Difference between revisions of "File:Acker2t400.jpg"

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($ -> \( ; description ; refs ; pre ; keywords)
 
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  +
{{oq|Acker2t400.jpg|Original file ‎(3,555 × 5,588 pixels, file size: 1.09 MB, MIME type: image/jpeg)|800}}
−
Importing image file
  
  +
  +
Fig.19.1 from page 255 of book «[[Superfunctions]]»<ref name="be">
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
D.Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>, 2020.
  +
  +
The same picture appears also as Рис.19.1 at page 262 of the Russian version
  +
«[[Суперфункции]]»<ref name="br">
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  +
</ref>, 2014.
  +
  +
The picture shows the comparison of the two notations for the [[Ackermann function]]s to base 2.
  +
  +
Thin curves: \(y=\mathcal A(m,x)=A_{2,m}(x\!+\!3)-3 ~\) for \(m=1,2,3,4\)
  +
  +
Thick curves: \(y=\mathcal A(m,x\!-\!3)+3=A_{2,m}(x) ~ \) for \(m=1,2,3,4\)
  +
  +
\(A_{2,1}=2+x=\mathcal A(1,x\!+\!3)-3=\mathcal A(1,x)\)
  +
  +
\(A_{2,2}=2x=\mathcal A(2,x\!+\!3)-3\)
  +
  +
\(A_{2,3}=\exp_2(x)=\mathcal A(3,x\!+\!3)-3=2^x\)
  +
  +
\(A_{2,4}=\mathrm{tet}_2(x)=\mathcal A(3,x\!+\!3)-3\)
  +
  +
==[[C++]] generator of curves==
  +
<pre>
  +
#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 "ado.cin"
  +
void ado(FILE *O, int X, int Y)
  +
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
  +
fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
  +
fprintf(O,"/M {moveto} bind def\n");
  +
fprintf(O,"/L {lineto} bind def\n");
  +
fprintf(O,"/S {stroke} bind def\n");
  +
fprintf(O,"/s {show newpath} bind def\n");
  +
fprintf(O,"/C {closepath} bind def\n");
  +
fprintf(O,"/F {fill} bind def\n");
  +
fprintf(O,"/o {.015 0 360 arc C F} bind def\n");
  +
fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
  +
fprintf(O,"/W {setlinewidth} bind def\n");
  +
fprintf(O,"/RGB {setrgbcolor} bind def\n");}
  +
/* end of routine */
  +
  +
// #include "fsexp.cin"
  +
// #include "fslog.cin"
  +
#include "fit1.cin"
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
FILE *o;o=fopen("acker2.eps","w"); ado(o,608,1008);
  +
fprintf(o,"304 304 translate\n 100 100 scale\n");
  +
#define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
  +
#define o(x,y) fprintf(o,"%8.4f %8.4f o\n",0.+x,0.+y);
  +
for(m=-3;m<4;m++) {M(m,-3)L(m,7)}
  +
for(n=-3;n<8;n++) {M( -3,n)L(3,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");
  +
  +
M(-3.02,-3.02+2.)L(3.02,3.02+2) fprintf(o,".02 W .3 0 .3 RGB S\n");
  +
M(-1.51, -1.51*2)L(3.02,3.02*2) fprintf(o,".02 W 0 .5 0 RGB S\n"); fprintf(o,"1 0 0 RGB\n");
  +
  +
DO(n,306){x=-3.02+.025*(n-.5);y=exp(log(2.)*x); o(x,y); if(y>7.1)break;} fprintf(o,".03 W 0 .8 0 RGB S\n");
  +
  +
//DO(n,150){x=-1.9+.04*n;y=Re(FSEXP(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 1 RGB S\n");
  +
//DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y));
  +
//if(n/2*2==n) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");
  +
  +
DO(n,330){x=-1.9+.018*n; y=Re(FIT1(log(2.),x)); if(n/2*2==n) M(x,y)else L(x,y); if(y>7.1)break;}
  +
fprintf(o,"0 setlinecap .03 W 0 0 .7 RGB S\n");
  +
  +
//DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");
  +
//DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");
  +
  +
//thin
  +
  +
//M(-3.02,-3.02+2.)L(3.02,3.02+2) fprintf(o,".007 W .3 0 .3 RGB S\n");
  +
M(-1.51-3, -1.51*2-3)L(5.02-3,5.02*2-3) fprintf(o,".005 W 0 .5 0 RGB S\n"); fprintf(o,"1 0 0 RGB\n");
