Newton-Raphson方法

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 x1=3/2=1.5. If we square x1=3/2, we obtain 9/4, which is bigger than 2. Consequently

.
If we now consider 2/x1=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 x2=17/12, we obtain 289/144, which is bigger than 2. Consequently

.
If we now consider 2/x2=24/17, its square 576/289 is of course smaller than 2, so

.
Let's take their average again:



x3 is a pretty good rational approximation to the square root of 2:



but if this is not good enough, we can just repeat the procedure again and again.
Newton and Raphson used ideas of the Calculus to generalize this ancient method to find the zeros of an arbitrary equation



Their underlying idea is the approximation of the graph of the function f(x)
by the tangent lines, which we discussed in detail in the previous pages.
Let r be a root (also called a "zero") of f(x), that is f(r) =0. Assume that

.
Let x1 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 (x1,f(x1)) has x2 as
its x-intercept.

From the above picture, we see that x2 is getting closer to r. Easy calculations give



Since we assumed

,
we will not have problems with the denominator being equal to 0. We continue this process and find x3 through
the equation



This process will generate a sequence of numbers

which
approximates r.
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



Let us start this process by taking x1 = 2.



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



In fact, looking at the graphs we can see that this equation has one solution.

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 x1=
1. The rest of the sequence is generated through the formula



We have