Newton’s Method Formula

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 $x_{0}$ 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 $x_{1}$.

In general solving an equation f(x) = 0 is not easy, though we can do it in simple cases like find 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($x_{0}$) is a function at $x_{0}$, f'(x) is the first derivative of the function at $x_{0}$, $x_{0}$ is the initial value.

Solved Examples

Question 1: Estimate the positive root of the equation x– 2 = 0 by using Newton’s method. Begin with x0 = 2 and compute x1

Solution:

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

Newton’s method formula is: x1 = x– $\frac{f(x-{0})}{f'(x_{0})}$

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

The formula becomes

x1 = 2 – $\frac{2}{4}$ = $\frac{6}{4}$ = $\frac{3}{2}$


Practise This Question

Tribhuvan Das is an ice-cream scientist and it is his job to make different yet delicious varieties of ice-creams. Before being frozen, ice cream mixtures are in the colloidal state. For months, Tribhuvan Das has been experimenting with new flavours but never got satisfactory results. On a Saturday night just before Diwali, he accidentally mixed up two different samples and got a delicious Sundae. Tribhuvan sent the sample to his boss for tasting, but the boss found it a bit saltier for his palate. It is Diwali time so Tribhuvan is away on leave. The boss is impatient and wants to taste the ice-cream but without the extra saltiness. What should he do?