Difference between revisions of "File:Itnem00plot.jpg"

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Iterates of the [[Nemtsov function]] at value of parameter $q\!=\!0$
−
Importing image file
  
  +
  +
  +
$y=\mathrm{Nem}_0^{\,n}(x)$
  +
  +
versus $x$ for various values of $n$.
  +
  +
==References==
  +
<references/>
  +
  +
==[[C++]] generator of curves ==
  +
  +
<poem><nomathjax><nowiki>
  +
#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 "conto.cin"
  +
  +
DB Q=2.; // Reassigned in the program
  +
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!
  +
  +
DB K=Q*Q; // Reassigned in the program
  +
DB A[37][19];
  +
int co(){
  +
#include "sunema.txt"
  +
return 0;}
  +
  +
z_type sun(z_type z){ z_type p[37],s; int m,M,n,N;
  +
z_type L=log(-z);
  +
z_type t=1./sqrt(-2.*z);
  +
M=36;
  +
for(m=2;m<M+1;m++){ N=m/2; s=A[m][N]*L; for(n=N-1;n>0;n--) { s+=A[m][n]; s*=L; }
  +
p[m]=A[m][0]+s;
  +
}
  +
s=p[M]*t; for(m=M-1; m>1;m--) { s+=p[m]; s*=t;}
  +
return t*( 1. + t* (-Q + s) );
  +
}
  +
  +
z_type sune(z_type z){ int m,n; DB x,y; z_type s;
  +
DB R=20.+20.*Q*Q;
  +
x=Re(z); if(x<-R) return sun(z);
  +
y=Im(z); if(fabs(y)>R) return sun(z);
  +
n=int(x+R);
  +
s=sun(z-(0.+n));
  +
DO(m,n) s=nem(s);
  +
return s;
  +
}
  +
  +
DB AUNE1=0;
  +
  +
z_type sunem(z_type z){ int m,n; DB x,y; z_type s;
  +
DB R=20.+20.*Q*Q;
  +
x=Re(z); if(x<-R) return sun(z);
  +
y=Im(z); if(fabs(y)>R) return sun(z);
  +
n=int(x+R);
  +
s=sun(z-(n-AUNE1));
  +
DO(m,n) s=nem(s);
  +
return s;
  +
}
  +
  +
#include"nembran.cin"
  +
z_type NemZo=nembra(Q);
  +
z_type ANemZo=nem(NemZo);
  +
DB tr=Re(ANemZo);
  +
DB ti=Im(ANemZo);
  +
  +
#include "arqnem.cin"
  +
  +
DB C[41];
  +
//DB K=Q*Q;
  +
int coa(){ //C[1]=(0.5 - K)*Q; C[2]=-0.625 + (-0.25 + K/2.)*K; ..
  +
#include "aunemco.txt"
  +
return 40;
  +
}
  +
  +
z_type aune0(z_type z){ z_type s; int n;
  +
s=C[40]*z; for(n=39;n>0;n--){s+=C[n];s*=z;}
  +
return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ;
  +
}
  +
  +
z_type aune1(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
  +
return aune0(z)+(0.+n);}
  +
  +
z_type aunem(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
  +
return aune0(z)+(n-AUNE1);}
  +
  +
z_type aune02(z_type z){ z_type s; int n; s=(C[2]*z+C[1])*z;
  +
return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; }
  +
z_type aune01(z_type z){ z_type s; int n; s=C[1]*z;
  +
return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) +s ; }
  +
  +
int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
  +
//FILE *o;o=fopen("aunemplo.eps","w");ado(o,4008,4008);
  +
//FILE *o;o=fopen("20.eps","w");ado(o,4008,4008);
  +
FILE *o;o=fopen("itnem00plo.eps","w");ado(o,4008,4008);
  +
fprintf(o,"4 4 translate\n 1000 1000 scale 2 setlinecap\n");
  +
for(m=0;m<41;m+=10){ M(.1*m,0)L(.1*m,4)}
  +
for(n=0;n<41;n+=10){ M(0,.1*n)L(4,.1*n)}
  +
fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");
  +
  +
// M(-1,ti)L(1,ti) M(-1,-ti)L(1,-ti) M(tr,-1)L(tr,1) fprintf(o,".002 W 0 0 0 RGB 2 setlinecap S\n");
  +
//M(-1,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
  +
//fprintf(o,".006 W 1 1 0 RGB 0 setlinecap S\n");
  +
  +
Q=1.e-6; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));
  +
// Q=1.; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));
  +
// Q=2.; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));
  +
  +
fprintf(o,"1 setlinejoin 1 setlinecap S\n");
