# Homework Help: Finding the roots through Newtons method

1. Aug 14, 2010

### Telemachus

1. The problem statement, all variables and given/known data
In each of the following items approximate the zeros of $$f$$ using Newton's method. Continue iterating until making two successive approximations differ at most in 0.001.

The Newton iteration: $$x_{n+1}=x_n-\displaystyle\frac{f(x_n)}{f'(x_n)}$$

Well, I have a doubt about this. I'm not sure if it's asking me to iterate till $$|x_{n+1}-x_n|\leq{0.001}|$$, or if I should apply some of this:

$$k_1>0$$, $$|f'(x)|\geq{k_1}$$ and $$|f''(x)|\leq{}k_2$$ for all $$x\in{}$$, then:

$$|x_{n+1}-r|<\displaystyle\frac{k_2}{2k_1}|x_n-r|^2$$

If $$r\in{}[r-\delta,r+\delta]\subset{[a,b]}$$, and $$\delta<2(\displaystyle\frac{k_1}{k_2})$$

$$|x_{n+1}-r|<\displaystyle\frac{2k_1}{k_2}(\displaystyle\frac{\delta}{\displaystyle\frac{2k_1}{k_2}})^2n$$

Last edited: Aug 14, 2010
2. Aug 14, 2010

### awkward

Stop when $$|x_{n+1} - x_n| < 0.001$$.