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Old   February 11, 2009, 05:49
Default Newton Raphson - strange results
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Bren
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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
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Old   February 11, 2009, 07:10
Default Re: Newton Raphson - strange results
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Robin
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http://www.bugman123.com/Fractals/Ne...hson-large.jpg
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Old   February 11, 2009, 07:14
Default Re: Newton Raphson - strange results
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Bren
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That's a nice picture but it really doesn't help me understand what's going on.
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Old   February 11, 2009, 09:40
Default Re: Newton Raphson - strange results
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Robin
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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."
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Old   February 11, 2009, 11:39
Default Re: Newton Raphson - strange results
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Bren
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Ok I take it back - that is very helpful )

Thanks Robin, that's an interesting find.
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Old   February 11, 2009, 11:47
Default Re: Newton Raphson - strange results
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Robin
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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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