Difference between revisions of "File:IterEq2plotU.png"

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[[Explicit plot]] of $c$th [[iteration]] of [[exponential]] to [[base sqrt(2)]] for various values of the number $c$ of iterations.
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Importing image file
  
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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^{c}(x)=F\big(c+G(x)\big)$
  +
  +
In the publication
  +
<ref>
  +
http://tori.ils.uec.ac.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.
  +
</ref>,
  +
these F and G are referred as $F_{4,5}$ and $F_{4,5}^{~-1}$, respectively.
  +
  +
==References==
  +
<references/>
  +
  +
==[[C++]] generator of curves==
  +
  +
// Files [[F45E.cin]], [[F45L.cin]] and [[ado.cin]] should be loaded in the working directory 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++)
  +
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.)
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#include "ado.cin"
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  +
#include "F45E.cin"
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#include "F45L.cin"
  +
  +
DB B=sqrt(2.);
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DB F(DB z) { return exp( exp( log(B)*z));}
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DB G(DB z) { return log( log(z) )/log(B);}
  +
  +
main(){ int m,n; double x,y,t; FILE *o;
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o=fopen("IterEq2plot.eps","w"); ado(o,1420,1420);
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fprintf(o,"1 1 translate 100 100 scale\n");
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#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
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#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
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for(m=0;m<15;m++) {M(m,0)L(m,14)}
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for(m=0;m<15;m++) {M(0,m)L(14,m)}
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fprintf(o,"2 setlinecap .01 W S\n");
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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");
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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 1 0 RGB S\n");
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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 1 0 1 RGB S\n");
  +
DO(m,131){x=1.421+.1*m;y=log(x)/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");
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for(n=-10;n<11;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;} }
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fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
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fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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system("epstopdf IterEq2plot.eps");
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system( "open IterEq2plot.pdf");
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getchar(); system("killall Preview");
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}
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==[[Latex]] generator of labels==
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%<nowiki> %<br>
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% file IterEq2plot.pdf should be generated with the code above in order to compile the Latex document below. %<br>
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% Copyleft 2012 by Dmitrii Kouznetsov <br> %
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\documentclass[12pt]{article} % <br>
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\usepackage{geometry} % <br>
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\usepackage{graphicx} % <br>
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\usepackage{rotating} % <br>
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\paperwidth 1418pt % <br>
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\paperheight 1418pt % <br>
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\topmargin -94pt % <br>
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\oddsidemargin -81pt % <br>
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\textwidth 1600pt % <br>
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\textheight 1600pt % <br>
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\pagestyle {empty} % <br>
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\newcommand \sx {\scalebox} % <br>
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\newcommand \rot {\begin{rotate}} % <br>
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\newcommand \ero {\end{rotate}} % <br>
