File:Vladi11.jpg

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 {|\mathrm{FSL}(\mathrm{FSE}(z)) - z|} {|\mathrm{FSL}(\mathrm{FSE}(z))| + |z|} \right) $

$ \displaystyle D_{\rm ta}(z)= -\ln\left( \frac {|\mathrm{FSE}(\mathrm{FSL}(z)) - z|} {|\mathrm{FSE}(\mathrm{FSL}(z))| + |z|} \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) ; the English version is in preparation in 2015.

First time published in the Vladikavkaz Matehmatical Journal .

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.

typedef std::complex z_type; //#include  //#define z_type complex
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//#include"superex.cin" //#include"superlo.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);
 * 1) include"fsexp.cin"
 * 2) include"fslog.cin"
 * 3) include "conto.cin"

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

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. typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//#include"superex.cin" //#include"superlo.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);
 * 1) include"fsexp.cin"
 * 2) include"fslog.cin"
 * 3) include "conto.cin"

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;
 * 1) 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}{$12\!\!14$}} \put(154,168){\sx{2.4}{$12\!\!14$}} %\put(204,373){\sx{1}{11}} %\put(204,226){\sx{1}{11}} \end{picture}} \end{document}