Difference between revisions of "File:KelleriteT.jpg"

Iterates of the Keller function.

\(y=\mathrm{Keller}^{n}(x) =\mathrm{Shoka}\Big(n+\mathrm{ArcShoka}(x)\Big)~\) versus \(x\) for various values of number \(n\) of the iterate.

This map is used as Fig.5.11 at page 56 of book «Superfunctions» ^[1][2].

C++ generator of curves

#include <math.h> 
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include"ado.cin"
z_type Shoka(z_type  z)  { return z + log(exp(-z)+(M_E-1.)); }
z_type ArcShoka(z_type z){ return z + log((1.-exp(-z))/(M_E-1.)) ;}  
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
int main(){ int j,k,m,n; DB x,y, a;
FILE *o;o=fopen("kellerite.eps","w");ado(o,408,412);
fprintf(o,"4 4 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n");
for(m=0;m<5;m++){ M(m,0)L(m,4)}
for(n=0;n<5;n++){ M(0,n)L(4,n)} fprintf(o,".002 W 0 0 0 RGB S\n");
M(0,0)L(4,4) fprintf(o,".02 W 1 0 1 RGB S\n");
M(0,0) DO(n,404){x=.005+.01*n ;y=Re(Shoka(5.+ArcShoka(x)));if(y<4.02) L(x,y) else break;} 
M(0,0) DO(n,404){x=.005+.01*n ;y=Re(Shoka(4.+ArcShoka(x)));if(y<4.02) L(x,y) else break;} 
M(0,0) DO(n,404){x=.005+.01*n ;y=Re(Shoka(3.+ArcShoka(x)));if(y<4.02) L(x,y) else break;} 
M(0,0) DO(n,404){x=.005+.01*n ;y=Re(Shoka(2.+ArcShoka(x)));if(y<4.02) L(x,y) else break;} 
M(0,0) DO(n,404){x=.005+.01*n ;y=Re(Shoka(1.+ArcShoka(x)));if(y<4.02) L(x,y) else break;} 
fprintf(o,".02 W 0 1 1 RGB S\n");

M(0,0) DO(n,101){x=.02+.04*n;y=Re(Shoka(-1.+ArcShoka(x))); L(x,y)}
M(0,0) DO(n,101){x=.02+.04*n;y=Re(Shoka(-2.+ArcShoka(x))); L(x,y)}
M(0,0) DO(n,101){x=.02+.04*n;y=Re(Shoka(-3.+ArcShoka(x))); L(x,y)}
M(0,0) DO(n,101){x=.02+.04*n;y=Re(Shoka(-4.+ArcShoka(x))); L(x,y)}
M(0,0) DO(n,101){x=.02+.04*n;y=Re(Shoka(-5.+ArcShoka(x))); L(x,y)}
fprintf(o,".02 W 1 .5 0 RGB S\n");

DO(m,81){ M(0,0) DO(n,401){x=.01*(n+1.); y=Re(Shoka(-4.+.1*m + ArcShoka(x))); if(y<4.02)L(x,y) else break; } }
fprintf(o,".002 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
    system("epstopdf kellerite.eps");
    system(    "open kellerite.pdf"); //these 2 commands may be specific for macintosh
getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 420pt
\paperheight 424pt
\topmargin -103pt
\oddsidemargin -83pt
\textwidth 1200pt
\textheight 600pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\begin{document}
\sx{1}{ \begin{picture}(410,410)
\put(1,9){\ing{kellerite}}
\put(-12,401){\sx{2.4}{$y$}}
\put(-12,304){\sx{2.4}{$3$}}
\put(-12,204){\sx{2.4}{$2$}}
\put(-12,104){\sx{2.4}{$1$}}
\put(-12, 05){\sx{2.4}{$0$}}
\put(0,-7){\sx{2.3}{$0$}}
\put(100,-7){\sx{2.3}{$1$}}
\put(201,-7){\sx{2.3}{$2$}}
\put(301,-7){\sx{2.3}{$3$}}
\put(394,-7){\sx{2.4}{$x$}}
 %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero}
\put( 27,343){\rot{76}\sx{2.8}{$n\!=\!5$}\ero}
\put( 51,343){\rot{67}\sx{2.8}{$n\!=\!4$}\ero}
\put( 96,343){\rot{57}\sx{2.8}{$n\!=\!3$}\ero}
\put(168,343){\rot{51}\sx{2.8}{$n\!=\!2$}\ero}
\put(254,343){\rot{48}\sx{2.8}{$n\!=\!1$}\ero}
\put(322,318){\rot{45}\sx{2.8}{$n\!=\!0$}\ero}
\put(327,229){\rot{43}\sx{2.8}{$n\!=\!-1$}\ero}
\put(327,146){\rot{39}\sx{2.7}{$n\!=\!-2$}\ero}
\put(324, 79){\rot{32}\sx{2.7}{$n\!=\!-3$}\ero}
\put(323,37){\rot{21}\sx{2.7}{$n\!=\!-4$}\ero}
\put(320, 17){\rot{10}\sx{2.7}{$n\!=\!-5$}\ero}
\end{picture}
}
\end{document}

References

Keywords

«ArcKeller», «ArcShoka», «Book», «Iteration», «Keller function», «Shoka function», «Superfunctions», «Transferfunction»,