quadcc.cc

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/*

Copyright (C) 2010-2015 Pedro Gonnet

This file is part of Octave.

Octave 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.

Octave 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 Octave; see the file COPYING.  If not, see
<http://www.gnu.org/licenses/>.

*/

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include "lo-ieee.h"
#include "oct-locbuf.h"

#include "defun.h"
#include "error.h"
#include "oct-obj.h"
#include "parse.h"
#include "utils.h"
#include "variables.h"

/* Extended debugging */
#define DEBUG_QUADCC 0

/* Define the minimum size of the interval heap. */
#define min_cquad_heapsize  200


/* Data of a single interval */
typedef struct
{
  double a, b;
  double c[64];
  double fx[33];
  double igral, err;
  int depth, rdepth, ndiv;
} cquad_ival;

/* Some constants and matrices that we'll need.  */

static const double xi[33] =
{
  -1., -0.99518472667219688624, -0.98078528040323044912,
  -0.95694033573220886493, -0.92387953251128675612,
  -0.88192126434835502970, -0.83146961230254523708,
  -0.77301045336273696082, -0.70710678118654752440,
  -0.63439328416364549822, -0.55557023301960222475,
  -0.47139673682599764857, -0.38268343236508977173,
  -0.29028467725446236764, -0.19509032201612826785,
  -0.098017140329560601995, 0., 0.098017140329560601995,
  0.19509032201612826785, 0.29028467725446236764, 0.38268343236508977173,
  0.47139673682599764857, 0.55557023301960222475, 0.63439328416364549822,
  0.70710678118654752440, 0.77301045336273696082, 0.83146961230254523708,
  0.88192126434835502970, 0.92387953251128675612, 0.95694033573220886493,
  0.98078528040323044912, 0.99518472667219688624, 1.
};

static const double bee[68] =
{
  0.00000000000000e+00, 2.28868854108532e-01, 0.00000000000000e+00,
  -8.15740215243451e-01, 0.00000000000000e+00, 5.31212715259731e-01,
  0.00000000000000e+00, 1.38538036812454e-02, 0.00000000000000e+00,
  3.74405228908818e-02, 0.00000000000000e+00, 2.12224115039342e-01,
  0.00000000000000e+00, -8.16362644507898e-01, 0.00000000000000e+00,
  5.35648426691481e-01, 0.00000000000000e+00, 1.52417902753662e-03,
  0.00000000000000e+00, 2.63058840550873e-03, 0.00000000000000e+00,
  4.15292106318904e-03, 0.00000000000000e+00, 6.97106011119775e-03,
  0.00000000000000e+00, 1.35535708431058e-02, 0.00000000000000e+00,
  3.52132898424856e-02, 0.00000000000000e+00, 2.06946714741884e-01,
  0.00000000000000e+00, -8.15674251283876e-01, 0.00000000000000e+00,
  5.38841175520580e-01, 0.00000000000000e+00, 1.84909689577590e-04,
  0.00000000000000e+00, 2.90936325007499e-04, 0.00000000000000e+00,
  3.84877750950089e-04, 0.00000000000000e+00, 4.86436656735046e-04,
  0.00000000000000e+00, 6.08688640346879e-04, 0.00000000000000e+00,
  7.66732830740331e-04, 0.00000000000000e+00, 9.82753336104205e-04,
  0.00000000000000e+00, 1.29359957505615e-03, 0.00000000000000e+00,
  1.76616363801885e-03, 0.00000000000000e+00, 2.53323433039089e-03,
  0.00000000000000e+00, 3.88872172121956e-03, 0.00000000000000e+00,
  6.58635106468291e-03, 0.00000000000000e+00, 1.30326736343254e-02,
  0.00000000000000e+00, 3.44353850696714e-02, 0.00000000000000e+00,
  2.05025409531915e-01, 0.00000000000000e+00, -8.14985893995401e-01,
  0.00000000000000e+00, 5.40679930965238e-01
};

static const double Lalpha[33] =
{
  5.77350269189626e-01, 5.16397779494322e-01, 5.07092552837110e-01,
  5.03952630678970e-01, 5.02518907629606e-01, 5.01745206004255e-01,
  5.01280411827603e-01, 5.00979432868120e-01, 5.00773395667191e-01,
  5.00626174321759e-01, 5.00517330712619e-01, 5.00434593736979e-01,
  5.00370233297676e-01, 5.00319182924304e-01, 5.00278009473803e-01,
  5.00244319584578e-01, 5.00216403386025e-01, 5.00193012939056e-01,
  5.00173220168024e-01, 5.00156323280355e-01, 5.00141783641018e-01,
  5.00129182278347e-01, 5.00118189340972e-01, 5.00108542278496e-01,
  5.00100030010004e-01, 5.00092481273333e-01, 5.00085755939229e-01,
  5.00079738458365e-01, 5.00074332862969e-01, 5.00069458915387e-01,
  5.00065049112355e-01, 5.00061046334395e-01, 5.00057401986298e-01
};

