Difference between revisions of "File:IterEq2plotT.jpg"

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(Explicit plot of $n$th iteration of exponential to base sqrt(2) for various values of the number $c$ of iterations. For evaluation of the non-integer iteration, the plotter uses the implementation through the superfunction $F$ of t...)
 
Line 14: Line 14:
   
 
==[[C++]] generator of curves==
  
==[[C++]] generator of curves==
−
<poem><nomathjax><nowiki>
 +
//<poem><nomathjax><nowiki>
  +
#include<math.h>
−
</nowiki></nomathjax>/<poem>
  
  +
#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"
  +
  +
#include "f45E.cin"
  +
#include "f45L.cin"
  +
  +
DB B=sqrt(2.);
  +
DB F(DB z) { return exp( exp( log(B)*z));}
  +
DB G(DB z) { return log( log(z) )/log(B);}
  +
  +
int main(){ int m,n; double x,y,t; FILE *o;
  +
o=fopen("IterEq2plot.eps","w"); ado(o,1420,1420);
  +
fprintf(o,"1 1 translate 100 100 scale\n");
  +
#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);
  +
  +
M(0,1.99)L(3.995,2.01)L(4.02,24)
  +
fprintf(o,"1 setlinecap 1 setlinejoin .03 W 0 .8 0 RGB S\n");
  +
M(1.99,0)L(2.01,3.995)L(14,4.02)
  +
fprintf(o,"1 setlinecap 1 setlinejoin .03 W .8 0 .8 RGB S\n");
  +
M(0,0)L(14,14)
  +
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 0 1 RGB S\n");
  +
  +
for(m=0;m<15;m++) {M(m,0)L(m,14)}
  +
for(m=0;m<15;m++) {M(0,m)L(14,m)}
  +
fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
  +
DO(m,82){x=0.001+.1*m;y=exp(log(B)*x); y=exp(log(B)*y); y=exp(log(B)*y); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y); if(y>15.1) break;}
  +
DO(m,82){x=0.001+.1*m; y=exp(log(B)*x);y=exp(log(B)*y); y=exp(log(B)*y); if(m==0) M(x,y) else L(x,y);if(y>15.1) break;} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
  +
DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
  +
DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 .9 0 RGB S\n");
  +
DO(m,141){x=0.001+.1*m; y=log(x)/log(B); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W .9 0 .9 RGB S\n");
  +
DO(m,131){x=1.41+.1*m;y=log(x)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}
  +
DO(m,131){x=1.63+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}
  +
DO(m,131){x=1.75+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}
  +
  +
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
  +
for(n=-20;n<21;n++){t=.1*n; M(2,2); DO(m,122){x=2.05+.1*m; y=Re(F45E(t+F45L(x+1.e-14*I))); L(x,y); if(y>14.1)break;} }
  +
fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf IterEq2plot.eps");
  +
system( "open IterEq2plot.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
  +
//</nowiki></nomathjax></poem>
   
   
 
==[[Latex]] generator of labels==
  
==[[Latex]] generator of labels==
 
<poem><nomathjax><nowiki>
  
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
−
</nowiki></nomathjax>/<poem>
  
