File:Vladi08.jpg

Fig.14.11 from page 200 of book «Superfunctions» ^[1], 2020

Agreement of approximations of natural tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.

The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.

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.

\(\displaystyle {\rm fse}(z)=\left\{ \begin{array}{ccrcccr} {\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\ {\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\ {\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\ {\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\ {\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5 \end{array} \right.\)

Mnemonics for the name of the approximation: Fast Super Exponent.

The agreement plotted is

\(\displaystyle D=D_{8}(z)=-\lg\left( \frac {|\mathrm{fse}(z)-F_{4}(z)|} {|\mathrm{fse}(z)|+|F_{4}(z)|} \right) \)

where \(F_4\) denotes the 4th ackermann, id set, natural tetration evaluated through the Cauchi integral ^[2].

The same image is used also as Рис.14.11 in the Russian version «Суперфункции»^[3], 2014.

First time published in the Vladikavkaz Matehmatical Journal ^[4], 2010.

The figure suggests that the fast implementation fsexp.cin with elementary functions happens to provide a little bit better precision that the initial implementation of natural tetration through the Cauchi integral ^[2] (2009).

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;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

//#include "superex.cin"
#include "fsexp.cin"
#include "f4natu.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.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=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("figcf4small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08a.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=fsexp(z);
        d=F4natu(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"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash  S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
//      system(    "gv fig08a.eps");
        system("epstopdf vladi08a.eps");
        system(    "open vladi08a.pdf"); // for linux
//      getchar(); system("killall Preview");// for macintosh
}

C++ generator of the second picture

/* vladif5c.cin also should be also loaded */

#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 "fsexp.cin"
//#include "f4natu.cin"
#include "vladif5c.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#include "superlo.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=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("figcf5small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08b.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=fsexp(z);
        d=F5(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"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash  S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi08b.eps");
        system(    "open vladi08b.pdf");//linux
//getchar(); system("killall Preview");
}

Latex combiner

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 378px
%\paperheight 134px 
\paperheight 180px 
\topmargin -104pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

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

\sx{2.2}{\begin{picture}(90,80)
%\put(0,0){\includegraphics{figsexpF4}}
%\put(0,0){\includegraphics{figcf4small}}
\put(0,0){\includegraphics{vladi08a}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\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(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17  ,6){\sx{.5}{$-\!2$}}
\put(25,62){\sx{.5}{$D\!>\!14$}}
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\sx{2.2}{\begin{picture}(80,80)
%\put(0,0){\includegraphics{figsexpF5}}
%\put(0,0){\includegraphics{figcf5small}}
\put(0,0){\includegraphics{vladi08b}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\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(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17  ,6){\sx{.5}{$-\!2$}}
\put(31,56){\sx{.5}{$D\!>\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\end{document}

Refereces

Keywords

«Agreement», «Approximation», «Natural tetration», «Superfunction», «Superfunctions», «Tetration»,