* Bill Thurston <wpt4@cornell.edu> [Sep 20. 2010 14:46]:
I haven't been following this thread, but I want to mention that Peter Doyle and Curt McMullen made a global analysis of [...]
This one(?): Peter Doyle, Curt McMullen: {Solving the quintic by iteration}, Acta Mathematica, vol.163, no.1, pp.151–180, (1989). http://www.math.harvard.edu/~ctm/papers/home/text/papers/icos/icos.pdf (Sadly much over my head). This is ref'd form iteration recipe at end of: http://en.wikipedia.org/wiki/Bring_radical
I think these methods are all super-exponential once they're in the attracting basin, but I don't remember
Bill
(Assuming the answer is 'yes', which I think it is): Still if you are very close to a boundary the method will _seem_ to fail. jj Possibly related: Scott Crass: "Solving the quintic by iteration in three dimensions" http://arxiv.org/abs/math/9903054