Newton's method in the real domain

Subsection 1.2.2 Newton's method in the real domain

Let's take a more general look at Newton's method. The problem is to estimate a root of a real, differentiable function \(f\text{,}\) i.e. a value of \(x\) such that \(f(x)=0\text{.}\) Suppose also that we have an initial guess \(x_0\) for the actual value of the root, which we denote \(r\text{.}\) Often, the value of \(x_0\) will be based on a graph. Since \(f\) is differentiable, we can estimate \(r\) with the root of the tangent line approximation to \(f\) at \(x_0\text{;}\) let's call this point \(x_1\text{.}\) When \(x_0\) is close to \(r\text{,}\) we often find that \(x_1\) is even closer. This process is illustrated in Figure 1.2.2 where it indeed appears that \(x_1\) is much closer to \(r\) than \(x_0\text{.}\)

Figure 1.2.2 One step in Newton's method

We now find a formula for \(x_1\) in terms of the given information. Recall that \(x_1\) is the root of the tangent line approximation to \(f\) at \(x_0\text{;}\) let's call this tangent line approximation \(\ell\text{.}\) Thus,

\begin{equation*} \ell(x) = f(x_0) + f'(x_0)(x-x_0) \end{equation*}

and \(\ell(x_1)=0\text{.}\) Thus, we must simply solve

\begin{equation*} f(x_0) + f'(x_0)(x-x_0) = 0 \end{equation*}

for \(x\) to get \(x_1 = x_0-f(x_0)/f'(x_0)\text{.}\)

Now, of course, \(x_1\) is again a point that is close \(r\text{.}\) Thus, we can repeat the process with \(x_1\) as the guess. The new, better estimate will then be

\begin{equation*} x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}. \end{equation*}

The process can then repeat. Thus, we can define a sequence \((x_n)\) recursively by

\begin{equation*} x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. \end{equation*}

This process is called iteration and the sequence it generates often converges to the root \(r\text{.}\)