Difference between revisions of "File:Norimap40.jpg"

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  +
[[complex map]] of function [[nori]],
−
Importing image file
  
  +
in some range of the complex plane,
  +
  +
$\mathrm{nori}(z)=\,$[[mori]]$\big(\sqrt{z}\big)^2$
  +
  +
$u\!+\!\mathrm i v=\mathrm{nori}(x\!+\!\mathrm i y)$
  +
  +
==[[C++]] generator of curves==
  +
Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[besselj0.cin]]
  +
should be loaded in order to compile the code below:
  +
<poem><nomathjax><nowiki>
  +
#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"
  +
#include "besselj0.cin"
  +
  +
DB L1= 2.404825557695773;
  +
DB L2= 5.5200781102863115;
  +
DB L3= 8.653727912911013;
  +
DB L4=11.791534439014281;
  +
z_type morin(z_type x){ return BesselJ0(L1*x)/(1.-x*x);} // naive representation fails at x=1.
  +
  +
z_type mori0(z_type x){ int n,m; z_type s, xx=x*x;
  +
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
  +
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
  +
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
  +
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
  +
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
  +
//with m=15, at |x|<2, the relative error is of order of 10^(-16)
  +
//In this sense, the result is accurate while |x|<2.
  +
m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;}
  +
return 1.+s;}
  +
z_type mori(z_type x){if(abs(x)<2.) return mori0(x);
  +
return morin(x);}
  +
  +
z_type no1(z_type x){ return BesselJ0(L1*sqrt(x))/(1.-x);} // naive representation fails at x=1.
  +
  +
z_type no0(z_type x){ int n,m; z_type s; //, xx=x*x;
  +
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
  +
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
  +
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
  +
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
  +
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
  +
//with m=15, at |x|<2, the relative error is of order of 10^(-16)
  +
//In this sense, the result is accurate while |x|<2.
  +
m=15; s=c[m]*x; for(n=m-1;n>0;n--){ s+=c[n]; s*=x;}
  +
return 1.+s;}
  +
  +
z_type no(z_type x){if(abs(x)<3.) return no0(x);
  +
return no1(x);}
  +
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=501,M1=M+1;
  +
int N=501,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("06.eps","w");ado(o,5020,5020);
  +
FILE *o;o=fopen("norima.eps","w");ado(o,5020,5020);
  +
fprintf(o,"1010 1010 translate\n 100 100 scale\n");
  +
DO(m,M1)X[m]=-10.+.1*(m-.5);
  +
DO(n,N1)Y[n]=-10.+.1*(n-.5);
  +
  +
for(m=-10;m<11;m++){if(m==0){M(m,-10.1)L(m,10.1)} else{M(m,-10)L(m,10)}}
  +
for(n=-10;n<11;n++){ M( -10,n)L(10,n)}
  +
fprintf(o,".01 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=99999; f[m*N1+n]=99999;}
  +
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
// c=BesselJ0(z);
  +
// c=mori(z);
