## Theory

The bisection method is the simplest and most reliable iterative method. Also known as binary chopping or half-interval method.

If f(x) is real and continuous, and f(x) & f(x2) are of opposite sign i.e. f(x1)*f(x2) < 0 then there is at least one root in the interval between x1 and x2 i.e. xm = (x1 + x2) / 2

Now, there exist the following three conditions.

- if f(xm) = 0, root is at xm
- if f(xm)*f(x1) < 0, root is betn xm & x2
- if f(xm)*f(x2) < 0, root is betn xm & x1

Then we further bisect the new further interval and continue the process until the root of desire accuracy is determined.

## Algorithm

- Start
- Define function f(x) and error E (stoping criteria)
- Read initial values x1 & x2 and funcational value f(x1) & f(x2)
- Check the bracketing condition
- if f(x1)*f(x2) > 0; goto step 3

- Find mid-point, xm = (x1 + x2) / 2 & find f(xm)
- If |f(xm)| <= E goto step 8
- If f(xm)*f(x1) < 0
- set x2 = xm
- else
- x1 = xm

- goto step 5

- Print root xm
- Stop

## Source Code

#include<iostream> #include<math.h> #define f(x) (x*x - 4*x - 10) #define E 0.0001 using namespace std; int main() { float x1, x2, xm; do { cout<<"Enter two points\n"; cin>>x1>>x2; }while(f(x1)*f(x2) > 0); do { xm = (x1 + x2) / 2; if(f(x1)*f(xm) < 0) { x2 = xm; } else { x1 = xm; } } while(fabs(f(xm))>E); cout<<"Root is "<<xm; return 0; }

## Output

Enter two points

1

-2

Root is -1.74166