  +
\usepackage{geometry}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\paperwidth 1408pt
  +
\paperheight 1408pt
  +
\topmargin -103pt
  +
\oddsidemargin -73pt
  +
\textwidth 1604pt
  +
\textheight 1604pt
  +
\pagestyle {empty}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\parindent 0pt% <br>
  +
\pagestyle{empty}
  +
\begin{document}
  +
\begin{picture}(1402,1402)
  +
%\put(10,10){\ing{IterPowPlot}}
  +
\put(1,1){\ing{IterEq2plot}}
  +
%\put(0,0){\ing{ZexIte}}
  +
\put(11,1384){\sx{4.4}{$y$}}
  +
\put(04,1290){\sx{4}{$13$}}
  +
\put(04,1190){\sx{4}{$12$}}
  +
\put(04,1090){\sx{4}{$11$}}
  +
\put(04,990){\sx{4}{$10$}}
  +
\put(11,890){\sx{4}{$9$}}
  +
\put(11,790){\sx{4}{$8$}}
  +
\put(11,690){\sx{4}{$7$}}
  +
\put(11,590){\sx{4}{$6$}}
  +
\put(11,490){\sx{4}{$5$}}
  +
\put(11,390){\sx{4}{$4$}}
  +
\put(11,290){\sx{4}{$3$}}
  +
\put(11,190){\sx{4}{$2$}}
  +
\put(11,090){\sx{4}{$1$}}
  +
  +
\put(91,6){\sx{4}{$1$}}
  +
\put(191,6){\sx{4}{$2$}}
  +
\put(291,6){\sx{4}{$3$}}
  +
\put(391,6){\sx{4}{$4$}}
  +
\put(492,6){\sx{4}{$5$}}
  +
\put(592,6){\sx{4}{$6$}}
  +
\put(693,6){\sx{4}{$7$}}
  +
\put(794,6){\sx{4}{$8$}}
  +
\put(894,6){\sx{4}{$9$}}
  +
\put(982,6){\sx{4}{$10$}}
  +
\put(1082,6){\sx{4}{$11$}}
  +
\put(1182,6){\sx{4}{$12$}}
  +
\put(1282,6){\sx{4}{$13$}}
  +
\put(1380,6){\sx{4.4}{$x$}}
  +
  +
\put(416,1158){\sx{5}{\rot{90}$n\!\rightarrow \! \infty$\ero}}
  +
\put(518,1250){\sx{5}{\rot{88}$n\!=\!3$\ero}}
  +
\put(590,1250){\sx{5}{\rot{84}$n\!=\!2$\ero}} %
  +
\put(750,1250){\sx{5}{\rot{78}$n\!=\!1$\ero}} %
  +
\put(800,1240){\sx{5}{\rot{74}$n\!=\!0.8$\ero}} %
  +
\put(872,1240){\sx{5}{\rot{71}$n\!=\!0.6$\ero}} %
  +
\put(961,1240){\sx{5}{\rot{64}$n\!=\!0.4$\ero}} %
  +
\put(1084,1240){\sx{5}{\rot{54}$n\!=\!0.2$\ero}} %
  +
\put(1172,1152){\sx{5.5}{\rot{44}$n\!=\!0$\ero}}
  +
\put(1230,1040){\sx{5}{\rot{34}$n\!=\!-0.2$\ero}} %
  +
\put(1210, 912){\sx{5}{\rot{26}$n\!=\!-0.4$\ero}} %
  +
\put(1204, 824){\sx{5}{\rot{19}$n\!=\!-0.6$\ero}} %
  +
\put(1200, 758){\sx{5}{\rot{15}$n\!=\!-0.8$\ero}} %
  +
\put(1210, 707){\sx{5}{\rot{11}$n\!=\!-1$\ero}} %
  +
\put(1234, 558){\sx{5}{\rot{4}$n\!=\!-2$\ero}} %
  +
\put(1234, 492){\sx{5}{\rot{2}$n\!=\!-3$\ero}} %
  +
%\put(560, 1032){\sx{6.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} %
  +
\put(694, 1032){\sx{6.6}{\rot{72}$y\!=\!b^x$\ero}} %
  +
\put(870,610){\sx{6}{\rot{16}$y\!=\!\log_b(x)$\ero}} %
  +
\put(825,510){\sx{6}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} %
  +
\put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}}
  +
\put(872,852){\sx{6}{\rot{44}$y\!=\!x$\ero}}
  +
\put(1180,374){\sx{5}{\rot{0.1}$n\!\rightarrow\!-\infty$\ero}}
  +
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
   
 
[[Category:Superesponential]]
  
[[Category:Superesponential]]

Revision as of 21:56, 27 September 2013

Explicit plot of $n$th iteration of exponential to base sqrt(2) for various values of the number $c$ of iterations.

For evaluation of the non-integer iteration, the plotter uses the implementation through the superfunction $F$ of the exponential to base $\sqrt{2}$, constructed at the fixed point $L\!=\!4$, and the corresponding Abel function $G$:

$ \exp_b^{n}(x)=F\big(n+G(x)\big)$

Note: In publication [1], these F and G are referred as $F_{4,5}$ and $F_{4,5}^{~-1}$, respectively.

References

  1. ↑ http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.

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"

 #include "f45E.cin"
 #include "f45L.cin"

 DB B=sqrt(2.);
 DB F(DB z) { return exp( exp( log(B)*z));}
 DB G(DB z) { return log( log(z) )/log(B);}

int main(){ int m,n; double x,y,t; FILE *o;
 o=fopen("IterEq2plot.eps","w"); ado(o,1420,1420);
 fprintf(o,"1 1 translate 100 100 scale\n");
 #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);

M(0,1.99)L(3.995,2.01)L(4.02,24)
fprintf(o,"1 setlinecap 1 setlinejoin .03 W 0 .8 0 RGB S\n");
M(1.99,0)L(2.01,3.995)L(14,4.02)
fprintf(o,"1 setlinecap 1 setlinejoin .03 W .8 0 .8 RGB S\n");
M(0,0)L(14,14)
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 0 1 RGB S\n");

 for(m=0;m<15;m++) {M(m,0)L(m,14)}
 for(m=0;m<15;m++) {M(0,m)L(14,m)}
 fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
 DO(m,82){x=0.001+.1*m;y=exp(log(B)*x); y=exp(log(B)*y); y=exp(log(B)*y); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y); if(y>15.1) break;}
 DO(m,82){x=0.001+.1*m; y=exp(log(B)*x);y=exp(log(B)*y); y=exp(log(B)*y); if(m==0) M(x,y) else L(x,y);if(y>15.1) break;} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
 DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
 DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 .9 0 RGB S\n");
 DO(m,141){x=0.001+.1*m; y=log(x)/log(B); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W .9 0 .9 RGB S\n");
 DO(m,131){x=1.41+.1*m;y=log(x)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}
 DO(m,131){x=1.63+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}
 DO(m,131){x=1.75+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);}

fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
 for(n=-20;n<21;n++){t=.1*n; M(2,2); DO(m,122){x=2.05+.1*m; y=Re(F45E(t+F45L(x+1.e-14*I))); L(x,y); if(y>14.1)break;} }
 fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf IterEq2plot.eps");
       system( "open IterEq2plot.pdf");
       getchar(); system("killall Preview");
 }

