How root converges in bisection method?

Answer 1

It need not necessarily do so. For example, consider f(x) = 1/(x-2)Suppose you start with x = 5 which gives f(x) = 0.33... and x = -5 which gives f(x) = -0.14286

Bisecting the interval (-5, 5) gives x = 0 and so f(x) = -0.5

which is further away from the previous value.

Answer 2

In the bisection method, the root convergence occurs by repeatedly dividing the interval that contains the root into smaller intervals. In each iteration, the method checks whether the midpoint of the interval is the root or if it lies on one side of the root. The method then selects the subinterval where the root lies and continues to divide it further until the desired level of accuracy is achieved. The convergence is guaranteed because the interval containing the root is halved in each iteration.