Re: Help Define a Constant I get from Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg118563] Re: Help Define a Constant I get from Mathematica
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Tue, 3 May 2011 05:45:04 -0400 (EDT)
Your 5,000 iterations are overkill, since it only takes 14 iterations (or
so) to get the same limit:
l = RandomReal[{0.044685172, 1}, WorkingPrecision -> 200];
list = FixedPointList[FromContinuedFraction@Convergents[#, 100] &, l,
50, SameTest -> (Abs[#1 - #2] < 10^-30 &)];
-1 + Length@list
N[Last@list, 30]
14
0.555753104278045912445405869381
Bobby
On Mon, 02 May 2011 05:53:41 -0500, Marvin Burns <marburns at umail.iu.edu>
wrote:
> Can you help define this constant I get from Mathematica?
>
> Here is the code I use:
>
> Table[{N[l = RandomReal[{0.044685172, 1}, WorkingPrecision -> 200]],
>
> Table[c = Convergents[l, 100];
>
> l = FromContinuedFraction[c], {n, 1, 50}];
>
> N[l, Floor[30]]}, {m, 1, 100}] // TableForm
>
>
>
> I get c=0.55575310427804591244540586938..., and no other value, every
> time I
> run it!
>
> I know basically Mathematica is using the convergents of a real number in
> the domain as terms to form a generalized continued fraction. Then it
> uses
> the convergents of that generalized continued fraction as terms to form
> another generalized continued fraction, and continues to repeat this
> process
> until it gets to where I told it to stop. Strangely, initial
> input<0.044685172 seems to have output that is extremely sensitive to
> variations. For example, .04468513845124463 gives 3.2765033850144244631
> but
> .044685138451244635 gives 0.55575310427804591245, and then
> .044685138451244635105 gives 3.276503385014424463 again.
>
> There must be more rigorous way to define it.
>
> Do you think it deserves a name? What would be a good name for it?
>
> Marvin Ray Burns
>
> Original investigator of the MRB constant (I didn't name that.)
--
DrMajorBob at yahoo.com