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MathGroup Archive 1998

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Re: any ideas on this ?



fenris@hotmail.com wrote:
> 
> I am having a problem with a set of 3 non-linear equations in three
> unkowns. They have to be solved with a non-linear method such as
> findroots. The reason that i am not bothering to post the exact
> equations is that i feel it is irelavant to the question i have. With a
> system of m non-linear equations in m unknowns, how does one go about
> determining the starting values the findroots requires for each of the
> m variables? I have looked through my numerical texts, they explain
> nicely how to solve the system, but no explanation on how to get the
> starting points for iteration? Any help on how to obtain the starting
> values in general would be appreciated.
> 
> Troy
> 
> s2700114     NOSPAM  @nickel.laurentian.ca


They don't talk about it because it is extremely difficult.  Most
numerical methods are only good for finding roots once you already know
about where they are anyway.  I usually try the graphic approach.  Have
you tried plotting the curves and seeing where they or thier
projections intersect?

The other way is a sort of fractal method.  I will try to describe it. 
Imaging a simple equation x^4==1.0.  There are four solutions to this
on the complex plane.  Take a method like fixed-point iteration.  Your
starting guess will determine which of the four roots the routine
converges to.  On a 2-D graph, color each point in the complex plane by
the following:  Each root gets a different color and the number of
iterations required to reach a certain level of accuracy gives the
brightness of the color.  The result is a fractal curve in the complex
plane.  If you do this in 3-D with the number of guesses being the
height of a surface, you see a kind of topography where the numerical
routine is attracted to certain roots depending on the starting guess.

Your problem with 3 equations and 3 unknowns can be thought of as some
kind of a big landscape (topography) with mountains and valleys.  At
the bottom of the valleys are the roots and you want to make an initial
guess somewhere within the valley and not in the planes or on the other
side of the mountains that lead away from the roots.

If you have absolutely no idea where the roots are, do the following: 
Set up a simple root finder yourself, like a Newtons Method, or a fixed
point, or even a secant method.  Make a course mesh of the complex
plane and set a low error tolerance for the roots.  Then make a 3-D
graph of numbers of guesses to convergence versus starting guess. 
Incorporate a safety valve so that if the numbers of guesses are above
a certain value, you set a height and go on.  You will have made a
topography of your problem and now can follow down the slopes toward
the valleys. -- 
Remove the _nospam_ in the return address to respond.



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