THE BISECTION METHOD

In general, if f(x) is real and continuous in the interval from Xl to XU and f(xl) and f(xU) have opposite signs, that is, . f(xl)*f(xU ) < 0 then there is at least one real root between Xl and Xu. Incremental search methods capitalize on this observation by locating un interval where the function changes sign.

 Then the location of the sign change (and consequently, the root) is identified more precisely by dividing the interval into a number of subintervals Each of these subinterval s is searched to locate the sign change. The process is repeated and the root estimate refined by dividing the subintervals into finer increments. 

The bisection method, which is alternatively called binary chopping, interval halving, or Bolzano’s method, is one type of incremental search method in which the interval is always divided in half. If a function changes sign over an interval, the function value at the midpoint is evaluated.

 The location of the root is then determined as lying at the midpoint of the subinterval within which the sign change occurs. The process is repeated to obtain refined estimates.