Difference between revisions of "File:Sqrt2uiimap80.jpg"

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  +
[[Complex map]] of iterate number i of exponent to base $\sqrt{2}$ constructed at its upper fixed point 4:
−
Importing image file
  
  +
  +
$u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm u}(x\!+\!\mathrm i y)$
  +
  +
Usage: this is figure 16.12 of the book [[Суперфункции]] (2014, In Russian) <ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
  +
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. Суперфункции. [[Lambert Academic Publishing]], 2014.
  +
</ref>; the English version is in preparation in 2015. (Numeration of figures in the English version may be different from that of the Russian version.)
  +
  +
The algorithm of the evaluation is also described in the article
  +
<ref>
  +
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint
  +
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
  +
</ref>.
  +
(top right map)
  +
  +
==Refereces==
  +
<references/>
  +
  +
==[[C++]] generator of the map==
  +
Files [[ado.cin]],
  +
[[conto.cin]],
  +
[[sqrt2f21e.cin]],
  +
[[sqrt2f21l.cin]],
  +
[[sqrt2f45e.cin]],
  +
[[sqrt2f45l.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()
  +
#include "conto.cin"
  +
  +
#include "sqrt2f45e.cin"
  +
#include "sqrt2f45l.cin"
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
int M=801,M1=M+1;
  +
int N=405,N1=N+1;
  +
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
  +
char v[M1*N1]; // v is working array
  +
FILE *o;o=fopen("sqrt2uiima.eps","w"); ado(o,202,202);
  +
fprintf(o,"101 101 translate\n 10 10 scale\n");
  +
DO(m,M1) X[m]=-10.+.025*(m-.5);
  +
//DO(n,N1) Y[n]=-10.+.04*(n-.5);
  +
// DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.);
  +
// DO(n,200) Y[n]=-10.+.05*(n-.5);
  +
// Y[200]=-.0001;
  +
// Y[201]= .0001;
  +
for(n=0;n<N1;n++) Y[n]=.25*sinh(4.33*(n-202.5)/200.);
  +
// for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-2);
  +
for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
  +
for(n=-10;n<11;n++) {M( -10,n)L(10,n)}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
fprintf(o," .006 W 0 0 0 RGB S\n");
  +
// z_type tm,tp,F[M1*N1];
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
  +
f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m];
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
c=F45L(z) + z_type(0.,1.); c=F45E(c);
  +
p=Re(c);
  +
q=Im(c);
  +
if(p>-201. && p<201. && q>-201. && q<201.
  +
// && fabs(p)>1.e-14
  +
// && fabs(q)>1.e-14
  +
) { g[m*N1+n]=p; f[m*N1+n]=q;}
  +
}}
  +
p=10; q=1;
  +
for(m=-10;m<10;m++)for(n=2 ;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".014 W 0 .7 0 RGB S\n");
  +
for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".014 W 1 0 0 RGB S\n");
  +
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".014 W 0 0 1 RGB S\n");
  +
for(m= 1;m<100;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
  +
for(m= 1;m<100;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n");
  +
for(m=-100;m<101;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
  +
  +
// #include "plofu.cin"
  +
M(-10,0)L(2,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n");
  +
for(n=0;n<27;n++){ M(2-.5*(n+.2),0) L(2-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n");
  +
for(n=0;n<27;n++){ M(2-.5*(n+.7),0) L(2-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
// system("epstopdf 03.eps");
  +
system("epstopdf sqrt2uiima.eps");
  +
system( "open sqrt2uiima.pdf"); //for macintosh
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
File generated with code above
  +
should be loaded in order to compile the code below.
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\paperwidth 2072px
  +
\paperheight 2076px
  +
\textwidth 2394px
  +
\textheight 2300px
  +
\topmargin -97px
  +
\oddsidemargin -78px
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\begin{document}\parindent 0pt
  +
\sx{10}{\begin{picture}(206,206)
  +
%\put(6,5){\ing{Esqrt2ite13Map}}
  +
%\put(6,5){\ing{03}}
  +
\put(6,5){\ing{sqrt2uiima}}
  +
\put(2,203.4){\sx{.7}{$y$}}
  +
\put(2,184){\sx{.6}{$8$}}
  +
\put(2,164){\sx{.6}{$6$}}
  +
\put(2,144){\sx{.6}{$4$}}
  +
\put(2,124){\sx{.6}{$2$}}
  +
\put(2,104){\sx{.6}{$0$}}
  +
\put(-2.2,84){\sx{.6}{$-2$}}
  +
\put(-2.2,64){\sx{.6}{$-4$}}
  +
\put(-2.2,44){\sx{.6}{$-6$}}
  +
\put(-2.2,24){\sx{.6}{$-8$}}
  +
\put(-2,-1){\sx{.7}{$-\!10$}}
  +
\put( 22,-1){\sx{.7}{$-8$}}
  +
\put( 42,-1){\sx{.7}{$-6$}}
  +
\put( 62,-1){\sx{.7}{$-4$}}
  +
\put( 82,-1){\sx{.7}{$-2$}}
  +
\put(106,-1){\sx{.7}{$0$}}
  +
\put(126,-1){\sx{.7}{$2$}}
  +
\put(146,-1){\sx{.7}{$4$}}
  +
\put(166,-1){\sx{.7}{$6$}}
  +
\put(186,-1){\sx{.7}{$8$}}
  +
\put(204,-1){\sx{.7}{$x$}}
  +
\put(024,103.5){\sx{.99}{\bf cut}}
  +
\put(118,173){\sx{.99}{\rot{-16}$v\!=\!3$\ero}}
  +
\put(146,135){\sx{.99}{\rot{-29}$v\!=\!2$\ero}}
  +
\put(148,117){\sx{.99}{\rot{-33}$v\!=\!1$\ero}}
  +
\put(149,105){\sx{.99}{\rot{-40}$v\!=\!0$\ero}}
  +
\put(149,096){\sx{.99}{\rot{-56}$v\!=\!-1$\ero}}
  +
%
  +
\put(57,140){\sx{.99}{\rot{-47}$u\!=\!1$\ero}}
  +
%
  +
\put(26,30){\sx{.99}{\rot{84}$v\!=\!100$\ero}}
  +
\put(37,57){\sx{.99}{\rot{-74}$v\!=\!0$\ero}}
  +
\put(39,83){\sx{.99}{\rot{-66}$v\!=\!-100$\ero}}
  +
%
  +
\put(51,99){\sx{.99}{\rot{-49}$u\!=\!-100$\ero}}
  +
\put(53,87){\sx{.99}{\rot{-64}$u\!=\!100$\ero}}
  +
%
  +
\put(115.6,180){\sx{.99}{\rot{74}$u\!=\!2$\ero}}
  +
\put(155,170){\sx{.99}{\rot{47}$u\!=\!3$\ero}}
  +
\put(174.4,147.6){\sx{.99}{\rot{36}$u\!=\!4$\ero}}
  +
\put(176,122){\sx{.99}{\rot{31}$u\!=\!5$\ero}}
  +
\put(176,104){\sx{.99}{\rot{26}$u\!=\!6$\ero}}
  +
\put(176,092){\sx{.99}{\rot{17}$u\!=\!7$\ero}}
  +
\put(176,082){\sx{.99}{\rot{12}$u\!=\!8$\ero}}
  +
\put(176,074){\sx{.99}{\rot{4}$u\!=\!9$\ero}}
  +
\put(176,067){\sx{.99}{\rot{-2}$u\!=\!10$\ero}}
  +
\put(176,055){\sx{.99}{\rot{-16}$u\!=\!12$\ero}}
  +
\put(176,044){\sx{.99}{\rot{-28}$u\!=\!14$\ero}}
  +
\put(176,033){\sx{.99}{\rot{-40}$u\!=\!16$\ero}}
  +
\put(169,027){\sx{.99}{\rot{-48}$u\!=\!18$\ero}}
  +
\put(159,025){\sx{.96}{\rot{-50}$u\!=\!20$\ero}}
  +
\end{picture}}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Complex map]]
  +
[[Category:Base sqrt2]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:Iterate]]
  +
[[Category:Tetration]]
  +
[[Category:Arctetration]]
  +
[[Category:SuExp]]
  +
[[Category:AuExp]]
  +
[[Category:Inverse function]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:52, 1 December 2018

