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Complex Newton Iteration

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The figure illustrates the convergence of the complex Newton iteration for the equation

$\displaystyle z^3 - 1 = 0 $

with the solutions $ z=\exp(2\pi$i$ k/3)$, $ k=0,1,2$. Different colors mark the domains of convergence for the corresponding solutions. Faster convergence is indicated by darker shades of color.

\includegraphics[width=\moimagesize]{z3color}

Apparently, the boundaries of the regions of convergence have a fractal character. This is typical for images generated with the aid of nonlinear iterations.

see also:

  automatisch erstellt am 15.  6. 2016