Bisection method:( Binary chopping or Bolzano's method) to find the roots of a equation

 Bisection method:( Binary chopping or Bolzano's method)

In general, if  f(x) is real and continuous in the interval Xl and Xu and f(Xl) and f(Xu) have opposite signs, that is f(Xl)*f(Xu)<0 then there exist one real root between Xl and Xu.

The location of the sign change is identified more precisely by dividing the interval into number of sub intervals.

Bisection method is one type of  incremental search method in which the interval is always divided in half. If a function changes changes sign over an interval, the function value at the mid point is evaluated. The location of the root is then determined as lying at the midpoint of  the sub interval within which the sign change occurs.

Steps involved in Bisection method:

1.  Choose lower and upper values of X (i.e Xl, Xu)insuch a way that the function sholud change it's sign over the interval.Ensure f(Xl)*f(Xu)<0
2. Now the estimated root is, Xm = ( Xl+ Xu)/2
3. Make the following evaluations to find in which sun interval the root lies 

• If  f(Xl)*f(Xm)<0, the root lies in the lower sub interval. Therefore take Xu=Xm and return to step 2.
• If  f(Xl)*f(Xm)>0, the root lies in the upper sub interval. Therefore take Xl=Xm and return to step 2.
• If  f(Xl)*f(Xm)=0, the root is Xm and stop the computation.