Difference between revisions of "File:Korifit76plot.jpg"

From TORI
Jump to: navigation, search
(Importing image file)
 
 
Line 1: Line 1:
  +
Comparison of function [[mori]] implemented through the definition
−
Importing image file
  
  +
  +
$y_1=\mathrm{mori}(x)=J_0(L_1\,x)/(1\!-\!x^2)$
  +
  +
to the approximation with
  +
  +
$\mathrm{kori76fit}=
  +
\prod_{n=2}^7\! \left(1-\frac{L_1^2}{L_n^2}x\right)~ \frac{1+\sum_{n=1}^6 a_n x^n}{1+\sum_{n=1}^6 b_n x^n}
  +
$
  +
  +
where $L_n=\mathrm{BesselJZero}[0,n]$ and coefficients $a$ and $b$ are:
  +
  +
$a_1=-0.04844698269548584 $<br>
  +
$a_2= 0.0010028289633265202 $<br>
  +
$a_3= -0.000011428855401098336$<br>
  +
$a_4= 7.61813379974462\times 10^{-8}$<br>
  +
$a_5= -2.8376606186641804\times 10^{-10}$<br>
  +
$a_6= 4.651275051439759\times 10^{-13}$
  +
  +
$b_1= 0.03223483760044156$<br>
  +
$b_2= 0.0004974915308429358$<br>
  +
$b_3= 4.7768603073237505\times 10^{-6}$<br>
  +
$b_4= 3.071615607112112\times 10^{-8}$<br>
  +
$b_5= 1.292095753771865\times 10^{-10}$<br>
  +
$b_6= 2.925186494186955\times 10^{-13}$
  +
  +
Approximation is the following:
  +
  +
$\displaystyle
  +
\mathrm{mori}(z) \approx
  +
\mathrm{korifit76}(z^2)$
  +
  +
Let $y_2=\mathrm{kori76fit}(x^2)$
  +
  +
the upper blue curve shows the agreement
  +
  +
$\mathrm{agreeA}(x)= - \log_{10}(|y_1\!-\!y_2|)$
  +
  +
scaled with factor 0.1; id est, $y=\frac{1}{10}\mathrm{agreeA}(x)$
  +
  +
the black intermediate curve shows the agreement
  +
  +
$\displaystyle
  +
\mathrm{agreeB}(x)= - \log_{10}\left(\frac{|y_1-y_2|}{|y_1| +| y_2|}\right)$
  +
  +
scaled with factor 0.1; id est, $y=\frac{1}{10}\mathrm{agreeAB}(x)$.
  +
  +
For example, agreement 10 (level $y=1$ in the graphic) indicates, that of order of 10 significant figures can be achieved with such a fit.
  +
  +
The lower red curve shows the approximated function
  +
  +
$y=\mathrm{mori}(x)$
  +
  +
However, the deviation of the fit from the approximated function is not seen in this curve.
  +
  +
==[[C++]] generator of curves==
  +
<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.)
  +
#define M(x,y) fprintf(o,"%6.4lf %6.4lf M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%6.4lf %6.4lf L\n",0.+x,0.+y);
  +
  +
#include "korifit76.cin"
  +
#include "ado.cin"
  +
  +
DB L1=2.404825557695773;
  +
DB L2=5.5200781102863115;
  +
DB L3=8.653727912911013;
  +
DB L4=11.791534439014281;
  +
DB L5=14.930917708487787;
  +
DB L6=18.071063967910924;
  +
DB L7=21.21163662987926;
  +
DB L8=24.352471530749302;
  +
  +
DB LN[9]={ 0., 2.404825557695773, 5.5200781102863115,
  +
8.653727912911013,11.791534439014281,14.930917708487787,
  +
18.071063967910924,21.21163662987926, 24.352471530749302};
  +
  +
int main(){int m,n,i; double x,y,z,t,u; FILE *o;
  +
  +
//o=fopen("23.eps","w");
