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-   -   Newton Raphson - strange results (http://www.cfd-online.com/Forums/main/16338-newton-raphson-strange-results.html)

Bren February 11, 2009 05:49

Newton Raphson - strange results
 
Hi Guys, I've been using the Newton-Raphson method to solve a rather nasty nonlinear equation. The method is converging quickly and is obtaining the same solution (to within machine accuracy) as a tried and tested method I used previously.

My problem is that there are two roots of the equation, one is real and the other complex. I can find the real one easily but I can only get close to the complex solution - successive iterations jump between:

x0 +/- i*x1

where x0 and x1 are constants. The imaginary component of the solution continaully flips back and forth between +/- on going from one step to the next although the magnitude stays the same. When I substitute the solution (using both + and -ve imaginary components) I find that the function I am trying to solve isn't equal to zero; it is reasonably close but is large enough (0.2 ish) that it isn't likely to be a machine-accuracy issue.

Does anyone know why this is happening? The complex answer isn't too important for what I'm doing but I'd like to know why I'm seeing this behaviour. Cheers, Bren

Robin February 11, 2009 07:10

Re: Newton Raphson - strange results
 
http://www.bugman123.com/Fractals/Ne...hson-large.jpg

Bren February 11, 2009 07:14

Re: Newton Raphson - strange results
 
That's a nice picture but it really doesn't help me understand what's going on.

Robin February 11, 2009 09:40

Re: Newton Raphson - strange results
 
http://en.wikipedia.org/wiki/Newton_fractal

in particular:

"However, for every polynomial of degree at least 2 there are points for which the Newton iteration does not converge to any root: examples are the boundaries of the basins of attraction of the various roots."

Bren February 11, 2009 11:39

Re: Newton Raphson - strange results
 
Ok I take it back - that is very helpful :o)

Thanks Robin, that's an interesting find.

Robin February 11, 2009 11:47

Re: Newton Raphson - strange results
 
Have been a big fan of non-linear dynamics+fractals for years. There are fascinating overlaps with turbulence but that would be for another thread.


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