File:PowIteT.jpg

Summary

Title / Description	Explicit plot of ${\displaystyle n}$th Iteration of the quadratic function, id est, power function for power 2; ${\displaystyle y=\mathrm {Pow} _{2}^{~n}(z)=T^{n}(x)~}$ for various values of number ${\displaystyle n}$ of iteration. Here, ${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (1)~~~T(z)=\mathrm {Pow} _{2}(z)=z^{2}=\exp {\Big (}\ln(z)\,2{\Big )}}$ The iteration of ${\displaystyle T}$ is implemented in the following way:  : ${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (2)~~~y=T^{n}(x)={\mathrm {Pow} _{2}}^{n}(z)=F{\Big (}n+G(x){\Big )}}$ where ${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (3)~~~F(z)=\exp {\Big (}\exp {\big (}\ln(2)\,z{\big )}{\Big )}}$ is superfunction for the transfer function ${\displaystyle T}$ and ${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (4)~~~G(z)=\ln {\Big (}\ln(z){\Big )}/\ln(2)}$ is the Abel Function. The thick lines represent ${\displaystyle y=x^{2}}$and ${\displaystyle y={\sqrt {x}}}$. FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to ${\displaystyle n\!=\!1}$, and the second overlaps with ${\displaystyle y=T^{-1}(x)}$ and corresponds to ${\displaystyle n\!=\!-1}$.
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2012.11.11
Country of first publication	Japan
Notes	You can edit this page and add notes here which may be useful to people who wish to re-use this media.
Other versions	http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
Using this image on CZ	Please click here to add the credit line, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary. {{Image|PowIteT.jpg|right|350px|Add image caption here.}}

Files used to generate the figure are supplied below.

C++ generator of curves

// file ado.cin should be loaded to 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"

DB B=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("PowIte.eps","w"); ado(o,1010,1010);
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<11;m++) {M(m,0)L(m,10)}
for(m=0;m<11;m++) {M(0,m)L(10,m)}
fprintf(o,"2 setlinecap .01 W S\n");
DO(m,402){x=0.001+.01*m; y=exp(2.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
DO(m,402){x=0.001+.01*m; y=exp(4.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
DO(m,402){x=0.001+.01*m; y=exp(8.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
DO(m,1002){x=0.001+.01*m; y=exp(.5*log(x)); 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,1002){x=0.001+.01*m; y=exp(.25*log(x)); 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,1002){x=0.001+.01*m; y=exp(.125*log(x)); 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; DO(m,182){x=1.+.05*m; y=F(t+G(x)); if(m==0) M(x,y) else L(x,y); if(y>10.1)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 PowIte.eps");    
      system(    "open PowIte.pdf");
      getchar(); system("killall Preview");
}

Latex generator of labels

% %<br> % file IterPowPlot.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 1008pt % <br> \paperheight 1008pt % <br> \topmargin -94pt % <br> \oddsidemargin -81pt % <br> \textwidth 1100pt % <br> \textheight 1100pt % <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}(1002,1002) % <br> %\put(10,10){\ing{IterPowPlot}} % <br> \put(10,10){\ing{PowIte}} % <br> %\put(11,976){\sx{4}{$y\!=\!\mathrm{pow}_2^c(x)$}} % <br> \put(11,976){\sx{4}{$y\!=\!x^{2^n}$}} % <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(990,16){\sx{4}{$x$}} % <br>  % <br> \put(140,870){\sx{3.6}{\rot{88}$n\!=\!3$\ero}} % <br> \put(179,870){\sx{3.6}{\rot{87}$n\!=\!2$\ero}} % <br> \put(325,921){\sx{3.6}{\rot{81}$n\!=\!1$\ero}} % <br> \put(461,893){\sx{3.6}{\rot{70}$n\!=\!0.6$\ero}} % <br> \put(509,894){\sx{3.6}{\rot{69}$n\!=\!0.5$\ero}} % <br> \put(563,895){\sx{3.6}{\rot{63}$n\!=\!0.4$\ero}} % <br> \put(630,897){\sx{3.6}{\rot{59}$n\!=\!0.3$\ero}} % <br> \put(713,900){\sx{3.6}{\rot{54}$n\!=\!0.2$\ero}} % <br> \put(826,913){\sx{3.6}{\rot{50}$n\!=\!0.1$\ero}} % <br>  % <br> \put(928,932){\sx{3.6}{\rot{45}$n\!=\!0$\ero}} % <br> % <br> \put(894,778){\sx{3.6}{\rot{39}$n\!=\!-0.1$\ero}} % <br> \put(888,676){\sx{3.6}{\rot{33}$n\!=\!-0.2$\ero}} % <br> \put(884,596){\sx{3.6}{\rot{28}$n\!=\!-0.3$\ero}} % <br> \put(877,528){\sx{3.6}{\rot{24}$n\!=\!-0.4$\ero}} % <br> \put(871,476){\sx{3.6}{\rot{19}$n\!=\!-0.5$\ero}} % <br> \put(870,429){\sx{3.6}{\rot{17}$n\!=\!-0.6$\ero}} % <br>  % <br> \put(890,298){\sx{3.6}{\rot{8}$n\!=\!-1$\ero}} % <br> \put(886,157){\sx{3.6}{\rot{1}$n\!=\!-2$\ero}} % <br> \put(886,115){\sx{3.6}{\rot{.3}$n\!=\!-3$\ero}} % <br> \end{picture} % <br> \end{document} % <br> %

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