File:PowIteT.jpg: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Dmitrii Kouznetsov
No edit summary
(== Summary == Importing file)
Tag: Server-side upload
 
Line 1: Line 1:
== 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 18:52, 11 March 2022

Summary

Importing file

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current18:52, 11 March 2022Thumbnail for version as of 18:52, 11 March 20222,093 × 2,093 (1.01 MB)Maintenance script (talk | contribs)== Summary == Importing file

You cannot overwrite this file.

The following 2 pages use this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.

If the file has been modified from its original state, some details may not fully reflect the modified file.

Retrieved from "https://citizendium.org/wiki/index.php?title=File:PowIteT.jpg&oldid=1560"