static int
qags (const gsl_function * f,
const double a, const double b,
const double epsabs, const double epsrel,
const size_t limit,
gsl_integration_workspace * workspace,
double *result, double *abserr,
gsl_integration_rule * q)
{
double area, errsum;
double res_ext, err_ext;
double result0, abserr0, resabs0, resasc0;
double tolerance;
double ertest = 0;
double error_over_large_intervals = 0;
double reseps = 0, abseps = 0, correc = 0;
size_t ktmin = 0;
int roundoff_type1 = 0, roundoff_type2 = 0, roundoff_type3 = 0;
int error_type = 0, error_type2 = 0;
size_t iteration = 0;
int positive_integrand = 0;
int extrapolate = 0;
int disallow_extrapolation = 0;
struct extrapolation_table table;
/* Initialize results */
initialise (workspace, a, b);
*result = 0;
*abserr = 0;
if (limit > workspace->limit)
{
GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
}
/* Test on accuracy */
if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
{
GSL_ERROR ("tolerance cannot be acheived with given epsabs and epsrel",
GSL_EBADTOL);
}
/* Perform the first integration */
q (f, a, b, &result0, &abserr0, &resabs0, &resasc0);
set_initial_result (workspace, result0, abserr0);
tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (result0));
if (abserr0 <= 100 * GSL_DBL_EPSILON * resabs0 && abserr0 > tolerance)
{
*result = result0;
*abserr = abserr0;
GSL_ERROR ("cannot reach tolerance because of roundoff error"
"on first attempt", GSL_EROUND);
}
else if ((abserr0 <= tolerance && abserr0 != resasc0) || abserr0 == 0.0)
{
*result = result0;
*abserr = abserr0;
return GSL_SUCCESS;
}
else if (limit == 1)
{
*result = result0;
*abserr = abserr0;
GSL_ERROR ("a maximum of one iteration was insufficient", GSL_EMAXITER);
}
/* Initialization */
initialise_table (&table);
append_table (&table, result0);
area = result0;
errsum = abserr0;
res_ext = result0;
err_ext = GSL_DBL_MAX;
positive_integrand = test_positivity (result0, resabs0);
iteration = 1;
do
{
size_t current_level;
double a1, b1, a2, b2;
double a_i, b_i, r_i, e_i;
double area1 = 0, area2 = 0, area12 = 0;
double error1 = 0, error2 = 0, error12 = 0;
double resasc1, resasc2;
double resabs1, resabs2;
double last_e_i;
/* Bisect the subinterval with the largest error estimate */
retrieve (workspace, &a_i, &b_i, &r_i, &e_i);
current_level = workspace->level[workspace->i] + 1;
a1 = a_i;
b1 = 0.5 * (a_i + b_i);
a2 = b1;
b2 = b_i;
iteration++;
q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
q (f, a2, b2, &area2, &error2, &resabs2, &resasc2);
area12 = area1 + area2;
error12 = error1 + error2;
last_e_i = e_i;
/* Improve previous approximations to the integral and test for
accuracy.
