Difference between revisions of "File:Penplot.jpg"

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[[Explicit plot]] of natural [[pentation]],
−
Importing image file
  
  +
  +
$y=\mathrm{pen}(x)$ is shown with thick black line.
  +
  +
==Other lines==
  +
The thin red line shows it asymptotic level $L\approx -1.8503545290271812$ is smallest real fixed point of [[natural tetration]].
  +
  +
  +
The thin blue line shows the asymptotic
  +
  +
$ y=L+\exp(k(x+x_1))$
  +
  +
where $k\approx 1.86573322821$
  +
  +
and $x_1 \approx 2.24817451898$
  +
  +
  +
The thin green line shown the deviation from the linear approximation
  +
  +
$\mathrm{linear}(x)=1+x$
  +
  +
The deviation is denoted as $~\delta(x)=\mathrm{pen}(x)-\mathrm{linear}(x)$
  +
  +
In the range $-2.1\!<\!x\!<\!1.1$, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, $y=10\delta(x)$ is plotted.
  +
  +
==Description of natural pentation==
  +
  +
Natiral [[pentation]] is specific superfunction of [[natural tetration]] (implementation [[fsexp.cin]] is available); so, the
  +
pentation safisfies the [[transfer equation]]
  +
  +
$\mathrm{pen}(z\!+\!1) = \mathrm{tet}\Big( \mathrm{pen}(z)\Big)$
  +
  +
The additional condition $\mathrm{pen}(1)=\mathrm e \approx 2.71$ is assumed.
  +
  +
Natural pentation is specific superfunction, it is constructed with [[regular iteration]] at the lowest real [[fixed point]] of [[natural tetration]], denoted with $L$.
  +
  +
  +
==[[C++]] generator of curves==
  +
Files [[ado.cin]], [[fsexp.cin]], [[fslog.cin]] should be loaded in order to compile the [[C++]] code below
  +
<poem><nomathjax><nowiki>
  +
#include <math.h>
  +
#include <stdio.h>
  +
#include <stdlib.h>
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "ado.cin"
  +
#include "fsexp.cin"
  +
#include "fslog.cin"
  +
  +
z_type pen0(z_type z){
  +
DB Lp=-1.8503545290271812;
  +
DB k,a,b;
  +
// k=1.86573322821; a=-.62632418; b=0.4827;
  +
k=1.86573322821; a=-.6263241; b=0.4827;
  +
  +
z_type e=exp(k*z);
  +
return Lp + e*(1.+e*(a+b*e));
  +
}
  +
  +
z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.));
  +
DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;}
  +
return z; }
  +
  +
z_type pen(z_type z){ DB x; int m,n;
  +
x=Re(z); if(x<= -4.) return pen0(z);
  +
m=int(x+5.);
  +
z-=DB(m);
  +
z=pen0(z);
  +
DO(n,m) z=FSEXP(z);
  +
return z;
  +
}
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
FILE *o;o=fopen("penplo.eps","w"); ado(o,608,1008);
  +
fprintf(o,"404 204 translate\n 100 100 scale\n");
  +
#define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
  +
for(m=-4;m<3;m++) {M(m,-2)L(m,8)}
  +
for(n=-2;n<11;n++) {M( -4,n)L(2,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");
  +
  +
DO(n,150){x=-4+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 0 RGB S\n");
  +
  +
DO(n,150){x=-2.2+.04*n;y=10.*(Re(pen7(x))-(1.+x)); if(n==0) M(x,y)else L(x,y); if(y>.3)break;} fprintf(o,".01 W 0 .5 0 RGB S\n");
  +
  +
