Newton's method for solving the equation $f\left(x\right)\=0$ is an iterative numeric technique defined by

$x_{n\+1}\=g\left(x_n\right)\equiv x_n-\frac{f\left(x_n\right)}{f\prime \left(x_n\right)}$ 

This "formula" expresses the simple idea that the $x$-intercept on a tangent line drawn at $x_k$ near an $x$-intercept for the graph of $f$, is generally closer to the intercept of $f$ than $x_k$. Indeed, the point-slope form of the equation of the line tangent at $x_k$ is

$y\=f\prime \left(x_k\right)\left(x-x_k\right)\+f\left(x_k\right)$ 

The $x$-intercept on this line is the solution for $x$ when $y\=0$. This $x$ becomes $x_{k\+1}$ by the calculation in Table 3.2.1.

In general, Newton's method converges rapidly to a root of $f\=0$ provided the initial point is chosen "close enough" to the root. However, the method is not infallible, and there are functions for which the method will converge more slowly, will cycle back and forth between two distinct iterates, will seem to converge but not to a root, and will fail to find an actual root. One of these anomalies is the subject of Example 3.2.3.$$