  +
  +
DO(n,306){x=-1.02+.025*(n-.5);y=exp(log(2.)*x); o(x-3,y-3); if(y-3>7.1)break;} fprintf(o,".05 W 0 .8 0 RGB S\n");
  +
  +
//DO(n,150){x=-1.9+.04*n;y=Re(FSEXP(x)); if(n==0) M(x-3,y-3)else L(x-3,y-3); if(y-3>8.)break;} fprintf(o,".02 W 0 0 1 RGB S\n");
  +
//DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y));
  +
//if(n/2*2==n) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");
  +
  +
DO(n,330){x=-.9+.018*n; y=Re(FIT1(log(2.),x)); if(n/2*2==n) M(x-3,y-3)else L(x-3,y-3); if(y-3>7.1)break;}
  +
fprintf(o,"0 setlinecap .018 W 0 0 .7 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf acker2.eps");
  +
system( "open acker2.pdf");
  +
}
  +
</pre>
  +
==[[Latex]] generator of labels==
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\paperwidth 640px
  +
\paperheight 1006px
  +
\textwidth 1394px
  +
\textheight 1300px
  +
\topmargin -104px
  +
\oddsidemargin -92px
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\begin{document}
  +
{\begin{picture}(608,1006) %\put(12,0){\ing{penma}}
  +
\put(0,0){\ing{acker2}}
  +
\put(281,988){\sx{3.}{$y$}}
  +
\put(279,895){\sx{3.}{$6$}}
  +
\put(279,795){\sx{3.}{$5$}}
  +
\put(279,694){\sx{3.}{$4$}}
  +
\put(279,594){\sx{3.}{$3$}}
  +
\put(279,468){\sx{3.}{$\mathrm e$}}
  +
\put(279,494){\sx{3.}{$2$}}
  +
\put(279,394){\sx{3.}{$1$}}
  +
\put(279,294){\sx{3.}{$0$}}
  +
\put(258,193){\sx{3.}{$-1$}}
  +
\put(258, 93){\sx{3.}{$-2$}}
  +
\put( 80,274){\sx{3.}{$-2$}}
  +
\put(180,274){\sx{3.}{$-1$}}
  +
\put(296,274){\sx{3.}{$0$}}
  +
\put(396,274){\sx{3.}{$1$}}
  +
\put(496,274){\sx{3.}{$2$}}
  +
\put(586,274){\sx{3.}{$x$}}
  +
%
  +
%\put(242,620){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_2(x\!+\!3)\!-\!3\!=\!\mathrm{tet}_2(x\!+\!3)\!-\!3$\ero}}
  +
\put(248,720){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_{2,4}(x\!+\!3)\!-\!3$\ero}}
  +
\put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}}
  +
\put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}}
  +
\put(314,526){\sx{1.8}{\rot{45}$y\!=\!\mathcal{A}(1,x)\!=\! A_{2,1}(x\!+\!3)\!-\!3$\ero}}
  +
  +
%\put(438,714){\sx{1.8}{\rot{85}$y\!=\!\mathrm{pen}(x)$\ero}}
  +
%\put(538,912){\sx{1.8}{\rot{82}$y\!=\!\mathrm{tet}_2(x)$\ero}}
  +
\put(526,822){\sx{1.8}{\rot{82}$y\!=\!A_{2,4}(x)\!=\!\mathrm{tet}_2(x)$\ero}}
  +
%\put(578,892){\sx{1.8}{\rot{73}$y\!=\!2^x$\ero}}
  +
\put(566,858){\sx{1.8}{\rot{73}$y\!=\!A_{2,3}(x)\!=\!2^x$\ero}}
  +
\put(566,792){\sx{1.8}{\rot{62}$y\!=\!A_{2,2}(x)\!=\!\mathrm{2}x$\ero}}
  +
%\put(478,628){\sx{1.8}{\rot{50}$y\!=\!\mathrm{e}\!+\!x$\ero}}
  +
\put(520,696){\sx{1.96}{\rot{44}$y\!=\!A_{2,1}(x)\!=\!2\!+\!x$\ero}}
  +
%
  +
%\put(86,222){\sx{1.9}{\rot{11}$y\!=\!\mathrm{exp}(x)$\ero}}
  +
%\put(20,30){\sx{1.9}{\rot{30}$y\!=\!\mathrm{pen}(x)$\ero}}
  +
\put(32,326){\sx{1.9}{\rot{3}$y\!=\!2^x$\ero}}
  +
\put(132,4){\sx{1.9}{\rot{81}$y\!=\!\mathrm{tet}_2(x)$\ero}}
  +
\put(178,8){\sx{1.9}{\rot{66}$y\!=\!\mathrm{2} x$\ero}}
  +
%
  +
%\put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}}
  +
\end{picture}
  +
\end{document}
  +
</pre>
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
  +
«[[Ackermann function]]»,
  +
«[[Explicit plot]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[Tetration]]»,
  +
«[[Tetration to base 2]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:Ackermann function]]
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:C++]]
  +
[[Category:Explicit plot]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:Tetration]]
  +
[[Category:Tetration to base 2]]