//


Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 1408pt
\paperheight 1408pt
\topmargin -103pt
\oddsidemargin -73pt
\textwidth 1604pt
\textheight 1604pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt% <br>
\pagestyle{empty}
\begin{document}
\begin{picture}(1402,1402)
%\put(10,10){\ing{IterPowPlot}}
\put(1,1){\ing{IterEq2plot}}
%\put(0,0){\ing{ZexIte}}
\put(11,1384){\sx{4.4}{$y$}}
\put(04,1290){\sx{4}{$13$}}
\put(04,1190){\sx{4}{$12$}}
\put(04,1090){\sx{4}{$11$}}
\put(04,990){\sx{4}{$10$}}
\put(11,890){\sx{4}{$9$}}
\put(11,790){\sx{4}{$8$}}
\put(11,690){\sx{4}{$7$}}
\put(11,590){\sx{4}{$6$}}
\put(11,490){\sx{4}{$5$}}
\put(11,390){\sx{4}{$4$}}
\put(11,290){\sx{4}{$3$}}
\put(11,190){\sx{4}{$2$}}
\put(11,090){\sx{4}{$1$}}
 
\put(91,6){\sx{4}{$1$}}
\put(191,6){\sx{4}{$2$}}
\put(291,6){\sx{4}{$3$}}
\put(391,6){\sx{4}{$4$}}
\put(492,6){\sx{4}{$5$}}
\put(592,6){\sx{4}{$6$}}
\put(693,6){\sx{4}{$7$}}
\put(794,6){\sx{4}{$8$}}
\put(894,6){\sx{4}{$9$}}
\put(982,6){\sx{4}{$10$}}
\put(1082,6){\sx{4}{$11$}}
\put(1182,6){\sx{4}{$12$}}
\put(1282,6){\sx{4}{$13$}}
\put(1380,6){\sx{4.4}{$x$}}

\put(416,1158){\sx{5}{\rot{90}$n\!\rightarrow \! \infty$\ero}}
\put(518,1250){\sx{5}{\rot{88}$n\!=\!3$\ero}}
\put(590,1250){\sx{5}{\rot{84}$n\!=\!2$\ero}} %
\put(750,1250){\sx{5}{\rot{78}$n\!=\!1$\ero}} %
\put(800,1240){\sx{5}{\rot{74}$n\!=\!0.8$\ero}} %
\put(872,1240){\sx{5}{\rot{71}$n\!=\!0.6$\ero}} %
\put(961,1240){\sx{5}{\rot{64}$n\!=\!0.4$\ero}} %
\put(1084,1240){\sx{5}{\rot{54}$n\!=\!0.2$\ero}} %
\put(1172,1152){\sx{5.5}{\rot{44}$n\!=\!0$\ero}}
\put(1230,1040){\sx{5}{\rot{34}$n\!=\!-0.2$\ero}} %
\put(1210, 912){\sx{5}{\rot{26}$n\!=\!-0.4$\ero}} %
\put(1204, 824){\sx{5}{\rot{19}$n\!=\!-0.6$\ero}} %
\put(1200, 758){\sx{5}{\rot{15}$n\!=\!-0.8$\ero}} %
\put(1210, 707){\sx{5}{\rot{11}$n\!=\!-1$\ero}} %
\put(1234, 558){\sx{5}{\rot{4}$n\!=\!-2$\ero}} %
\put(1234, 492){\sx{5}{\rot{2}$n\!=\!-3$\ero}} %
%\put(560, 1032){\sx{6.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} %
\put(694, 1032){\sx{6.6}{\rot{72}$y\!=\!b^x$\ero}} %
\put(870,610){\sx{6}{\rot{16}$y\!=\!\log_b(x)$\ero}} %
\put(825,510){\sx{6}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} %
\put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}}
\put(872,852){\sx{6}{\rot{44}$y\!=\!x$\ero}}
\put(1180,374){\sx{5}{\rot{0.1}$n\!\rightarrow\!-\infty$\ero}}

\end{picture}
\end{document}

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current21:53, 27 September 2013Thumbnail for version as of 21:53, 27 September 20132,922 × 2,922 (1.35 MB)T (talk | contribs)Explicit plot of $n$th iteration of exponential to base sqrt(2) for various values of the number $c$ of iterations. For evaluation of the non-integer iteration, the plotter uses the implementation through the superfunction $F$ of t...

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