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cquad.c

/* integration/cquad.c
 * 
 * Copyright (C) 2010 Pedro Gonnet
 * 
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 3 of the License, or (at
 * your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * General Public License for more details.
 * 
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 */

#include <config.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_integration.h>

#include "cquad_const.c"


/* Allocates a workspace for the given maximum number of intervals.
    Note that if the workspace gets filled, the intervals with the
    lowest error estimates are dropped. The maximum number of
    intervals is therefore not the maximum number of intervals
    that will be computed, but merely the size of the buffer.
    */

gsl_integration_cquad_workspace *
gsl_integration_cquad_workspace_alloc (const size_t n)
{

  gsl_integration_cquad_workspace *w;

  /* Check inputs */
  if (n < 3)
    GSL_ERROR_VAL ("workspace size n must be at least 3", GSL_EDOM, 0);

  /* Allocate first the workspace struct */
  if ((w =
       (gsl_integration_cquad_workspace *)
       malloc (sizeof (gsl_integration_cquad_workspace))) == NULL)
    GSL_ERROR_VAL ("failed to allocate space for workspace struct",
               GSL_ENOMEM, 0);

  /* Allocate the intervals */
  if ((w->ivals =
       (gsl_integration_cquad_ival *)
       malloc (sizeof (gsl_integration_cquad_ival) * n)) == NULL)
    {
      free (w);
      GSL_ERROR_VAL ("failed to allocate space for the intervals", GSL_ENOMEM,
                 0);
    }

  /* Allocate the max-heap indices */
  if ((w->heap = (size_t *) malloc (sizeof (size_t) * n)) == NULL)
    {
      free (w->ivals);
      free (w);
      GSL_ERROR_VAL ("failed to allocate space for the heap", GSL_ENOMEM, 0);
    }

  /* Remember the size of the workspace */
  w->size = n;

  /* Return the result */
  return w;

}


/* Liberates the workspace memory.
    */

void
gsl_integration_cquad_workspace_free (gsl_integration_cquad_workspace * w)
{

  /* Nothing to be done? */
  if (w == NULL)
    return;

  /* Free the intervals first */
  if (w->ivals != NULL)
    free (w->ivals);

  /* Free the heap */
  if (w->heap != NULL)
    free (w->heap);

  /* Free the structure */
  free (w);

}


/* Compute the product of the fx with one of the inverse
    Vandermonde-like matrices. */

static void
Vinvfx (const double *fx, double *c, const int d)
{

  int i, j;

  switch (d)
    {
    case 0:
      for (i = 0; i <= 4; i++)
      {
        c[i] = 0.0;
        for (j = 0; j <= 4; j++)
          c[i] += V1inv[i * 5 + j] * fx[j * 8];
      }
      break;
    case 1:
      for (i = 0; i <= 8; i++)
      {
        c[i] = 0.0;
        for (j = 0; j <= 8; j++)
          c[i] += V2inv[i * 9 + j] * fx[j * 4];
      }
      break;
    case 2:
      for (i = 0; i <= 16; i++)
      {
        c[i] = 0.0;
        for (j = 0; j <= 16; j++)
          c[i] += V3inv[i * 17 + j] * fx[j * 2];
      }
      break;
    case 3:
      for (i = 0; i <= 32; i++)
      {
        c[i] = 0.0;
        for (j = 0; j <= 32; j++)
          c[i] += V4inv[i * 33 + j] * fx[j];
      }
      break;
    }

}


/* Downdate the interpolation given by the n coefficients c
    by removing the nodes with indices in nans. */

static void
downdate (double *c, int n, int d, int *nans, int nnans)
{

  static const int bidx[4] = { 0, 6, 16, 34 };
  double b_new[34], alpha;
  int i, j;

  for (i = 0; i <= n + 1; i++)
    b_new[i] = bee[bidx[d] + i];
  for (i = 0; i < nnans; i++)
    {
      b_new[n + 1] = b_new[n + 1] / Lalpha[n];
      b_new[n] = (b_new[n] + xi[nans[i]] * b_new[n + 1]) / Lalpha[n - 1];
      for (j = n - 1; j > 0; j--)
      b_new[j] =
        (b_new[j] + xi[nans[i]] * b_new[j + 1] -
         Lgamma[j + 1] * b_new[j + 2]) / Lalpha[j - 1];
      for (j = 0; j <= n; j++)
      b_new[j] = b_new[j + 1];
      alpha = c[n] / b_new[n];
      for (j = 0; j < n; j++)
      c[j] -= alpha * b_new[j];
      c[n] = 0;
      n--;
    }

