File:Vladi11.jpg

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Agreements at the substituiton of the numerical implementations of natural tetration tet and that of the natural arctetration ate into equations

$\mathrm{ate}(\mathrm{tet}(z))=z$

$\mathrm{tet}(\mathrm{ate}(z))=z$

Tetration is implemented as FSE.

Arctetration is implemented as FSL.

The agreements are

$ \displaystyle D_{\rm at}(z)= -\ln\left( \frac

\right) $ $ \displaystyle D_{\rm ta}(z)= -\ln\left( \frac
\right) $ Maps $D=D_{\rm at}(x\!+\!\mathrm i y)$ is shown at left, and map of $D=D_{\rm at}(x\!+\!\mathrm i y)$ is shown at right. In the central regions, the agreement with more than 14 digits takes place. Usage: this is figure 15.3 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 11.


C++ generator of the left hand side map

ado.cin, conto.cin, fslog.cin fsexp.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"superex.cin"
//#include"superlo.cin"
#include"fsexp.cin"
#include"fslog.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

int M=200,M1=M+1;
int N=300,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("figslosexp.eps","w");ado(o,0,0,404,604);
FILE *o;o=fopen("vladi11a.eps","w");ado(o,404,604);
fprintf(o,"202 302 translate\n 50 50 scale\n");

DO(m,M1) X[m]=-4+.04*m;
DO(n,N1) Y[n]=-6+.04*(n+.5);

for(m=-4;m<5;m++) { if(m==0){M(m,-6.1)L(m,6.1)}
                        else {M(m,-6)L(m,6)} }
for(n=-6;n<7;n++) {M( -4,n)L(4,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("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=FSEXP(z);
        if(abs(c)<1.e128){ c=FSLOG(c);
                                d=z;
                                p=abs(c-d);
                                p=-log(p)/log(10.);
                        if(p>-99 && p<99) g[m*N1+n]=p;
                        }
        }}

p=40;q=8;
#include "plodi.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi11a.eps");
        system( "open vladi11a.pdf");// linux
//getchar(); system("killall Preview");//macintosh
}

C++ generator of the right hand side map

ado.cin, conto.cin, fslog.cin fsexp.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;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

//#include"superex.cin"
//#include"superlo.cin"
#include"fsexp.cin"
#include"fslog.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

int M=200,M1=M+1;
int N=300,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("figslosexp.eps","w");ado(o,0,0,404,604);
FILE *o;o=fopen("vladi11b.eps","w");ado(o,404,604);
fprintf(o,"202 302 translate\n 50 50 scale\n");

DO(m,M1) X[m]=-4+.04*m;
DO(n,N1) Y[n]=-6+.04*(n+.5);

for(m=-4;m<5;m++) { if(m==0){M(m,-6.1)L(m,6.1)}
                        else {M(m,-6)L(m,6)} }
for(n=-6;n<7;n++) {M( -4,n)L(4,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("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=FSLOG(z);
        if(abs(c)<1.e128){ c=FSEXP(c);
                                d=z;
                                p=abs(c-d);
                                p=-log(p)/log(10.);
                        if(p>-99 && p<99) g[m*N1+n]=p;
                        }
        }}

p=40;q=8;
#include "plodi.cin"

//M(Re(Zo),Im(Zo)) L(-2.02,Im(Zo)) M(Re(Zo),-Im(Zo)) L(-2.02,-Im(Zo))
//fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,14){ M(Re(Zo)-.31*(m ), Im(Zo)) L(Re(Zo)-.31*(m+.5), Im(Zo)) }
DO(m,14){ M(Re(Zo)-.31*(m ),-Im(Zo)) L(Re(Zo)-.31*(m+.5),-Im(Zo)) }
fprintf(o,"0 setlinecap .08 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi11b.eps");
        system( "open vladi11b.pdf");// linux
//getchar(); system("killall Preview");//macintosh
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 418px
%\paperheight 134px
\paperheight 296px
\topmargin -96pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

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


\newcommand \vladiEL {
\put(-18,590){\sx{2.3}{$y$}}
\put(-18,544){\sx{2.3}{4}}
\put(-18,494){\sx{2.3}{4}}
\put(-18,444){\sx{2.3}{3}}
\put(-18,394){\sx{2.3}{2}}
\put(-18,344){\sx{2.3}{1}}
\put(-18,294){\sx{2.3}{0}}
\put(-36,244){\sx{2.3}{$-1$}}
\put(-36,194){\sx{2.3}{$-2$}}
\put(-36,144){\sx{2.3}{$-3$}}
\put(-36, 94){\sx{2.3}{$-4$}}
\put(-36, 44){\sx{2.3}{$-5$}}
%\put(-20, -20){\sx{2.3}{$-4$}}
\put( 30, -20){\sx{2.3}{$-3$}}
\put( 80, -20){\sx{2.3}{$-2$}}
\put(130, -20){\sx{2.3}{$-1$}}
\put(196, -20){\sx{2.3}{$0$}}
\put(246, -20){\sx{2.3}{$1$}}
\put(296, -20){\sx{2.3}{$2$}}
\put(346, -20){\sx{2.3}{$3$}}
\put(391, -20){\sx{2.3}{$x$}}
}

%\begin{figure}[t]
%\begin{center}
\vskip 16pt
\hskip 16pt
%\sx{.54}{\begin{picture}(472,580)
\sx{.47}{\begin{picture}(456,580)
%\put(0,0){\includegraphics{figslosexp}}
\put(0,0){\includegraphics{vladi11a}}
\vladiEL
\put(110,520){\sx{2.8}{$10\!<\!D\!<\!12$}}
\put(154,420){\sx{2.4}{$12\!<D\!<\!14$}}
\put(160,310){\sx{2.8}{$D\!>\!14$}}
\put(154,168){\sx{2.4}{$12\!<D\!<\!14$}}
\put(110,64){\sx{2.8}{$10\!<\!D\!<\!12$}}
\put(20, 320){\sx{2.8}{$D\!<\!1$}}
\put(310,560){\sx{2.8}{$D\!<\!1$}}
\put(310,20){\sx{2.8}{$D\!<\!1$}}
\end{picture}}
\sx{.47}{\begin{picture}(388,580)
%\put(0,0){\includegraphics{figsexpslo}}
\put(0,0){\includegraphics{vladi11b}}
\vladiEL
\put( 20,379){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put( 20,210){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put(200,310){\sx{3}{$D\!>\!14$}}
%\put(204,373){\sx{1}{11}}
%\put(204,226){\sx{1}{11}}
\end{picture}}
\end{document}

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