Fixed-Point Iteration Method

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

ALGORITHM:

1. Given an equation f(x) = 0.
2. Convert f(x) = 0 into the form x = g(x).
3. Let the initial guess be x_i.
4. Do x_i+1 = g(x_i).
5. If |x_i+1 - x_i| < tolerance, root = x_{i+1, Else go to step 4.}
6. _STOP.

PROGRAM in C:

#include<stdio.h> 

#include<conio.h>

#include<math.h>

float f(float x)

{

  return ((pow(x,2)+2)/3); /* x2 -3x+2 = 0 */

}

void main()

{

  float x0,x1=0,tmp,eps=0.001;

  int iteration=0;

  printf("Enter initial guess: ");

  scanf("%f",&x0);

  do

  {

    tmp=x1;

    x1=f(x0);

    iteration++;

    printf("\n%d\t%f",iteration,x1);

    x0=x1;

  }while(fabs(tmp-x1)>eps);

  printf("\n\n Real root=%0.4f\n Iteration=%d",x1,iteration);

  getch();

}

Horner's Method >>