  +
o=fopen("korifit76plo.eps","w");
  +
ado(o,920,220);
  +
fprintf(o,"10 10 translate 100 100 scale 2 setlinecap 1 setlinejoin\n");
  +
  +
for(m=0;m<10;m++) {M(m,0 )L(m,1.5)}
  +
for(n=0;n<16;n+=5){M(0,.1*n)L(9,.1*n)}
  +
fprintf(o,"0.01 W S\n");
  +
  +
for(m=2;m<8;m++) { M(LN[m]/L1,0) L(LN[m]/L1,1.5)}
  +
fprintf(o,"0.005 W S\n");
  +
  +
M(0,1.66);
  +
for(n=1;n<4600;n++){x=.002*n; y=j0(L1*x)/(1.-x*x); z=Re(korifit76(x*x)); t=z-y;
  +
u=fabs(t)/(fabs(y)+fabs(z));u=-log(u)/log(10);
  +
printf("%5.2lf %19.16lf %19.16lf %19.16lf %5.2lf\n",x,y,z,t,u);
  +
if(u>0. && u<20.){ L(x,.1*u) }
  +
}
  +
fprintf(o,"0 0 0 RGB .02 W S\n");
  +
  +
M(0,1.66);
  +
for(n=1;n<4600;n++){x=.002*n; y=j0(L1*x)/(1.-x*x); z=Re(korifit76(x*x)); t=z-y;
  +
u=fabs(t);
  +
u=-log(u)/log(10);
  +
if(u>0. && u<30.){ L(x,.1*u) }
  +
}
  +
fprintf(o,"0 0 1 RGB .02 W S\n");
  +
  +
DO(n,92){x=.1*n; y=j0(L1*x)/(1.-x*x);
  +
//z=Re(korifit76(x*x));
  +
if(y>-99 && y<99){ if(n==0) M(x,y) else L(x,y) }
  +
}
  +
fprintf(o,"1 0 0 RGB .02 W S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o);
  +
system("epstopdf korifit76plo.eps");
  +
system( "open korifit76plo.pdf");
  +
return 0;}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\paperwidth 923pt
  +
\paperheight 208pt
  +
\textwidth 420pt
  +
\textheight 300pt
  +
\topmargin -108pt
  +
\oddsidemargin -73pt
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {\mathrm{e}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\pagestyle{empty}
  +
\parindent 0pt
  +
\begin{document}
  +
\begin{picture}(614,197)
  +
\put(0,0){\ing{korifit76plo}}
  +
\put(1,190){\sx{1.25}{$y$}}
  +
\put(.6,156){\sx{1.4}{$\frac{3}{2}$}}
  +
\put(.6,105){\sx{1.3}{$1$}}
  +
\put(.6,56){\sx{1.4}{$\frac{1}{2}$}}
  +
\put(.6, 5){\sx{1.3}{$0$}}
  +
\put(107,-2){\sx{1.3}{$1$}}
  +
\put(207,-2){\sx{1.3}{$2$}}
  +
\put(233,-3){\sx{1.2}{$\frac{L_2}{L_1}$}}
  +
\put(308,-2){\sx{1.3}{$3$}}
  +
\put(364,-3){\sx{1.2}{$\frac{L_3}{L_1}$}}
  +
\put(408,-2){\sx{1.3}{$4$}}
  +
\put(494,-3){\sx{1.2}{$\frac{L_4}{L_1}$}}
  +
\put(509,-2){\sx{1.3}{$5$}}
  +
\put(609,-2){\sx{1.3}{$6$}}
  +
\put(627,-3){\sx{1.2}{$\frac{L_5}{L_1}$}}
  +
\put(709,-2){\sx{1.3}{$7$}}
  +
\put(759,-3){\sx{1.2}{$\frac{L_6}{L_1}$}}
  +
\put(810,-2){\sx{1.3}{$8$}}
  +
\put(889.6,-3){\sx{1.2}{$\frac{L_7}{L_1}$}}
  +
\put(914,-1.6){\sx{1.3}{$x$}}
  +
\put(112,58){\rot{-33}\sx{1.4}{$y\!=\! \mathrm{mori}(x)$}\ero}
  +
\put(778,103){\rot{-16}\sx{1.3}{$y\!=\! \frac{1}{10}\mathrm{agreeA}(x)$}\ero}
  +
\put(766,60){\rot{-16}\sx{1.3}{$y\!=\! \frac{1}{10}\mathrm{agreeB}(x)$}\ero}
  +
\end{picture}\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Approximation]]
  +
[[Category:Bessel function]]
  +
[[Category:Explicit plot]]
  +
[[Category:Kori function]]
  +
[[Category:Morinaga function]]

