Difference between revisions of "File:SuZexPlot511T.jpg"

Explicit plot of function SuZex in comparison to its transfer function zex\((z)\!=\!z\,\exp(z)\):

\(y\!=\!\mathrm{SuZex}(x)\) and

\(y\!=\!\mathrm{zex}(x)\)

versus \(x\)

This plot is used as Fig.11.5 at page 142 of book «Superfunctions», 2020 ^[1][2]
in order to compare functions zex and SuZex.

C++ generator of curves

// Files ado.cin. SuZexCoef20.cin, SuZexTay2008co.cin, SuZexTay0co.cin, SuZex.cin //should be loaded to the working directory 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++)
 using namespace std;
 #include<complex>
 typedef complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)

/*
 z_type zex(z_type z) { return z*exp(z);}

 z_type suzexo(z_type z){ int n,m; z_type L=log(-z); z_type c[21]; z_type s;
 #include "SuZexCoef20.cin"
 c[0]=-1.; c[1]=.5*L;
 for(n=2;n<21;n++){ s=a[n][n]; 
                    for(m=n-1; m>=0; m--){ s*=L; s+=a[n][m]; }
                    c[n]=s; }
 s=c[20]/z; for(n=19;n>=0;n--){ s+=c[n]; s/=z;}
 return s; }

  z_type suzex2008t12(z_type z){ int n,m=80; z_type s;
  #include "SuZexTay2008co.cin"
  s=c[m]; for(n=m-1;n>=0;n--){ s*=z; s+=c[n];}
  return s;}

  z_type SuZexTay0(z_type z){ int n,m=96; z_type s;
  #include "SuZexTay0co.cin"
  s=SuZexTay0co[m]; for(n=m-1;n>=0;n--){ s*=z; s+=SuZexTay0co[n];}
  return s;}

 z_type suzex(z_type z){int m,n; z_type s;
 if( abs(z) < 1.6 )                   return SuZexTay0(z) ; // I made the Taylor expansion for this case
 if( Re(z)>0 && fabs(Im(z))<1.5){n=int(Re(z)+.5); s=SuZexTay0(z-(0.+n)); DO(m,n) s=zex(s); return s;}
  z+=-1.1259817765745026;     // WARNING! ARGUEMENT CHANGES ITS VALUE!
 if( abs(z+12.) < 8.1 )               return suzex2008t12(z+12.) ; // I made the Taylor expansion for this case
 if( Re(z)<-12. || fabs(Im(z)) > 8. ) return suzexo(z); // definitely, |z|>8
 n= int(Re(z)+12.); s=suzex2008t12(z+12.-(0.+n)); DO(m,n) s=zex(s); return s;
 } 
 // Copyleft 2012 by Dmitrii Kouznetsov
 */

 #include "SuZex.cin"
 
 #include "ado.cin"
 #define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 DB x1=-1.1259817765745026; //DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
 FILE *o;o=fopen("SuZexPlot511.eps","w");  ado(o,504,1104);
 fprintf(o,"303 101 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 for(m=-3;m<3;m++) {M(m,-1)L(m,16)}
 for(n=-1;n<17;n++) {M(-3,n)L(2,n)}
 M(0,M_E)L(1,M_E)
 M(0,-1./M_E)L(-1,-1./M_E)
 fprintf(o,".006 W S\n");
 //
 DO(m,130){x=-3.02 +.1*m; y=Re(suzex(x)); if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".03 W 0 0 1 RGB S\n");
 // DO(m,130){x=-10.+.1*m; y=-1./(x+x1);   if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W 0 0 0 RGB S\n");
 // DO(m,130){x=-2.+.1*m; y=exp(x);       if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W .5 0 0 RGB S\n");
  DO(m,130){x=-3.02+.1*m; y=Re(zex(x));    if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W 0 0 0 RGB S\n");
 //
 fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
       system("epstopdf SuZexPlot511.eps"); 
       system(    "open SuZexPlot511.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex generator of labels

% file SuZexPlot.pdf should be generated with the code above in order to compile the Latex document below. %<br>
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
\documentclass[12pt]{article}  % <br> 
\usepackage{geometry}  % <br> 
\usepackage{graphicx} % <br> 
\usepackage{rotating} % <br> 
\paperwidth 508pt % <br> 
\paperheight 1104pt % <br> 
\topmargin -96pt % <br> 
\oddsidemargin -81pt % <br> 
\textwidth 1200pt % <br> 
\textheight 1200pt % <br> 
\pagestyle {empty} % <br> 
\newcommand \sx {\scalebox} % <br> 
\newcommand \rot {\begin{rotate}} % <br> 
\newcommand \ero {\end{rotate}} % <br> 
\newcommand \ing {\includegraphics} % <br> 
\parindent 0pt%  <br> 
\pagestyle{empty} % <br> 
\begin{document}  % <br> 
\begin{picture}(1202,1102) % <br> 
%\put(10,10){\ing{ExpQ2plot}} % <br> 
\put(322,1090){\sx{4}{$y$}} % <br> 
\put(292,372){\sx{4}{$\mathrm e$}} % <br>
\put(315, 62){\sx{4}{$-\!1/\mathrm e$}} % <br>  
\put(325,998){\sx{4}{$9$}} % <br> 
\put(325,898){\sx{4}{$8$}} % <br> 
\put(325,798){\sx{4}{$7$}} % <br> 
\put(325,698){\sx{4}{$6$}} % <br> 
\put(325,598){\sx{4}{$5$}} % <br> 
\put(325,498){\sx{4}{$4$}} % <br> 
\put(325,398){\sx{4}{$3$}} % <br> 
\put(325,298){\sx{4}{$2$}} % <br> 
\put(325,198){\sx{4}{$1$}} % <br> 
%\put(325,098){\sx{4}{$0$}} % <br> 
 % <br> 
\put(078,116){\sx{4}{$-\!2$}} % <br> 
\put(178,116){\sx{4}{$-\!1$}} % <br> 
\put(304,116){\sx{4}{$0$}} % <br> 
\put(404,116){\sx{4}{$1$}} % <br> 
\put(490,116){\sx{4}{$x$}} % <br> 
 % <br> 
\put(12,188){\sx{4}{\rot{0}$y\!=\!\mathrm{SuZex}(x)$\ero}} % <br> 
\put(358,162){\sx{4}{\rot{0}$y\!=\!x\mathrm{e}^x$\ero}} % <br> 
%\put(10,10){\ing{ExpQ2plot}} % <br> 
\put(10,10){\ing{SuZexPlot511}} % <br> 
\end{picture} % <br> 
\end{document} % <br> 
%

References

Keywords

«ArcLambertW», «Exotic iteration», «Fixed point», «LambertW», «ProductLog», «Superfunction», «Superfunctions», «SuZex», «Zex»,

«Суперфункции»,