File:PowIteT.jpg: Difference between revisions

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== Summary ==
== Summary ==
{{Image_Details|user
Importing file
|description  = [[Explicit plot]] of <math>n</math>th Iteration of the quadratic function, id est, [[power function]] for power 2; <math>y=\mathrm{Pow}_2^{~n}(z)=T^n(x)~</math> for various values of number <math>n</math> of iteration. Here, 
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)
</math>
 
The iteration of <math>T</math> is implemented in the following way:  :
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)
</math>
 
where 
 
<math>\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)</math>
 
is [[superfunction]] for the transfer function
<math>T</math>
and
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)
</math>
 
is the [[Abel Function]].  The thick lines represent
<math>
y=x^2
</math>and
<math>
y=\sqrt{x}</math>.
FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to <math>n\!=\!1</math>, and the second overlaps with <math>y=T^{-1}(x)</math> and corresponds to  <math>n\!=\!-1</math>.
|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|date-created = 2012.11.11
|pub-country  = Japan
|notes        =
|versions    = http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
}}
 
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==
 
 
%<nowiki> %<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>
%</nowiki>
 
== Licensing ==
{{CC|by|3.0}}

Latest revision as of 19:52, 11 March 2022

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