The Newton Raphson Method is an iterative method for finding roots.

Suppose we know the rough location of a rootto a functionwe know it is close to some valueWe can expand the functionin a Taylor series about a value of close to

Sinceis close to 0 we can say

Generalizing we have

This is illustrated in the diagram below.

Each iteration gives a value closer to the true root, which is used in the next iteration to find a value closer to the true root.

Example: Find the rootof the equationstarting from

We could go on but these two values agree to three decimal places, so to 3 decimal places,