Newton's Method on Systems
 The following are examples of running Newton's method on systems of equations. All the examples are based upon systems which have the fifth roots of unity paired in various ways as solutions. Work with Mark Motyka led to [1]. We present some variants on those constructions, some new examples and give some annimations. The equations that we consider are:
In each case, the color in the image corresponds to the basin of attraction and the shade corresponds to the number of steps to convergence, mod 3. If the position in the image corresponds to w, then the initial point was selected as x0=(w,w) or x0=(w,c) as indicated in the examples.
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System (1) with x0=(w,i)
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System (1) with x0=(w,0.5)
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System (1) with x0=(w,w)
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System (3) with x0=(w,1)
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System (4) with x0=(w,0.01)
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System (1) with x0=(w,w)
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Animation of System (1) with x0=(w,c), with 0.1< c< 0.7
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Animation of System (4) with x0=(w,c), with 0.001< c< 1
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Animation of System (5) with x0=(w,w), zooming into the origin
Scripts
  1. A J script newt_sys.ijs to create the first image of this type.
  2. It requires raster5.ijs as well (installed in the fvj2 subdirectory of the J directory).

References
  1. Mark A. Motyka and Clifford A. Reiter, Chaos and Newton's Method on Systems, Computers & Graphics, 14 1 (1990),131-134.