Mathematica has no problem with any of the polynomials I get from any sequences of the kind we’ve been discussing, but I suspect they have tens of thousands of lines of code, which switches between dozens of possible approaches depending on a detailed analysis of the polynomial. Of course on top of all the issues of convergence, there is the problem of how to implement these various algorithms with floating point numbers.

]]>• Peter Doyle and Curtis McMullen, Solving the quintic by iteration, *Acta Mathematica*, **163** (1989), 151–180.

They show a polynomial equation is solvable by a ‘tower’ of generally convergent iterative systems iff its Galois group is **nearly solvable**: it’s built from abelian groups and A_{5} by iterated extensions. This includes any quintic but not, say, a typical sextic.

57, 71, 86, 100, 100, 100, 100, 114,114, 114, 114, 128, 128, 128, 128, 143, 143, 157, 157, 186.

There are still no guarantees of finding the roots every time, for this sequence or any other, but by allowing the program to fall back progressively to less stringent criteria for accepting a candidate root, it seems to be able to solve a wider range of polynomials.

]]>While I like this picture, I don’t get what it has to do with the book’s theme. Maybe it’s the cover they use on this whole series!

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