Newton’s Method

 This lesson will be an introduction to Newton's method of approximating roots and will address some of its nuances. As the name suggests, Newton's Method is a method of approximating roots of functions. If you've heard of it before, you may have seen the following formula

and wondered what it meant and how to get there. We can begin to find out by discovering Newton's method ourselves.

Imagine we want to find the root r of a function f. We might start with an initial guess x0, but how should we proceed from there? One thought that may come to mind is instead of finding the root of the function, we can find the root of a linear approximation of the function. It'll be close enough and it's much more doable. So we proceed:

For the function and initial guess in our animation, our approach seems to have worked! Since it has merit, we should write down our findings.

The slope of f at the point x₀ is simply f'(x₀), and in point-slope form, the equation of the linear approximation of f at x₀ is

To find the root of the linear approximation, we set y equal to 0 and solve for x:

Look familiar?

While we might stop here, the truth is that our approximation for r isn't all that good. However, we did get closer than our initial guess x₀. So, common sense dictates that if we use the root of our linear approximation as a new guess, then we should get an even closer approximation, right? Let's try calling this approximation x₁ and going through the same procedure.

And would you look at that! Our new approximation (let's call it x₂) is even closer than x₁! Solving the equation of the new linear approximation for x₂ as we did with x₁ before, we have

Now we're getting somewhere. Let's try it again one more time, just for good measure.

That one was absurdly close! And once again, we have

If we keep using our approximation as the new starting point to get closer and closer to the root, we can generalize this as

How about that?

Now, we'll delve into some discussion on some of the limitations of Newton's method.

Consider x to the fourth power and try to use some initial guess and Newton's method to approximate its root at 0.

As the animation makes obvious, Newton's method takes ages to get anywhere near the root. Visually, this can be seen from how steep the function becomes. What causes both of these, however, is the multiplicity of the root at 0. It's good to keep in mind that the higher the multiplicity of a root, the less effective Newton's method will be at approximating it.

Next, let's consider the cube root of x and again try to approximate its root at 0.

In a shocking turn of events, Newton's method fails entirely despite an extremely close initial guess, actually taking us farther away from the root with each iteration! In this case, Newton's method fails because the derivative of f(x) does not exist at x = 0. In fact, for any function not differentiable at the root being approximated, Newton's method will always fail.