# 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/S0025-5718-1991-1079009-6/… does not look to be scalable, even if you hitch it to a sequence transformation algorithm. – J. M. is not a mathematician Nov 26 '10 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 '10 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 '10 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. is not a mathematician Nov 26 '10 at 14:51
• maybe directly emailing Bailey or Borwein or Broadhurst might help? – Suvrit Nov 26 '10 at 18:15

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).

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.

David Broadhurst computed some 15 years ago those constants to high precision (in particular $\delta$ to 1018 digits) using Keith Briggs' method http://www.plouffe.fr/simon/constants/feigenbaum.txt

Note: the link provided by Andrej is not valid anymore, sorry if my answer is just a duplicate (today Wikipedia gives a link to OEIS.org for a value with about 103 digits).

• I wish mathematicians knew their computations ought to be verifiable and repeatable. The link you provided, how can I tell whether those results are correct? – Andrej Bauer Feb 6 '14 at 17:35
• Perhaps Keith Briggs' thesis would have more information relevant to Andrej's query? – Todd Trimble Feb 6 '14 at 17:54