Difference between revisions of "File:Vladi07.jpg"

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Maps of agreement of approximations of natural [[tetration]] with elementary functions [[fima]] and [[maclo]], used in the implementation [[fsexp.cin]].
−
Importing image file
  
  +
  +
Left: $D=D_6(x\!+\!\mathrm i y)$;
  +
  +
$\displaystyle
  +
D_6(z)= - \ln \left( \frac
  +
{|\mathrm{tai}(z) - \mathrm{fima}(z)|}
  +
{|\mathrm{tai}(z)|+|\mathrm{fima}(z)|}
  +
\right)$
  +
  +
right: $D=D_7(x\!+\!\mathrm i y)$;
  +
  +
$\displaystyle
  +
D_7(z)= - \ln \left( \frac
  +
{|\mathrm{tai}(z) - \mathrm{maclo}(z)|}
  +
{|\mathrm{tai}(z)|+|\mathrm{maclo}(z)|}
  +
\right)$
  +
  +
Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines.
  +
  +
Usage: this is figure 14.10 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.
  +
  +
First time published in the [[Vladikavkaz Matehmatical Journal]]
  +
<ref>
  +
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
  +
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
Figure 7.
  +
</ref>.
  +
  +
==Refereces==
  +
<references/>
  +
  +
==[[C++]] generator of the first picture==
  +
[[Fsexp.cin]],
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[plodi.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++)
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
//#include <complex.h>
  +
//#define z_type complex<double>
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
  +
#include "fsexp.cin"
  +
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
  +
int M=100,M1=M+1;
  +
int N=101,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("vladi07a.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1) X[m]=-4.+.08*(m-.5);
  +
DO(n,N1)Y[n]= -1 +.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n=0;n<7;n++) { M( -3,n)L(3,n)}
  +
fprintf(o,".006 W 0 0 0 RGB S\n");
  +
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
c=tai3(z);
  +
// d=fima1(z);
  +
d=fima(z);
  +
p=abs(c-d)/(abs(c)+abs(d));
  +
p=-log(p)/log(10.);
  +
// p=Re(c); q=Im(c);
  +
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
  +
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
  +
}}
  +
  +
#include"plodi.cin"
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf vladi07a.eps");
  +
system( "open vladi07a.pdf");//macintosh
  +
// getchar(); system("killall Preview"); //macintosh
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[C++]] generator of the second picture==
  +
<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++)
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
//#include <complex.h>
  +
//#define z_type complex<double>
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
  +
#include "fsexp.cin"
  +
  +
#include "conto.cin"
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
  +
int M=100,M1=M+1;
  +
int N=101,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("vladi07b.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1) X[m]=-4.+.08*(m-.5);
  +
DO(n,N1)Y[n]= -1 +.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n=0;n<7;n++) { M( -3,n)L(3,n)}
  +
fprintf(o,".006 W 0 0 0 RGB S\n");
  +
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
c=tai3(z);
  +
d=maclo(z);
  +
// d=fima1(z);
  +
p=abs(c-d)/(abs(c)+abs(d));
  +
p=-log(p)/log(10.);
  +
// p=Re(c); q=Im(c);
  +
if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
  +
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
  +
}}
  +
  +
#include"plodi.cin"
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf vladi07b.eps");
  +
system( "open vladi07b.pdf");//macintosh
  +
// getchar(); system("killall Preview");//macintosh
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] combiner==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\usepackage{geometry}
  +
\paperwidth 356px
  +
%\paperheight 134px
  +
\paperheight 184px
  +
\topmargin -98pt
  +
\oddsidemargin -94pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
  +
\newcommand \tafiax {
  +
\put(7,68){\sx{.5}{$y$}}
  +
%\put(5,74){\sx{.5}{$\Im(z)$}}
  +
%\put(7,69){\sx{.5}{$6$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put( 7 ,6){\sx{.5}{$-\!3$}}
  +
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(27 ,6){\sx{.5}{$-\!1$}}
  +
\put(40 , 6){\sx{.5}{$0$}}
  +
\put(50 , 6){\sx{.5}{$1$}}
  +
\put(60 , 6){\sx{.5}{$2$}}
  +
%\put(70 , 6){\sx{.5}{$3$}}
  +
%\put(78 , 6){\sx{.5}{$\Re(z)$}}
  +
\put(70 , 6){\sx{.5}{$x$}}
  +
}
  +
%\sx{2.33}{\begin{picture}(96,75)
  +
\sx{2.33}{\begin{picture}(80,75)
  +
%\put(0,0){\includegraphics{figtaifima}}
  +
\put(0,0){\includegraphics{vladi07a}}
  +
\tafiax
  +
\put(31,54){\sx{.45}{$D\!>\!14$}}
  +
\end{picture}}
  +
\sx{2.33}{\begin{picture}(84,70)
  +
%\put(0,0){\includegraphics{figtaimaclo}}
  +
\put(0,0){\includegraphics{vladi07b}}
  +
\tafiax
  +
\put(33,55){\sx{.52}{$D\!<\!1$}}
  +
\put(33,24){\sx{.45}{$D\!>\!14$}}
  +
\end{picture}}
  +
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Agreement]]
  +
[[Category:Book]]
  +
[[Category:Agreement]]
  +
[[Category:Complex map]]
  +
[[Category:BookMap]]
  +
[[Category:Tetration]]
  +
[[Category:C++]]
  +
[[Category:Latex]]

