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}.