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\newcommand \ing {\includegraphics} % <br>
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\parindent 0pt% <br>
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\pagestyle{empty} % <br>
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\begin{document} % <br>
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\begin{picture}(1412,1412) % <br>
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%\put(10,10){\ing{IterPowPlot}} % <br>
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\put(10,10){\ing{IterEq2plot}} % <br>
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\put(11,1374){\sx{4.7}{$y\!=\!\exp_{b}^{~ c}(x)$}} % <br>
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\put(11,1298){\sx{4}{$13$}} % <br>
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\put(11,1198){\sx{4}{$12$}} % <br>
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\put(11,1098){\sx{4}{$11$}} % <br>
  +
\put(11,998){\sx{4}{$10$}} % <br>
  +
\put(11,898){\sx{4}{$9$}} % <br>
  +
\put(11,798){\sx{4}{$8$}} % <br>
  +
\put(11,698){\sx{4}{$7$}} % <br>
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\put(11,598){\sx{4}{$6$}} % <br>
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\put(11,498){\sx{4}{$5$}} % <br>
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\put(11,398){\sx{4}{$4$}} % <br>
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\put(11,298){\sx{4}{$3$}} % <br>
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\put(11,198){\sx{4}{$2$}} % <br>
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\put(11,098){\sx{4}{$1$}} % <br>
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% <br>
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\put(100,16){\sx{4}{$1$}} % <br>
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\put(200,16){\sx{4}{$2$}} % <br>
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\put(301,16){\sx{4}{$3$}} % <br>
  +
\put(401,16){\sx{4}{$4$}} % <br>
  +
\put(502,16){\sx{4}{$5$}} % <br>
  +
\put(602,16){\sx{4}{$6$}} % <br>
  +
\put(703,16){\sx{4}{$7$}} % <br>
  +
\put(803,16){\sx{4}{$8$}} % <br>
  +
\put(903,16){\sx{4}{$9$}} % <br>
  +
\put(993,16){\sx{4}{$10$}} % <br>
  +
\put(1093,16){\sx{4}{$11$}} % <br>
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\put(1193,16){\sx{4}{$12$}} % <br>
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\put(1293,16){\sx{4}{$13$}} % <br>
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\put(1390,16){\sx{4}{$x$}} % <br>
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% <br>
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\put(595,1250){\sx{5}{\rot{84}$c\!=\!2$\ero}} % <br>
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\put(754,1250){\sx{5}{\rot{78}$c\!=\!1$\ero}} % <br>
  +
\put(809,1250){\sx{5}{\rot{74}$c\!=\!0.8$\ero}} % <br>
  +
\put(882,1250){\sx{5}{\rot{72}$c\!=\!0.6$\ero}} % <br>
  +
\put(971,1250){\sx{5}{\rot{64}$c\!=\!0.4$\ero}} % <br>
  +
\put(1094,1250){\sx{5}{\rot{54}$c\!=\!0.2$\ero}} % <br>
  +
% <br>
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\put(1282,1262){\sx{5}{\rot{44}$c\!=\!0$\ero}} % <br>
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% <br>
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\put(1256,1060){\sx{5}{\rot{34}$c\!=\!-0.2$\ero}} % <br>
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\put(1235, 926){\sx{5}{\rot{26}$c\!=\!-0.4$\ero}} % <br>
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\put(1230, 840){\sx{5}{\rot{19}$c\!=\!-0.6$\ero}} % <br>
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\put(1220, 770){\sx{5}{\rot{15}$c\!=\!-0.8$\ero}} % <br>
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\put(1230, 702){\sx{5}{\rot{11}$c\!=\!-1$\ero}} % <br>
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\put(1234, 542){\sx{5}{\rot{5}$c\!=\!-2$\ero}} % <br>
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%<br>
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\put(560, 1032){\sx{5.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} % <br>
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\put(674, 1062){\sx{5}{\rot{72}$y\!=\!b^x$\ero}} % <br>
  +
\put(890,610){\sx{5}{\rot{12}$y\!=\!\log_b(x)$\ero}} % <br>
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\put(825,484){\sx{5}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} % <br>
  +
\put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}}
  +
\end{picture} % <br>
  +
\end{document} % <br>
  +
%</nowiki>
  +
  +
[[Category:Iteration]]
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[[Category:SuperFunctions]]
  +
[[Category:AbelFunctions]]
  +
[[Category:Explicit plot]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Revision as of 09:43, 21 June 2013

