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(x**, we conclude that_{0}) = 0**x**is the root. Otherwise root lies between_{0}**x**and_{0}**b**(if**f(x**is positive) or between_{0})**a**and**x**(if_{0}**f(x**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_{0})^{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).*

An excellent beginning!!! All the best

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