  +
c=no(z); c*=c;
  +
p=Re(c); q=Im(c);
  +
if(p>-999. && p<999. && q>-999. && q<999
  +
)
  +
{g[m*N1+n]=p;
  +
f[m*N1+n]=q;
  +
}
  +
}}
  +
//#include "plodi.cin"
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=40;q=1;
  +
for(m=-4;m<4;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<4;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<4;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<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .9 0 0 RGB S\n");
  +
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".04 W .6 0 .6 RGB S\n");
  +
for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
  +
//#include "plofu.cin"
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf norima.eps");
  +
system( "open norima.pdf");
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\paperheight 5120px
  +
\paperwidth 5100px
  +
\textwidth 5094px
  +
\textheight 5200px
  +
\topmargin -80px
  +
\oddsidemargin -72px
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
%\usepackage{color}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\begin{document}\parindent 0pt
  +
\begin{picture}(5100,5100)
  +
\put(80,80){\ing{norima}}
  +
%\put(0,0){\sx{8}{-10}}
  +
\put(32,2068){\sx{8}{$y$}}
  +
\put(32,1868){\sx{8}{$8$}}
  +
\put(32,1668){\sx{8}{$6$}}
  +
\put(32,1468){\sx{8}{$4$}}
  +
\put(32,1268){\sx{8}{$2$}}
  +
\put(32,1068){\sx{8}{$0$}}
  +
\put(-32,868){\sx{8}{$-2$}}
  +
\put(-32,668){\sx{8}{$-4$}}
  +
\put(-32,468){\sx{8}{$-6$}}
  +
\put(-32,268){\sx{8}{$-8$}}
  +
\put(230,8){\sx{8}{$-8$}}
  +
\put(430,8){\sx{8}{$-6$}}
  +
\put(630,8){\sx{8}{$-4$}}
  +
\put(830,8){\sx{8}{$-2$}}
  +
\put(1070,8){\sx{8}{$0$}}
  +
\put(1270,8){\sx{8}{$2$}}
  +
\put(1470,8){\sx{8}{$4$}}
  +
\put(1670,8){\sx{8}{$6$}}
  +
\put(1870,8){\sx{8}{$8$}}
  +
\put(2070,8){\sx{8}{$x$}}
  +
%
  +
\put(120,1100){\rot{2}\sx{8}{$v\!=\!-8$}\ero}
  +
\put(160,1032){\rot{-2}\sx{8}{$v\!=\!8$}\ero}
  +
\put(120,800){\rot{8}\sx{8}{$u\!=\!8$}\ero}
  +
\put(120,732){\rot{3}\sx{8}{$u\!=\!-8$}\ero}
  +
%
  +
\put(840,910){\rot{50}\sx{8}{$u\!=\!4$}\ero}
  +
\put(890,862){\rot{6}\sx{8}{$u\!=\!0$}\ero}
  +
\put(760,830){\rot{-32}\sx{8}{$u\!=\!-4$}\ero}
  +
%
  +
\put(1340,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero}
  +
\put(1906,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero}
  +
%
  +
\put(1450,840){\rot{38}\sx{8}{$v\!=\!0$}\ero}
  +
\put(1560,740){\rot{50}\sx{8}{$u\!=\!0$}\ero}
  +
\put(1680,610){\rot{50}\sx{8}{$v\!=\!0$}\ero}
  +
\put(1844,480){\rot{51}\sx{8}{$u\!=\!0$}\ero}
  +
\put(1980,290){\rot{50}\sx{8}{$v\!=\!0$}\ero}
  +
\end{picture}
  +
\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Bessel function]]
  +
[[Category:C++]]
  +
[[Category:Complex map]]
  +
[[Category:Latex]]
  +
[[Category:Makoto Morinaga]]
  +
[[Category:Mori]]
  +
[[Category:Mori function]]
  +
[[Category:Nori]]
  +
[[Category:Nori function]]

