imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Explicit plot]] of <math>n</math>th Iteration of the quadratic function, id est, [[power function]] for power 2; <math>y=\mathrm{Pow}_2^{~n}(z)=T^n(x)~</math> for various values of number <math>n</math> of iteration. Here,
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)
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| </math>
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| The iteration of <math>T</math> is implemented in the following way: :
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)
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| </math>
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| where
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| <math>\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)</math>
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| is [[superfunction]] for the transfer function
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| <math>T</math>
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| and
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)
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| </math>
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| is the [[Abel Function]]. The thick lines represent
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| <math>
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| y=x^2
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| </math>and
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| <math>
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| y=\sqrt{x}</math>.
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| FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to <math>n\!=\!1</math>, and the second overlaps with <math>y=T^{-1}(x)</math> and corresponds to <math>n\!=\!-1</math>.
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2012.11.11
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| |pub-country = Japan
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| |notes =
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| |versions = http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
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| }}
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| Files used to generate the figure are supplied below.
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| ==[[C++]] generator of curves==
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| // file [[ado.cin]] should be loaded to the working directory in order to compile the [[C++]] code below.
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| #include<math.h>
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| #include<stdio.h>
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| #include<stdlib.h>
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| #define DB double
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| #define DO(x,y) for(x=0;x<y;x++)
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| using namespace std;
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| #include <complex>
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| typedef complex<double> z_type;
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| #define Re(x) x.real()
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| #define Im(x) x.imag()
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| #define I z_type(0.,1.)
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| #include "ado.cin"
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| DB B=2.;
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| DB F(DB z) { return exp( exp( log(B)*z));}
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| DB G(DB z) { return log( log(z) )/log(B);}
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| main(){ int m,n; double x,y,t; FILE *o;
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| o=fopen("PowIte.eps","w"); ado(o,1010,1010);
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| fprintf(o,"1 1 translate 100 100 scale\n");
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| #define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
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| #define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
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| for(m=0;m<11;m++) {M(m,0)L(m,10)}
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| for(m=0;m<11;m++) {M(0,m)L(10,m)}
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| fprintf(o,"2 setlinecap .01 W S\n");
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| DO(m,402){x=0.001+.01*m; y=exp(2.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
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| DO(m,402){x=0.001+.01*m; y=exp(4.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
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| DO(m,402){x=0.001+.01*m; y=exp(8.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;}
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| fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
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| DO(m,1002){x=0.001+.01*m; y=exp(.5*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
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| DO(m,1002){x=0.001+.01*m; y=exp(.25*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
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| DO(m,1002){x=0.001+.01*m; y=exp(.125*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
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| for(n=-20;n<21;n++){t=.1*n; DO(m,182){x=1.+.05*m; y=F(t+G(x)); if(m==0) M(x,y) else L(x,y); if(y>10.1)break;} }
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| fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| system("epstopdf PowIte.eps");
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| system( "open PowIte.pdf");
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| getchar(); system("killall Preview");
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| }
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| == Licensing ==
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| {{CC|by|3.0}}
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