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Re: Help Define a Constant I get from Mathematica

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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@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@yahoo.com

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Re: Help Define a Constant I get from Mathematica DrMajorBob <btreat1@austin.rr.com> - 2011-05-03 09:45 +0000

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