Latest revision as of 08:44, 1 December 2018

complex map of function nori, in some range of the complex plane,

$\mathrm{nori}(z)=\,$mori$\big(\sqrt{z}\big)^2$

$u\!+\!\mathrm i v=\mathrm{nori}(x\!+\!\mathrm i y)$

C++ generator of curves

Files ado.cin, conto.cin, besselj0.cin should be loaded in order to compile the code below:


#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"
#include "besselj0.cin"

DB L1= 2.404825557695773;
DB L2= 5.5200781102863115;
DB L3= 8.653727912911013;
DB L4=11.791534439014281;
z_type morin(z_type x){ return BesselJ0(L1*x)/(1.-x*x);} // naive representation fails at x=1.

z_type mori0(z_type x){ int n,m; z_type s, xx=x*x;
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
//with m=15, at |x|<2, the relative error is of order of 10^(-16)
//In this sense, the result is accurate while |x|<2.
m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;}
return 1.+s;}
z_type mori(z_type x){if(abs(x)<2.) return mori0(x);
                        return morin(x);}

z_type no1(z_type x){ return BesselJ0(L1*sqrt(x))/(1.-x);} // naive representation fails at x=1.

z_type no0(z_type x){ int n,m; z_type s; //, xx=x*x;
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
//with m=15, at |x|<2, the relative error is of order of 10^(-16)
//In this sense, the result is accurate while |x|<2.
m=15; s=c[m]*x; for(n=m-1;n>0;n--){ s+=c[n]; s*=x;}
return 1.+s;}

z_type no(z_type x){if(abs(x)<3.) return no0(x);
                        return no1(x);}


int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=501,M1=M+1;
int N=501,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("06.eps","w");ado(o,5020,5020);
FILE *o;o=fopen("norima.eps","w");ado(o,5020,5020);
fprintf(o,"1010 1010 translate\n 100 100 scale\n");
DO(m,M1)X[m]=-10.+.1*(m-.5);
DO(n,N1)Y[n]=-10.+.1*(n-.5);

for(m=-10;m<11;m++){if(m==0){M(m,-10.1)L(m,10.1)} else{M(m,-10)L(m,10)}}
for(n=-10;n<11;n++){ M( -10,n)L(10,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=99999; f[m*N1+n]=99999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=BesselJ0(z);
// c=mori(z);
      c=no(z); c*=c;
     p=Re(c); q=Im(c);
     if(p>-999. && p<999. && q>-999. && q<999
     )
{g[m*N1+n]=p;
 f[m*N1+n]=q;
}
                     }}
//#include "plodi.cin"
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=40;q=1;
for(m=-4;m<4;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<4;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<4;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<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .9 0 0 RGB S\n");
for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".04 W .6 0 .6 RGB S\n");
for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
    m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".04 W 0 0 0 RGB S\n");
for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n");
//#include "plofu.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf norima.eps");
     system( "open norima.pdf");
}

Latex generator of labels


\documentclass[12pt]{article}
\paperheight 5120px
\paperwidth 5100px
\textwidth 5094px
\textheight 5200px
\topmargin -80px
\oddsidemargin -72px
\usepackage{graphics}
\usepackage{rotating}
%\usepackage{color}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}\parindent 0pt
\begin{picture}(5100,5100)
\put(80,80){\ing{norima}}
%\put(0,0){\sx{8}{-10}}
\put(32,2068){\sx{8}{$y$}}
\put(32,1868){\sx{8}{$8$}}
\put(32,1668){\sx{8}{$6$}}
\put(32,1468){\sx{8}{$4$}}
\put(32,1268){\sx{8}{$2$}}
\put(32,1068){\sx{8}{$0$}}
\put(-32,868){\sx{8}{$-2$}}
\put(-32,668){\sx{8}{$-4$}}
\put(-32,468){\sx{8}{$-6$}}
\put(-32,268){\sx{8}{$-8$}}
\put(230,8){\sx{8}{$-8$}}
\put(430,8){\sx{8}{$-6$}}
\put(630,8){\sx{8}{$-4$}}
\put(830,8){\sx{8}{$-2$}}
\put(1070,8){\sx{8}{$0$}}
\put(1270,8){\sx{8}{$2$}}
\put(1470,8){\sx{8}{$4$}}
\put(1670,8){\sx{8}{$6$}}
\put(1870,8){\sx{8}{$8$}}
\put(2070,8){\sx{8}{$x$}}
%
\put(120,1100){\rot{2}\sx{8}{$v\!=\!-8$}\ero}
\put(160,1032){\rot{-2}\sx{8}{$v\!=\!8$}\ero}
\put(120,800){\rot{8}\sx{8}{$u\!=\!8$}\ero}
\put(120,732){\rot{3}\sx{8}{$u\!=\!-8$}\ero}
%
\put(840,910){\rot{50}\sx{8}{$u\!=\!4$}\ero}
\put(890,862){\rot{6}\sx{8}{$u\!=\!0$}\ero}
\put(760,830){\rot{-32}\sx{8}{$u\!=\!-4$}\ero}
%
\put(1340,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero}
\put(1906,1070){\rot{0}\sx{8}{$v\!=\!0$}\ero}
%
\put(1450,840){\rot{38}\sx{8}{$v\!=\!0$}\ero}
\put(1560,740){\rot{50}\sx{8}{$u\!=\!0$}\ero}
\put(1680,610){\rot{50}\sx{8}{$v\!=\!0$}\ero}
\put(1844,480){\rot{51}\sx{8}{$u\!=\!0$}\ero}
\put(1980,290){\rot{50}\sx{8}{$v\!=\!0$}\ero}
\end{picture}
\end{document}

References

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