Difference between revisions of "File:Sunem1ht.jpg"

Central picture of Figure 13.6 at page 167 of book Superfunctions^[1], 2020 :

Complex map of function SuNem with parameter \( q\!=\!1 \):

\( u\!+\!\mathrm i v=\mathrm{SuNem}_{1}(x\!+\!\mathrm i y) \)

SuNem is superfunction for the Nemtsov function,

\( \mathrm{nem}_q(z)=z+z^3+qz^4 \)

The map is symmetric with respect to reflection from the real axis; so, the only upper half of the complex plane is shown.

The picture is generated with the C++ and Latex codes below.

C++ generator of the map

Files ado.cin, conto.cin, sune.cin, sunema.txt should be loaded for the compilation of 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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"

DB Q=1.;
z_type nem(z_type z){ return z*(1.+z*z*(1.+z*Q)); }
z_type nem1(z_type z){ return 1.+z*z*(3.+z*(4.*Q)); }   // WARNING: Q is global!

#include "sune.cin"

DB SUNo=0; // should be reassigned
z_type sunem(z_type z){ return sune(z + SUNo);}

int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
// DB rr,ti;
co(); // Evaluate soefficients of the expansion

x=0;  DO(n,60) { y=Re(sune(x)); x-= y-1.; printf("%19.16lf %19.16lf\n", x,y);}
SUNo=x;
printf("A[2][0]= %9.5lf , A[2][1]=%9.5lf\n",A[2][0],A[2][1]);
printf("A[3][0]= %9.5lf , A[3][1]=%9.5lf\n",A[3][0],A[3][1]);
 int M=501,M1=M+1;
 int N=251,N1=N+1;
DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
//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("sunem1h.eps","w");ado(o,1008,508);
fprintf(o,"504 4 translate\n 100 100 scale 2 setlinecap\n");
DO(m,M1) X[m]=-5.+.02*(m-.5);
DO(n,N1) Y[n]=0.+.02*(n-.5);
for(m=-5;m<6;m+=1){ M(m,0)L(m,5)}
for(n=0;n<6;n+=1){ M(-5,n)L(5,n)}
fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

DO(m,M1)DO(n,N1){g[m*N1+n]=9999999; f[m*N1+n]=9999999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
c=sunem(z);
//d=sune(z);
//p=abs(c-d)/abs(c+d); p=-log(p)/log(10.); 
        p=Re(c); q=Im(c);
//if(p>-85 && p<85) g[m*N1+n]=p;
        if(p>-1001 && p<1001 && 
        q >-1001 && q<1001 ) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

//M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
//fprintf(o,".002 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=6.;q=.6;
//#include"plofu.cin"

for(m=-5;m<5;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".01 W .9 0 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".01 W 0 0 .9 RGB S\n");

for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".023 W .8 0 0 RGB S\n");
for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".023 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".023 W .5 0 .5 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".023 W 0 0 0 RGB S\n");

/*
conto(o,g,w,v,X,Y,M,N,15.5,-1,1);fprintf(o,".02 W 1 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".04 W 0 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".02 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".02 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".04 W 0 0 .7 RGB S\n");
conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,9.,-p,p);fprintf(o,".04 W 0 .6 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N,8.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,7.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,6.,-p,p);fprintf(o,".04 W 0 .6 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,5.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,4.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,3.,-p,p);fprintf(o,".04 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,2.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,1.,-p,p);fprintf(o,".05 W .5 0 0 RGB S\n");
*/

//M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
//fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf sunem1h.eps");
        system(    "open sunem1h.pdf"); //mac

//for(m=-40;m<0;m++) {x=m; y=Re(sun(x)); t=Re(nem(sun(x-1.))); printf("%6.1lf %18.15lf %18.15lf %18.15lf\n", x,y,t,y-t);}
return 0;
}

Latex generator of the labels

%\documentclass[12pt]{article}
\documentclass{mcom-l}
 \usepackage{graphics}
 \paperwidth 1030pt
 \paperheight 530pt
 \usepackage{geometry}
 \usepackage{rotating}
 \textwidth 1260pt
 \textheight 1260pt
 \topmargin -96pt
 \oddsidemargin -82pt
 \parindent 0pt
 \pagestyle{empty}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(1038,530)
%\put(50,40){\ing{sunem10q10ma6s}}
%\put(24,20){\ing{nem120ma}}
\put(30,20){\ing{sunem1h}}
\put(12,514){\sx{3.1}{$y$}}
\put(12,417){\sx{3}{$4$}}
\put(12,317){\sx{3}{$3$}}
\put(12,217){\sx{3}{$2$}}
\put(12,117){\sx{3}{$1$}}
\put(12,017){\sx{3}{$0$}}
\put( 9, 0){\sx{3}{$-5$}}
\put(109, 0){\sx{3}{$-4$}}
\put(209, 0){\sx{3}{$-3$}}
\put(309, 0){\sx{3}{$-2$}}
\put(409, 0){\sx{3}{$-1$}}
\put(530, 0){\sx{3}{$0$}}
\put(630, 0){\sx{3}{$1$}}
\put(730, 0){\sx{3}{$2$}}
\put(830, 0){\sx{3}{$3$}}
\put(930, 0){\sx{3}{$4$}}
\put(1022,0){\sx{3.2}{$x$}}
%
\put(820,344){\sx{3}{\rot{49} $u\!=\!0.1$\ero}}
\put(552,514){\sx{3}{\rot{-65} $u\!=\!0.2$\ero}}
\put(54,134){\sx{3}{\rot{64} $u\!=\!0.3$\ero}}
\put(260, 38){\sx{3}{\rot{74} $u\!=\!0.4$\ero}}
\put(364, 36){\sx{3}{\rot{66} $u\!=\!0.5$\ero}}
\put(426, 34){\sx{3}{\rot{60} $u\!=\!0.6$\ero}}
\put(468, 32){\sx{3}{\rot{56} $u\!=\!0.7$\ero}}
\put(494, 30){\sx{3}{\rot{54} $u\!=\!0.8$\ero}}
\put(534, 26){\sx{3}{\rot{56} $u\!=\!1$\ero}}
%
\put(640,376){\sx{3}{\rot{-10} $v\!=\!0.3$\ero}}
\put(396,390){\sx{3}{\rot{70} $v\!=\!0.2$\ero}}
\put( 96,400){\sx{3}{\rot{-62} $v\!=\!0.1$\ero}}
\put(146, 18){\sx{3}{\rot{0} $v\!=\!0$\ero}}
\put(856, 18){\sx{3}{\rot{0} \bf huge values\ero}}
%\put(100,126){\sx{3}{\rot{61} $v\!=\!-0.1$\ero}}
%\put(380, 170){\sx{3}{\rot{-72} $v\!=\!-0.2$\ero}}
%
\end{picture}
\end{document}

References

Keywords

«Nemtsov function», «Boris Nemtsov», «Complex map», «C++», «Exotic iterate», «Iterate», «Latex», «Nemtsov function», «SuNem», «Superfunction», «Superfunctions»,