In this post we discuss about** Newton’s Raphson Method **for finding roots. Previously we also discussed various important topics of Numerical techniques like BiSection Method In Numerical Techniques, Regula Falsi Method In Numerical Method, Errors In Numerical Techniques and many more.

## What is Newton’s Raphson Method ?

**Newton’s method **is also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, it is a root-finding algorithm which produces better approximations to the roots of a real-valued function.

This method is generally used to improve the result obtained by one of the previous methods. Let x_{0} be an approximate root of f(x) = 0 and let x_{1}=x_{0}+h be the correct root so that f(x_{1})= 0. Expanding f(x_{0} + h) by Taylor’s series, we obtain

Neglecting the second and higher order derivatives, we have,

**F(x _{0}) + hf^{’ }(x_{0}) =0**

Which gives:

A better approximation than to is therefore given by x, where,

Successive approximations are given by x_{1},x_{2},…..x_{n+1} , where

Geometrically the method consists in replacing the path curve between the point [x_{0} f(x_{0})] and the x-axis by means of the tangent to the curve at the point . it can be used when the roots are algebraic and transcendental and it can also be used when the roots are complex.

**Using the Newton-Raphson method to find a root of the equation x**^{3} -2x -5 = 0

^{3}-2x -5 = 0

From this question we found f(x) = x^{3} – 2x – 5

f’(x) is the derivative of f(x) so f’(x) = 3x^{2}-2

hence the **Newton-Raphson method** equation become

Choosing x_{0} = 2, we obtain f(x_{0}) = -1 and f'(x_{0})= 10. Putting n = 0 , we obtain

This example demonstrates that Newton-Raphson method converges more rapidly than the methods described in regula false, since this requires fewer iterations to obtain a specified accuracy. But since two function evaluations are required for each iteration, Newton-Raphson method requires more computing time.

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