static const double Lgamma[33] =
{
  0.0, 0.0, 5.16397779494322e-01, 5.07092552837110e-01, 5.03952630678970e-01,
  5.02518907629606e-01, 5.01745206004255e-01, 5.01280411827603e-01,
  5.00979432868120e-01, 5.00773395667191e-01, 5.00626174321759e-01,
  5.00517330712619e-01, 5.00434593736979e-01, 5.00370233297676e-01,
  5.00319182924304e-01, 5.00278009473803e-01, 5.00244319584578e-01,
  5.00216403386025e-01, 5.00193012939056e-01, 5.00173220168024e-01,
  5.00156323280355e-01, 5.00141783641018e-01, 5.00129182278347e-01,
  5.00118189340972e-01, 5.00108542278496e-01, 5.00100030010003e-01,
  5.00092481273333e-01, 5.00085755939229e-01, 5.00079738458365e-01,
  5.00074332862969e-01, 5.00069458915387e-01, 5.00065049112355e-01,
  5.00061046334395e-01
};

static const double V1inv[5 * 5] =
{
  .47140452079103168293e-1, .37712361663282534635, .56568542494923801952,
  .37712361663282534635, .47140452079103168293e-1,
  -.81649658092772603273e-1, -.46188021535170061160, 0,
  .46188021535170061160, .81649658092772603273e-1, .15058465048420853962,
  .12046772038736683169, -.54210474174315074262, .12046772038736683169,
  .15058465048420853962, -.21380899352993950775, .30237157840738178177, -0.,
  -.30237157840738178177, .21380899352993950775, .10774960475223581324,
  -.21549920950447162648, .21549920950447162648, -.21549920950447162648,
  .10774960475223581324
};

static const double V2inv[9 * 9] =
{
  .11223917161691230546e-1, .10339219839658349826, .19754094204576565761,
  .25577315077753587922, .27835314560994251755, .25577315077753587922,
  .19754094204576565761, .10339219839658349826, .11223917161691230546e-1,
  -.19440394783993476970e-1, -.16544884625069155470, -.24193725566041460608,
  -.16953338808305493604, 0.0, .16953338808305493604, .24193725566041460608,
  .16544884625069155470, .19440394783993476970e-1, .26466393115406349388e-1,
  .17766815796285469394, .11316664642449611462, -.16306601003711325980,
  -.30847037493128779631, -.16306601003711325980, .11316664642449611462,
  .17766815796285469394, .26466393115406349388e-1,
  -.32395302049990834508e-1, -.15521142532414866547,
  .88573492664788602740e-1, .29570405784974857322, 0.0,
  -.29570405784974857322, -.88573492664788602740e-1, .15521142532414866547,
  .32395302049990834508e-1, .41442155673936851246e-1,
  .98186757907405608245e-1, -.23056908429499411784,
  -.68047008326360625520e-1, .31797435808002456774,
  -.68047008326360625520e-1, -.23056908429499411784,
  .98186757907405608245e-1, .41442155673936851246e-1,
  -.49981120317798783134e-1, -.24861810572835756217e-1,
  .23561326072010832539, -.24472785656448415351, 0.0, .24472785656448415351,
  -.23561326072010832539, .24861810572835756217e-1,
  .49981120317798783134e-1, .79691635865674781228e-1,
  -.95725617891693941833e-1, -.57957553356854386344e-1,
  .21164072460540271452, -.27529837844505833514, .21164072460540271452,
  -.57957553356854386344e-1, -.95725617891693941833e-1,
  .79691635865674781228e-1,
  -.10894869830716590913, .20131094491947531782, -.15407672674888869038,
  .83385723639789791384e-1, 0.0, -.83385723639789791384e-1,
  .15407672674888869038, -.20131094491947531782, .10894869830716590913,
  .54581057089643838221e-1, -.10916211417928767644, .10916211417928767644,
  -.10916211417928767644, .10916211417928767644, -.10916211417928767644,
  .10916211417928767644, -.10916211417928767644, .54581057089643838221e-1
};

