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] :Program used :
Legend of the colors :
- green : 0 attraction basin
- blue : -1 attraction basin
- red : 1 attraction basin
Then we zoom :
And another 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 !...
The program has been written in the Matlab programming language, with the help of .Three ways of understanding :
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 :
- go to the directory where you put basin.m
- type basin
Is Matlab wrong, or is it right ?
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...).
- Newton's method in R2 :
- Newton's method applied to the polynomial :
- Julia's set :
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 :
For the first zoom :
For the last one :
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...