The Newton-Raphson Method

Already the Babylonians knew how to approximate square roots. Let's consider the example of how they found approximations to .

Let's start with a close approximation, say x 1 =3/2=1.5. If we square x 1 =3/2, we obtain 9/4, which is bigger than 2. Consequently . If we now consider 2/x 1 =4/3, its square 16/9 is of course smaller than 2, so .

We will do better if we take their average:



If we square x 2 =17/12, we obtain 289/144, which is bigger than 2. Consequently . If we now consider 2/x 2 =24/17, its square 576/289 is of course smaller than 2, so .

Let's take their average again:



x 3 is a pretty good rational approximation to the square root of 2:



Newton and Raphson used ideas of the Calculus to generalize this ancient method to find the zeros of an arbitrary equation



Let r be a root (also called a "zero") of f(x), that is f(r) =0. Assume that . Let x 1 be a number close to r (which may be obtained by looking at the graph of f(x)). The tangent line to the graph of f(x) at (x 1 ,f(x 1 )) has x 2 as its x-intercept.

From the above picture, we see that x 2 is getting closer to r. Easy calculations give



3

This technique of successive approximations of real zeros is called Newton's method, or the Newton-Raphson Method.

Example. Let us find an approximation to to ten decimal places.

Note that is an irrational number. Therefore the sequence of decimals which defines will not stop. Clearly is the only zero of f(x) = x2 - 5 on the interval [1,3]. See the Picture.

Let be the successive approximations obtained through Newton's method. We have



1





It is quite remarkable that the results stabilize for more than ten decimal places after only 5 iterations!

Example. Let us approximate the only solution to the equation



This solution is also the only zero of the function . So now we see how Newton's method may be used to approximate r. Since r is between 0 and , we set x 1 = 1. The rest of the sequence is generated through the formula

