Problem 8 - Newton Method

Answer the following questions concerning solving an equation $f(x)=\frac{1}{x^2}-a=0$ by the Newton method.

1.

Give an iterative formula.

2.

Find a region of initial values to guarantee its convergence.

3.

Compute the order of convergence.

1.

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n + \frac{\frac{1}{x_n^2}-a}{2\frac{1}{x_n^3}} = \frac{3}{2}x_n - \frac{a}{2}x_n^3$

2.

3.

$e_{n+1} = \left\vert \frac{-3 x_n\sqrt{a}+x_n^3 a\sqrt{a}+2}{-2\sqrt{a}} \righ... ...t = \left\vert \frac{(x_n\sqrt{a}-1)^2(x_n\sqrt{a}+2)}{-2\sqrt{a}} \right\vert$
$\quad = \left\vert \frac{a\left(x_n - \frac{1}{\sqrt{a}} \right)^2 (x_n\sqrt{a}+2)}{-2\sqrt{a}} \right\vert$
$\frac{e_{n+1}}{e_n^2} = \left\vert \frac{ a(x_n\sqrt{a}+2) }{-2\sqrt{a}} \right\vert$
$\lim_{n\rightarrow \infty} \frac{e_{n+1}}{e_n^2} = \left\vert -\frac{3a}{2\sqrt{a}} \right\vert$
Thus, the order of convergence is 2. This answers the third question.