static const double V3inv[17 * 17] =
{
  .27729677693590098996e-2, .26423663180333065153e-1,
  .53374068493933898312e-1, .77007854739523195947e-1,
  .98257061072911596869e-1, .11538049741786835604, .12832134344120884559,
  .13612785914022865001, .13888293186236181317, .13612785914022865001,
  .12832134344120884559, .11538049741786835604, .98257061072911596869e-1,
  .77007854739523195947e-1, .53374068493933898312e-1,
  .26423663180333065153e-1, .27729677693590098996e-2,
  -.48029210642807413690e-2, -.44887724635478800254e-1,
  -.85409520147301089416e-1, -.11090267822061423050, -.12033983162705862441,
  -.11102786862182788886, -.85054870109799336515e-1,
  -.45998467987742225160e-1, 0.0, .45998467987742225160e-1,
  .85054870109799336515e-1, .11102786862182788886, .12033983162705862441,
  .11090267822061423050, .85409520147301089416e-1, .44887724635478800254e-1,
  .48029210642807413690e-2, .62758546879582030087e-2,
  .55561297093529155869e-1,
  .93281491021051539742e-1, .92320151237493695139e-1,
  .55077987469605684531e-1,
  -.96998141716497488255e-2, -.80285961895427405567e-1,
  -.13496839655913850224,
  -.15512521776684524331, -.13496839655913850224, -.80285961895427405567e-1,
  -.96998141716497488255e-2, .55077987469605684531e-1,
  .92320151237493695139e-1, .93281491021051539742e-1,
  .55561297093529155869e-1, .62758546879582030087e-2,
  -.74850969394858555939e-2, -.61751608943839234096e-1,
  -.82974150437304275958e-1, -.38437763431942633378e-1,
  .45745502025779701366e-1, .12369235652734542162, .14720439712852868239,
  .98768034347019704401e-1, 0.0,
  -.98768034347019704401e-1, -.14720439712852868239, -.12369235652734542162,
  -.45745502025779701366e-1, .38437763431942633378e-1,
  .82974150437304275958e-1, .61751608943839234096e-1,
  .74850969394858555939e-2, .86710099994384056338e-2,
  .64006230103659573344e-1, .58517426396091675690e-1,
  -.29743410528985802680e-1,
  -.11934127779157114754, -.12686773515361299409, -.30729137153877447035e-1,
  .97307836256600731568e-1, .15635811574451401023, .97307836256600731568e-1,
  -.30729137153877447035e-1, -.12686773515361299409, -.11934127779157114754,
  -.29743410528985802680e-1, .58517426396091675690e-1,
  .64006230103659573344e-1, .86710099994384056338e-2,
  -.97486395666294840165e-2, -.62995604908060224672e-1,
  -.24373234450275529219e-1, .87760984413626872730e-1,
  .12205204576993351394,
  .16216004196864002088e-1, -.12422320942156845775, -.13682714580929614678,
  0.0, .13682714580929614678, .12422320942156845775,
  -.16216004196864002088e-1, -.12205204576993351394,
  -.87760984413626872730e-1, .24373234450275529219e-1,
  .62995604908060224672e-1, .97486395666294840165e-2,
  .10956271233750488468e-1, .58613204255294358939e-1,
  -.13306063940736618859e-1, -.11606666444978454399,
  -.52059598001115805639e-1, .10868540217796151849, .12594452879014618005,
  -.44678658254872910434e-1, -.15617684362128533405,
  -.44678658254872910434e-1, .12594452879014618005, .10868540217796151849,
  -.52059598001115805639e-1, -.11606666444978454399,
  -.13306063940736618859e-1, .58613204255294358939e-1,
  .10956271233750488468e-1, -.12098893000863087230e-1,
  -.51626244709126208453e-1, .48919433304746979330e-1,
  .10467644465949427090,
  -.48729879523084673782e-1, -.13668732103524749234, .28190838706814496438e-1,
  .15434223333238741600, 0.0, -.15434223333238741600,
  -.28190838706814496438e-1, .13668732103524749234,
  .48729879523084673782e-1, -.10467644465949427090,
  -.48919433304746979330e-1, .51626244709126208453e-1,
  .12098893000863087230e-1, .13542668300437944822e-1,
  .41712033418258689308e-1,