Latest revision as of 08:56, 1 December 2018

Maps of agreement of approximations of natural tetration with elementary functions fima and maclo, used in the implementation fsexp.cin.

Left: $D=D_6(x\!+\!\mathrm i y)$;

$\displaystyle D_6(z)= - \ln \left( \frac

\right)$ right: $D=D_7(x\!+\!\mathrm i y)$; $\displaystyle D_7(z)= - \ln \left( \frac
\right)$ Levels $D=\mathrm{const}$ are drawn with step 2, but the exception is dome for level $D=1$, this level is drawn with thick lines. Usage: this is figure 14.10 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015. First time published in the Vladikavkaz Matehmatical Journal[2].

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://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. Figure 7.

C++ generator of the first picture

Fsexp.cin, ado.cin, conto.cin, plodi.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++)
#include <complex>
typedef std::complex<double> z_type;
//#include <complex.h>
//#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

#include "fsexp.cin"

#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;

  int M=100,M1=M+1;
  int N=101,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("vladi07a.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]= -1 +.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n=0;n<7;n++) { M( -3,n)L(3,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=tai3(z);
// d=fima1(z);
        d=fima(z);
        p=abs(c-d)/(abs(c)+abs(d));
        p=-log(p)/log(10.);
        // p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}

#include"plodi.cin"

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi07a.eps");
        system( "open vladi07a.pdf");//macintosh
// getchar(); system("killall Preview"); //macintosh
}

C++ generator of the second picture



#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex>
typedef std::complex<double> z_type;
//#include <complex.h>
//#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

#include "fsexp.cin"

#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;

  int M=100,M1=M+1;
  int N=101,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("vladi07b.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]= -1 +.08*(n-.5);

for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
for(n=0;n<7;n++) { M( -3,n)L(3,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");

DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=tai3(z);
        d=maclo(z);
// d=fima1(z);
        p=abs(c-d)/(abs(c)+abs(d));
        p=-log(p)/log(10.);
        // p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}

#include"plodi.cin"

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi07b.eps");
        system( "open vladi07b.pdf");//macintosh
// getchar(); system("killall Preview");//macintosh
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 356px
%\paperheight 134px
\paperheight 184px
\topmargin -98pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\newcommand \tafiax {
\put(7,68){\sx{.5}{$y$}}
%\put(5,74){\sx{.5}{$\Im(z)$}}
%\put(7,69){\sx{.5}{$6$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put( 7 ,6){\sx{.5}{$-\!3$}}
\put(17 ,6){\sx{.5}{$-\!2$}}
\put(27 ,6){\sx{.5}{$-\!1$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(50 , 6){\sx{.5}{$1$}}
\put(60 , 6){\sx{.5}{$2$}}
%\put(70 , 6){\sx{.5}{$3$}}
%\put(78 , 6){\sx{.5}{$\Re(z)$}}
\put(70 , 6){\sx{.5}{$x$}}
}
%\sx{2.33}{\begin{picture}(96,75)
\sx{2.33}{\begin{picture}(80,75)
%\put(0,0){\includegraphics{figtaifima}}
\put(0,0){\includegraphics{vladi07a}}
\tafiax
\put(31,54){\sx{.45}{$D\!>\!14$}}
\end{picture}}
\sx{2.33}{\begin{picture}(84,70)
%\put(0,0){\includegraphics{figtaimaclo}}
\put(0,0){\includegraphics{vladi07b}}
\tafiax
\put(33,55){\sx{.52}{$D\!<\!1$}}
\put(33,24){\sx{.45}{$D\!>\!14$}}
\end{picture}}

\end{document}

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