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/* integration/qagp.c
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*
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* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Brian Gough
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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#include <config.h>
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#include <stdlib.h>
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#include <gsl/gsl_errno.h>
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#include <gsl/gsl_integration.h>
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static int
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qagp (const gsl_function *f,
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const double *pts, const size_t npts,
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const double epsabs, const double epsrel, const size_t limit,
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gsl_integration_workspace * workspace,
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double *result, double *abserr,
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gsl_integration_rule * q);
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#include "initialise.c"
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#include "qpsrt.c"
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#include "util.c"
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#include "append.c"
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#include "reset.c"
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#include "qelg.c"
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#include "qpsrt2.c"
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#include "ptsort.c"
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#include "positivity.c"
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int
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gsl_integration_qagp (const gsl_function *f,
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double * pts, size_t npts,
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double epsabs, double epsrel, size_t limit,
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gsl_integration_workspace * workspace,
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double * result, double * abserr)
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{
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int status = qagp (f, pts, npts,
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epsabs, epsrel, limit,
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workspace,
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result, abserr,
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&gsl_integration_qk21) ;
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return status ;
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}
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static int
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qagp (const gsl_function * f,
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const double *pts, const size_t npts,
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const double epsabs, const double epsrel,
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const size_t limit,
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gsl_integration_workspace * workspace,
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double *result, double *abserr,
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gsl_integration_rule * q)
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{
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double area, errsum;
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double res_ext, err_ext;
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double result0, abserr0, resabs0;
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double tolerance;
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double ertest = 0;
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double error_over_large_intervals = 0;
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double reseps = 0, abseps = 0, correc = 0;
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size_t ktmin = 0;
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int roundoff_type1 = 0, roundoff_type2 = 0, roundoff_type3 = 0;
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int error_type = 0, error_type2 = 0;
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size_t iteration = 0;
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int positive_integrand = 0;
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int extrapolate = 0;
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int disallow_extrapolation = 0;
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struct extrapolation_table table;
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const size_t nint = npts - 1; /* number of intervals */
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size_t *ndin = workspace->level; /* temporarily alias ndin to level */
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size_t i;
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/* Initialize results */
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*result = 0;
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*abserr = 0;
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/* Test on validity of parameters */
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if (limit > workspace->limit)
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{
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GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
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}
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if (npts > workspace->limit)
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{
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GSL_ERROR ("npts exceeds size of workspace", GSL_EINVAL);
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}
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if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
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{
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GSL_ERROR ("tolerance cannot be achieved with given epsabs and epsrel",
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GSL_EBADTOL);
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}
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/* Check that the integration range and break points are an
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ascending sequence */
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for (i = 0; i < nint; i++)
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{
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if (pts[i + 1] < pts[i])
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{
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GSL_ERROR ("points are not in an ascending sequence", GSL_EINVAL);
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}
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}
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/* Perform the first integration */
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result0 = 0;
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abserr0 = 0;
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resabs0 = 0;
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initialise (workspace, 0.0, 0.0) ;
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for (i = 0; i < nint; i++)
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{
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double area1, error1, resabs1, resasc1;
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const double a1 = pts[i];
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const double b1 = pts[i + 1];
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q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
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result0 = result0 + area1;
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abserr0 = abserr0 + error1;
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resabs0 = resabs0 + resabs1;
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append_interval (workspace, a1, b1, area1, error1);
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if (error1 == resasc1 && error1 != 0.0)
