On 7/23/13, Mike Stay <metaweta@gmail.com> wrote:
... Actually, it has to do with "cubic" continued fractions. See, e.g. Gupta & Mittal, "Bifurcating Continued Fractions" and Mittal & Gupta "Bifurcating Continued Fractions II". Both are available on the arxiv or here:
http://www.cs.auckland.ac.nz/~mike/maths/continued%20fractions/BCF1.pdf http://www.cs.auckland.ac.nz/~mike/maths/continued%20fractions/BCF2.pdf --
[Oops--- there goes the carpet again.] The authors seem completely unaware of the existing literature on MDCF (Multidimensional Continued Fractions), Lattice Basis Reduction, Diophantine Approximation etc. There are a number of algorithms besides LLL, with implementations built into Mathematica, Maple etc. See eg. http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lat... http://ocw.mit.edu/courses/mathematics/18-409-topics-in-theoretical-computer... I haven't managed to make sense of the "bifurcating" scripts at a quick glance; but it's worth remembering that "MDCF" algorithms --- which attempt directly to generalise continued fractions in some kind of mechanistic fashion --- invariably fail to converge in general. Fred Lunnon