Difference between revisions of "File:Tet5loplot.jpg"

From TORI
Jump to navigation Jump to search
(Importing image file)
 
 
Line 1: Line 1:
  +
Graphical search for the real [[fixed point]]s of [[tetration]]:
−
Importing image file
  
  +
  +
$y=\mathrm{tet}_b(x)$ for various $b$ and line $y\!=\!x$.
  +
  +
Notations:
  +
  +
$\mathrm e\!=\!\exp(1)\!\approx\!2.71$ is base of the natural logarithm,
  +
  +
$L_{\mathrm e,0}\approx -1.8503545290271812$ is the only real fixed point of the [[natural tetration]], $b\!=\!\mathrm e$
  +
  +
$\tau\!\approx\! 1.63532$ is crytical base; at $b\!=\!\tau$, tetration has 2 real fixed points:
  +
  +
Regular one at $L_{\tau,0}\!\approx\! -1.7$ and exotic one $L_{\tau,1}\!\approx\!3.087$
  +
  +
At smaller base $b$, function $\mathrm{tet}_b$ has 3 regular real fixed points; in this sense, variety of [[supertetration]]s is richer, than that of
  +
[[superexponential]]s.
  +
  +
An $b$ decreases, approaching the [[Henryk constant]] $\eta=\exp(1/\mathrm e)$, the biggest fixed point runs to infinity;
  +
and at $1\!<\!b\!\le\! \eta$, function $\mathrm{tet}_b$ has two real regular fixed points.
  +
  +
==[[C++]] generator of curves==
  +
Files
  +
[[ado.cin]] and [[fit1.cin]] should be loaded in order to compile the code below:
  +
  +
<poem><nomathjax><nowiki>
  +
#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 std::complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
//b=10
  +
//#include "f4ten.cin"
  +
#include "fit1.cin"
  +
#include "ado.cin"
  +
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  +
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
  +
int main(){ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d;
  +
FILE *o;o=fopen("tet5loplo.eps","w");ado(o,708,708);
  +
fprintf(o,"204 204 translate\n 100 100 scale\n");
  +
fprintf(o,"2 setlinecap\n");
  +
for(m=-2;m<6;m++){if(m!=0){M(m,-2)L(m,5)}}
  +
for(n=-2;n<6;n++){if(n!=0){M(-2,n)L(5,n)}} fprintf(o,".006 W 0 0 0 RGB S\n");
  +
M(-2,0)L(5.1,0) M(0, -2)L(0,5.1) fprintf(o,".01 W 0 0 0 RGB S\n");
  +
M(0,M_E)L(1.,M_E) fprintf(o,".006 W 0 0 0 RGB S\n");
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n");
  +
DO(m,300){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
  +
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.63532),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
  +
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
  +
DO(m,300){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(x>5.1 || y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
  +
DO(m,300){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W 0 0 .7 RGB S\n");
  +
DO(m,300){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .8 0 0 RGB S\n");
  +
