MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## How to calculate the Feigenbaum constant to high precision?

I am looking for a reference on how to calculate the Feigenbaum constant $\delta$ to high precision. It seems that naive methods do not work well because they lead to solving very high degree polynomials.

I have searched Math Reviews, Googled around, and the best I found was Appendix A of Keith Brigg's thesis. Wikipedia gives a link which contains $\delta$ to high precision, but does not explain how it was computed.

-
 I'll only note that the method in ams.org/journals/mcom/1991-57-195/… does not look to be scalable, even if you hitch it to a sequence transformation algorithm. – J. M. Nov 26 2010 at 14:02 what if one implements the algo in Brigg's thesis using mpfr or some other high precision library, is that process too slow or too error prone?? – suVRit Nov 26 2010 at 14:36 i don't know what i am talking about, but i am curious whether the following paper is relevant? arxiv.org/pdf/1008.4608 – suVRit Nov 26 2010 at 14:39 @Suvrit: As noted in the appendix, 1. it's only linearly convergent (sequence transformation might help, but I haven't conducted the requisite experiments) and 2. it requires the solution of a high-degree polynomial with clustered zeroes. FWIW, Briggs does mention his use of a high-precision package (Bailey's mpfun). – J. M. Nov 26 2010 at 14:51 maybe directly emailing Bailey or Borwein or Broadhurst might help? – suVRit Nov 26 2010 at 18:15

One idea for computing the Feigenbaum constant is as follows: $\delta$ is the unique expanding eigenvalue for period-doubling renormalization at its fixed point $F$. So take any real-analytic 1-parameter family of univalent maps that is transverse to the stable manifold of renormalization (for example, the real quadratic family). Then the iterated renormalizations of this 1-parameter family will converge to the unstable manifold of renormalization. The unstable manifold is of course mapped to itself by renormalization and the derivative at the fixed point will be $\delta$.

It should be possible to implement this numerically, by keeping track of the power series for a real-analytic family of real-analytic univalent map, and applying the renormalization by replacing $f_\lambda$ with $f_\lambda \circ f_\lambda$ (and suitably rescaling). Because renormalization acts as a contraction on these 1-parameter families, this procedure should be computationally stable, and provide a number of digits proportional to the number of times one renormalizes.

One classical reference for the stable/unstable manifold picture for period doubling renormalization is Iterated Maps on the Interval as Dynamical Systems by Collet and Eckmann.

-

As far as I know O.E. Lanford III was the first to compute the Feigenbaum constant around 1980 using interval arithmetics. That is to say, he represented real numbers not by floating point arithmetics but rather using intervals containing the sought number. If I remember correctly he had at least fifty decimals at some point (he told me orally, so I might be completely wrong).

If you enter Author="Lanford" and Anywhere="Feigenbaum" in MathSciNet, you should find the relevant articles and some general remarks on using computers in rigorous proofs (I don't know how to post links here).

-
 MR0934342 [1] seems most relevant, but the reference is not easily obtainable. [1] ams.org/mathscinet/search/… – Andrej Bauer Nov 26 2010 at 12:57