Newton's Method: an Introduction

The most important image in calculus is the movie below. This movie shows a section of a curved mirror with a light ray coming in and bouncing off. As the movie plays, the scene is magnified and under high magnification we see that the curved mirror looks flat. This is the key feature of a differentiable function

y = f(x).

Missing movie Zooming in on a light ray bouncing off a curved mirror.

Because such curves look flat under high magnification, when we are looking at a phenomenon -- like a light ray bouncing off a mirror -- that is a "local" phenomenon -- that is, that happens in a small neighborhood of a point on a curve -- we are able to think of the curve as being flat. Because flat curves are easy to understand, this is often exactly the lever that enables us to solve or at least begin solving a problem that at first seems very difficult. We exploit this idea here to develop a method, called Newton's Method, for estimating solutions of equations.

Newton's Method illustrates both the power and the limitations of this idea. It works well when we are working in a small neighborhood of a point on a curve but it can break down when we are looking at a bigger portion of the curve. For this reason we will see that Newton's Method can break down unless we begin with a good estimate for a solution of our equation.

This module is about estimating solutions of equations. For example, we may want to solve the equation

cos(x) - x  = 0,

or even something as simple as

x3 - 3 = 0.

For this second equation we could and should use properties of exponents to solve -- that is, x = 3(1/3). However, the solution, x*, of cos(x) - x = 0 is not so easily found. One approach would be to carefully graph the function

f(x) = cos(x) - x

and observe where the graph intersects the x-axis.

Graph of cos x - x = 0

This method would yield a rough approximation to x* that could be quite adequate for many applications. What limits the accuracy of this graphical method is the resolution of the computer screeen and our eyes. We will use calculus to construct a method which approximates x* as closely as we desire.

To do this recall that if f has a derivative at x0, then the slope of the line tangent to the graph of f at x0 (let's call this tangent line L) is f'(x0). This means that L and the graph of f(x) itself will stay quite close to each other near x0, and we see this behavior in the figure below. In particular, notice how this forces the intersection of L with the x-axis to be quite close to the intersection of the graph of f(x) with the x-axis.

We now exploit this to approximate as closely as we like the solution we are seeking. First, we find the equation of the line L and use this equation to find the intersection of L with the x-axis. The equation of L is:

y - f(x0) = f '(x0) (x - x0).

At the intersection point with the x-axis, y in the preceding equation is 0. So we solve

0 - f(x0) = f '(x0) (x - x0).

for x. We call this solution x1 and it is

.

Our second step is to see if x1 is a close enough approximation to x*. If it is, we are done. If it isn't, repeat the whole process using x1 in place of x0, to get

.

We now give a general formula for Newton's Method. We get:

Given x0,

Since there is nothing terribly special about f(x) = cos(s) - x, we can use this procedure to solve f(x) = 0 for any differentiable f(x). To see how the process "works" view Newton's Method in Action -- An animation.

Now, use your CAS window to find estimates of the solution(s) of

f(x) = 0

for each choice of f given below:

  1. f(x) = cos(x) - x

    Answer

  2. f(x) = x5 - 7x2 + 2

    Answer

  3. f(x) = x2cos(x) - x

    Answer

How do we know when Newton's Method Works?
Approximation and Convergence Issues

Copyright c 1996 by Steve Hetzler and Bob Tardiff Department of Mathematics and Computer Science, Salisbury State University, Salisbury, MD 21801.