}


/* The actual integration routine.
    */

int
gsl_integration_cquad (const gsl_function * f, double a, double b,
                   double epsabs, double epsrel,
                   gsl_integration_cquad_workspace * ws,
                   double *result, double *abserr, size_t * nevals)
{

  /* Some constants that we will need. */
  static const int n[4] = { 4, 8, 16, 32 };
  static const int skip[4] = { 8, 4, 2, 1 };
  static const int idx[4] = { 0, 5, 14, 31 };
  static const double w = M_SQRT2 / 2;
  static const int ndiv_max = 20;

  /* Actual variables (as opposed to constants above). */
  double m, h, temp;
  double igral, err, igral_final, err_final, err_excess;
  int nivals, neval = 0;
  int i, j, d, split, t;
  int nnans, nans[32];
  gsl_integration_cquad_ival *iv, *ivl, *ivr;
  double nc, ncdiff;

  /* Check the input arguments. */
  if (f == NULL)
    GSL_ERROR ("function pointer shouldn't be NULL", GSL_EINVAL);
  if (result == NULL)
    GSL_ERROR ("result pointer shouldn't be NULL", GSL_EINVAL);
  if (ws == NULL)
    GSL_ERROR ("workspace pointer shouldn't be NULL", GSL_EINVAL);


  /* Check for unreasonable accuracy demands */
  if (epsabs < 0.0 || epsrel < 0.0)
    GSL_ERROR ("tolerances may not be negative", GSL_EBADTOL);
  if (epsabs <= 0 && epsrel < GSL_DBL_EPSILON)
    GSL_ERROR ("unreasonable accuracy requirement", GSL_EBADTOL);


  /* Create the first interval. */
  iv = &(ws->ivals[0]);
  m = (a + b) / 2;
  h = (b - a) / 2;
  nnans = 0;
  for (i = 0; i <= n[3]; i++)
    {
      iv->fx[i] = GSL_FN_EVAL (f, m + xi[i] * h);
      neval++;
      if (!finite (iv->fx[i]))
      {
        nans[nnans++] = i;
        iv->fx[i] = 0.0;
      }
    }
  Vinvfx (iv->fx, &(iv->c[idx[0]]), 0);
  Vinvfx (iv->fx, &(iv->c[idx[3]]), 3);
  Vinvfx (iv->fx, &(iv->c[idx[2]]), 2);
  for (i = 0; i < nnans; i++)
    iv->fx[i] = GSL_NAN;
  iv->a = a;
  iv->b = b;
  iv->depth = 3;
  iv->rdepth = 1;
  iv->ndiv = 0;
  iv->igral = 2 * h * iv->c[idx[3]] * w;
  nc = 0.0;
  for (i = n[2] + 1; i <= n[3]; i++)
    {
      temp = iv->c[idx[3] + i];
      nc += temp * temp;
    }
  ncdiff = nc;
  for (i = 0; i <= n[2]; i++)
    {
      temp = iv->c[idx[2] + i] - iv->c[idx[3] + i];
      ncdiff += temp * temp;
      nc += iv->c[idx[3] + i] * iv->c[idx[3] + i];
    }
  ncdiff = sqrt (ncdiff);
  nc = sqrt (nc);
  iv->err = ncdiff * 2 * h;
  if (ncdiff / nc > 0.1 && iv->err < 2 * h * nc)
    iv->err = 2 * h * nc;


  /* Initialize the heaps. */
  for (i = 0; i < ws->size; i++)
    ws->heap[i] = i;


  /* Initialize some global values. */
  igral = iv->igral;
  err = iv->err;
  nivals = 1;
  igral_final = 0.0;
  err_final = 0.0;
  err_excess = 0.0;


