Difference between revisions of "File:Vladi07.jpg"

Fig.14.10 from page 199 of book «Superfunctions», 2020 ^[1].

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

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.

This picture is used also as Рис.14.10 in the Russian version Суперфункции ^[2], 2014.

First time published in the Vladikavkaz Matehmatical Journal ^[3], 2010.

C++ generator of the first picture

/* Fsexp.cin, ado.cin, conto.cin, plodi.cin should be loaded in order to compile the codes 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}

References

Keywords

«[[]]», «Agreement», «Approximation», «Book», «BookMap», «Complex map», «C++», «fsexp.cin», «Latex», «Natural tetration», «Tetration»,