Latest revision as of 08:40, 1 December 2018

Comparison of function mori implemented through the definition

$y_1=\mathrm{mori}(x)=J_0(L_1\,x)/(1\!-\!x^2)$

to the approximation with

$\mathrm{kori76fit}= \prod_{n=2}^7\! \left(1-\frac{L_1^2}{L_n^2}x\right)~ \frac{1+\sum_{n=1}^6 a_n x^n}{1+\sum_{n=1}^6 b_n x^n} $

where $L_n=\mathrm{BesselJZero}[0,n]$ and coefficients $a$ and $b$ are:

$a_1=-0.04844698269548584 $
$a_2= 0.0010028289633265202 $
$a_3= -0.000011428855401098336$
$a_4= 7.61813379974462\times 10^{-8}$
$a_5= -2.8376606186641804\times 10^{-10}$
$a_6= 4.651275051439759\times 10^{-13}$

$b_1= 0.03223483760044156$
$b_2= 0.0004974915308429358$
$b_3= 4.7768603073237505\times 10^{-6}$
$b_4= 3.071615607112112\times 10^{-8}$
$b_5= 1.292095753771865\times 10^{-10}$
$b_6= 2.925186494186955\times 10^{-13}$

Approximation is the following:

$\displaystyle \mathrm{mori}(z) \approx \mathrm{korifit76}(z^2)$

Let $y_2=\mathrm{kori76fit}(x^2)$

the upper blue curve shows the agreement

$\mathrm{agreeA}(x)= - \log_{10}(|y_1\!-\!y_2|)$

scaled with factor 0.1; id est, $y=\frac{1}{10}\mathrm{agreeA}(x)$

the black intermediate curve shows the agreement

$\displaystyle \mathrm{agreeB}(x)= - \log_{10}\left(\frac{|y_1-y_2|}{|y_1| +| y_2|}\right)$

scaled with factor 0.1; id est, $y=\frac{1}{10}\mathrm{agreeAB}(x)$.

For example, agreement 10 (level $y=1$ in the graphic) indicates, that of order of 10 significant figures can be achieved with such a fit.

The lower red curve shows the approximated function

$y=\mathrm{mori}(x)$

However, the deviation of the fit from the approximated function is not seen in this curve.

C++ generator of curves


#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.)
#define M(x,y) fprintf(o,"%6.4lf %6.4lf M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4lf %6.4lf L\n",0.+x,0.+y);

#include "korifit76.cin"
#include "ado.cin"

DB L1=2.404825557695773;
DB L2=5.5200781102863115;
DB L3=8.653727912911013;
DB L4=11.791534439014281;
DB L5=14.930917708487787;
DB L6=18.071063967910924;
DB L7=21.21163662987926;
DB L8=24.352471530749302;

DB LN[9]={ 0., 2.404825557695773, 5.5200781102863115,
 8.653727912911013,11.791534439014281,14.930917708487787,
18.071063967910924,21.21163662987926, 24.352471530749302};

int main(){int m,n,i; double x,y,z,t,u; FILE *o;

//o=fopen("23.eps","w");
o=fopen("korifit76plo.eps","w");
ado(o,920,220);
fprintf(o,"10 10 translate 100 100 scale 2 setlinecap 1 setlinejoin\n");

for(m=0;m<10;m++) {M(m,0 )L(m,1.5)}
for(n=0;n<16;n+=5){M(0,.1*n)L(9,.1*n)}
fprintf(o,"0.01 W S\n");

for(m=2;m<8;m++) { M(LN[m]/L1,0) L(LN[m]/L1,1.5)}
fprintf(o,"0.005 W S\n");