  +
  +
DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(nem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(x)))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){x=.01+.02*m;y=Re(nem(nem(x))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){x=.01+.02*m;y=Re(nem(x)); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
fprintf(o,".02 W 0 .7 1 RGB S\n");
  +
  +
DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(arqnem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(x)))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(x))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
DO(m,201){ x=.01+.02*m; y=Re(arqnem(x)); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
  +
fprintf(o,".02 W 1 .5 0 RGB S\n");
  +
  +
M(0,0)L(4.1,4.1)
  +
fprintf(o,".02 W .1 1 .1 RGB S\n");
  +
for(n=-20;n<21;n++){
  +
DO(m,200){ x=.02+.02*m; y=Re(sunem(.1*n+aunem(x))); if(m==0) M(x,y) else L(x,y); /* printf("%9.2lf %9.2lf\n",x,y);*/ if(y>4) break; }
  +
}
  +
  +
fprintf(o,".004 W 0 0 0 RGB S\n");
  +
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf itnem00plo.eps");
  +
system( "open itnem00plo.pdf"); //mac
  +
return 0;}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels ==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphics}
  +
\paperwidth 4020pt
  +
\paperheight 4020pt
  +
\usepackage{geometry}
  +
\usepackage{rotating}
  +
\textwidth 4400pt
  +
\textheight 4400pt
  +
\topmargin -97pt
  +
\oddsidemargin -74pt
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
\begin{picture}(4008,4008)
  +
%\put(20,3840){\sx{20}{$y\!=\!\mathrm{AuNem}_q(x)$}}
  +
%\put(20,3840){\sx{20}{$y\!=\!\mathrm{nem}_2^n(x)$}}
  +
\put(20,3900){\sx{20}{$y$}}
  +
\put(20,2944){\sx{20}{$3$}}
  +
\put(20,1940){\sx{20}{$2$}}
  +
\put(20, 940){\sx{20}{$1$}}
  +
%\put(10, 936){\sx{20}{$-1$}}
  +
\put( 946,30){\sx{20}{$1$}}
  +
\put(1950,30){\sx{20}{$2$}}
  +
\put(2954,30){\sx{20}{$3$}}
  +
\put(3870,30){\sx{20}{$x$}}
  +
%
  +
%\put(0,0){\ing{aunemplo}}
  +
\put(0,0){\ing{itnem00plo}}
  +
%
  +
\put( 470,3450){\sx{16}{\rot{89} $n\!=\!4$\ero}}
  +
\put( 630,3450){\sx{16}{\rot{88} $n\!=\!3$\ero}}
  +
\put( 800,3450){\sx{16}{\rot{87} $n\!=\!2$\ero}}
  +
\put(1356,3480){\sx{16}{\rot{80} $n\!=\!1$\ero}}
  +
\put(1960,3420){\sx{16}{\rot{70} $n\!=\!0.5$\ero}}
  +
\put(2150,3410){\sx{16}{\rot{66} $n\!=\!0.4$\ero}}
  +
\put(2390,3400){\sx{16}{\rot{63} $n\!=\!0.3$\ero}}
  +
\put(2640,3356){\sx{16}{\rot{56} $n\!=\!0.2$\ero}}
  +
\put(2870,3200){\sx{16}{\rot{52} $n\!=\!0.1$\ero}}
  +
\put(3100,3020){\sx{18}{\rot{45} $n\!=\!0$\ero}}
  +
\put(3194,2730){\sx{16}{\rot{38} $n\!=\!-0.1$\ero}}
  +
\put(3280,2470){\sx{16}{\rot{32} $n\!=\!-0.2$\ero}}
  +
\put(3336,2240){\sx{16}{\rot{26} $n\!=\!-0.3$\ero}}
  +
\put(3350,2020){\sx{16}{\rot{22} $n\!=\!-0.4$\ero}}
  +
\put(3360,1834){\sx{16}{\rot{18} $n\!=\!-0.5$\ero}}
  +
\put(3480,1256){\sx{16}{\rot{7} $n\!=\!-1$\ero}}
  +
\put(3480,750){\sx{16}{\rot{2} $n\!=\!-2$\ero}}
  +
\put(3480,550){\sx{16}{\rot{1} $n\!=\!-3$\ero}}
  +
\put(3480,400){\sx{16}{\rot{0} $n\!=\!-4$\ero}}
  +
%
  +
\put(1058,2020){\sx{16}{\rot{78} $y\!=\!\mathrm{Nem}_0(x)$\ero}}
  +
\put(1570,1500){\sx{17}{\rot{45} $y\!=\!x$\ero}}
  +
\put(1810,920){\sx{16}{\rot{12} $y\!=\!\mathrm{ArqNem}_0(x)$\ero}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:C++]]
  +
[[Category:Iterate]]
  +
[[Category:Latex]]
  +
[[Category:Nemtsov function]]

