# Newtons method

1. Dec 8, 2006

### robierob

so I was helping my friend today and ran into a problem.

the problem was to use newtons method to approx. the intersection points of two graphs.

f=x^2
g=cosx

so I subtracted f-g, found the derivative and pluged in some guesses.

Except all of my guesses just blew upwards in values instead of zooming in on a point. whats the deal?

Rob

2. Dec 8, 2006

### GDogg

Your guesses were too far from the root. To ensure convergence, the distance between your initial guess and the root has to be less than $2 \delta \over M$ where

$$|f'(x)| \geq \delta > 0$$

$$|f''(x)| \leq M$$

for all $x \epsilon [a,b]$ in the interval $[a,b]$ you're considering

Last edited: Dec 8, 2006
3. Dec 8, 2006

### robierob

Your totally right. Thats one tight interval to make a guess for though. I was only about .2 away from a working guess. Thanks

4. Dec 20, 2006

### NTUENG

Hi Robierob,

When u compute f(x)-g(x), u should get x² - cos x on which u can apply Newton's method. Actually u can ensure that your guess is not far from the actual root by finding when your function changes from positive to negative or vice versa. There will be a root when the function changes sign provided that it is continuous for the interval where the root lies. For example, there is a root between x=a and x=b when a and b are consecutive and f(a) f(b) < 0. U can deduce the values of a and b by substitution and use a starting value between a and b for your approximation. That should help u arrive at your root quickly.