M(0,1.66);
for(n=1;n<4600;n++){x=.002*n; y=j0(L1*x)/(1.-x*x); z=Re(korifit76(x*x)); t=z-y;
u=fabs(t)/(fabs(y)+fabs(z));u=-log(u)/log(10);
printf("%5.2lf %19.16lf %19.16lf %19.16lf %5.2lf\n",x,y,z,t,u);
 if(u>0. && u<20.){ L(x,.1*u) }
}
fprintf(o,"0 0 0 RGB .02 W S\n");

M(0,1.66);
for(n=1;n<4600;n++){x=.002*n; y=j0(L1*x)/(1.-x*x); z=Re(korifit76(x*x)); t=z-y;
u=fabs(t);
u=-log(u)/log(10);
if(u>0. && u<30.){ L(x,.1*u) }
}
fprintf(o,"0 0 1 RGB .02 W S\n");

DO(n,92){x=.1*n; y=j0(L1*x)/(1.-x*x);
//z=Re(korifit76(x*x));
if(y>-99 && y<99){ if(n==0) M(x,y) else L(x,y) }
}
fprintf(o,"1 0 0 RGB .02 W S\n");

fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o);
system("epstopdf korifit76plo.eps");
system( "open korifit76plo.pdf");
return 0;}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 923pt
\paperheight 208pt
\textwidth 420pt
\textheight 300pt
\topmargin -108pt
\oddsidemargin -73pt
\newcommand \ds {\displaystyle}
\newcommand \sx {\scalebox}
\newcommand \rme {\mathrm{e}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\pagestyle{empty}
\parindent 0pt
\begin{document}
\begin{picture}(614,197)
\put(0,0){\ing{korifit76plo}}
\put(1,190){\sx{1.25}{$y$}}
\put(.6,156){\sx{1.4}{$\frac{3}{2}$}}
\put(.6,105){\sx{1.3}{$1$}}
\put(.6,56){\sx{1.4}{$\frac{1}{2}$}}
\put(.6, 5){\sx{1.3}{$0$}}
\put(107,-2){\sx{1.3}{$1$}}
\put(207,-2){\sx{1.3}{$2$}}
\put(233,-3){\sx{1.2}{$\frac{L_2}{L_1}$}}
\put(308,-2){\sx{1.3}{$3$}}
\put(364,-3){\sx{1.2}{$\frac{L_3}{L_1}$}}
\put(408,-2){\sx{1.3}{$4$}}
\put(494,-3){\sx{1.2}{$\frac{L_4}{L_1}$}}
\put(509,-2){\sx{1.3}{$5$}}
\put(609,-2){\sx{1.3}{$6$}}
\put(627,-3){\sx{1.2}{$\frac{L_5}{L_1}$}}
\put(709,-2){\sx{1.3}{$7$}}
\put(759,-3){\sx{1.2}{$\frac{L_6}{L_1}$}}
\put(810,-2){\sx{1.3}{$8$}}
\put(889.6,-3){\sx{1.2}{$\frac{L_7}{L_1}$}}
\put(914,-1.6){\sx{1.3}{$x$}}
\put(112,58){\rot{-33}\sx{1.4}{$y\!=\! \mathrm{mori}(x)$}\ero}
\put(778,103){\rot{-16}\sx{1.3}{$y\!=\! \frac{1}{10}\mathrm{agreeA}(x)$}\ero}
\put(766,60){\rot{-16}\sx{1.3}{$y\!=\! \frac{1}{10}\mathrm{agreeB}(x)$}\ero}
\end{picture}\end{document}

References

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current06:12, 1 December 2018Thumbnail for version as of 06:12, 1 December 20183,831 × 863 (318 KB)Maintenance script (talk | contribs)Importing image file
  • You cannot overwrite this file.

The following page links to this file:

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:Korifit76plot.jpg&oldid=19724"