  -.76190463272803434388e-1, -.58303943170068132010e-1, .12158068748245606853,
  .42121099930651007882e-1, -.14684425840766337756,
  -.16108203535058647043e-1, .15698075850757976092,
  -.16108203535058647043e-1, -.14684425840766337756,
  .42121099930651007882e-1, .12158068748245606853,
  -.58303943170068132010e-1, -.76190463272803434388e-1,
  .41712033418258689308e-1, .13542668300437944822e-1,
  -.14939634995117694417e-1, -.30047246373341564039e-1,
  .91624635082546425678e-1, -.79133374319110026377e-2,
  -.12292558212072233355, .90013382617762643524e-1,
  .84013717196539593395e-1, -.14813033309980695856, 0.0,
  .14813033309980695856, -.84013717196539593395e-1,
  -.90013382617762643524e-1,
  .12292558212072233355, .79133374319110026377e-2, -.91624635082546425678e-1,
  .30047246373341564039e-1, .14939634995117694417e-1,
  .16986031342807474208e-1,
  .15760203882617033601e-1, -.91494054040950941996e-1,
  .70082459207876130806e-1,
  .53390713710144539104e-1, -.14340746778352039430, .84048122493418898508e-1,
  .72456667788091316868e-1, -.15564535320096811360,
  .72456667788091316868e-1, .84048122493418898508e-1,
  -.14340746778352039430, .53390713710144539104e-1,
  .70082459207876130806e-1, -.91494054040950941996e-1,
  .15760203882617033601e-1,
  .16986031342807474208e-1, -.18994065631858742028e-1,
  -.82901821370405592927e-3, .77239669773015192888e-1,
  -.10850735431039424680, .47524484622086496464e-1,
  .69148184871588737021e-1, -.14829314646228194928, .11992057742398672066,
  0.0, -.11992057742398672066, .14829314646228194928,
  -.69148184871588737021e-1, -.47524484622086496464e-1,
  .10850735431039424680, -.77239669773015192888e-1,
  .82901821370405592927e-3, .18994065631858742028e-1,
  .22761703826371535132e-1, -.17728848711449643358e-1,
  -.47496371572480503788e-1, .10659958402328690063, -.11696013966166296514,
  .63073750910894244526e-1, .32928881123602721303e-1,
  -.12280950532497593683, .15926189077282729505, -.12280950532497593683,
  .32928881123602721303e-1, .63073750910894244526e-1,
  -.11696013966166296514, .10659958402328690063, -.47496371572480503788e-1,
  -.17728848711449643358e-1, .22761703826371535132e-1,
  -.26493215276042203434e-1, .35579780856128386192e-1,
  .10447309718398935122e-1, -.68616154085314996709e-1,
  .11775363082763954214, -.13918901977011837274, .12312819418827395690,
  -.72053565748259077905e-1, 0.0, .72053565748259077905e-1,
  -.12312819418827395690, .13918901977011837274, -.11775363082763954214,
  .68616154085314996709e-1, -.10447309718398935122e-1,
  -.35579780856128386192e-1,
  .26493215276042203434e-1, .40742523354399706918e-1,
  -.73124912999529117195e-1, .49317266444153837821e-1,
  -.13686605413876015320e-1, -.28342624942191100464e-1,
  .70371855298258216249e-1, -.10600251632853603875, .12981016288391131812,
  -.13817029659318161476, .12981016288391131812, -.10600251632853603875,
  .70371855298258216249e-1, -.28342624942191100464e-1,
  -.13686605413876015320e-1,
  .49317266444153837821e-1, -.73124912999529117195e-1,
  .40742523354399706918e-1, -.54944368958699908688e-1,
  .10777725663147408190, -.10152395581538265428, .91369146312596428468e-1,
  -.77703071757424700773e-1, .61050911730999815031e-1,
  -.42052599404498348871e-1, .21438229266251454773e-1, 0.0,
  -.21438229266251454773e-1, .42052599404498348871e-1,
  -.61050911730999815031e-1, .77703071757424700773e-1,
  -.91369146312596428468e-1,
  .10152395581538265428, -.10777725663147408190, .54944368958699908688e-1,
  .27485608464748840573e-1, -.54971216929497681146e-1,
  .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .54971216929497681146e-1,
  -.54971216929497681146e-1, .27485608464748840573e-1
};

