Neat!

]]>The Ordered Sums page uses the Weierstrass–Durand–Kerner method, which is an application of Newton’s method. I tried Aberth’s method, but that did much worse.

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.

]]>Is Mathematica better at finding roots of high-degree polynomials? (It should be, because they’re paying a lot of people to solve such problems.) Do you have any idea what tricks they use? If you use something like Newton’s method, it’s an interesting problem in dynamical systems. Smale posed the question of which polynomials can be solved by ‘generally convergent iterative systems’ (which are a class of methods more sure-fire than yours), and this paper discusses how they work for quintics:

• 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.

After a few more tweaks to the Ordered Sums software, it can now (usually) solve for the sequence:

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.

]]>I strongly suspect, but cannot prove, that eukaryotes could still have evolved with hydrogen atoms that are twice as heavy. I think they’re just too finely adapted to the chemistry of our world to survive a sudden change to this alternative world.

]]>Thanks for the comprehensive answer. A followup question (possibly not answerable) is what would happen if life evolved in an environment of D2O instead of H2O? It would not stop the prokaryotes, but would the eukaryotes have been inhibited from evolving, or could they simply have adjusted their biochemistry slightly to adapt to living in D2O? The latter sounds more likely but the former would be more interesting for fine-tuning arguments.

]]>Thanks John. That a couple of effective-mass adjustments being 1/3600 instead of the usual 1/1800 can kill off complex creatures is really striking to me.

]]>FWIW, I’ve tweaked the program so it now recognises any non-trivial GCD in the exponents of the characteristic polynomial, and uses that to solve the lower-degree polynomial whose roots can be used to construct those of the original. So the mass sequence 62,78,94,106,110,110,110,122,122,126,126,138,138,

142,142,146,158,170,174,198 now (usually) gets a direct answer.