Newton Basin
Chanteperdrix Guilhem(mfn04)

RESULTS AND DISCUSSION








Results on the polynomial z->z^3-z :

Here, we present first the result obtained in iterating the Newton's method : for all z_0, a complex number in a square centered on the origin,  the program calculate the roots obtained. A first figure has been computed on the square [-2,2]*[-2,2] :

First view






Legend of the colors :


Then we zoom :

First zoom

And another zoom :

Second zoom

We can understand now what happened with the real polynomial...
Note that if you go on zooming around the real axis, you will always obtain the same figure, only colors exchange each other (here : red and blue) : we can speak about fractal !...
 

Program used :
The program has been written in the Matlab programming language, with the help of [2].
The way use is the Julia's set one (see next paragraph).

The source is available here. If you want to test it, copy the program on your disk to basin.m, launch Matlab, then :

Three ways of understanding :
The two last points of view are not completely equivalent because if you cannot calculate explicitly the derivative, you cannot obtain the function to iterate to obtain the corresponding Julia's set. In the case of polynomial (our case), these two notions are of course equivalent, and that's why I choose the Julia's point of view to write my program (and also because it is the fastest way...).
 
 
 
 
Is Matlab wrong, or is it right ?
 
To know if my results are "true", I went there, I click on generation form, and the three associated results are :

For the first view :

First view computed on the Web

For the first zoom :

First zoom computed on the Web

For the last one :

Second zoom computed on the Web




These three figure have been obtain with the same number of pixels (500*500) and under the same conditions (same segments, and same maximum number of iterations (20)) than those obtained by my program. The only difference is, I think, the distance chosen to consider that the limit obtained is this or that roots.

These figures allows me to think too that my program is not so false...



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