  /* Main loop. */
  while (nivals > 0 && err > 0.0 &&
       !(err <= fabs (igral) * epsrel || err <= epsabs)
       && !(err_final > fabs (igral) * epsrel
            && err - err_final < fabs (igral) * epsrel)
       && !(err_final > epsabs && err - err_final < epsabs))
    {

      /* Put our finger on the interval with the largest error. */
      iv = &(ws->ivals[ws->heap[0]]);
      m = (iv->a + iv->b) / 2;
      h = (iv->b - iv->a) / 2;

/*      printf
        ("cquad: processing ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
         ws->heap[0], nivals, iv->a, iv->b, iv->igral, iv->err, iv->depth);
*/
      /* Should we try to increase the degree? */
      if (iv->depth < 3)
      {

        /* Keep tabs on some variables. */
        d = ++iv->depth;

        /* Get the new (missing) function values */
        for (i = skip[d]; i <= 32; i += 2 * skip[d])
          {
            iv->fx[i] = GSL_FN_EVAL (f, m + xi[i] * h);
            neval++;
          }
        nnans = 0;
        for (i = 0; i <= 32; i += skip[d])
          {
            if (!finite (iv->fx[i]))
            {
              nans[nnans++] = i;
              iv->fx[i] = 0.0;
            }
          }

        /* Compute the new coefficients. */
        Vinvfx (iv->fx, &(iv->c[idx[d]]), d);

        /* Downdate any NaNs. */
        if (nnans > 0)
          {
            downdate (&(iv->c[idx[d]]), n[d], d, nans, nnans);
            for (i = 0; i < nnans; i++)
            iv->fx[i] = GSL_NAN;
          }

        /* Compute the error estimate. */
        nc = 0.0;
        for (i = n[d - 1] + 1; i <= n[d]; i++)
          {
            temp = iv->c[idx[d] + i];
            nc += temp * temp;
          }
        ncdiff = nc;
        for (i = 0; i <= n[d - 1]; i++)
          {
            temp = iv->c[idx[d - 1] + i] - iv->c[idx[d] + i];
            ncdiff += temp * temp;
            nc += iv->c[idx[d] + i] * iv->c[idx[d] + i];
          }
        ncdiff = sqrt (ncdiff);
        nc = sqrt (nc);
        iv->err = ncdiff * 2 * h;

        /* Compute the local integral. */
        iv->igral = 2 * h * w * iv->c[idx[d]];

        /* Split the interval prematurely? */
        split = (nc > 0 && ncdiff / nc > 0.1);

      }

      /* Maximum degree reached, just split. */
      else
      {
        split = 1;
      }


      /* Should we drop this interval? */
      if ((m + h * xi[0]) >= (m + h * xi[1])
        || (m + h * xi[31]) >= (m + h * xi[32])
        || iv->err < fabs (iv->igral) * GSL_DBL_EPSILON * 10)
      {

/*          printf
            ("cquad: dumping ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
             ws->heap[0], nivals, iv->a, iv->b, iv->igral, iv->err,
             iv->depth);
*/
        /* Keep this interval's contribution */
        err_final += iv->err;
        igral_final += iv->igral;

        /* Swap with the last element on the heap */
        t = ws->heap[nivals - 1];
        ws->heap[nivals - 1] = ws->heap[0];
        ws->heap[0] = t;
        nivals--;

        /* Fix up the heap */
        i = 0;
        while (2 * i + 1 < nivals)
          {

            /* Get the kids */
            j = 2 * i + 1;

            /* If the j+1st entry exists and is larger than the jth,
               use it instead. */
            if (j + 1 < nivals
              && ws->ivals[ws->heap[j + 1]].err >=
              ws->ivals[ws->heap[j]].err)
            j++;

            /* Do we need to move the ith entry up? */
            if (ws->ivals[ws->heap[j]].err <= ws->ivals[ws->heap[i]].err)
            break;
            else
            {
              t = ws->heap[j];
              ws->heap[j] = ws->heap[i];
              ws->heap[i] = t;
              i = j;
            }
          }

      }

      /* Do we need to split this interval? */
      else if (split)
      {

        /* Some values we will need often... */
        d = iv->depth;

