# The Algorithm for Newton's Method for Approximating Roots

We will now look at the algorithm for Newton's Method for approximating roots to functions.

Obtain a function $f$ and assume that a root $\alpha$ exists. Obtain an initial approximation $x_0$ to this root. Also obtain a maximum number of iterations allowed and an error tolerance $\epsilon$.

For $i = 1, 2, ...$ up to the maximum number of iterations prescribed:

**Step 1:** Obtain the successive approximations of the root $\alpha$ with the following formula:

**Step 2:** Check the error tolerance:

If the above inequality does not hold, then continue to obtain successive approximations for $\alpha$. If the above inequality is true, then further verify the accuracy of $x_n$. Check that to see whether or not:

(3)If the signs of $f(x_n + \epsilon)$ and $f(x_n - \epsilon)$ are opposites of each other, then the accuracy of $x_n$ is verified and stop the algorithm. $x_n$ is a good approximation to $\alpha$. If the signs of $f(x_n + \epsilon)$ and $f(x_n - \epsilon)$ are the same, then $\alpha$ is not contained in the small interval $[x_n - \epsilon, x_n + \epsilon]$. Print out an error message.

If the maximum number of iterations is reached, then print our a failure message.