void
gsl_integration_qk (const int n,
const double xgk[], const double wg[], const double wgk[],
double fv1[], double fv2[],
const gsl_function * f, double a, double b,
double *result, double *abserr,
double *resabs, double *resasc)
{
const double center = 0.5 * (a + b);
const double half_length = 0.5 * (b - a);
const double abs_half_length = fabs (half_length);
const double f_center = GSL_FN_EVAL (f, center);
double result_gauss = 0;
double result_kronrod = f_center * wgk[n - 1];
double result_abs = fabs (result_kronrod);
double result_asc = 0;
double mean = 0, err = 0;
int j;
if (n % 2 == 0)
{
result_gauss = f_center * wg[n / 2 - 1];
}
for (j = 0; j < (n - 1) / 2; j++)
{
const int jtw = j * 2 + 1; /* j=1,2,3 jtw=2,4,6 */
const double abscissa = half_length * xgk[jtw];
const double fval1 = GSL_FN_EVAL (f, center - abscissa);
const double fval2 = GSL_FN_EVAL (f, center + abscissa);
const double fsum = fval1 + fval2;
fv1[jtw] = fval1;
fv2[jtw] = fval2;
result_gauss += wg[j] * fsum;
result_kronrod += wgk[jtw] * fsum;
result_abs += wgk[jtw] * (fabs (fval1) + fabs (fval2));
}
for (j = 0; j < n / 2; j++)
{
int jtwm1 = j * 2;
const double abscissa = half_length * xgk[jtwm1];
const double fval1 = GSL_FN_EVAL (f, center - abscissa);
const double fval2 = GSL_FN_EVAL (f, center + abscissa);
fv1[jtwm1] = fval1;
fv2[jtwm1] = fval2;
result_kronrod += wgk[jtwm1] * (fval1 + fval2);
result_abs += wgk[jtwm1] * (fabs (fval1) + fabs (fval2));
};
mean = result_kronrod * 0.5;
result_asc = wgk[n - 1] * fabs (f_center - mean);
for (j = 0; j < n - 1; j++)
{
result_asc += wgk[j] * (fabs (fv1[j] - mean) + fabs (fv2[j] - mean));
}
/* scale by the width of the integration region */
err = (result_kronrod - result_gauss) * half_length;
result_kronrod *= half_length;
result_abs *= abs_half_length;
result_asc *= abs_half_length;
*result = result_kronrod;
*resabs = result_abs;
*resasc = result_asc;
*abserr = rescale_error (err, result_abs, result_asc);
}
int
gsl_integration_qng (const gsl_function *f,
double a, double b,
double epsabs, double epsrel,
double * result, double * abserr, size_t * neval)
{
double fv1[5], fv2[5], fv3[5], fv4[5];
double savfun[21]; /* array of function values which have been computed */
double res10, res21, res43, res87; /* 10, 21, 43 and 87 point results */
double result_kronrod, err ;
double resabs; /* approximation to the integral of abs(f) */
double resasc; /* approximation to the integral of abs(f-i/(b-a)) */
const double half_length = 0.5 * (b - a);
const double abs_half_length = fabs (half_length);
const double center = 0.5 * (b + a);
const double f_center = GSL_FN_EVAL(f, center);
int k ;
if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
{
* result = 0;
* abserr = 0;
* neval = 0;
GSL_ERROR ("tolerance cannot be acheived with given epsabs and epsrel",
GSL_EBADTOL);
};
/* Compute the integral using the 10- and 21-point formula. */
res10 = 0;
res21 = w21b[5] * f_center;
resabs = w21b[5] * fabs (f_center);
for (k = 0; k < 5; k++)
{
const double abscissa = half_length * x1[k];
const double fval1 = GSL_FN_EVAL(f, center + abscissa);
const double fval2 = GSL_FN_EVAL(f, center - abscissa);
const double fval = fval1 + fval2;
res10 += w10[k] * fval;
res21 += w21a[k] * fval;
resabs += w21a[k] * (fabs (fval1) + fabs (fval2));
savfun[k] = fval;
fv1[k] = fval1;
fv2[k] = fval2;
}
for (k = 0; k < 5; k++)
{
const double abscissa = half_length * x2[k];
const double fval1 = GSL_FN_EVAL(f, center + abscissa);
const double fval2 = GSL_FN_EVAL(f, center - abscissa);
const double fval = fval1 + fval2;
res21 += w21b[k] * fval;
resabs += w21b[k] * (fabs (fval1) + fabs (fval2));
savfun[k + 5] = fval;
fv3[k] = fval1;
fv4[k] = fval2;
}
resabs *= abs_half_length ;
{
const double mean = 0.5 * res21;
resasc = w21b[5] * fabs (f_center - mean);
for (k = 0; k < 5; k++)
{
resasc +=
(w21a[k] * (fabs (fv1[k] - mean) + fabs (fv2[k] - mean))
+ w21b[k] * (fabs (fv3[k] - mean) + fabs (fv4[k] - mean)));
}
resasc *= abs_half_length ;
}
result_kronrod = res21 * half_length;
err = rescale_error ((res21 - res10) * half_length, resabs, resasc) ;
/* test for convergence. */
if (err < epsabs || err < epsrel * fabs (result_kronrod))
{
* result = result_kronrod ;
* abserr = err ;
* neval = 21;
return GSL_SUCCESS;
}
/* compute the integral using the 43-point formula. */
res43 = w43b[11] * f_center;
for (k = 0; k < 10; k++)
{
res43 += savfun[k] * w43a[k];
}
for (k = 0; k < 11; k++)
{
const double abscissa = half_length * x3[k];
const double fval = (GSL_FN_EVAL(f, center + abscissa)
+ GSL_FN_EVAL(f, center - abscissa));
res43 += fval * w43b[k];
savfun[k + 10] = fval;
}
/* test for convergence */
result_kronrod = res43 * half_length;
err = rescale_error ((res43 - res21) * half_length, resabs, resasc);
if (err < epsabs || err < epsrel * fabs (result_kronrod))
{
* result = result_kronrod ;
* abserr = err ;
* neval = 43;
return GSL_SUCCESS;
}
/* compute the integral using the 87-point formula. */
res87 = w87b[22] * f_center;
for (k = 0; k < 21; k++)
{
res87 += savfun[k] * w87a[k];
}
for (k = 0; k < 22; k++)
{
const double abscissa = half_length * x4[k];
res87 += w87b[k] * (GSL_FN_EVAL(f, center + abscissa)
+ GSL_FN_EVAL(f, center - abscissa));
}
/* test for convergence */
result_kronrod = res87 * half_length ;
err = rescale_error ((res87 - res43) * half_length, resabs, resasc);
if (err < epsabs || err < epsrel * fabs (result_kronrod))
{
* result = result_kronrod ;
* abserr = err ;
* neval = 87;
return GSL_SUCCESS;
}
/* failed to converge */
* result = result_kronrod ;
* abserr = err ;
* neval = 87;
GSL_ERROR("failed to reach tolerance with highest-order rule", GSL_ETOL) ;
}
