Cayley used an example to introduce the problem of finding the basins of attraction of the roots of a function. |
The complex function |
Denoting the basins of attraction of these roots by A(+1) and A(-1), that is |
  A(+1) is all z0 from which Newton's method converges to |
and |
  A(-1) is all z0 from which Newton's method converges to |
Cayley proved |
  A(-1) is the left half of the complex plane, and |
  A(+1) is the right half of the complex plane. |
In the picture, all points painted black converge to +1, all points painted white converge to -1. |
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We might expect that the points on the red vertical line are equally attracted to both +1 and -1, and consequently they will not converge at all. |
This is true, and on this line iterating Newton's method is a chaotic process. |
That is, on the boundary of the basins of attraction, the dynamics are chaotic. |
Return to Newton's Method Basins of Attraction.