We write these expressions in the same way as the original
QUADPACK code so that the rounding errors are the same, which
makes testing easier. */
errsum = errsum + error12 - e_i;
area = area + area12 - r_i;
tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (area));
if (resasc1 != error1 && resasc2 != error2)
{
double delta = r_i - area12;
if (fabs (delta) <= 1.0e-5 * fabs (area12) && error12 >= 0.99 * e_i)
{
if (!extrapolate)
{
roundoff_type1++;
}
else
{
roundoff_type2++;
}
}
if (iteration > 10 && error12 > e_i)
{
roundoff_type3++;
}
}
/* Test for roundoff and eventually set error flag */
if (roundoff_type1 + roundoff_type2 >= 10 || roundoff_type3 >= 20)
{
error_type = 2; /* round off error */
}
if (roundoff_type2 >= 5)
{
error_type2 = 1;
}
/* set error flag in the case of bad integrand behaviour at
a point of the integration range */
if (subinterval_too_small (a1, a2, b2))
{
error_type = 4;
}
/* append the newly-created intervals to the list */
update (workspace, a1, b1, area1, error1, a2, b2, area2, error2);
if (errsum <= tolerance)
{
goto compute_result;
}
if (error_type)
{
break;
}
if (iteration >= limit - 1)
{
error_type = 1;
break;
}
if (iteration == 2) /* set up variables on first iteration */
{
error_over_large_intervals = errsum;
ertest = tolerance;
append_table (&table, area);
continue;
}
if (disallow_extrapolation)
{
continue;
}
error_over_large_intervals += -last_e_i;
if (current_level < workspace->maximum_level)
{
error_over_large_intervals += error12;
}
if (!extrapolate)
{
/* test whether the interval to be bisected next is the
smallest interval. */
if (large_interval (workspace))
continue;
extrapolate = 1;
workspace->nrmax = 1;
}
if (!error_type2 && error_over_large_intervals > ertest)
{
if (increase_nrmax (workspace))
continue;
}
/* Perform extrapolation */
append_table (&table, area);
qelg (&table, &reseps, &abseps);
ktmin++;
if (ktmin > 5 && err_ext < 0.001 * errsum)
{
error_type = 5;
}
if (abseps < err_ext)
{
ktmin = 0;
err_ext = abseps;
res_ext = reseps;
correc = error_over_large_intervals;
ertest = GSL_MAX_DBL (epsabs, epsrel * fabs (reseps));
if (err_ext <= ertest)
break;
}
/* Prepare bisection of the smallest interval. */
if (table.n == 1)
{
disallow_extrapolation = 1;
}
if (error_type == 5)
{
break;
}
/* work on interval with largest error */
reset_nrmax (workspace);
extrapolate = 0;
error_over_large_intervals = errsum;
}
while (iteration < limit);
*result = res_ext;
*abserr = err_ext;
if (err_ext == GSL_DBL_MAX)
goto compute_result;
if (error_type || error_type2)
{
if (error_type2)
{
err_ext += correc;
}
if (error_type == 0)
error_type = 3;
if (res_ext != 0.0 && area != 0.0)
{
if (err_ext / fabs (res_ext) > errsum / fabs (area))
goto compute_result;
}
else if (err_ext > errsum)
{
goto compute_result;
}
else if (area == 0.0)
{
goto return_error;
}
}
/* Test on divergence. */
{
double max_area = GSL_MAX_DBL (fabs (res_ext), fabs (area));
if (!positive_integrand && max_area < 0.01 * resabs0)
goto return_error;
}
{
double ratio = res_ext / area;
if (ratio < 0.01 || ratio > 100.0 || errsum > fabs (area))
error_type = 6;
}
goto return_error;
compute_result:
*result = sum_results (workspace);
*abserr = errsum;
return_error:
if (error_type > 2)
error_type--;
if (error_type == 0)
{
return GSL_SUCCESS;
}
else if (error_type == 1)
{
GSL_ERROR ("number of iterations was insufficient", GSL_EMAXITER);
}
else if (error_type == 2)
{
GSL_ERROR ("cannot reach tolerance because of roundoff error",
GSL_EROUND);
}
else if (error_type == 3)
{
GSL_ERROR ("bad integrand behavior found in the integration interval",
GSL_ESING);
}
else if (error_type == 4)
{
GSL_ERROR ("roundoff error detected in the extrapolation table",
GSL_EROUND);
}
else if (error_type == 5)
{
GSL_ERROR ("integral is divergent, or slowly convergent",
GSL_EDIVERGE);
}
else
{
GSL_ERROR ("could not integrate function", GSL_EFAILED);
}
}
int