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{
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ndin[i] = 1;
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}
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else
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{
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ndin[i] = 0;
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}
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}
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/* Compute the initial error estimate */
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errsum = 0.0;
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for (i = 0; i < nint; i++)
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{
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if (ndin[i])
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{
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workspace->elist[i] = abserr0;
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}
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errsum = errsum + workspace->elist[i];
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}
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for (i = 0; i < nint; i++)
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{
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workspace->level[i] = 0;
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}
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/* Sort results into order of decreasing error via the indirection
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array order[] */
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sort_results (workspace);
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/* Test on accuracy */
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tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (result0));
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if (abserr0 <= 100 * GSL_DBL_EPSILON * resabs0 && abserr0 > tolerance)
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{
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*result = result0;
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*abserr = abserr0;
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GSL_ERROR ("cannot reach tolerance because of roundoff error"
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"on first attempt", GSL_EROUND);
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}
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else if (abserr0 <= tolerance)
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{
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*result = result0;
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*abserr = abserr0;
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return GSL_SUCCESS;
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}
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else if (limit == 1)
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{
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*result = result0;
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*abserr = abserr0;
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GSL_ERROR ("a maximum of one iteration was insufficient", GSL_EMAXITER);
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}
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/* Initialization */
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initialise_table (&table);
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append_table (&table, result0);
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area = result0;
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res_ext = result0;
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err_ext = GSL_DBL_MAX;
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error_over_large_intervals = errsum;
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ertest = tolerance;
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positive_integrand = test_positivity (result0, resabs0);
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iteration = nint - 1;
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do
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{
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size_t current_level;
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double a1, b1, a2, b2;
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double a_i, b_i, r_i, e_i;
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double area1 = 0, area2 = 0, area12 = 0;
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double error1 = 0, error2 = 0, error12 = 0;
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double resasc1, resasc2;
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double resabs1, resabs2;
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double last_e_i;
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/* Bisect the subinterval with the largest error estimate */
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retrieve (workspace, &a_i, &b_i, &r_i, &e_i);
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current_level = workspace->level[workspace->i] + 1;
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a1 = a_i;
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b1 = 0.5 * (a_i + b_i);
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a2 = b1;
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b2 = b_i;
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iteration++;
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q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
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q (f, a2, b2, &area2, &error2, &resabs2, &resasc2);
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area12 = area1 + area2;
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error12 = error1 + error2;
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last_e_i = e_i;
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/* Improve previous approximations to the integral and test for
|
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accuracy.
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We write these expressions in the same way as the original
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QUADPACK code so that the rounding errors are the same, which
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makes testing easier. */
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errsum = errsum + error12 - e_i;
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area = area + area12 - r_i;
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tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (area));
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|
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if (resasc1 != error1 && resasc2 != error2)
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{
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double delta = r_i - area12;
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if (fabs (delta) <= 1.0e-5 * fabs (area12) && error12 >= 0.99 * e_i)
|
|
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{
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if (!extrapolate)
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{
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roundoff_type1++;
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}
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else
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{
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roundoff_type2++;
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}
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}
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if (i > 10 && error12 > e_i)
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{
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roundoff_type3++;
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}
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|
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}
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|
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|
|
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/* Test for roundoff and eventually set error flag */
|
|
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|
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if (roundoff_type1 + roundoff_type2 >= 10 || roundoff_type3 >= 20)
