Problem 7 - Newton Method

Consider the Newton method for $f(x)=\frac{1}{x}-a=0\; (a>0)$.

1.

Give the Newton iteration for this equation.

2.

Give the range of the initial value for which the Newton method converges to the solution.

3.

Show that this method converges quadratically.

1.

2.

$e_n = \vert x_n - \frac{1}{a}\vert$


Thus, the Newton method converges for , i.e., for

3.

Thus, the Newton method converges quadratically.