Explicit plot of $c$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^{c}(x)=F\big(c+G(x)\big)$

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

References

  1. ↑ http://tori.ils.uec.ac.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

// Files F45E.cin, F45L.cin and ado.cin should be loaded in the working directory 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++)
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);}

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);
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 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 1 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 1 0 1 RGB S\n");
DO(m,131){x=1.421+.1*m;y=log(x)/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=-10;n<11;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

% %<br> % file IterEq2plot.pdf should be generated with the code above in order to compile the Latex document below. %<br> % Copyleft 2012 by Dmitrii Kouznetsov <br> % \documentclass[12pt]{article} % <br> \usepackage{geometry} % <br> \usepackage{graphicx} % <br> \usepackage{rotating} % <br> \paperwidth 1418pt % <br> \paperheight 1418pt % <br> \topmargin -94pt % <br> \oddsidemargin -81pt % <br> \textwidth 1600pt % <br> \textheight 1600pt % <br> \pagestyle {empty} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \parindent 0pt% <br> \pagestyle{empty} % <br> \begin{document} % <br> \begin{picture}(1412,1412) % <br> %\put(10,10){\ing{IterPowPlot}} % <br> \put(10,10){\ing{IterEq2plot}} % <br> \put(11,1374){\sx{4.7}{$y\!=\!\exp_{b}^{~ c}(x)$}} % <br> \put(11,1298){\sx{4}{$13$}} % <br> \put(11,1198){\sx{4}{$12$}} % <br> \put(11,1098){\sx{4}{$11$}} % <br> \put(11,998){\sx{4}{$10$}} % <br> \put(11,898){\sx{4}{$9$}} % <br> \put(11,798){\sx{4}{$8$}} % <br> \put(11,698){\sx{4}{$7$}} % <br> \put(11,598){\sx{4}{$6$}} % <br> \put(11,498){\sx{4}{$5$}} % <br> \put(11,398){\sx{4}{$4$}} % <br> \put(11,298){\sx{4}{$3$}} % <br> \put(11,198){\sx{4}{$2$}} % <br> \put(11,098){\sx{4}{$1$}} % <br> % <br> \put(100,16){\sx{4}{$1$}} % <br> \put(200,16){\sx{4}{$2$}} % <br> \put(301,16){\sx{4}{$3$}} % <br> \put(401,16){\sx{4}{$4$}} % <br> \put(502,16){\sx{4}{$5$}} % <br> \put(602,16){\sx{4}{$6$}} % <br> \put(703,16){\sx{4}{$7$}} % <br> \put(803,16){\sx{4}{$8$}} % <br> \put(903,16){\sx{4}{$9$}} % <br> \put(993,16){\sx{4}{$10$}} % <br> \put(1093,16){\sx{4}{$11$}} % <br> \put(1193,16){\sx{4}{$12$}} % <br> \put(1293,16){\sx{4}{$13$}} % <br> \put(1390,16){\sx{4}{$x$}} % <br> % <br> \put(595,1250){\sx{5}{\rot{84}$c\!=\!2$\ero}} % <br> \put(754,1250){\sx{5}{\rot{78}$c\!=\!1$\ero}} % <br> \put(809,1250){\sx{5}{\rot{74}$c\!=\!0.8$\ero}} % <br> \put(882,1250){\sx{5}{\rot{72}$c\!=\!0.6$\ero}} % <br> \put(971,1250){\sx{5}{\rot{64}$c\!=\!0.4$\ero}} % <br> \put(1094,1250){\sx{5}{\rot{54}$c\!=\!0.2$\ero}} % <br> % <br> \put(1282,1262){\sx{5}{\rot{44}$c\!=\!0$\ero}} % <br> % <br> \put(1256,1060){\sx{5}{\rot{34}$c\!=\!-0.2$\ero}} % <br> \put(1235, 926){\sx{5}{\rot{26}$c\!=\!-0.4$\ero}} % <br> \put(1230, 840){\sx{5}{\rot{19}$c\!=\!-0.6$\ero}} % <br> \put(1220, 770){\sx{5}{\rot{15}$c\!=\!-0.8$\ero}} % <br> \put(1230, 702){\sx{5}{\rot{11}$c\!=\!-1$\ero}} % <br> \put(1234, 542){\sx{5}{\rot{5}$c\!=\!-2$\ero}} % <br> %<br> \put(560, 1032){\sx{5.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} % <br> \put(674, 1062){\sx{5}{\rot{72}$y\!=\!b^x$\ero}} % <br> \put(890,610){\sx{5}{\rot{12}$y\!=\!\log_b(x)$\ero}} % <br> \put(825,484){\sx{5}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} % <br> \put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}} \end{picture} % <br> \end{document} % <br> %

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