        /* Generate the interval on the left */
        ivl = &(ws->ivals[ws->heap[nivals++]]);
        ivl->a = iv->a;
        ivl->b = m;
        ivl->depth = 0;
        ivl->rdepth = iv->rdepth + 1;
        ivl->fx[0] = iv->fx[0];
        ivl->fx[32] = iv->fx[16];
        for (i = skip[0]; i < 32; i += skip[0])
          {
            ivl->fx[i] =
            GSL_FN_EVAL (f, (ivl->a + ivl->b) / 2 + xi[i] * h / 2);
            neval++;
          }
        nnans = 0;
        for (i = 0; i <= 32; i += skip[0])
          {
            if (!finite (ivl->fx[i]))
            {
              nans[nnans++] = i;
              ivl->fx[i] = 0.0;
            }
          }
        Vinvfx (ivl->fx, ivl->c, 0);
        if (nnans > 0)
          {
            downdate (ivl->c, n[0], 0, nans, nnans);
            for (i = 0; i < nnans; i++)
            ivl->fx[i] = GSL_NAN;
          }
        for (i = 0; i <= n[d]; i++)
          {
            ivl->c[idx[d] + i] = 0.0;
            for (j = i; j <= n[d]; j++)
            ivl->c[idx[d] + i] += Tleft[i * 33 + j] * iv->c[idx[d] + j];
          }
        ncdiff = 0.0;
        for (i = 0; i <= n[0]; i++)
          {
            temp = ivl->c[i] - ivl->c[idx[d] + i];
            ncdiff += temp * temp;
          }
        for (i = n[0] + 1; i <= n[d]; i++)
          {
            temp = ivl->c[idx[d] + i];
            ncdiff += temp * temp;
          }
        ncdiff = sqrt (ncdiff);
        ivl->err = ncdiff * h;

        /* Check for divergence. */
        ivl->ndiv = iv->ndiv + (fabs (iv->c[0]) > 0
                          && ivl->c[0] / iv->c[0] > 2);
        if (ivl->ndiv > ndiv_max && 2 * ivl->ndiv > ivl->rdepth)
          {
              /* need copysign(INFINITY, igral) */
            *result = (igral >= 0) ? GSL_POSINF : GSL_NEGINF;  
            if (nevals != NULL)
            *nevals = neval;
            return GSL_EDIVERGE;
          }

        /* Compute the local integral. */
        ivl->igral = h * w * ivl->c[0];


        /* Generate the interval on the right */
        ivr = &(ws->ivals[ws->heap[nivals++]]);
        ivr->a = m;
        ivr->b = iv->b;
        ivr->depth = 0;
        ivr->rdepth = iv->rdepth + 1;
        ivr->fx[0] = iv->fx[16];
        ivr->fx[32] = iv->fx[32];
        for (i = skip[0]; i < 32; i += skip[0])
          {
            ivr->fx[i] =
            GSL_FN_EVAL (f, (ivr->a + ivr->b) / 2 + xi[i] * h / 2);
            neval++;
          }
        nnans = 0;
        for (i = 0; i <= 32; i += skip[0])
          {
            if (!finite (ivr->fx[i]))
            {
              nans[nnans++] = i;
              ivr->fx[i] = 0.0;
            }
          }
        Vinvfx (ivr->fx, ivr->c, 0);
        if (nnans > 0)
          {
            downdate (ivr->c, n[0], 0, nans, nnans);
            for (i = 0; i < nnans; i++)
            ivr->fx[i] = GSL_NAN;
          }
        for (i = 0; i <= n[d]; i++)
          {
            ivr->c[idx[d] + i] = 0.0;
            for (j = i; j <= n[d]; j++)
            ivr->c[idx[d] + i] += Tright[i * 33 + j] * iv->c[idx[d] + j];
          }
        ncdiff = 0.0;
        for (i = 0; i <= n[0]; i++)
          {
            temp = ivr->c[i] - ivr->c[idx[d] + i];
            ncdiff += temp * temp;
          }
        for (i = n[0] + 1; i <= n[d]; i++)
          {
            temp = ivr->c[idx[d] + i];
            ncdiff += temp * temp;
          }
        ncdiff = sqrt (ncdiff);
        ivr->err = ncdiff * h;

