Newton's Method with an Algorithm

[Violagirl]

Hi, I've been having a hard time with trying to solve this problem. The algorithm in it throws me off and if someone could point me towards the right direction, I'd greatly appreciate it.

Apply Newton's method to the equation x[sup:2o0cq9ll]2[/sup:2o0cq9ll]-a=0 to derive the following square-root algorithm (used by the ancient Babylonians to compute a[sup:2o0cq9ll]1/2[/sup:2o0cq9ll]:


X[sub:2o0cq9ll]x+1[/sub:2o0cq9ll] = 1/2(X[sub:2o0cq9ll]n[/sub:2o0cq9ll]+a/X[sub:2o0cq9ll]n[/sub:2o0cq9ll])

I know that for the first equation, f'(x)=2x but other then that not sure what to do from there. Thanks!

 

[galactus]

There's not that much to it. You are on the right track.

By Newton's Method: \(\displaystyle x_{n}-\frac{x_{n}^{2}-a}{2x_{n}}\)

Factor out the 1/2. Remember, \(\displaystyle \frac{x_{n}^{2}}{x^{n}}=x_{n}\)

\(\displaystyle =x_{n}-\frac{1}{2}\left(x_{n}-\frac{a}{x_{n}}\right)\)

and there it is with just a wee bit of algebra.