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