Newton’s Raphson Method

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 x0 be an approximate root of f(x) = 0 and let x1=x0+h be the correct root so that f(x1)= 0. Expanding f(x0 + h) by Taylor’s series, we obtain

Neglecting the second and higher order derivatives, we have,

F(x0) + hf(x0) =0

Which gives:

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

Successive approximations are given by x1,x2,…..xn+1 , where

Which is the Newton-Raphson formula.

Geometrically  the method consists in replacing the path curve between the point [x0 f(x0)] 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 x3 -2x -5 = 0

From this question we found f(x) = x3 – 2x – 5

f’(x) is the derivative of f(x) so f’(x) = 3x2-2

hence the Newton-Raphson method equation become

Choosing x0 = 2, we obtain f(x0) = -1 and f'(x0)= 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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