## Newton's Method for Finding Roots

Read this section. Work through practice problems 1-6.

### Iteration

We have been emphasizing the geometric nature of Newton's method, but Newton's method is also an example of iterating a function. If $\mathrm{N}(\mathrm{x})=\mathrm{x}-\frac{\mathrm{f}(\mathrm{x})}{\mathrm{f}^{\prime}(\mathrm{x})}$, the "pattern" in the algorithm, then

$\mathrm{x}_{1}=\mathrm{x}_{0}-\frac{\mathrm{f}\left(\mathrm{x}_{0}\right)}{\mathrm{f}^{\prime}\left(\mathrm{x}_{0}\right)}=\mathrm{N}\left(\mathrm{x}_{0}\right)$,

$\mathrm{x}_{2}=\mathrm{x}_{1}-\frac{\mathrm{f}\left(\mathrm{x}_{1}\right)}{\mathrm{f}^{\prime}\left(\mathrm{x}_{1}\right)}=\mathrm{N}\left(\mathrm{x}_{1}\right)=\mathrm{N}\left(\mathrm{N}\left(\mathrm{x}_{0}\right)\right)=\mathrm{N} \circ \mathrm{N}\left(\mathrm{x}_{0}\right)$,

$\mathrm{x}_{3}=\mathrm{N}\left(\mathrm{x}_{2}\right)=\mathrm{N} \circ \mathrm{N} \circ \mathrm{N}\left(\mathrm{x}_{0}\right)$, and, in general,

$\mathrm{x}_{\mathrm{n}}=\mathrm{N}\left(\mathrm{x}_{\mathrm{n}-1}\right)=\mathrm{n}^{\text {th }}$ iteration of $\mathrm{N}$ starting with $\mathrm{x}_{0} .$

At each step, we are using the output from the function $\mathrm{N}$ as the next input into $\mathrm{N}$.