Difference between revisions of "File:Sutraamap.jpg"

Fig.20.8 from page 282 of book «Superfunctions»^[1], 2020.

It appears also as Рис.20.8 at page 292 of the Russian version «Суперфункции»^[2], 2014.

The same picture appears also as Figure 4 at page 6536 of publication ^[3] in the Applied Mathematical Sciences, 2013.

The picture shows the map of agreement for the primary approximation \(z\mapsto F_M(z\!+\!x_0)\) of function SuTra \[ \mathcal A(z)= - \lg\! \left( \frac { | F_M(z\!+\!x_0)- \mathrm {SuTra}(z) | } { | F_M(z\!+\!x_0) | + |\mathrm {SuTra}(z) | } \right) \] Levels of \(\mathcal A(x\!+\!\mathrm i y)\) are drawn in the \(x\), \(y\) plane.

Function SuTra appears as the superfunction of the elementary Trappmann function \(\mathrm{tra}= z \mapsto z+\exp(z) \); \[ \mathrm{SuTra}(z+1)=\mathrm{tra}(\mathrm{SuTra}(z)) \]

The additional condition \(\mathrm{SuTra}(0)=0\) is assumed; it determines the constant \(x_0\) above.

C++ generator of map

/* Files ado.cin, conto.cin, sutran.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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"

z_type tra(z_type z){ return exp(z)+z;}

#include"sutran.cin"

// z_type sutrap(z_type z) { int n; z_type c=sutra0(z-36.); DO(n,36) c=tra(c); return c;}

int 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();
//DB x1=-1.1259817765745026; DO(n,18){ y=Re(sutrap(x1)); x1+=-1.3*y; printf("%18.16f %18.16f\n", x1,y);} getchar();

int M=441,M1=M+1;
int N=201,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
//FILE *o;o=fopen("SuTraMap.eps","w"); ado(o,4402,2002);
FILE *o;o=fopen("SuTrapag.eps","w"); ado(o,4402,2002);
fprintf(o,"2001 1 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-20+.1*(m-.5);
DO(n,N1) Y[n]=0+.1*(n-.5); 
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-20;m<21;m++){M(m,0) L(m,20) }
for(n=0;n<21;n++){M( -20,n) L(20,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
                       f[m*N1+n]=999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
  c=sutran(z);
  d=sutra0(z);
  p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); if(p>0 && p<18) g[m*N1+n]=p;
// p=Re(c); q=Im(c); if(p>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/*
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
// fprintf(o,"0 setlinejoin 0 setlinecap\n");
// M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .02 W S\n");
//#include "plofu.cin"
*/

DB angle;
M(24,6)
for(n=20;n<160;n++){ angle=M_PI/180. * (n-.5); x=5.+17*cos(angle); y=18*sin(angle); L(x,y)}
L(-11,6) L(-11,0)
fprintf(o,".2 W 1 .5 0 RGB S\n");
        
conto(o,g,w,v,X,Y,M,N, (1. ),-2.,2.); fprintf(o,".08 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (2. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (3. ),-2.,2.); fprintf(o,".08 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (4. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (5. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (6. ),-2.,2.); fprintf(o,".08 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (7. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (8. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (9. ),-2.,2.); fprintf(o,".08 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (10. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (11. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (12. ),-2.,2.); fprintf(o,".06 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (13. ),-2.,2.); fprintf(o,".02 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (14. ),-2.,2.); fprintf(o,".03 W 0 .5 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N, (15. ),-2.,2.); fprintf(o,".06 W 0 0 1 RGB S\n");
//conto(o,g,w,v,X,Y,M,N, (15.4 ),-8.,8.); fprintf(o,".04 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o); free(f); free(g); free(w);
      system("epstopdf sutrapag.eps"); 
      system( "open sutrapag.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt]{article}
\paperwidth 2190px
\paperheight 1044px
\textwidth 2394px
\textheight 1300px
\topmargin -94px
\oddsidemargin -72px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
\parindent 0pt
\sx{.5}{\begin{picture}(4408,1982)
\put(0,0){\ing{sutrapag}}
\put(44,1955){\sx{8}{$y$}}
\put(04,1775){\sx{8}{$18$}}
\put(04,1575){\sx{8}{$16$}}
\put(04,1375){\sx{8}{$14$}}
\put(04,1175){\sx{8}{$12$}}
\put(04, 975){\sx{8}{$10$}}
\put(040, 775){\sx{8}{$8$}}
\put(040, 575){\sx{8}{$6$}}
\put(040, 375){\sx{8}{$4$}}
\put(040, 175){\sx{8}{$2$}}
%\put(020, 80){\sx{8}{$1$}}
\put(040, -25){\sx{8}{$0$}}
\put(0108,-77){\sx{7}{$-18$}}
\put(0308,-77){\sx{7}{$-16$}}
\put(0508,-77){\sx{7}{$-14$}}
\put(0708,-77){\sx{7}{$-12$}}
\put(0908,-77){\sx{7}{$-10$}}
\put(1140,-80){\sx{8}{$-8$}}
\put(1340,-80){\sx{8}{$-6$}}
\put(1540,-80){\sx{8}{$-4$}}
\put(1740,-80){\sx{8}{$-2$}}
\put(1992,-80){\sx{8}{$0$}}
\put(2192,-80){\sx{8}{$2$}}
\put(2392,-80){\sx{8}{4}}
\put(2592,-80){\sx{8}{6}}
\put(2792,-80){\sx{8}{8}}
\put(2970,-80){\sx{8}{10}}
\put(3170,-80){\sx{8}{12}}
\put(3370,-80){\sx{8}{14}}
\put(3570,-80){\sx{8}{16}}
\put(3770,-80){\sx{8}{18}}
\put(3970,-80){\sx{8}{20}}
\put(4192,-80){\sx{8}{$x$}}
\put(180,1860){\sx{16}{Range used for approximation of SuTra}}
\put(400,1200){\sx{19}{$\mathcal A \!>\! 15$}}
\put(2020,1520){\sx{18}{$\mathcal A \!=\!15$}}
\put(2020,1250){\sx{18}{$\mathcal A \!=\!14$}}
\put(2020,1040){\sx{18}{$\mathcal A \!=\!13$}}
\put(2020,0840){\sx{18}{$\mathcal A \!=\!12$}}
\put(2030,0488){\sx{16}{$\mathcal A \!=\!9$}}
\end{picture}}
\end{document}

References

Keywords

«ado.cin», «conto.cin», «Agreement», «Approximation», «Asymptotic», «Entire Function with Logarithmic Asymptotic», «Implementation», «Primary approximation», «Special function», «SuTra», «Sutran.cin», «Superfunction», «Superfunctions», «Trappmann function»,

«Суперфункции»,