Chapter 4, Problem 9RWDT

To determine

The objective is to show that using the old algorithm is exactly same as using Newton’s method to solve the equation $x^2-N=0.$

Answer to Problem 9RWDT

It is shown that using the old algorithm is exactly same as using Newton’s method.

Explanation of Solution

Given information:

The old algorithm for estimating $\sqrt{N}$ is to make a guess, $G$ by using the average of $G$ and $\frac{N}{G}$ as your next guess (and so on).

New guess:

  $G_1=\frac{G+\frac{N}{G}}{2}$

Concept used:

Newton’s Method:

Guess a first approximation to a solution of $f\left(x\right)=0$ . Use this approximation to get a second, the second to third and so on by using the formula

  $x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

Consider $x^2-N=0.$

Here, $f(x)=x^2-N$

  $f\text{'}\left(x\right)=2x$

Then,

  $\begin{array}{c}{x}_{n+1}=x_n-\frac{x_n{}^2-N}{2x_n}\\ =\frac{2x_n{}^2-x_n{}^2+N}{2x_n}\\ =\frac{x_n{}^2+N}{2x_n}\\ =\frac{x_n+\frac{N}{x_n}}{2}\end{array}$

which is the similar as the old algorithm.