static const double V4inv[33 * 33] =
{
  .69120897476690862600e-3, .66419939766331555194e-2,
  .13600665164323186111e-1, .20122785860913684493e-1,
  .26583214101668429944e-1, .32712713318999268739e-1,
  .38576221976287138036e-1, .44033030938268925133e-1,
  .49092709529622799673e-1, .53657949874312515646e-1,
  .57724533144734311859e-1, .61219564530655179096e-1,
  .64138907503837875026e-1, .66427905189318792009e-1,
  .68088956652280022887e-1, .69083051391555695878e-1,
  .69422738116739271449e-1, .69083051391555695878e-1,
  .68088956652280022887e-1, .66427905189318792009e-1,
  .64138907503837875026e-1, .61219564530655179096e-1,
  .57724533144734311859e-1, .53657949874312515646e-1,
  .49092709529622799673e-1, .44033030938268925133e-1,
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};

static const double Tleft[33 * 33] =
{
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};

static const double Tright[33 * 33] =
{
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  .14899342093408253335e-7, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
  0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
  0., 0., .23283064365386962891e-9
};

/* 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.
    */

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

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. */

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.  */

DEFUN (quadcc, args, nargout,
       "-*- texinfo -*-\n\
@deftypefn  {Function File} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b})\n\
@deftypefnx {Function File} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b}, @var{tol})\n\
@deftypefnx {Function File} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b}, @var{tol}, @var{sing})\n\
@deftypefnx {Function File} {[@var{q}, @var{err}, @var{nr_points}] =} quadcc (@dots{})\n\
Numerically evaluate the integral of @var{f} from @var{a} to @var{b}\n\
using doubly-adaptive @nospell{Clenshaw-Curtis} quadrature.\n\
\n\
@var{f} is a function handle, inline function, or string containing the name\n\
of the function to evaluate.  The function @var{f} must be vectorized and\n\
must return a vector of output values if given a vector of input values.\n\
For example,\n\
\n\
@example\n\
f = @@(x) x .* sin (1./x) .* sqrt (abs (1 - x));\n\
@end example\n\
\n\
@noindent\n\
which uses the element-by-element ``dot'' form for all operators.\n\
\n\
@var{a} and @var{b} are the lower and upper limits of integration.  Either\n\
or both limits may be infinite.  @code{quadcc} handles an inifinite limit\n\
by substituting the variable of integration with @code{x = tan (pi/2*u)}.\n\
\n\
The optional argument @var{tol} defines the relative tolerance used to stop\n\
the integration procedure.  The default value is @math{1e^{-6}}.\n\
\n\
The optional argument @var{sing} contains a list of points where the\n\
integrand has known singularities, or discontinuities\n\
in any of its derivatives, inside the integration interval.\n\
For the example above, which has a discontinuity at x=1, the call to\n\
@code{quadcc} would be as follows\n\
\n\
@example\n\
int = quadcc (f, a, b, 1.0e-6, [ 1 ]);\n\
@end example\n\
\n\
The result of the integration is returned in @var{q}.\n\
\n\
@var{err} is an estimate of the absolute integration error.\n\
\n\
@var{nr_points} is the number of points at which the integrand was evaluated.\n\
\n\
If the adaptive integration did not converge, the value of @var{err} will be\n\
larger than the requested tolerance.  Therefore, it is recommended to verify\n\
this value for difficult integrands.\n\
\n\
@code{quadcc} is capable of dealing with non-numeric values of the integrand\n\
such as @code{NaN} or @code{Inf}.  If the integral diverges, and\n\
@code{quadcc} detects this, then a warning is issued and @code{Inf} or\n\
@code{-Inf} is returned.\n\
\n\
Note: @code{quadcc} is a general purpose quadrature algorithm and, as such,\n\
may be less efficient for a smooth or otherwise well-behaved integrand than\n\
other methods such as @code{quadgk}.\n\
\n\
The algorithm uses @nospell{Clenshaw-Curtis} quadrature rules of increasing\n\
degree in each interval and bisects the interval if either the function does\n\
not appear to be smooth or a rule of maximum degree has been reached.  The\n\
error estimate is computed from the L2-norm of the difference between two\n\
successive interpolations of the integrand over the nodes of the respective\n\
quadrature rules.\n\
\n\
Reference: @nospell{P. Gonnet}, @cite{Increasing the Reliability of Adaptive\n\
Quadrature Using Explicit Interpolants}, ACM Transactions on\n\
Mathematical Software, Vol. 37, Issue 3, Article No. 3, 2010.\n\
@seealso{quad, quadv, quadl, quadgk, trapz, dblquad, triplequad}\n\
@end deftypefn")
{
  octave_value_list retval;

  /* 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;

  /* Arguments left and right */
  int nargin = args.length ();
  octave_function *fcn;
  double a, b, tol, *sing;

  /* Variables needed for transforming the integrand. */
  bool wrap = false;
  double xw;

  /* Stuff we will need to call the integrand. */
  octave_value_list fargs, fvals;

  /* Actual variables (as opposed to constants above). */
  double m, h, ml, hl, mr, hr, temp;
  double igral, err, igral_final, err_final;
  int nivals, neval = 0;
  int i, j, d, split, t;
  int nnans, nans[33];
  cquad_ival *iv, *ivl, *ivr;
  double nc, ncdiff;