Latest revision as of 19:50, 3 January 2026


Fig.19.1 from page 255 of book «Superfunctions»[1], 2020.

The same picture appears also as Рис.19.1 at page 262 of the Russian version «Суперфункции»[2], 2014.

The picture shows the comparison of the two notations for the Ackermann functions to base 2.

Thin curves: \(y=\mathcal A(m,x)=A_{2,m}(x\!+\!3)-3 ~\) for \(m=1,2,3,4\)

Thick curves: \(y=\mathcal A(m,x\!-\!3)+3=A_{2,m}(x) ~ \) for \(m=1,2,3,4\)

\(A_{2,1}=2+x=\mathcal A(1,x\!+\!3)-3=\mathcal A(1,x)\)

\(A_{2,2}=2x=\mathcal A(2,x\!+\!3)-3\)

\(A_{2,3}=\exp_2(x)=\mathcal A(3,x\!+\!3)-3=2^x\)

\(A_{2,4}=\mathrm{tet}_2(x)=\mathcal A(3,x\!+\!3)-3\)

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 "ado.cin"
void ado(FILE *O, int X, int Y)
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.015 0 360 arc C F} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}
/* end of routine */

// #include "fsexp.cin"
// #include "fslog.cin"
#include "fit1.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
FILE *o;o=fopen("acker2.eps","w"); ado(o,608,1008);
 fprintf(o,"304 304 translate\n 100 100 scale\n");
#define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%8.4f %8.4f o\n",0.+x,0.+y);
 for(m=-3;m<4;m++) {M(m,-3)L(m,7)}
 for(n=-3;n<8;n++) {M( -3,n)L(3,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");

M(-3.02,-3.02+2.)L(3.02,3.02+2) fprintf(o,".02 W .3 0 .3 RGB S\n");
M(-1.51, -1.51*2)L(3.02,3.02*2) fprintf(o,".02 W 0 .5 0 RGB S\n"); fprintf(o,"1 0 0 RGB\n");

DO(n,306){x=-3.02+.025*(n-.5);y=exp(log(2.)*x); o(x,y); if(y>7.1)break;} fprintf(o,".03 W 0 .8 0 RGB S\n");

//DO(n,150){x=-1.9+.04*n;y=Re(FSEXP(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 1 RGB S\n");
//DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y));
//if(n/2*2==n) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");

DO(n,330){x=-1.9+.018*n; y=Re(FIT1(log(2.),x)); if(n/2*2==n) M(x,y)else L(x,y); if(y>7.1)break;} 
fprintf(o,"0 setlinecap .03 W 0 0 .7 RGB S\n");

//DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");
//DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");

//thin

//M(-3.02,-3.02+2.)L(3.02,3.02+2) fprintf(o,".007 W .3 0 .3 RGB S\n");
M(-1.51-3, -1.51*2-3)L(5.02-3,5.02*2-3) fprintf(o,".005 W 0 .5 0 RGB S\n"); fprintf(o,"1 0 0 RGB\n");

DO(n,306){x=-1.02+.025*(n-.5);y=exp(log(2.)*x); o(x-3,y-3); if(y-3>7.1)break;} fprintf(o,".05 W 0 .8 0 RGB S\n");