Latest revision as of 08:38, 1 December 2018

Iterates of the Nemtsov function at value of parameter $q\!=\!0$


$y=\mathrm{Nem}_0^{\,n}(x)$

versus $x$ for various values of $n$.

References


C++ generator of curves


#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 "conto.cin"

DB Q=2.; // Reassigned in the program
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!

DB K=Q*Q; // Reassigned in the program
DB A[37][19];
int co(){
#include "sunema.txt"
return 0;}

z_type sun(z_type z){ z_type p[37],s; int m,M,n,N;
z_type L=log(-z);
z_type t=1./sqrt(-2.*z);
M=36;
for(m=2;m<M+1;m++){ N=m/2; s=A[m][N]*L; for(n=N-1;n>0;n--) { s+=A[m][n]; s*=L; }
                p[m]=A[m][0]+s;
                }
s=p[M]*t; for(m=M-1; m>1;m--) { s+=p[m]; s*=t;}
        return t*( 1. + t* (-Q + s) );
}

z_type sune(z_type z){ int m,n; DB x,y; z_type s;
DB R=20.+20.*Q*Q;
x=Re(z); if(x<-R) return sun(z);
y=Im(z); if(fabs(y)>R) return sun(z);
n=int(x+R);
s=sun(z-(0.+n));
DO(m,n) s=nem(s);
return s;
}

DB AUNE1=0;

z_type sunem(z_type z){ int m,n; DB x,y; z_type s;
DB R=20.+20.*Q*Q;
x=Re(z); if(x<-R) return sun(z);
y=Im(z); if(fabs(y)>R) return sun(z);
n=int(x+R);
s=sun(z-(n-AUNE1));
DO(m,n) s=nem(s);
return s;
}

#include"nembran.cin"
z_type NemZo=nembra(Q);
z_type ANemZo=nem(NemZo);
DB tr=Re(ANemZo);
DB ti=Im(ANemZo);

#include "arqnem.cin"

DB C[41];
//DB K=Q*Q;
int coa(){ //C[1]=(0.5 - K)*Q; C[2]=-0.625 + (-0.25 + K/2.)*K; ..
#include "aunemco.txt"
return 40;
}

z_type aune0(z_type z){ z_type s; int n;
s=C[40]*z; for(n=39;n>0;n--){s+=C[n];s*=z;}
return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ;
}

z_type aune1(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
return aune0(z)+(0.+n);}

z_type aunem(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
return aune0(z)+(n-AUNE1);}

z_type aune02(z_type z){ z_type s; int n; s=(C[2]*z+C[1])*z;
                return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; }
z_type aune01(z_type z){ z_type s; int n; s=C[1]*z;
                return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) +s ; }

int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;

//FILE *o;o=fopen("aunemplo.eps","w");ado(o,4008,4008);
//FILE *o;o=fopen("20.eps","w");ado(o,4008,4008);
FILE *o;o=fopen("itnem00plo.eps","w");ado(o,4008,4008);
fprintf(o,"4 4 translate\n 1000 1000 scale 2 setlinecap\n");
for(m=0;m<41;m+=10){ M(.1*m,0)L(.1*m,4)}
for(n=0;n<41;n+=10){ M(0,.1*n)L(4,.1*n)}
fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

// M(-1,ti)L(1,ti) M(-1,-ti)L(1,-ti) M(tr,-1)L(tr,1) fprintf(o,".002 W 0 0 0 RGB 2 setlinecap S\n");
//M(-1,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
//fprintf(o,".006 W 1 1 0 RGB 0 setlinecap S\n");

        Q=1.e-6; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));
// Q=1.; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));
// Q=2.; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co(); coa(); AUNE1=Re(aune1(1.));

fprintf(o,"1 setlinejoin 1 setlinecap S\n");

DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(nem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(x)))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){x=.01+.02*m;y=Re(nem(nem(x))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){x=.01+.02*m;y=Re(nem(x)); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
fprintf(o,".02 W 0 .7 1 RGB S\n");

DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(arqnem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(x)))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(x))); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
DO(m,201){ x=.01+.02*m; y=Re(arqnem(x)); if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; }
fprintf(o,".02 W 1 .5 0 RGB S\n");

M(0,0)L(4.1,4.1)
fprintf(o,".02 W .1 1 .1 RGB S\n");
for(n=-20;n<21;n++){
DO(m,200){ x=.02+.02*m; y=Re(sunem(.1*n+aunem(x))); if(m==0) M(x,y) else L(x,y); /* printf("%9.2lf %9.2lf\n",x,y);*/ if(y>4) break; }
}

fprintf(o,".004 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf itnem00plo.eps");
        system( "open itnem00plo.pdf"); //mac
return 0;}

Latex generator of labels


 \documentclass[12pt]{article}
 \usepackage{graphics}
 \paperwidth 4020pt
 \paperheight 4020pt
 \usepackage{geometry}
 \usepackage{rotating}
 \textwidth 4400pt
 \textheight 4400pt
 \topmargin -97pt
 \oddsidemargin -74pt
 \parindent 0pt
 \pagestyle{empty}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(4008,4008)
%\put(20,3840){\sx{20}{$y\!=\!\mathrm{AuNem}_q(x)$}}
%\put(20,3840){\sx{20}{$y\!=\!\mathrm{nem}_2^n(x)$}}
\put(20,3900){\sx{20}{$y$}}
\put(20,2944){\sx{20}{$3$}}
\put(20,1940){\sx{20}{$2$}}
\put(20, 940){\sx{20}{$1$}}
%\put(10, 936){\sx{20}{$-1$}}
\put( 946,30){\sx{20}{$1$}}
\put(1950,30){\sx{20}{$2$}}
\put(2954,30){\sx{20}{$3$}}
\put(3870,30){\sx{20}{$x$}}
%
%\put(0,0){\ing{aunemplo}}
\put(0,0){\ing{itnem00plo}}
%
\put( 470,3450){\sx{16}{\rot{89} $n\!=\!4$\ero}}
\put( 630,3450){\sx{16}{\rot{88} $n\!=\!3$\ero}}
\put( 800,3450){\sx{16}{\rot{87} $n\!=\!2$\ero}}
\put(1356,3480){\sx{16}{\rot{80} $n\!=\!1$\ero}}
\put(1960,3420){\sx{16}{\rot{70} $n\!=\!0.5$\ero}}
\put(2150,3410){\sx{16}{\rot{66} $n\!=\!0.4$\ero}}
\put(2390,3400){\sx{16}{\rot{63} $n\!=\!0.3$\ero}}
\put(2640,3356){\sx{16}{\rot{56} $n\!=\!0.2$\ero}}
\put(2870,3200){\sx{16}{\rot{52} $n\!=\!0.1$\ero}}
\put(3100,3020){\sx{18}{\rot{45} $n\!=\!0$\ero}}
\put(3194,2730){\sx{16}{\rot{38} $n\!=\!-0.1$\ero}}
\put(3280,2470){\sx{16}{\rot{32} $n\!=\!-0.2$\ero}}
\put(3336,2240){\sx{16}{\rot{26} $n\!=\!-0.3$\ero}}
\put(3350,2020){\sx{16}{\rot{22} $n\!=\!-0.4$\ero}}
\put(3360,1834){\sx{16}{\rot{18} $n\!=\!-0.5$\ero}}
\put(3480,1256){\sx{16}{\rot{7} $n\!=\!-1$\ero}}
\put(3480,750){\sx{16}{\rot{2} $n\!=\!-2$\ero}}
\put(3480,550){\sx{16}{\rot{1} $n\!=\!-3$\ero}}
\put(3480,400){\sx{16}{\rot{0} $n\!=\!-4$\ero}}
%
\put(1058,2020){\sx{16}{\rot{78} $y\!=\!\mathrm{Nem}_0(x)$\ero}}
\put(1570,1500){\sx{17}{\rot{45} $y\!=\!x$\ero}}
\put(1810,920){\sx{16}{\rot{12} $y\!=\!\mathrm{ArqNem}_0(x)$\ero}}
\end{picture}
\end{document}

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