  /* Parse the input arguments. */
  if (nargin < 3)
    {
      print_usage ();
      return retval;
    }

  if (args(0).is_function_handle () || args(0).is_inline_function ())
    fcn = args(0).function_value ();
  else
    {
      std::string fcn_name = unique_symbol_name ("__quadcc_fcn__");
      std::string fname = "function y = ";
      fname.append (fcn_name);
      fname.append ("(x) y = ");
      fcn = extract_function (args(0), "quadcc", fcn_name, fname,
                              "; endfunction");
    }

  if (! args(1).is_real_scalar ())
    {
      error ("quadcc: lower limit of integration (A) must be a single real scalar");
      return retval;
    }
  else
    a = args(1).double_value ();

  if (! args(2).is_real_scalar ())
    {
      error ("quadcc: upper limit of integration (B) must be a single real scalar");
      return retval;
    }
  else
    b = args(2).double_value ();

  if (nargin < 4 || args(3).is_empty ())
    tol = 1.0e-6;
  else if (! args(3).is_real_scalar () || args(3).double_value () <= 0)
    {
      error ("quadcc: tolerance (TOL) must be a single real scalar > 0");
      return retval;
    }
  else
    tol = args(3).double_value ();

  if (nargin < 5)
    {
      nivals = 1;
    }
  else if (!(args(4).is_real_scalar () || args(4).is_real_matrix ()))
    {
      error ("quadcc: list of singularities (SING) must be a vector of real values");
      return retval;
    }
  else
    {
      nivals = 1 + args(4).numel ();
    }

  int cquad_heapsize = (nivals >= min_cquad_heapsize ? nivals + 1
                                                     : min_cquad_heapsize);
  /* The interval heap. */
  OCTAVE_LOCAL_BUFFER (cquad_ival, ivals, cquad_heapsize);
  OCTAVE_LOCAL_BUFFER (double, iivals, cquad_heapsize);
  OCTAVE_LOCAL_BUFFER (int, heap, cquad_heapsize);

  if (nivals == 1)
    {
      iivals[0] = a;
      iivals[1] = b;
    }
  else
    {
      // Intervals around singularities
      sing = args(4).array_value ().fortran_vec ();
      iivals[0] = a;
      for (i = 0; i < nivals - 1; i++)
        iivals[i + 1] = sing[i];
      iivals[nivals] = b;
    }

  /* If a or b are +/-Inf, transform the integral. */
  if (xisinf (a) || xisinf (b))
    {
      wrap = true;
      for (i = 0; i < nivals + 1; i++)
        if (xisinf (iivals[i]))
          iivals[i] = gnulib::copysign (1.0, iivals[i]);
        else
          iivals[i] = 2.0 * atan (iivals[i]) / M_PI;
    }


  /* Initialize the heaps. */
  for (i = 0; i < cquad_heapsize; i++)
    heap[i] = i;

  /* Create the first interval(s). */
  igral = 0.0;
  err = 0.0;
  for (j = 0; j < nivals; j++)
    {

      /* Initialize the interval. */
      iv = &(ivals[heap[j]]);
      m = (iivals[j] + iivals[j + 1]) / 2;
      h = (iivals[j + 1] - iivals[j]) / 2;
      nnans = 0;
      ColumnVector ex (33);
      if (wrap)
        {
          for (i = 0; i <= n[3]; i++)
            ex(i) = tan (M_PI / 2 * (m + xi[i] * h));
        }
      else
        {
          for (i = 0; i <= n[3]; i++)
            ex(i) = m + xi[i] * h;
        }
      fargs(0) = ex;
      fvals = feval (fcn, fargs, 1);
      if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
        {
          error ("quadcc: integrand F must return a single, real-valued vector");
          return retval;
        }
      Matrix effex = fvals(0).matrix_value ();
      if (effex.length () != ex.length ())
        {
          error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
          return retval;
        }
      for (i = 0; i <= n[3]; i++)
        {
          iv->fx[i] = effex(i);
          if (wrap)
            {
              xw = ex(i);
              iv->fx[i] *= (1.0 + xw * xw) * M_PI / 2;
            }
          neval++;
          if (! xfinite (iv->fx[i]))
            {
              nans[nnans++] = i;
              iv->fx[i] = 0.0;
            }
        }
      Vinvfx (iv->fx, &(iv->c[idx[3]]), 3);
      Vinvfx (iv->fx, &(iv->c[idx[2]]), 2);
      Vinvfx (iv->fx, &(iv->c[0]), 0);
      for (i = 0; i < nnans; i++)
        iv->fx[nans[i]] = octave_NaN;
      iv->a = iivals[j];
      iv->b = iivals[j + 1];
      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;

      /* Tabulate this interval's data. */
      igral += iv->igral;
      err += iv->err;

      /* Sift it up the heap. */
      i = j;
      while (i > 0 && ivals[heap[i / 2]].err < ivals[heap[i]].err)
        {
          temp = heap[i];
          heap[i] = heap[i / 2];
          heap[i / 2] = temp;
          i /= 2;
        }

    }


  /* Initialize some global values. */
  igral_final = 0.0;
  err_final = 0.0;


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

      /* Allow the user to interrupt. */
      OCTAVE_QUIT;

