GLxw2048.inc

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// GLxw2048.inc

s text file, containing nodes and weights of the Gauss-Legendre (or Legenrde-Gauss) quadrature formula [1][2][3] with 2048 nodes.

Below, the file is provided and its generator. The update do another number of points is straightforward. For moderate number of nodes (maximum 32), the Efunda online service can be used [4].

GLxw2048.inc can be included "as is" to the C++ code for the numerical implementation of integration. This file is used in the generators of figures for the Cauchi integral implementation of the tetration to complex base and to base $b>\exp(1/\mathrm{e})$, and, in particular to $b=\mathrm e\approx 2.71$ described in Mathematics of Computation [5].

GLxw2048.inc

// The content of this section should be saved as GLxw2048.inc
// or (preferably) generated with the code from the next section

int NPO=2048;
DB GLx[2048]={
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};

GLxw2048.cc

// The code below generates the text in the previous section.
// If you plan to use some codes from TORI, the generation of the nodes and weights // with the code below is better, than copypast from the previous section.

#include <stdio.h>
#include <math.h>
#include<stdlib.h>
#define DB double
#define EPS 1.0e-18
// Be careful: numeration begins with UNITY !!!!
void gauleg(long double x1, long double x2, long double x[], long double w[],  int n)
{ int m,j,i; long double z1,z,xm,xl,pp,p3,p2,p1;
 m=(n+1)/2;
 xm=0.5*(x2+x1);
 xl=0.5*(x2-x1);
 for(i=1;i<=m;i++)
 { z=cos(M_PI*(i-0.25)/(n+0.5));
   do{	p1=1.0;
	p2=0.0;
	for(j=1;j<=n;j++) 
	{ p3=p2; p2=p1;
	  p1=((2.*j-1.)*z*p2-(j-1.)*p3)/j;
	}
	pp=n*(z*p1-p2)/(z*z-1.0);
	z1=z;
	z=z1-p1/pp;
      } while (fabs(z-z1) > EPS);
    x[i]=xm-xl*z;
    x[n+1-i]=xm+xl*z;
    w[i]=2.0*xl/((1.0-z*z)*pp*pp);
    w[n+1-i]=w[i];
  }
}
#undef EPS
//#include "gauleg.cin"
#define NPOINT 2048
#define X1 -1.0
#define X2 1.0
int main(void){ int i; long double xx=0.0; long double *x,*w;
x=(long double *)malloc((size_t)(NPOINT+1)*sizeof(long double));
w=(long double *)malloc((size_t)(NPOINT+1)*sizeof(long double));
gauleg(X1,X2,x,w,NPOINT);
printf("\n%2s %10s %12s\n","#","x[i]","w[i]");
for (i=1;i<=NPOINT;i++) {	printf("%4d ",i);
				printf("%18.16Lf ", x[i]);
				printf("%18.16Lf\n",w[i]);
			}
FILE *o; o=fopen("GLxw2048.inc","w");
fprintf(o,"int NPO=%3d;\n",NPOINT);
fprintf(o,"DB GLx[%3d]={\n",NPOINT);
for (i=1;i<=NPOINT;i++)
{	if(i>1)	fprintf(o,","); else fprintf(o," ");
	fprintf(o,"%24.21Lf\n",x[i]);}
fprintf(o,"};\n");
fprintf(o,"DB GLw[%3d]={\n",NPOINT);
for (i=1;i<=NPOINT;i++)
{	if(i>1)	fprintf(o,","); else fprintf(o," ");
	fprintf(o,"%24.22Lf\n",w[i]);}
fprintf(o,"};\n");
fclose(o);
//small test:
	for (i=1;i<=NPOINT;i++)
		xx += (w[i]*cos(x[i]));
	printf("from GAULEG: %32.30Lf\n",xx);
	printf("should be  : %32.30f\n",	2*sin(1.));
	printf("diff       : % 9.3Le\n",	xx-2*sin(1.));
	printf("failure in last diguits? Why are not they different?\n");
       free((char*)(w));
       free((char*)(x));
	return 0;
}

Keywords

Integration, Tetration, C++

References

  1. http://en.citizendium.org/wiki/Legendre-Gauss_Quadrature_formula
  2. http://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Legendre_quadrature
  3. http://mathworld.wolfram.com/Legendre-GaussQuadrature.html
  4. http://www.efunda.com/math/num_integration/findgausslegendre.cfm
  5. http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex plane.. Mathematics of Computation, 78: 1647-1670.