The basin of attraction of a particular solution to
an equation, such as a cubic equation, under a solution
method, such as Newton's Method, is the collection of
starting points that end at that solution.
For the cubic equation in the complex plane the basins
of attraction are fractals.
Here we have coloured each point according to the root
that Newton's Method converges to. When close to a
root, that's where you end up. However, when nearly
equidistant from two roots the derivative is close to
zero, so the tangent can take us a long way away. Then
we are starting the process from an effectively random
point.
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