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

#if (DEBUG_QUADCC)
      printf ("quadcc: processing ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
              heap[0], nivals, iv->a, iv->b, iv->igral, iv->err, iv->depth);
#endif

      /* 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 */
          {
            ColumnVector ex (n[d] / 2);
            if (wrap)
              {
                for (i = 0; i < n[d] / 2; i++)
                  ex(i) = tan (M_PI / 2 * (m + xi[(2 * i + 1) * skip[d]] * h));
              }
            else
              {
                for (i = 0; i < n[d] / 2; i++)
                  ex(i) = m + xi[(2 * i + 1) * skip[d]] * h;
              }
            fargs(0) = ex;
            fvals = feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector");
                return retval;
              }
            Matrix effex = fvals(0).matrix_value ();
            if (effex.length () != ex.length ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
                return retval;
              }
            neval += effex.length ();
            for (i = 0; i < n[d] / 2; i++)
              {
                j = (2 * i + 1) * skip[d];
                iv->fx[j] = effex(i);
                if (wrap)
                  {
                    xw = ex(i);
                    iv->fx[j] *= (1.0 + xw * xw) * M_PI / 2;
                  }
              }
          }
          nnans = 0;
          for (i = 0; i <= 32; i += skip[d])
            {
              if (! xfinite (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[nans[i]] = octave_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)
                       * std::numeric_limits<double>::epsilon () * 10)
        {

#if (DEBUG_QUADCC)
          printf ("quadcc: dropping ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
                  heap[0], nivals, iv->a, iv->b, iv->igral, iv->err, iv->depth);
#endif

          /* Keep this interval's contribution */
          err_final += iv->err;
          igral_final += iv->igral;
          /* Swap with the last element on the heap */
          t = heap[nivals - 1];
          heap[nivals - 1] = heap[0];
          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
                  && ivals[heap[j + 1]].err >= ivals[heap[j]].err)
                j++;
              /* Do we need to move the ith entry up? */
              if (ivals[heap[j]].err <= ivals[heap[i]].err)
                break;
              else
                {
                  t = heap[j];
                  heap[j] = heap[i];
                  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 = &(ivals[heap[nivals++]]);
          ivl->a = iv->a;
          ivl->b = m;
          ml = (ivl->a + ivl->b) / 2;
          hl = h / 2;
          ivl->depth = 0;
          ivl->rdepth = iv->rdepth + 1;
          ivl->fx[0] = iv->fx[0];
          ivl->fx[32] = iv->fx[16];
          {
            ColumnVector ex (n[0] - 1);
            if (wrap)
              {
                for (i = 0; i < n[0] - 1; i++)
                  ex(i) = tan (M_PI / 2 * (ml + xi[(i + 1) * skip[0]] * hl));
              }
            else
              {
                for (i = 0; i < n[0] - 1; i++)
                  ex(i) = ml + xi[(i + 1) * skip[0]] * hl;
              }
            fargs(0) = ex;
            fvals = feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector");
                return retval;
              }
            Matrix effex = fvals(0).matrix_value ();
            if (effex.length () != ex.length ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
                return retval;
              }
            neval += effex.length ();
            for (i = 0; i < n[0] - 1; i++)
              {
                j = (i + 1) * skip[0];
                ivl->fx[j] = effex(i);
                if (wrap)
                  {
                    xw = ex(i);
                    ivl->fx[j] *= (1.0 + xw * xw) * M_PI / 2;
                  }
              }
          }
          nnans = 0;
          for (i = 0; i <= 32; i += skip[0])
            {
              if (! xfinite (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[nans[i]] = octave_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)
            {
              igral = gnulib::copysign (octave_Inf, igral);
              warning ("quadcc: divergent integral detected");
              break;
            }

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


          /* Generate the interval on the right */
          ivr = &(ivals[heap[nivals++]]);
          ivr->a = m;
          ivr->b = iv->b;
          mr = (ivr->a + ivr->b) / 2;
          hr = h / 2;
          ivr->depth = 0;
          ivr->rdepth = iv->rdepth + 1;
          ivr->fx[0] = iv->fx[16];
          ivr->fx[32] = iv->fx[32];
          {
            ColumnVector ex (n[0] - 1);
            if (wrap)
              {
                for (i = 0; i < n[0] - 1; i++)
                  ex(i) = tan (M_PI / 2 * (mr + xi[(i + 1) * skip[0]] * hr));
              }
            else
              {
                for (i = 0; i < n[0] - 1; i++)
                  ex(i) = mr + xi[(i + 1) * skip[0]] * hr;
              }
            fargs(0) = ex;
            fvals = feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector");
                return retval;
              }
            Matrix effex = fvals(0).matrix_value ();
            if (effex.length () != ex.length ())
              {
                error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
                return retval;
              }
            neval += effex.length ();
            for (i = 0; i < n[0] - 1; i++)
              {
                j = (i + 1) * skip[0];
                ivr->fx[j] = effex(i);
                if (wrap)
                  {
                    xw = ex(i);
                    ivr->fx[j] *= (1.0 + xw * xw) * M_PI / 2;
                  }
              }
          }
          nnans = 0;
          for (i = 0; i <= 32; i += skip[0])
            {
              if (! xfinite (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[nans[i]] = octave_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)
            {
              igral = gnulib::copysign (octave_Inf, igral);
              warning ("quadcc: divergent integral detected");
              break;
            }

