Newton's Method and convergence

[Guest]

I was running Newton's method on Matlab for different functions. For x^3-3x+2. X0 =1.2.

The question is run Newton's method........Does the convergence seem quadratic and why?

I am really not sure how to tell if it's quadratic. I ran Newtons and got the following values....
1.1030
1.0524
1.0264
1.0133

Thanks.

 

[Count Iblis]

I was running Newton's method on Matlab for different functions. For x^3-3x+2. X0 =1.2.

The question is run Newton's method........Does the convergence seem quadratic and why?

I am really not sure how to tell if it's quadratic. I ran Newtons and got the following values....
1.1030
1.0524
1.0264
1.0133

Thanks.

Quadratic convergence means that once you are close to the root, the error at the next step becomes the square of the error in the previous step. Ths means that the number of correct decimal digits will double at each step. Newton's method generically has quadratic convergence, but there are exceptions. In this case, you can clearly see that the convergence is not quadratic. The reason is that the root at x= 1 is a "double" root:


x^3-3x+2 = (x-1)^2(x+2)

This means that close to x = 1, the function is not well approximated by the local tangent line. Around x = 1 the leading behavior is quadratic, while Newton's methods assumes linear behavior. It is, however, not difficult to modify Newton's method to deal with such cases.

 

[galactus]

Here's an animated graph of Newton for fun:

See what the Count means at x=1?. The tangent line is horizontal as x=1.

 

[Guest]

THANKS! That helps a lot!