Global maximum of ?(x)-x: a related problem : PARI code

[Gerald McGarvey]

Here's some PARI code for Pierce(x):

pierce(x)=cf=contfrac(x);V1=vector(length(cf)-1);V2=vector(length(cf)-1);s=0;for(k=1,length(cf)-1,s=s+cf[k+1];V1[k]=s);p=1;for(k=1,length(cf)-1,p=p*V1[k];V2[k]=p);return(sum(k=1,length(cf)-1,(-1.)^(k-1)/V2[k]))

It checks out ok for phi^{-1} -and sqrt(2)-1.

Using PV=vector(999);for(n=1,999,PV[n]=pierce(n/1000)) and then a write of 
the vector to a file,
I then imported the file into Excel and created a gif file of a graph of 
Pierce(x)-x.
I didn't attach it because I don't know if that is allowed.

For 0.57187500456342 I get a value of 0.60050000465994...
Maybe if someone else codes the Pierce function we can compare.
I think that might be just a local maximum of Pierce(x)-x, e.g. I get for x 
= .57193
Pierce(x)-x = .02862566538151677470562919479328148...

- Gerald

At 02:54 PM 6/14/2006, franktaw at netscape.net wrote:
>Here's a little more info.
>
>The solution Pierce(x) = x near 0.355 is approximately 0.35446930411246, 
>with continued fraction  2,1,4,1,1,2,3,1,2,1,1,1,... (and hence Pierce 
>expansion 2,3,7,8,9,11,14,15,17,18,19,20,...).  The global maximum appears 
>to be near 0.57187500456342, with continued fraction 1,1,2,1,44,1,2138,... 
>(I'm not 100% sure that that isn't just a local maximum, and the 2138 
>might be off.  The value at that point is about 0.60050000466366.
>
>Franklin T. Adams-Watters
>
>
>-----Original Message-----
>From: franktaw at netscape.net
>
>Correction: 0.428 is the apparent global minimum of Pierce(x) - x; 0.572 
>would then be the global maximum.
>
>Franklin T. Adams-Watters
>
>-----Original Message-----
>From: franktaw at netscape.net
>
>You folks might try also looking at another, similar function I discovered 
>recently. This relates to the very nice Pierce expansion. Every irrational 
>number between 0 and 1 has a unique representation as an alternating sum:
>
>x = 1/a_1 - 1/(a_1 a_2) + 1/(a_1 a_2 a_3) - ...,
>
>where the sequence of a_i's are strictly increasing. Rational numbers in 
>this range have two representations, one ending in a-1, a and the other 
>with just a.
>
>So take a number between zero and one, get its continued fraction, take 
>the cumulative sums, and regard that as a Pierce expansion. Voila - an 
>increasing, one-to-one function from [0,1] to [0,1]. I propose calling 
>this the Pierce function.
>
>Some particular values:
>
>phi^{-1} -> 1-e^{-1}
>sqrt(2)-1 -> 1-e^{-1/2}
>
>1/n and 1-1/n are fixed points, but there are others, including one at 
>about 0.355 (and, of course, another at about 0.645). The global maximum 
>for Pierce(x)-x appears to be near 0.428.
>
>Franklin T. Adams-Watters
>
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