DB L=-1.8503545290271812;
  +
DB K=1.86573322821;
  +
DB a=-.6263241;
  +
DB b=0.4827;
  +
DO(n,80){x=-4.+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e;
  +
if(n==0) M(x,y) else L(x,y); if(y>8.) break;}
  +
fprintf(o,".01 W 0 0 1 RGB S\n");
  +
  +
/*
  +
DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a));
  +
if(n==0) M(x,y) else L(x,y); if(y>8.||y<-2.) break;}
  +
*/
  +
M(-4,L)L(0,L)
  +
fprintf(o,".01 W 1 0 0 RGB S\n");
  +
  +
  +
/*
  +
DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a+e*b));
  +
if(n==0) M(x,y) else L(x,y); if(y<-2.) break;}
  +
fprintf(o,".01 W 1 0 0 RGB S\n");
  +
*/
  +
  +
DB t2=M_PI/1.86573322821;
  +
DB tx=-2.32;
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
  +
printf("pen7(-1)=%18.14f\n", Re(pen7(-1.)));
  +
printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821);
  +
  +
system("epstopdf penplo.eps");
  +
system( "open penplo.pdf");
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\paperwidth 608px
  +
\paperheight 1008px
  +
\textwidth 1394px
  +
\textheight 1300px
  +
\topmargin -104px
  +
\oddsidemargin -90px
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\begin{document}
  +
{\begin{picture}(608,1008)
  +
%\put(12,0){\ing{24}}
  +
%\put(12,0){\ing{penma}}
  +
\put(0,0){\ing{penplo}}
  +
\put(377,994){\sx{3.2}{$y$}}
  +
\put(377,895){\sx{3.2}{$7$}}
  +
\put(377,795){\sx{3.2}{$6$}}
  +
\put(377,695){\sx{3.2}{$5$}}
  +
\put(377,594){\sx{3.2}{$4$}}
  +
\put(377,494){\sx{3.2}{$3$}}
  +
\put(377,394){\sx{3.2}{$2$}}
  +
\put(377,294){\sx{3.2}{$1$}}
  +
\put(377,194){\sx{3.2}{$0$}}
  +
\put(358, 93){\sx{3.2}{$-1$}}
  +
\put(80,174){\sx{3.2}{$-3$}}
  +
\put(180,174){\sx{3.2}{$-2$}}
  +
\put(280,174){\sx{3.2}{$-1$}}
  +
\put(396,174){\sx{3.2}{$0$}}
  +
\put(496,174){\sx{3.2}{$1$}}
  +
\put(590,174){\sx{3.2}{$x$}}
  +
\put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
  +
%
  +
%\put(560,510){\sx{3.6}{\rot{84}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
  +
%\put(532,708){\sx{3.6}{\rot{86}$y\!=\!\mathrm{pen}(x)$\ero}}
  +
\put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}}
  +
%\put(366,236){\sx{2.3}{$y\!=\!10(\mathrm{pen}(x)\!-\!1\!-\!x)$}}
  +
%\put(416,239){\sx{3.4}{$y=10\,\delta(x)$}}
  +
\put(8,236){\sx{3.3}{$y=10\,\delta(x)$}}
  +
%\put(366, 8){\sx{3.3}{$L$}}
  +
\put(312, 9){\sx{3.2}{$y\!=\!L$}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.
  +
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
  +
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014 (In Russian), page 269, figure 19.3.
  +
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf
  +
http://mizugadro.mydns.jp/PAPERS/2014acker.pdf
  +
D.Kouznetsov. Holomorphic ackermanns. 2014, under consideration
  +
  +
[[Category:Ackermann]]
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:Explicit plot]]
  +
[[Category:Natural pentation]]
  +
[[Category:Pentation]]
  +
[[Category:Superfunction]]
  +
[[Category:Tetriation]]
  +
[[Category:AMS]]