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|
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{
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error_type = 2; /* round off error */
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}
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if (roundoff_type2 >= 5)
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|
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{
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|
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error_type2 = 1;
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}
|
|
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|
|
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/* set error flag in the case of bad integrand behaviour at
|
|
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a point of the integration range */
|
|
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67cb25 |
|
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if (subinterval_too_small (a1, a2, b2))
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{
|
|
Packit |
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error_type = 4;
|
|
Packit |
67cb25 |
}
|
|
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67cb25 |
|
|
Packit |
67cb25 |
/* append the newly-created intervals to the list */
|
|
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67cb25 |
|
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update (workspace, a1, b1, area1, error1, a2, b2, area2, error2);
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|
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67cb25 |
|
|
Packit |
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if (errsum <= tolerance)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
goto compute_result;
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|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
break;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (iteration >= limit - 1)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
error_type = 1;
|
|
Packit |
67cb25 |
break;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (disallow_extrapolation)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
continue;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
error_over_large_intervals += -last_e_i;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (current_level < workspace->maximum_level)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
error_over_large_intervals += error12;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (!extrapolate)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
/* test whether the interval to be bisected next is the
|
|
Packit |
67cb25 |
smallest interval. */
|
|
Packit |
67cb25 |
if (large_interval (workspace))
|
|
Packit |
67cb25 |
continue;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
extrapolate = 1;
|
|
Packit |
67cb25 |
workspace->nrmax = 1;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* The smallest interval has the largest error. Before
|
|
Packit |
67cb25 |
bisecting decrease the sum of the errors over the larger
|
|
Packit |
67cb25 |
intervals (error_over_large_intervals) and perform
|
|
Packit |
67cb25 |
extrapolation. */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (!error_type2 && error_over_large_intervals > ertest)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
if (increase_nrmax (workspace))
|
|
Packit |
67cb25 |
continue;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* Perform extrapolation */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
append_table (&table, area);
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (table.n < 3)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
goto skip_extrapolation;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
qelg (&table, &reseps, &abseps);
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
ktmin++;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (ktmin > 5 && err_ext < 0.001 * errsum)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
error_type = 5;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (abseps < err_ext)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
ktmin = 0;
|
|
Packit |
67cb25 |
err_ext = abseps;
|
|
Packit |
67cb25 |
res_ext = reseps;
|
|
Packit |
67cb25 |
correc = error_over_large_intervals;
|
|
Packit |
67cb25 |
ertest = GSL_MAX_DBL (epsabs, epsrel * fabs (reseps));
|
|
Packit |
67cb25 |
if (err_ext <= ertest)
|
|
Packit |
67cb25 |
break;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* Prepare bisection of the smallest interval. */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (table.n == 1)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
disallow_extrapolation = 1;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type == 5)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
break;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
skip_extrapolation:
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
reset_nrmax (workspace);
|
|
Packit |
67cb25 |
extrapolate = 0;
|
|
Packit |
67cb25 |
error_over_large_intervals = errsum;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
while (iteration < limit);
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
*result = res_ext;
|
|
Packit |
67cb25 |
*abserr = err_ext;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (err_ext == GSL_DBL_MAX)
|
|
Packit |
67cb25 |
goto compute_result;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type || error_type2)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
if (error_type2)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
err_ext += correc;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type == 0)
|
|
Packit |
67cb25 |
error_type = 3;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (result != 0 && area != 0)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
if (err_ext / fabs (res_ext) > errsum / fabs (area))
|
|
Packit |
67cb25 |
goto compute_result;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (err_ext > errsum)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
goto compute_result;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (area == 0.0)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
goto return_error;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* Test on divergence. */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
double max_area = GSL_MAX_DBL (fabs (res_ext), fabs (area));
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (!positive_integrand && max_area < 0.01 * resabs0)
|
|
Packit |
67cb25 |
goto return_error;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
double ratio = res_ext / area;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (ratio < 0.01 || ratio > 100 || errsum > fabs (area))
|
|
Packit |
67cb25 |
error_type = 6;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
goto return_error;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
compute_result:
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
*result = sum_results (workspace);
|
|
Packit |
67cb25 |
*abserr = errsum;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
return_error:
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type > 2)
|
|
Packit |
67cb25 |
error_type--;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if (error_type == 0)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (error_type == 1)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("number of iterations was insufficient", GSL_EMAXITER);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (error_type == 2)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("cannot reach tolerance because of roundoff error",
|
|
Packit |
67cb25 |
GSL_EROUND);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (error_type == 3)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("bad integrand behavior found in the integration interval",
|
|
Packit |
67cb25 |
GSL_ESING);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (error_type == 4)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("roundoff error detected in the extrapolation table",
|
|
Packit |
67cb25 |
GSL_EROUND);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if (error_type == 5)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("integral is divergent, or slowly convergent",
|
|
Packit |
67cb25 |
GSL_EDIVERGE);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
GSL_ERROR ("could not integrate function", GSL_EFAILED);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|