File:Norifit76fragment.jpg

Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Norifit76plot.jpg

Analysis of the approximation of nori function through function korifit76,

nori$(x)=\, $kori$(x)^2\approx\, $korifit76$(x)^2$

The free agreements are shown:

$\mathrm{A}(x)= - \lg\! \Big( \big|\mathrm{korifit76}(x)^2-\mathrm{nori}(x)\big|\Big)$

$\mathrm{B}(x)= - \lg\! \Big( \big|\mathrm{korifit76}(x)-\mathrm{kori}(x)\big| \Big)$

$\displaystyle \mathrm{C}(x)= - \lg \left(\frac {\big|\mathrm{korifit76}(x)-\mathrm{kori}(x)\big|}{\big|\mathrm{korifit76}(x)\big| + \big|\mathrm{kori}(x)\big|}\,\right)$

where functions nori and kori are evaluated through the straightforward definitions using the C++ built-in Bessel function double j0(double ).

kori$(x) =\,$mori$\big(\sqrt{x}\big)$ $\,= \displaystyle \frac{ J_0\big(L_1 \sqrt{x} \big)}{1-x}$

where $L_n =\,$BesselJZero$[0,n]~$ is $n$th zero of function BesselJ0.

For reference, at the bottom, functions nori and kori, scaled with factor 10, are shown with green and red curves.

The additional vertical grid lines show abscissas $\displaystyle M_n=\left(\frac{L_n}{L_1}\right)^2~ $ for $~n=2\,..\,5$

korifit76
Function korifit76 is defined in the following way:

$\mathrm{kori76fit}= \displaystyle \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:

$\begin{array}{l} a_1=-0.04844698269548584 \\ a_2= 0.0010028289633265202 \\ a_3= -0.000011428855401098336\\ a_4= 7.61813379974462\times 10^{-8}\\ a_5= -2.8376606186641804\!\times\! 10^{-10}~\\ a_6= 4.651275051439759\times 10^{-13} \end{array}$ $\begin{array}{l} 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} \end{array} $

C++ generator of curves
//using namespace std; typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) define DB double
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) define M(x,y) fprintf(o,"%6.4lf %6.4lf M\n",0.+x,0.+y);
 * 5) define L(x,y) fprintf(o,"%6.4lf %6.4lf L\n",0.+x,0.+y);


 * 1) include "korifit76.cin"
 * 2) 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("norifit76plo.eps","w"); ado(o,520,220); fprintf(o,"10 10 translate 10 10 scale 2 setlinecap 1 setlinejoin\n");

for(m=0;m<51;m+=1) {M(m,0)L(m,15)} for(n=0;n<16;n+=1){M(0,n)L(50,n)} fprintf(o,"0.04 W S\n");

for(m=2;m<6;m++){ x=LN[m]/LN[1]; x*=x; M(x,0)L(x,20) } fprintf(o,"0.03 W 1 0 1 RGB S\n");

for(n=1;n<5001;n+=5){ x=.01*n; y=j0(L1*sqrt(x))/(1.-x); y*=y; z=Re(korifit76(x)); z*=z; t=y-z; u=-log(fabs(t))/log(10); printf("%6.2lf %19.16lf %19.16lf %19.16lf %8.2lf\n",x,y,z,t,u); if(n==1) M(x,u) else if(u>0 && u <20)L(x,u) } fprintf(o,"0 .7 0 RGB .1 W S\n");

for(n=1;n<5001;n+=5){ x=.01*n; y=j0(L1*sqrt(x))/(1.-x); z=Re(korifit76(x)); t=y-z; u=-log(fabs(t))/log(10); printf("%6.2lf %19.16lf %19.16lf %19.16lf %8.2lf\n",x,y,z,t,u); if(n==1) M(x,u) else if(u>0 && u <20)L(x,u) } fprintf(o,"1 0 0 RGB .1 W S\n");

for(n=1;n<5001;n+=5){ x=.01*n; y=j0(L1*sqrt(x))/(1.-x); y*=y; z=Re(korifit76(x)); z*=z; t=y-z; u=-log(fabs(t)/(fabs(y)+fabs(z)))/log(10); printf("%6.2lf %19.16lf %19.16lf %19.16lf %8.2lf\n",x,y,z,t,u); if(n==1) M(x,u) else if(u>0 && u <20)L(x,u) } fprintf(o,"0 0 0 RGB .1 W S\n");

for(n=1;n<5001;n+=5){ x=.01*n; y=j0(L1*sqrt(x))/(1.-x); if(n==1) M(x,10*y) else if(y>-1 && y <2)L(x,10*y) } fprintf(o,"1 0 0 RGB .1 W S\n");

for(n=1;n<5001;n+=5){ x=.01*n; y=j0(L1*sqrt(x))/(1.-x); y*=y; if(n==1) M(x,10*y) else if(y>-1 && y <2)L(x,10*y) } fprintf(o,"0 .6 0 RGB .1 W S\n");

fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf norifit76plo.eps"); system( "open norifit76plo.pdf"); return 0;}