Latest revision as of 08:46, 1 December 2018

Explicit plot of natural pentation,

$y=\mathrm{pen}(x)$ is shown with thick black line.

Other lines

The thin red line shows it asymptotic level $L\approx -1.8503545290271812$ is smallest real fixed point of natural tetration.


The thin blue line shows the asymptotic

$ y=L+\exp(k(x+x_1))$

where $k\approx 1.86573322821$

and $x_1 \approx 2.24817451898$


The thin green line shown the deviation from the linear approximation

$\mathrm{linear}(x)=1+x$

The deviation is denoted as $~\delta(x)=\mathrm{pen}(x)-\mathrm{linear}(x)$

In the range $-2.1\!<\!x\!<\!1.1$, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, $y=10\delta(x)$ is plotted.

Description of natural pentation

Natiral pentation is specific superfunction of natural tetration (implementation fsexp.cin is available); so, the pentation safisfies the transfer equation

$\mathrm{pen}(z\!+\!1) = \mathrm{tet}\Big( \mathrm{pen}(z)\Big)$

The additional condition $\mathrm{pen}(1)=\mathrm e \approx 2.71$ is assumed.

Natural pentation is specific superfunction, it is constructed with regular iteration at the lowest real fixed point of natural tetration, denoted with $L$.


C++ generator of curves

Files ado.cin, fsexp.cin, fslog.cin should be loaded in order to compile the C++ code below


 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
#include <complex>
typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "ado.cin"
 #include "fsexp.cin"
 #include "fslog.cin"

z_type pen0(z_type z){
DB Lp=-1.8503545290271812;
DB k,a,b;
// k=1.86573322821; a=-.62632418; b=0.4827;
        k=1.86573322821; a=-.6263241; b=0.4827;

z_type e=exp(k*z);
return Lp + e*(1.+e*(a+b*e));
}

z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.));
DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;}
return z; }

z_type pen(z_type z){ DB x; int m,n;
x=Re(z); if(x<= -4.) return pen0(z);
m=int(x+5.);
z-=DB(m);
z=pen0(z);
DO(n,m) z=FSEXP(z);
return z;
}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
FILE *o;o=fopen("penplo.eps","w"); ado(o,608,1008);
 fprintf(o,"404 204 translate\n 100 100 scale\n");
#define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
 for(m=-4;m<3;m++) {M(m,-2)L(m,8)}
 for(n=-2;n<11;n++) {M( -4,n)L(2,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");

DO(n,150){x=-4+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 0 RGB S\n");

DO(n,150){x=-2.2+.04*n;y=10.*(Re(pen7(x))-(1.+x)); if(n==0) M(x,y)else L(x,y); if(y>.3)break;} fprintf(o,".01 W 0 .5 0 RGB S\n");

DB L=-1.8503545290271812;
DB K=1.86573322821;
DB a=-.6263241;
DB b=0.4827;
DO(n,80){x=-4.+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e;
        if(n==0) M(x,y) else L(x,y); if(y>8.) break;}
fprintf(o,".01 W 0 0 1 RGB S\n");

/*
DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a));
        if(n==0) M(x,y) else L(x,y); if(y>8.||y<-2.) break;}
*/
M(-4,L)L(0,L)
fprintf(o,".01 W 1 0 0 RGB S\n");


/*
DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a+e*b));
        if(n==0) M(x,y) else L(x,y); if(y<-2.) break;}
fprintf(o,".01 W 1 0 0 RGB S\n");
*/

DB t2=M_PI/1.86573322821;
DB tx=-2.32;
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);

printf("pen7(-1)=%18.14f\n", Re(pen7(-1.)));
printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821);

        system("epstopdf penplo.eps");
        system( "open penplo.pdf");
}

Latex generator of labels


\documentclass[12pt]{article}
\paperwidth 608px
\paperheight 1008px
\textwidth 1394px
\textheight 1300px
\topmargin -104px
\oddsidemargin -90px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
{\begin{picture}(608,1008)
%\put(12,0){\ing{24}}
%\put(12,0){\ing{penma}}
\put(0,0){\ing{penplo}}
\put(377,994){\sx{3.2}{$y$}}
\put(377,895){\sx{3.2}{$7$}}
\put(377,795){\sx{3.2}{$6$}}
\put(377,695){\sx{3.2}{$5$}}
\put(377,594){\sx{3.2}{$4$}}
\put(377,494){\sx{3.2}{$3$}}
\put(377,394){\sx{3.2}{$2$}}
\put(377,294){\sx{3.2}{$1$}}
\put(377,194){\sx{3.2}{$0$}}
\put(358, 93){\sx{3.2}{$-1$}}
\put(80,174){\sx{3.2}{$-3$}}
\put(180,174){\sx{3.2}{$-2$}}
\put(280,174){\sx{3.2}{$-1$}}
\put(396,174){\sx{3.2}{$0$}}
\put(496,174){\sx{3.2}{$1$}}
\put(590,174){\sx{3.2}{$x$}}
\put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
%
%\put(560,510){\sx{3.6}{\rot{84}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
%\put(532,708){\sx{3.6}{\rot{86}$y\!=\!\mathrm{pen}(x)$\ero}}
\put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}}
%\put(366,236){\sx{2.3}{$y\!=\!10(\mathrm{pen}(x)\!-\!1\!-\!x)$}}
%\put(416,239){\sx{3.4}{$y=10\,\delta(x)$}}
\put(8,236){\sx{3.3}{$y=10\,\delta(x)$}}
%\put(366, 8){\sx{3.3}{$L$}}
\put(312, 9){\sx{3.2}{$y\!=\!L$}}
\end{picture}
\end{document}

References

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014 (In Russian), page 269, figure 19.3.

http://www.ils.uec.ac.jp/~dima/PAPERS/2014acker.pdf http://mizugadro.mydns.jp/PAPERS/2014acker.pdf D.Kouznetsov. Holomorphic ackermanns. 2014, under consideration

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