        /* Check for divergence. */
        ivr->ndiv = iv->ndiv + (fabs (iv->c[0]) > 0
                          && ivr->c[0] / iv->c[0] > 2);
        if (ivr->ndiv > ndiv_max && 2 * ivr->ndiv > ivr->rdepth)
          {
              /* need copysign(INFINITY, igral) */
            *result = (igral >= 0) ? GSL_POSINF : GSL_NEGINF;  
            if (nevals != NULL)
            *nevals = neval;
            return GSL_EDIVERGE;
          }

        /* Compute the local integral. */
        ivr->igral = h * w * ivr->c[0];


        /* Fix-up the heap: we now have one interval on top
           that we don't need any more and two new, unsorted
           ones at the bottom. */

        /* Flip the last interval to the top of the heap and
           sift down. */
        t = ws->heap[nivals - 1];
        ws->heap[nivals - 1] = ws->heap[0];
        ws->heap[0] = t;
        nivals--;

        /* Sift this interval back down the heap. */
        i = 0;
        while (2 * i + 1 < nivals - 1)
          {
            j = 2 * i + 1;
            if (j + 1 < nivals - 1
              && ws->ivals[ws->heap[j + 1]].err >=
              ws->ivals[ws->heap[j]].err)
            j++;
            if (ws->ivals[ws->heap[j]].err <= ws->ivals[ws->heap[i]].err)
            break;
            else
            {
              t = ws->heap[j];
              ws->heap[j] = ws->heap[i];
              ws->heap[i] = t;
              i = j;
            }
          }

        /* Now grab the last interval and sift it up the heap. */
        i = nivals - 1;
        while (i > 0)
          {
            j = (i - 1) / 2;
            if (ws->ivals[ws->heap[j]].err < ws->ivals[ws->heap[i]].err)
            {
              t = ws->heap[j];
              ws->heap[j] = ws->heap[i];
              ws->heap[i] = t;
              i = j;
            }
            else
            break;
          }


      }

      /* Otherwise, just fix-up the heap. */
      else
      {
        i = 0;
        while (2 * i + 1 < nivals)
          {
            j = 2 * i + 1;
            if (j + 1 < nivals
              && ws->ivals[ws->heap[j + 1]].err >=
              ws->ivals[ws->heap[j]].err)
            j++;
            if (ws->ivals[ws->heap[j]].err <= ws->ivals[ws->heap[i]].err)
            break;
            else
            {
              t = ws->heap[j];
              ws->heap[j] = ws->heap[i];
              ws->heap[i] = t;
              i = j;
            }
          }

      }

      /* If the heap is about to overflow, remove the last two
         intervals. */
      while (nivals > ws->size - 2)
      {
        iv = &(ws->ivals[ws->heap[nivals - 1]]);

/*          printf
            ("cquad: dumping ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
             ws->heap[0], nivals, iv->a, iv->b, iv->igral, iv->err,
             iv->depth);
*/
        err_final += iv->err;
        igral_final += iv->igral;
        nivals--;
      }

      /* Collect the value of the integral and error. */
      igral = igral_final;
      err = err_final;
      for (i = 0; i < nivals; i++)
      {
        igral += ws->ivals[ws->heap[i]].igral;
        err += ws->ivals[ws->heap[i]].err;
      }

    }

  /* Dump the contents of the heap. */
/*  for (i = 0; i < nivals; i++)
    {
      iv = &(ws->ivals[ws->heap[i]]);
      printf
        ("cquad: ival %i (%i) with [%e,%e], int=%e, err=%e, depth=%i, rdepth=%i\n",
         i, ws->heap[i], iv->a, iv->b, iv->igral, iv->err, iv->depth,
         iv->rdepth);
    }
*/
  /* Clean up and present the results. */
  *result = igral;
  if (abserr != NULL)
    *abserr = err;
  if (nevals != NULL)
    *nevals = neval;

  /* All is well that ends well. */
  return GSL_SUCCESS;

}

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