gsl_integration_qawf (gsl_function * f,
const double a,
const double epsabs,
const size_t limit,
gsl_integration_workspace * workspace,
gsl_integration_workspace * cycle_workspace,
gsl_integration_qawo_table * wf,
double *result, double *abserr)
{
double area, errsum;
double res_ext, err_ext;
double correc, total_error = 0.0, truncation_error;
size_t ktmin = 0;
size_t iteration = 0;
struct extrapolation_table table;
double cycle;
double omega = wf->omega;
const double p = 0.9;
double factor = 1;
double initial_eps, eps;
int error_type = 0;
/* Initialize results */
initialise (workspace, a, a);
*result = 0;
*abserr = 0;
if (limit > workspace->limit)
{
GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
}
/* Test on accuracy */
if (epsabs <= 0)
{
GSL_ERROR ("absolute tolerance epsabs must be positive", GSL_EBADTOL) ;
}
if (omega == 0.0)
{
if (wf->sine == GSL_INTEG_SINE)
{
/* The function sin(w x) f(x) is always zero for w = 0 */
*result = 0;
*abserr = 0;
return GSL_SUCCESS;
}
else
{
/* The function cos(w x) f(x) is always f(x) for w = 0 */
int status = gsl_integration_qagiu (f, a, epsabs, 0.0,
cycle_workspace->limit,
cycle_workspace,
result, abserr);
return status;
}
}
if (epsabs > GSL_DBL_MIN / (1 - p))
{
eps = epsabs * (1 - p);
}
else
{
eps = epsabs;
}
initial_eps = eps;
area = 0;
errsum = 0;
res_ext = 0;
err_ext = GSL_DBL_MAX;
correc = 0;
cycle = (2 * floor (fabs (omega)) + 1) * M_PI / fabs (omega);
gsl_integration_qawo_table_set_length (wf, cycle);
initialise_table (&table);
for (iteration = 0; iteration < limit; iteration++)
{
double area1, error1, reseps, erreps;
double a1 = a + iteration * cycle;
double b1 = a1 + cycle;
double epsabs1 = eps * factor;
int status = gsl_integration_qawo (f, a1, epsabs1, 0.0, limit,
cycle_workspace, wf,
&area1, &error1);
append_interval (workspace, a1, b1, area1, error1);
factor *= p;
area = area + area1;
errsum = errsum + error1;
/* estimate the truncation error as 50 times the final term */
truncation_error = 50 * fabs (area1);
total_error = errsum + truncation_error;
if (total_error < epsabs && iteration > 4)
{
goto compute_result;
}
if (error1 > correc)
{
correc = error1;
}
if (status)
{
eps = GSL_MAX_DBL (initial_eps, correc * (1.0 - p));
}
if (status && total_error < 10 * correc && iteration > 3)
{
goto compute_result;
}
append_table (&table, area);
if (table.n < 2)
{
continue;
}
qelg (&table, &reseps, &erreps);
ktmin++;
if (ktmin >= 15 && err_ext < 0.001 * total_error)
{
error_type = 4;
}
if (erreps < err_ext)
{
ktmin = 0;
err_ext = erreps;
res_ext = reseps;
if (err_ext + 10 * correc <= epsabs)
break;
if (err_ext <= epsabs && 10 * correc >= epsabs)
break;
}
}
if (iteration == limit)
error_type = 1;
if (err_ext == GSL_DBL_MAX)
goto compute_result;
err_ext = err_ext + 10 * correc;
*result = res_ext;
*abserr = err_ext;
if (error_type == 0)
{
return GSL_SUCCESS ;
}
if (res_ext != 0.0 && area != 0.0)
{
if (err_ext / fabs (res_ext) > errsum / fabs (area))
goto compute_result;
}
else if (err_ext > errsum)
{
goto compute_result;
}
else if (area == 0.0)
{
goto return_error;
}
if (error_type == 4)
{
err_ext = err_ext + truncation_error;
}
goto return_error;
compute_result:
*result = area;
*abserr = total_error;
return_error:
if (error_type > 2)
error_type--;
if (error_type == 0)
{
return GSL_SUCCESS;
}
else if (error_type == 1)
{
GSL_ERROR ("number of iterations was insufficient", GSL_EMAXITER);
}
else if (error_type == 2)
{
GSL_ERROR ("cannot reach tolerance because of roundoff error",
GSL_EROUND);
}
else if (error_type == 3)
{
GSL_ERROR ("bad integrand behavior found in the integration interval",
GSL_ESING);
}
else if (error_type == 4)
{
GSL_ERROR ("roundoff error detected in the extrapolation table",
GSL_EROUND);
}
else if (error_type == 5)
{
GSL_ERROR ("integral is divergent, or slowly convergent",
GSL_EDIVERGE);
}
else
{
GSL_ERROR ("could not integrate function", GSL_EFAILED);
}
}
