Bisection Method

If a function f(x) is continuous from a to b and f(a) and f(b) are of opposite signs, then there exists at least one x such that f(x) = 0. This fact is used in finding the root of the given equation by bisection method. Here, let us assume that f(a) is positive and f(b) is negative. Then the root lies between a and b and let us approximate it as

If f(x₀) = 0, we conclude that x₀ is the root. Otherwise root lies between x₀ and b (if f(x₀) is positive) or between a and x₀ (if f(x₀) is negative) and accordingly we have to choose the interval. We have to repeat the process till the required accuracy is reached. In each iteration the width of the interval becomes half of the previous iteration. So width of the interval after n^th iteration is given by

If e is the desired accuracy then we have,

 On simplification, 

Using this inequality, one can estimate the number of iterations required to achieve accuracy e.

Graphical representation of bisection method

References:

1.         Introductory Methods of Numerical Analysis, S.S. Sastry, Prentice Hall of India (1983).

2.         Numerical Methods, E Balagurusamy, Tata McGraw-Hill Publishing Company Ltd (1999).