Convergence of Bisection Method

The rate of convergence of the Bisection method is linear and slow but it is guaranteed to converge if function is real and continuous in an interval bounded by given two initial guess.

Accuracy of bisection method is very good and this method is more reliable than other open methods like Secant, Newton Raphson method etc.

Despite being slower to converge, accuracy of this method increases as number of iterations increases.

In Bisection method, error is reduced by factor of ½ after each iterations, so we can write: e_n+1/e_n = 1/2. Which gives:

e_n+1 = e_n/2

Or, e_n+1 = 0.5 e_n ----- (1)

Here e_n+1 is error at n+1^th iteration and e_n is error at n^th iteration.

From equation (1) it can be concluded that, error at n+1^th iteration is linearly related to error at n^th iteration i.e. e_n+1 ∝ e_n

For this reason, Bisection method is said to have linear rate of convergence.