| Are there more failures more serious than starting exactly on a basin boundary? |
| Are there large sets of starting points from which Newton's method does not converge to a root? |
| For example, can Newton's method ever have attracting cycles? |
| Curry, Garnett, and Sullivan experimented with the cubic polynomials fc(z) |
| fc(z) = z3 + (c - 1)z - c |
| Classical results show if there is an attracting cycle, it would have to attract z0 = 0. |
| So the basic experiment was to apply Newton's method for each point c in the complex plane, always starting with z0 = 0, and see if the sequence converges to a root. |
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