          /* 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 = heap[nivals - 1];
          heap[nivals - 1] = heap[0];
          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
                  && ivals[heap[j + 1]].err >= ivals[heap[j]].err)
                j++;
              if (ivals[heap[j]].err <= ivals[heap[i]].err)
                break;
              else
                {
                  t = heap[j];
                  heap[j] = heap[i];
                  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 (ivals[heap[j]].err < ivals[heap[i]].err)
                {
                  t = heap[j];
                  heap[j] = heap[i];
                  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
                  && ivals[heap[j + 1]].err >= ivals[heap[j]].err)
                j++;
              if (ivals[heap[j]].err <= ivals[heap[i]].err)
                break;
              else
                {
                  t = heap[j];
                  heap[j] = heap[i];
                  heap[i] = t;
                  i = j;
                }
            }

        }

      /* If the heap is about to overflow, remove the last two intervals. */
      while (nivals > cquad_heapsize - 2)
        {
          iv = &(ivals[heap[nivals - 1]]);
#if (DEBUG_QUADCC)
          printf ("quadcc: dropping ival %i (of %i) with [%e,%e] int=%e, err=%e, depth=%i\n",
                  heap[0], nivals, iv->a, iv->b, iv->igral, iv->err, iv->depth);
#endif
          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 += ivals[heap[i]].igral;
          err += ivals[heap[i]].err;
        }

    }

  /* Dump the contents of the heap. */
#if (DEBUG_QUADCC)
  for (i = 0; i < nivals; i++)
    {
      iv = &(ivals[heap[i]]);
      printf ("quadcc: ival %i (%i) with [%e,%e], int=%e, err=%e, depth=%i, rdepth=%i, ndiv=%i\n",
              i, heap[i], iv->a, iv->b, iv->igral, iv->err, iv->depth,
              iv->rdepth, iv->ndiv);
    }
#endif

  /* Clean up and present the results. */
  if (nargout > 2)
    retval(2) = neval;
  if (nargout > 1)
    retval(1) = err;
  retval(0) = igral;
  /* All is well that ends well. */
  return retval;
}


/*
%!assert (quadcc (@sin, -pi, pi), 0, 1e-6)
%!assert (quadcc (inline ("sin"),- pi, pi), 0, 1e-6)
%!assert (quadcc ("sin", -pi, pi), 0, 1e-6)

%!assert (quadcc (@sin, -pi, 0), -2, 1e-6)
%!assert (quadcc (@sin, 0, pi), 2, 1e-6)
%!assert (quadcc (@(x) 1./sqrt (x), 0, 1), 2, 1e-6)
%!assert (quadcc (@(x) 1./(sqrt (x).*(x+1)), 0, Inf), pi, 1e-6)

%!assert (quadcc (@(x) exp (-x .^ 2), -Inf, Inf), sqrt (pi), 1e-6)
%!assert (quadcc (@(x) exp (-x .^ 2), -Inf, 0), sqrt (pi)/2, 1e-6)

## Test function with NaNs in interval
%!function y = __nansin (x)
%!  nan_locs = [-3*pi/4, -pi/4, 0, pi/3, pi/2, pi];
%!  y = sin (x);
%!  idx = min (abs (bsxfun (@minus, x(:), nan_locs)), [], 2);
%!  y(idx < 1e-10) = NaN;
%!endfunction

%!test
%! [q, err, npoints] = quadcc ("__nansin", -pi, pi);
%! assert (q, 0, 1e-6);
%! assert (err, 0, 15*eps);

%% Test input validation
%!error (quadcc ())
%!error (quadcc (@sin))
%!error (quadcc (@sin, 0))
%!error (quadcc (@sin, ones (2), pi))
%!error (quadcc (@sin, -i, pi))
%!error (quadcc (@sin, 0, ones (2)))
%!error (quadcc (@sin, 0, i))
%!error (quadcc (@sin, 0, pi, 0))
%!error (quadcc (@sin, 0, pi, 1e-6, [ i ]))
*/