Newton's Method

The iteration method in the previous section can work very slowly. However, it provides the basis for Newton's method , which is usually much faster than direct iteration. Newton's method is based on the observation that the tangent line is a good local approximation to the graph of a function.

Let $\left(\vphantom{ x_0,f(x_0)}\right.$x0, f (x0)$\left.\vphantom{ x_0,f(x_0)}\right)$ be a point on the graph of the function f. The tangent line is given by the equation

y - f (x0) = f′(x0)(x - x0)

This line crosses the x-axis when y = 0. The corresponding value of x is given by

x = x0 - $\displaystyle {\frac{{f(x_0)}}{{f^{\prime }(x_0)}}}$

In general, given an approximation xn to a zero of a function f (x), the tangent line at the point $\left(\vphantom{ x_n,f(x_n)}\right.$xn, f (xn)$\left.\vphantom{ x_n,f(x_n)}\right)$ crosses the x-axis at the point $\left(\vphantom{ x_{n+1},0}\right.$xn+1, 0$\left.\vphantom{ x_{n+1},0}\right)$ where

xn+1 = xn - $\displaystyle {\frac{{f(x_n)}}{{f^{\prime }(x_n)}}}$

Given x0, Newton's method produces a list x1, x2, …, xn of approximations to a zero of f.

In the following graph, f (x) = x - x3, x0 = 0.44, x1 $\approx$ - 0.41, x2 $\approx$ 0.27, and x3 $\approx$ - 0.048.

dtbpFU3in2.0003in0ptx - x3

This figure can be generated by plotting x - x3 as usual, zooming in to change the viewing rectangle, then selecting the matrix

$\displaystyle \left[\vphantom{ \begin{array}{ll} .44 & 0 \\ .44 & f(.44) ... ...\ .27 & f(.27) \\ -.048 & 0 \\ -.048 & f(-.048) \end{array} }\right.$$\displaystyle \begin{array}{ll} .44 & 0 \\ .44 & f(.44) \\ -.41 & 0 \\ ... ... .27 & 0 \\ .27 & f(.27) \\ -.048 & 0 \\ -.048 & f(-.048) \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{ll} .44 & 0 \\ .44 & f(.44) ... ...\ .27 & f(.27) \\ -.048 & 0 \\ -.048 & f(-.048) \end{array} }\right]$

and dragging it to the frame.

The Newton iteration function for a function f is the function g defined by

g(x) = x - $\displaystyle {\frac{{f(x)}}{{f^{\prime }(x)}}}$


\begin{example} This example uses Newton's method to solve the same equation $x... ...166\right) =\allowbreak .73908513321516064165 \end{displaymath} \end{example}


\begin{example} Make a graph of $y=\cos x$\ and $y=x$\ to see the approximate s... ...ive the response \lq\lq Solution is: $% \left\{ x=.73909\right\} $.'' \end{example}


\begin{example} Consider the function $f(x)=x(2-x^{2})$, a function that has ze... ...ts 49;axesstyle ''normal'';xis \TEXUX{x};var1name \TEXUX{$x$};}} \end{example}