
        error=fabs(x1-x0);
        if(count==0)
        {
                if(gd(x0)>1){
                        printf("\nThe equation is not convergent");
                        iffound=1;
                        break;
                }
                else{
                        printf("\n   i    xi      f(xi)          error");
                        printf("\n-------------------------------------------");
                }
        }
        printf("\n %3d  %3.5f    %3.5f  %3.5f",count,x0,f(x0),count==0?0:error);
        x0=x1;
        count++;
}while(error>0.0005 && count < MAX_COUNT);

if(!iffound)
        printf("\nThe required root is: %f\n",x0);
return 0;
}

Output

Please enter the initial value: 2

  i      xi      f(xi)          error
-------------------------------------------
  0    2.00000  -4.00000       -1.00000
  1    4.00000  6.00000        1.50000
  2    2.50000  -2.25000       0.90000
  3    3.40000  2.16000        0.63529
  4    2.76471  -1.12111       0.40551
  5    3.17021  0.88004        0.27760
  6    2.89262  -0.52538       0.18163
  7    3.07425  0.37674        0.12255
  8    2.95170  -0.23918       0.08103
  9    3.03273  0.16471        0.05431
 10    2.97842  -0.10745       0.03608
 11    3.01449  0.07268        0.02411
 12    2.99038  -0.04799       0.01605
 13    3.00643  0.03220        0.01071
 14    2.99572  -0.02137       0.00713
 15    3.00286  0.01429        0.00476
 16    2.99810  -0.00951       0.00317
 17    3.00127  0.00635        0.00211
 18    2.99915  -0.00423       0.00141
 19    3.00056  0.00282        0.00094
 20    2.99962  -0.00188       0.00063
 21    3.00025  0.00125        0.00042
The required root is: 2.999833