In numerical analysis, Newton’s method is named after Isaac Newton and Joseph Raphson. This method is to find successively better approximations to the roots (or zeroes) of a real-valued function.

The method starts with a function f defined over the real numbers x, the function’s derivative f’, and an initial guess

for a root of the function f. If the function satisfies the assumptions made in the derivation of the formula and the initial guess is close, then a better approximation will occur.

In general, solving an equation f(x) = 0 is not easy, though we can do it in simple cases like finding roots of quadratics. If the function is complicated we can approximate the solution using an iterative procedure also known as a numerical method. One simple method is called Newton’s Method.

The formula for Newton’s method is given as,

\[\large x_{1}=x_{0}-\frac{f(x_{0})}{{f}'{(x_{0})}}\]

Where,
f(

) is a function at ,

f'(

) is the first derivative of the function at ,

is the initial value.

Solved Example

Question: Estimate the positive root of the equation x² – 2 = 0 by using Newton’s method. Begin with x₀ = 2 and compute x₁.

Solution:

Given measures are,
f(x) = x² – 2 = 0, x₀ = 2

Newton’s method formula is: x₁ = x₀ – 

To calculate this we have to find out the first derivative f'(x)
f'(x) = 2x
So, at x₀ = 2,
f(x₀) = 2² – 2 = 4 – 2 = 2
f'(x₀) = 2

2 = 4

Substituting these values in the formula,

x₁ = 2 – 

=

\(\begin{array}{l}\frac{6}{4}\end{array} \)

=

\(\begin{array}{l}\frac{3}{2}\end{array} \)