Complex map of iterate number i of exponent to base $\sqrt{2}$ constructed at its upper fixed point 4:

$u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm u}(x\!+\!\mathrm i y)$

Usage: this is figure 16.12 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015. (Numeration of figures in the English version may be different from that of the Russian version.)

The algorithm of the evaluation is also described in the article [2]. (top right map)

Refereces

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
    http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

C++ generator of the map

Files ado.cin, conto.cin, sqrt2f21e.cin, sqrt2f21l.cin, sqrt2f45e.cin, sqrt2f45l.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()
 #include "conto.cin"

#include "sqrt2f45e.cin"
#include "sqrt2f45l.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=801,M1=M+1;
 int N=405,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("sqrt2uiima.eps","w"); ado(o,202,202);
 fprintf(o,"101 101 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-10.+.025*(m-.5);
 //DO(n,N1) Y[n]=-10.+.04*(n-.5);
 // DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.);
// DO(n,200) Y[n]=-10.+.05*(n-.5);
// Y[200]=-.0001;
// Y[201]= .0001;
for(n=0;n<N1;n++) Y[n]=.25*sinh(4.33*(n-202.5)/200.);
// for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-2);
 for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
 for(n=-10;n<11;n++) {M( -10,n)L(10,n)}
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 fprintf(o," .006 W 0 0 0 RGB S\n");
 // z_type tm,tp,F[M1*N1];
 DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
                        f[m*N1+n]=9999;}
 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);
                c=F45L(z) + z_type(0.,1.); c=F45E(c);
                p=Re(c);
                q=Im(c);
                if(p>-201. && p<201. && q>-201. && q<201.
 // && fabs(p)>1.e-14
 // && fabs(q)>1.e-14
                ) { g[m*N1+n]=p; f[m*N1+n]=q;}
        }}
 p=10; q=1;
 for(m=-10;m<10;m++)for(n=2 ;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".014 W 0 .7 0 RGB S\n");
 for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".014 W 1 0 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".014 W 0 0 1 RGB S\n");
 for(m= 1;m<100;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
 for(m= 1;m<100;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n");
 for(m=-100;m<101;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