//DO(n,150){x=-1.9+.04*n;y=Re(FSEXP(x)); if(n==0) M(x-3,y-3)else L(x-3,y-3); if(y-3>8.)break;} fprintf(o,".02 W 0 0 1 RGB S\n");
//DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y));
//if(n/2*2==n) M(x,y)else L(x,y); if(y>6.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");

DO(n,330){x=-.9+.018*n; y=Re(FIT1(log(2.),x)); if(n/2*2==n) M(x-3,y-3)else L(x-3,y-3); if(y-3>7.1)break;} 
fprintf(o,"0 setlinecap .018 W 0 0 .7 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf acker2.eps"); 
        system(    "open acker2.pdf");
}

Latex generator of labels

\documentclass[12pt]{article}
\paperwidth 640px
\paperheight 1006px
\textwidth 1394px
\textheight 1300px
\topmargin -104px
\oddsidemargin -92px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
{\begin{picture}(608,1006) %\put(12,0){\ing{penma}}
\put(0,0){\ing{acker2}}
\put(281,988){\sx{3.}{$y$}}
\put(279,895){\sx{3.}{$6$}}
\put(279,795){\sx{3.}{$5$}}
\put(279,694){\sx{3.}{$4$}}
\put(279,594){\sx{3.}{$3$}}
\put(279,468){\sx{3.}{$\mathrm e$}}
\put(279,494){\sx{3.}{$2$}}
\put(279,394){\sx{3.}{$1$}}
\put(279,294){\sx{3.}{$0$}}
\put(258,193){\sx{3.}{$-1$}}
\put(258, 93){\sx{3.}{$-2$}}
\put( 80,274){\sx{3.}{$-2$}}
\put(180,274){\sx{3.}{$-1$}}
\put(296,274){\sx{3.}{$0$}}
\put(396,274){\sx{3.}{$1$}}
\put(496,274){\sx{3.}{$2$}}
\put(586,274){\sx{3.}{$x$}}
%
%\put(242,620){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_2(x\!+\!3)\!-\!3\!=\!\mathrm{tet}_2(x\!+\!3)\!-\!3$\ero}}
\put(248,720){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_{2,4}(x\!+\!3)\!-\!3$\ero}}
\put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}}
\put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}}
\put(314,526){\sx{1.8}{\rot{45}$y\!=\!\mathcal{A}(1,x)\!=\! A_{2,1}(x\!+\!3)\!-\!3$\ero}}

%\put(438,714){\sx{1.8}{\rot{85}$y\!=\!\mathrm{pen}(x)$\ero}}
%\put(538,912){\sx{1.8}{\rot{82}$y\!=\!\mathrm{tet}_2(x)$\ero}}
\put(526,822){\sx{1.8}{\rot{82}$y\!=\!A_{2,4}(x)\!=\!\mathrm{tet}_2(x)$\ero}}
%\put(578,892){\sx{1.8}{\rot{73}$y\!=\!2^x$\ero}}
\put(566,858){\sx{1.8}{\rot{73}$y\!=\!A_{2,3}(x)\!=\!2^x$\ero}}
\put(566,792){\sx{1.8}{\rot{62}$y\!=\!A_{2,2}(x)\!=\!\mathrm{2}x$\ero}}
%\put(478,628){\sx{1.8}{\rot{50}$y\!=\!\mathrm{e}\!+\!x$\ero}}
\put(520,696){\sx{1.96}{\rot{44}$y\!=\!A_{2,1}(x)\!=\!2\!+\!x$\ero}}
%
%\put(86,222){\sx{1.9}{\rot{11}$y\!=\!\mathrm{exp}(x)$\ero}}
%\put(20,30){\sx{1.9}{\rot{30}$y\!=\!\mathrm{pen}(x)$\ero}}
\put(32,326){\sx{1.9}{\rot{3}$y\!=\!2^x$\ero}}
\put(132,4){\sx{1.9}{\rot{81}$y\!=\!\mathrm{tet}_2(x)$\ero}}
\put(178,8){\sx{1.9}{\rot{66}$y\!=\!\mathrm{2} x$\ero}}
%
%\put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}}
\end{picture}
\end{document}

References

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