DO(m,340){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>5.) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 0 RGB S\n");
  +
M(-2,-2)L(5,5) fprintf(o,".01 W 0 0 0 RGB S\n");
  +
x=3.087; M(x,0) L(x,x) L(0,x) fprintf(o,".001 W 0 0 0 RGB S\n");
  +
DB Lt=-1.8503545290271812;
  +
M(Lt,0) L(Lt,Lt) L(0,Lt); fprintf(o,".001 W 0 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf tet5loplo.eps");
  +
system( "open tet5loplo.pdf"); //mac
  +
getchar(); system("killall Preview");// mac
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry} % See geometry.pdf
  +
\geometry{letterpaper} % ... or a4paper or a5paper or ... ??
  +
\usepackage{graphicx}
  +
\usepackage{amssymb}
  +
\usepackage{hyperref}
  +
\usepackage{rotating}
  +
\usepackage[utf8x]{inputenc}
  +
\usepackage[english,russian]{babel} %some packages are not used
  +
\usepackage{color}
  +
\definecolor{red}{rgb}{1,0.1,0.1}
  +
\definecolor{black}{rgb}{0,0,0}
  +
\definecolor{white}{rgb}{1,1,1}
  +
\definecolor{yellow}{rgb}{1,.93,0}
  +
\definecolor{bluedark}{rgb}{0,0,.87}
  +
\paperwidth 712pt
  +
\paperheight 716pt
  +
\topmargin -98pt
  +
\oddsidemargin -72pt
  +
\textwidth 810pt
  +
\textheight 870pt
  +
  +
\newcommand \sx {\scalebox}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \tet {\mathrm{tet}}
  +
\newcommand \pen {\mathrm{pen}}
  +
\newcommand \bC {\mathbb C}
  +
\newcommand \fac {\mathrm {Factorial}}
  +
\newcommand \rme {\mathrm e}
  +
\newcommand \rmi {\mathrm i}
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
  +
\begin{document}
  +
\parindent 0pt
  +
{\normalsize
  +
\begin{picture}(702,700)\put(0,0){\ing{tet5loplo}}
  +
\put( 172,680){\sx{4}{$y$}}
  +
\put( 172,590){\sx{4}{$4$}}
  +
\put( 182,467){\sx{4}{e}}
  +
\put( 172,390){\sx{4}{$2$}}
  +
\put( 172,190){\sx{4}{$0$}}
  +
\put( 194,164){\sx{4}{$0$}}
  +
\put( 394,164){\sx{4}{$2$}}
  +
\put( 594,164){\sx{4}{$4$}}
  +
\put( 686,166){\sx{4}{$x$}}
  +
\put(344,570){\sx{4}{\rot{80} $b\!=\! \rme$ \ero } }
  +
\put(522,574){\sx{4}{\rot{66} $b\!=\!1.7$ \ero } }
  +
\put(600,603){\sx{4}{\rot{64} $b\!=\!\tau$ \ero } }
  +
\put(550,492){\sx{4}{\rot{42} $b\!=\!1.6$ \ero } }
  +
%\put(574,422){\sx{4}{\rot{13} $b\!=\!1.5$ \ero } }
  +
\put(580,438){\sx{4}{\rot{14} $b\!=\!1.5$ \ero } }
  +
\put(592,400){\sx{4}{\rot{7} $b\!=\! \eta$ \ero}}
  +
\put(574,345){\sx{4}{\rot{2}$b\!=\!\sqrt{2}$\ero}}
  +
\put(130,504){\sx{4}{$L_{\tau,1}$}}
  +
\put(500,170){\sx{4}{$L_{\tau,1}$}}
  +
\put(-2,219){\sx{4}{$L_{\mathrm e,0}$}}
  +
\put(202,6){\sx{4}{$L_{\mathrm e,0}$}}
  +
\end{picture}}
  +
\end{document}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Explicit plot]]
  +
[[Category:Tetration]]
  +
[[Category:Fixed point]]
  +
[[Category:Pentation]]

