In this post, we discuss 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 that produces better approximations of the roots of a real-valued function.

This technique is typically used to enhance the outcome of a preceding technique. 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

Geometrically, the method replaces 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 the 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, the Newton-Raphson method requires more computing time.

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