Solve the bisection method equation using C

The bisection method is based on the following result from calculus:

The Intermediate Value Theorem: Assume f : IR→IR is a continuous function and there are two real numbers a and b such that f (a)f (b) < 0. Then f (x) has at least one zero between a and b.

In other words, if a continuous function has different signs at two points, it has to go through zero somewhere in between!

The bisection method consists of finding two such numbers a and b, then halving the interval [a,b] and keeping the half on which f (x) changes sign and repeating the procedure until this interval shrinks to give the required accuracy for the root.

An algorithm could be defined as follows. Suppose we need a root for f (x) = 0 and we have an error tolerance of ε (the absolute error in calculating the root must be less that ε).

Bisection Algorithm:

1. Find two numbers a and b at which f has different signs. 
2. Define c = a+b  
3. If b−c ≤ε then accept c as the root and stop. 
4. If f (a)f (c)≤0 then set c as the new b. Otherwise, set c as the new a. Return to step 1.

Error bounds
Let α be the value of the root, a≤α≤b. Let an, bn and cn be the values of a, b and c on the nth iteration of the algorithm.

Then the error bound for cn is given by
|α−c_n| ≤  (1/2ⁿ) (b−a)

This inequality can give us the number of iterations needed for a required accuracy ε
n ≥  ((log(b−a)/ε)) / log2

Advantages and disadvantages of the bisection method

1. The method is guaranteed to converge.
2. The error bound decreases by half with each iteration.
3. The bisection method converges very slowly.
4. The bisection method cannot detect multiple roots

Example Code:

Bisection Method x²-5

root between 2 and 3

File name : bisection.c

#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>
float f(float x)
{
 return (x*x)-5;
}
main()
{
 float x0,x1,x2,E;
 int count=1;
 charch;
 clrscr();
 xyz:
 printf("\nEnter value of x0");
 scanf("%f",&x0);
 printf("\nEnter value of x1");
 scanf("%f",&x1);
 printf("\nEnter value of error interation:");
 scanf("%f",&E);
 clrscr();
 printf("\n NO\tx0\tx1\tx2\tf(x0)\tf(x1)\tf(x2)");
 if(f(x0)*f(x1)<0)
 {
  x2=(x0+x1)/2;
  while(fabs(x1-x0)>E)
  {
   printf("\n %d\t%.3f\t%.3f\t%.3f\t%.3f\t%.3f\t%.3f",count,x0,x1,x2,f(x0),f(x1

),f(x2));
   if(f(x0)*f(x2)<0)
   {
    x1=x2;
   }
   else
   {
    x0=x2;
   }

   count++;
   x2=(x0+x1)/2;
  }
  printf("\nRoot is %f",x2);
 }
 else
 {
  clrscr();
  printf("\nYour value is incorrect...\nPlease enter a new value then Press 'Y' key otherwise any key press to exit!....");
  fflush(stdin);
  scanf("%c",&ch);
  if(ch=='y'||ch=='Y')
  {
   goto xyz;
  }
  else
  { 
   exit(0);
  }
 }
 getch();
}