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 Expansion of base-e pentation sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 10/31/2010, 01:56 AM The analytic extension of Pentation wobbles. pentation(-2)=-1 pentation(-1)=0 pentation(0)=1 Here, I'm graphing d/dx(pentation(x)), which wobbles in the neighborhood of z=-2..z=0, with two inflection points where the derivative has a local maxima, and a local minima. I calculated the Taylor series around -1 in the previous post, pentaylor(-1,1,0), which converges very nicely in the neighborhood of pent(-2)..pent(0), with approximately 21 digits of accuracy. Here, I am plotting the derivative of the paprx(x) Taylor series. So the pentation Taylor series apparently doesn't have the inherent beauty of the sexp Taylor series, for which all odd derivatives are positive for all real numbers>-2.     By the way, the wobble is quite a bit worst for smaller bases. Here is the graph for pentation base=1.7, from -6 to 6, where the wobble is visible without taking the first derivative. I think this has been pointed out before, but there is a lower limit base, for which pentation has a parabolic upper fixed point, in addition to the lower fixed point. That base is somewhere near B=1.6355. Again, the graph for pentation base 1.7 from -6 to 6. - Sheldon     sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 11/02/2010, 03:37 PM (This post was last modified: 11/02/2010, 04:13 PM by sheldonison.) I updated the pentation.gp code. The first thing the pentation code does is generate the sexp fixed point. The sexp fixed point gets very close to -2 as the base increases, and since there is a singularity at sexp(-2), that was breaking the previous code for bases>3. I decided to patch the slog function, and make a few other changes so that pentation at least works for bases<=10. It also works for all bases>=1.47, bases for which kneser.gp works. This code includes all of the kneser.gp code, with the penation.gp code in one program, to make things simpler. I also included the "sexpupfixed" routing, to iteratively generate the sexp base with a parabolic upper fixed point. The "sexpupfixed" routine requires about a dozen iterations of "init(B);loop" to get results accurate to 32 decimal digits, which takes about 15 minutes; I posted results for the value of the fixed point in this thread.. - Sheldon Code:\r pentation.gp init(2);loop;  /* generate kneser.gp sexp series */ genpent; Attached Files   pentation.gp (Size: 27.26 KB / Downloads: 657) sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 07/01/2011, 10:38 PM (This post was last modified: 11/21/2011, 09:29 PM by sheldonison.) (11/02/2010, 03:37 PM)sheldonison Wrote: I updated the pentation.gp code. The first thing the pentation code does is generate the sexp fixed point. The sexp fixed point gets very close to -2 as the base increases, and since there is a singularity at sexp(-2), that was breaking the previous code for bases>3. I decided to patch the slog function, and make a few other changes so that pentation at least works for bases<=10. It also works for all bases>=1.47, bases for which kneser.gp works. This code includes all of the kneser.gp code, with the penation.gp code in one program, to make things simpler. I also included the "sexpupfixed" routing, to iteratively generate the sexp base with a parabolic upper fixed point. The "sexpupfixed" routine requires about a dozen iterations of "init(B);loop" to get results accurate to 32 decimal digits, which takes about 15 minutes; I posted results for the value of the fixed point in this thread.. - Sheldon Code:\r pentation.gp init(2);  /* generate kneser.gp sexp series */ genpent; This is an update, with a fix for $b=\exp(\pi/2)$, which was reported by Cherrina_Pixie in this thread. Its also includes a very new version of the core kneser.gp sexp code, which runs 5x-7x faster, mostly by generating a polynomial series for the Schroder and inverse Schroder functions for the superfunction and inverse superfunction routines. This eliminates many of the repeated iterations of log(log(log....) or exp(exp(exp))). The other thing I did was switch to using pari-gp built in polynomial handling. So that pentaylor, and sexptaylor, and slogtaylor all return pari-gp polynomials. Internal, the xtheta(x) and the xsexp(x) are stored as iteratively generated polynomials, as are the xsuperf, and xisuperf polynomails. Because of the kneser.gp sexp(z) improvements included, the sexpupfixed example, that used to take a little over 15 minutes to generate the tetra-euler constant, now takes about 2 minutes and 15 seconds. The bigger improvement occurs on smaller bases. Once I have a few more minor functions for bases

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