Newton's method

georgech · Poincare Conjecture Offline Joined: 30 Jul 2005 Posts: 134

Hello, i am having some difficulties solving the following problems using newton's method:

1)What does the sequence defined by
$x_0=1,\;\;\;\;\;x_{k+1}=\frac{1}{2}x_k+\frac{1}{x_k}$
converge to?
and

2)Devise a subroutine using only simple operations that finds, using newton's method, the cube root of some input number z.

Do u have any ideas?
Thanks

**Posted:** Sat Oct 31, 2009 10:13 am

georgech · Poincare Conjecture Offline Joined: 30 Jul 2005 Posts: 134

i only need some help to start..i know how to use newton to find roots of equations, but i haven't seen such problems before...

**Posted:** Sun Nov 01, 2009 1:37 am

JBL

If I say that " $a$ is a cube root of $z$ ," what's the corresponding equation?

**Posted:** Sun Nov 01, 2009 7:51 am

_________________
Joel
Hi Deeps! <3

georgech · Poincare Conjecture Offline Joined: 30 Jul 2005 Posts: 134

Thanks, i have done the second question. But i am not sure how to proceed with the first one, which needs the convgence of the sequence. How can i do something like tha without using matlab? thanks

**Posted:** Sun Nov 01, 2009 1:11 pm

atomicwedgie

**Posted:** Sun Nov 01, 2009 1:48 pm

JBL

That was a pretty impressive reverse-engineering of the question, atomicwedgie. The natural way to do that question has nothing to do with Newton's method: if $x_k \to L$ then $x_{k + 1} \to L$ and so $L = \frac {1}{2}L + \frac {1}{L}$ .

**Posted:** Sun Nov 01, 2009 4:35 pm

_________________
Joel
Hi Deeps! <3

J.Y.Choi

Think about graph $y = \frac {1}{2}x + \frac {1}{x}$ and $y = x$ . The limit we want to find is the fixed point.

This is one well-known method, but I don't know this is the Newton's method. The book [Mathematical Methods for Physicists] by Arfken will help.

**Posted:** Sun Nov 01, 2009 4:54 pm