The Newton Raphson Method is an iterative method for finding roots.
Suppose we know the rough location of a root
to a function
we know it is close to some value
We can expand the function
in a Taylor series about a value of
close to
Since
is 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 equation
starting from
We could go on but these two values agree to three decimal places, so to 3 decimal places,