The Newton Raphson method finds successively better approximations to roots of the real valued function \( f(x) = 0 \).
Refer to the above image. We start off with the approximate root , find the value of , take the tangent to the curve, which results in the approximate root . We continue this process, find the value of , take the tangent to the curve, which results in the approximate root . By continuing the process, we should reach the actual root , to a great degree of accuracy.
Let's write this down in equation form. We start off with a first approximation, which is denoted by . Then, we consider the point , which has a slope of , hence the equation of the tangent is
and the intersection with the x-axis occurs when , or that
We repeat this process with
until a sufficiently accurate value is reached.
Find the value of as the root of the equation , using the starting value of . Note that all values of will be rational numbers, hence we get a sequence of rationals which approximate the value of the irrational number .