 // #include "plofu.cin"
  M(-10,0)L(2,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n");
 for(n=0;n<27;n++){ M(2-.5*(n+.2),0) L(2-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n");
 for(n=0;n<27;n++){ M(2-.5*(n+.7),0) L(2-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
// system("epstopdf 03.eps");
   system("epstopdf sqrt2uiima.eps");
   system( "open sqrt2uiima.pdf"); //for macintosh
 }

Latex generator of labels

File generated with code above should be loaded in order to compile the code below.


\documentclass[12pt]{article}
\paperwidth 2072px
\paperheight 2076px
\textwidth 2394px
\textheight 2300px
\topmargin -97px
\oddsidemargin -78px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\parindent 0pt
\pagestyle{empty}
\begin{document}\parindent 0pt
\sx{10}{\begin{picture}(206,206)
%\put(6,5){\ing{Esqrt2ite13Map}}
%\put(6,5){\ing{03}}
\put(6,5){\ing{sqrt2uiima}}
\put(2,203.4){\sx{.7}{$y$}}
\put(2,184){\sx{.6}{$8$}}
\put(2,164){\sx{.6}{$6$}}
\put(2,144){\sx{.6}{$4$}}
\put(2,124){\sx{.6}{$2$}}
\put(2,104){\sx{.6}{$0$}}
\put(-2.2,84){\sx{.6}{$-2$}}
\put(-2.2,64){\sx{.6}{$-4$}}
\put(-2.2,44){\sx{.6}{$-6$}}
\put(-2.2,24){\sx{.6}{$-8$}}
\put(-2,-1){\sx{.7}{$-\!10$}}
\put( 22,-1){\sx{.7}{$-8$}}
\put( 42,-1){\sx{.7}{$-6$}}
\put( 62,-1){\sx{.7}{$-4$}}
\put( 82,-1){\sx{.7}{$-2$}}
\put(106,-1){\sx{.7}{$0$}}
\put(126,-1){\sx{.7}{$2$}}
\put(146,-1){\sx{.7}{$4$}}
\put(166,-1){\sx{.7}{$6$}}
\put(186,-1){\sx{.7}{$8$}}
\put(204,-1){\sx{.7}{$x$}}
\put(024,103.5){\sx{.99}{\bf cut}}
\put(118,173){\sx{.99}{\rot{-16}$v\!=\!3$\ero}}
\put(146,135){\sx{.99}{\rot{-29}$v\!=\!2$\ero}}
\put(148,117){\sx{.99}{\rot{-33}$v\!=\!1$\ero}}
\put(149,105){\sx{.99}{\rot{-40}$v\!=\!0$\ero}}
\put(149,096){\sx{.99}{\rot{-56}$v\!=\!-1$\ero}}
%
\put(57,140){\sx{.99}{\rot{-47}$u\!=\!1$\ero}}
%
\put(26,30){\sx{.99}{\rot{84}$v\!=\!100$\ero}}
\put(37,57){\sx{.99}{\rot{-74}$v\!=\!0$\ero}}
\put(39,83){\sx{.99}{\rot{-66}$v\!=\!-100$\ero}}
%
\put(51,99){\sx{.99}{\rot{-49}$u\!=\!-100$\ero}}
\put(53,87){\sx{.99}{\rot{-64}$u\!=\!100$\ero}}
%
\put(115.6,180){\sx{.99}{\rot{74}$u\!=\!2$\ero}}
\put(155,170){\sx{.99}{\rot{47}$u\!=\!3$\ero}}
\put(174.4,147.6){\sx{.99}{\rot{36}$u\!=\!4$\ero}}
\put(176,122){\sx{.99}{\rot{31}$u\!=\!5$\ero}}
\put(176,104){\sx{.99}{\rot{26}$u\!=\!6$\ero}}
\put(176,092){\sx{.99}{\rot{17}$u\!=\!7$\ero}}
\put(176,082){\sx{.99}{\rot{12}$u\!=\!8$\ero}}
\put(176,074){\sx{.99}{\rot{4}$u\!=\!9$\ero}}
\put(176,067){\sx{.99}{\rot{-2}$u\!=\!10$\ero}}
\put(176,055){\sx{.99}{\rot{-16}$u\!=\!12$\ero}}
\put(176,044){\sx{.99}{\rot{-28}$u\!=\!14$\ero}}
\put(176,033){\sx{.99}{\rot{-40}$u\!=\!16$\ero}}
\put(169,027){\sx{.99}{\rot{-48}$u\!=\!18$\ero}}
\put(159,025){\sx{.96}{\rot{-50}$u\!=\!20$\ero}}
\end{picture}}
\end{document}

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