Latest revision as of 08:53, 1 December 2018

Graphical search for the real fixed points of tetration:

$y=\mathrm{tet}_b(x)$ for various $b$ and line $y\!=\!x$.

Notations:

$\mathrm e\!=\!\exp(1)\!\approx\!2.71$ is base of the natural logarithm,

$L_{\mathrm e,0}\approx -1.8503545290271812$ is the only real fixed point of the natural tetration, $b\!=\!\mathrm e$

$\tau\!\approx\! 1.63532$ is crytical base; at $b\!=\!\tau$, tetration has 2 real fixed points:

Regular one at $L_{\tau,0}\!\approx\! -1.7$ and exotic one $L_{\tau,1}\!\approx\!3.087$

At smaller base $b$, function $\mathrm{tet}_b$ has 3 regular real fixed points; in this sense, variety of supertetrations is richer, than that of superexponentials.

An $b$ decreases, approaching the Henryk constant $\eta=\exp(1/\mathrm e)$, the biggest fixed point runs to infinity; and at $1\!<\!b\!\le\! \eta$, function $\mathrm{tet}_b$ has two real regular fixed points.

C++ generator of curves

Files ado.cin and fit1.cin should be loaded in order to compile the 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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
//b=10
//#include "f4ten.cin"
#include "fit1.cin"
#include "ado.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
int main(){ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d;
FILE *o;o=fopen("tet5loplo.eps","w");ado(o,708,708);
fprintf(o,"204 204 translate\n 100 100 scale\n");
fprintf(o,"2 setlinecap\n");
for(m=-2;m<6;m++){if(m!=0){M(m,-2)L(m,5)}}
for(n=-2;n<6;n++){if(n!=0){M(-2,n)L(5,n)}} fprintf(o,".006 W 0 0 0 RGB S\n");
M(-2,0)L(5.1,0) M(0, -2)L(0,5.1) fprintf(o,".01 W 0 0 0 RGB S\n");
M(0,M_E)L(1.,M_E) fprintf(o,".006 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 1 setlinecap\n");
DO(m,300){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.63532),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(x>5.1 || y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W 0 0 .7 RGB S\n");
DO(m,300){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .8 0 0 RGB S\n");
DO(m,340){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>5.) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 0 RGB S\n");
M(-2,-2)L(5,5) fprintf(o,".01 W 0 0 0 RGB S\n");
x=3.087; M(x,0) L(x,x) L(0,x) fprintf(o,".001 W 0 0 0 RGB S\n");
DB Lt=-1.8503545290271812;
M(Lt,0) L(Lt,Lt) L(0,Lt); fprintf(o,".001 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
       system("epstopdf tet5loplo.eps");
       system( "open tet5loplo.pdf"); //mac
getchar(); system("killall Preview");// mac
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry} % See geometry.pdf
\geometry{letterpaper} % ... or a4paper or a5paper or ... ??
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{rotating}
\usepackage[utf8x]{inputenc}
\usepackage[english,russian]{babel} %some packages are not used
\usepackage{color}
\definecolor{red}{rgb}{1,0.1,0.1}
\definecolor{black}{rgb}{0,0,0}
\definecolor{white}{rgb}{1,1,1}
\definecolor{yellow}{rgb}{1,.93,0}
\definecolor{bluedark}{rgb}{0,0,.87}
\paperwidth 712pt
\paperheight 716pt
\topmargin -98pt
\oddsidemargin -72pt
\textwidth 810pt
\textheight 870pt

\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\newcommand \tet {\mathrm{tet}}
\newcommand \pen {\mathrm{pen}}
\newcommand \bC {\mathbb C}
\newcommand \fac {\mathrm {Factorial}}
\newcommand \rme {\mathrm e}
\newcommand \rmi {\mathrm i}
\newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\begin{document}
\parindent 0pt
{\normalsize
\begin{picture}(702,700)\put(0,0){\ing{tet5loplo}}
\put( 172,680){\sx{4}{$y$}}
\put( 172,590){\sx{4}{$4$}}
\put( 182,467){\sx{4}{e}}
\put( 172,390){\sx{4}{$2$}}
\put( 172,190){\sx{4}{$0$}}
\put( 194,164){\sx{4}{$0$}}
\put( 394,164){\sx{4}{$2$}}
\put( 594,164){\sx{4}{$4$}}
\put( 686,166){\sx{4}{$x$}}
\put(344,570){\sx{4}{\rot{80} $b\!=\! \rme$ \ero } }
\put(522,574){\sx{4}{\rot{66} $b\!=\!1.7$ \ero } }
\put(600,603){\sx{4}{\rot{64} $b\!=\!\tau$ \ero } }
\put(550,492){\sx{4}{\rot{42} $b\!=\!1.6$ \ero } }
%\put(574,422){\sx{4}{\rot{13} $b\!=\!1.5$ \ero } }
\put(580,438){\sx{4}{\rot{14} $b\!=\!1.5$ \ero } }
\put(592,400){\sx{4}{\rot{7} $b\!=\! \eta$ \ero}}
\put(574,345){\sx{4}{\rot{2}$b\!=\!\sqrt{2}$\ero}}
\put(130,504){\sx{4}{$L_{\tau,1}$}}
\put(500,170){\sx{4}{$L_{\tau,1}$}}
\put(-2,219){\sx{4}{$L_{\mathrm e,0}$}}
\put(202,6){\sx{4}{$L_{\mathrm e,0}$}}
\end{picture}}
\end{document}

References

File history

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

Date/TimeThumbnailDimensionsUserComment
current06:14, 1 December 2018Thumbnail for version as of 06:14, 1 December 20181,477 × 1,486 (283 KB)Maintenance script (talk | contribs)Importing image file

You cannot overwrite this file.

The following page uses 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://mizugadro.mydns.jp/t/index.php?title=File:Tet5loplot.jpg&oldid=21847"