Newton's Method for Approximating Roots

# Newton's Method for Approximating Roots

We will now look at another method for approximating roots of functions. Suppose that $y = f(x)$, and let $\alpha$ be a root of $f$, and suppose that $x_0$ is a first approximation of the root $\alpha$ of $f$. Now consider the tangent line to the graph of $f$ at the point $(x_0, f(x_0))$. Provided that this tangent line does not have slope $0$, then this tangent line will have a root of its own that will approximately be equal to $\alpha$. The equation of this tangent line can be given by the following equation:

(1)
\begin{align} \quad p_1(x) = f(x_0) + f'(x_0)(x - x_0) \end{align}

If we set $p_1(x) = 0$ and solve for $x_1$ as the $x$-intercept of the tangent line $p_1(x)$, then we obtain:

(2)
\begin{align} \quad p_1(x) = f(x_0) + f'(x_0)(x - x_0) \\ \quad 0 = f(x_0) + f'(x_0)(x_1 - x_0) \\ \quad x_1 = x_0 -\frac{f(x_0)}{f'(x_0)} \end{align}

We now take $x_1$ (the root of the first tangent line) to be an approximation of $\alpha$. If we now look at the tangent line at $(x_1, f(x_1))$ on $f$, then we obtain a new tangent line, and provided that the slope of this tangent line is not $0$, then this tangent line has a root of its own that is an even better approximation of $\alpha$. The equation of this tangent line can be given by the following equation:

(3)
\begin{align} \quad p_2(x) = f(x_1) + f'(x_1)(x - x_1) \end{align}

Once again, if we set $p_1(x) = 0$ and solve for $x_2$ as the x-intercept of the tangent line $p_2(x)$, then we obtain:

(4)
\begin{align} \quad x_2 = x_1 -\frac{f(x_1)}{f'(x_1)} \end{align}

In fact, the more we repeat this procedure, the closer and closer our approximation gets to $\alpha$. For $n + 1$ iterations of this procedure and provided that $f'(x_i) \neq 0$ for $i = 1, 2, ..., n$, we obtain the following general formula for the $x$-intercepts of the corresponding tangent lines as approximations to $\alpha$:

(5)
\begin{align} \quad x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \end{align}
 Theorem 1 (Newton's Method): Suppose that $f$ is a differentiable function that contains the root $\alpha$, and $x_0$ is an approximation of $\alpha$. Step 1: A better approximation of $x_1$ can be obtained as $x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$ provided that $f'(x_0) \neq 0$. Step n + 1: A better approximation of $x_{n}$ can be obtained as $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ provided that $f'(x_n) \neq 0$.

One advantage of Newton's Method is that the sequence of approximations $\{ x_n \}$ tend to converge much more quickly towards the root $\alpha$. The major disadvantage of Newton's Method is that the sequence of approximations $\{ x_n \}$ may not converge to $\alpha$ if we do not choose an initial approximation $x_0$ that is sufficiently close to $\alpha$. We discuss this potential problem on the Error Analysis of Newton's Method for Approximating Roots page.