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. |
Return to Universality of the Mandelbrot set.