quadcc.cc

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////////////////////////////////////////////////////////////////////////
//
// Copyright (C) 2010-2021 The Octave Project Developers
//
// See the file COPYRIGHT.md in the top-level directory of this
// distribution or <https://octave.org/copyright/>.
//
// 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
// <https://www.gnu.org/licenses/>.
//
////////////////////////////////////////////////////////////////////////
 
#if defined (HAVE_CONFIG_H)
#  include "config.h"
#endif
 
#include <cmath>
 
#include <algorithm>
 
#include "lo-ieee.h"
#include "oct-locbuf.h"
 
#include "defun.h"
#include "error.h"
#include "interpreter-private.h"
#include "ovl.h"
#include "parse.h"
#include "utils.h"
#include "variables.h"
 
// Extended debugging.
#define DEBUG_QUADCC 0
 
// Define the minimum size of the interval heap.
static const int 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;
 
// Define relative tolerance used when deciding to drop an interval.
static const double DROP_RELTOL = std::numeric_limits<double>::epsilon () * 10;
 
// 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,
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};
 
static const double Tleft[33 * 33] =
{
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static const double Tright[33 * 33] =
{
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  .28867244235852488244e-7, .43982811713864556957e-6, 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., .46566128730773925781e-9,
  .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 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.
 
DEFMETHOD (quadcc, interp, args, ,
           doc: /* -*- texinfo -*-
@deftypefn  {} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b})
@deftypefnx {} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b}, @var{tol})
@deftypefnx {} {@var{q} =} quadcc (@var{f}, @var{a}, @var{b}, @var{tol}, @var{sing})
@deftypefnx {} {[@var{q}, @var{err}, @var{nr_points}] =} quadcc (@dots{})
Numerically evaluate the integral of @var{f} from @var{a} to @var{b} using
doubly-adaptive @nospell{Clenshaw-Curtis} quadrature.
 
@var{f} is a function handle, inline function, or string containing the name
of the function to evaluate.  The function @var{f} must be vectorized and
must return a vector of output values if given a vector of input values.
For example,
 
@example
f = @@(x) x .* sin (1./x) .* sqrt (abs (1 - x));
@end example
 
@noindent
which uses the element-by-element ``dot'' form for all operators.
 
@var{a} and @var{b} are the lower and upper limits of integration.  Either or
both limits may be infinite.  @code{quadcc} handles an infinite limit by
substituting the variable of integration with @code{x = tan (pi/2*u)}.
 
The optional argument @var{tol} is a 1- or 2-element vector that specifies the
desired accuracy of the result.  The first element of the vector is the desired
absolute tolerance, and the second element is the desired relative tolerance.
To choose a relative test only, set the absolute tolerance to zero.  To choose
an absolute test only, set the relative tolerance to zero.  The default
absolute tolerance is 1e-10 (1e-5 for single), and the default relative
tolerance is 1e-6 (1e-4 for single).
 
The optional argument @var{sing} contains a list of points where the integrand
has known singularities, or discontinuities in any of its derivatives, inside
the integration interval.  For the example above, which has a discontinuity at
x=1, the call to @code{quadcc} would be as follows
 
@example
int = quadcc (f, a, b, [], [ 1 ]);
@end example
 
The result of the integration is returned in @var{q}.
 
@var{err} is an estimate of the absolute integration error.
 
@var{nr_points} is the number of points at which the integrand was evaluated.
 
If the adaptive integration did not converge, the value of @var{err} will be
larger than the requested tolerance.  Therefore, it is recommended to verify
this value for difficult integrands.
 
@code{quadcc} is capable of dealing with non-numeric values of the integrand
such as @code{NaN} or @code{Inf}.  If the integral diverges, and @code{quadcc}
detects this, then a warning is issued and @code{Inf} or @code{-Inf} is
returned.
 
Note: @code{quadcc} is a general purpose quadrature algorithm and, as such,
may be less efficient for a smooth or otherwise well-behaved integrand than
other methods such as @code{quadgk}.
 
The algorithm uses @nospell{Clenshaw-Curtis} quadrature rules of increasing
degree in each interval and bisects the interval if either the function does
not appear to be smooth or a rule of maximum degree has been reached.  The
error estimate is computed from the L2-norm of the difference between two
successive interpolations of the integrand over the nodes of the respective
quadrature rules.
 
Reference: @nospell{P. Gonnet}, @cite{Increasing the Reliability of Adaptive
Quadrature Using Explicit Interpolants}, @nospell{ACM} Transactions on
Mathematical Software, Vol.@: 37, Issue 3, Article No.@: 3, 2010.
@seealso{quad, quadv, quadl, quadgk, trapz, dblquad, triplequad}
@end deftypefn */)
{
  // 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_value fcn;
  double a, b, abstol, reltol, *sing;
  bool issingle;
 
  // 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;
  int neval = 0;
  int i, j, d, split;
  int nnans, nans[33];
  cquad_ival *iv, *ivl, *ivr;
  double nc, ncdiff;
 
  // Parse the input arguments.
  if (nargin < 3)
    print_usage ();
 
  fcn = octave::get_function_handle (interp, args(0), "x");
 
  if (! args(1).is_real_scalar ())
    error ("quadcc: lower limit of integration (A) must be a real scalar");
  a = args(1).double_value ();
  issingle = args(1).is_single_type ();
 
  if (! args(2).is_real_scalar ())
    error ("quadcc: upper limit of integration (B) must be a real scalar");
  b = args(2).double_value ();
  issingle = (issingle || args(2).is_single_type ());
 
  if (nargin < 4 || args(3).isempty ())
    {
      if (issingle)
        {
          abstol = 1.0e-5;
          reltol = 1.0e-4;
        }
      else
        {
          abstol = 1.0e-10;
          reltol = 1.0e-6;
        }
    }
  else if (! args(3).isnumeric () || args(3).numel () > 2)
    error ("quadcc: TOL must be a 1- or 2-element vector [AbsTol, RelTol]");
  else
    {
      NDArray tol = args(3).array_value ();
 
      abstol = tol(0);
      if (abstol < 0)
        error ("quadcc: absolute tolerance must be >=0");
 
      if (tol.numel () == 1)
        {
          if (issingle)
            reltol = 1.0e-4;
          else
            reltol = 1.0e-6;
        }
      else
        {
          reltol = tol(1);
          if (reltol < 0)
            error ("quadcc: relative tolerance must be >=0");
        }
    }
 
  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");
  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;
      std::copy_n (sing, nivals-1, iivals+1);
      iivals[nivals] = b;
    }
 
  // If a or b are +/-Inf, transform the integral.
  if (octave::math::isinf (a) || octave::math::isinf (b))
    {
      wrap = true;
      for (i = 0; i < nivals + 1; i++)
        if (octave::math::isinf (iivals[i]))
          iivals[i] = std::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 = octave::feval (fcn, fargs, 1);
      if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
        error ("quadcc: integrand F must return a single, real-valued vector");
 
      Matrix effex = fvals(0).matrix_value ();
      if (effex.numel () != ex.numel ())
        error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
 
      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 (! octave::math::isfinite (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::numeric_limits<double>::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;
          temp = iv->c[idx[3] + i];
          nc += temp * temp;
        }
      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)
        {
          std::swap (heap[i], heap[i / 2]);
          i /= 2;
        }
    }
 
  // Initialize some global values.
  igral_final = 0.0;
  err_final = 0.0;
 
  // Main loop.
  while (nivals > 0
         && err > std::max (abstol, fabs (igral) * reltol)
         && ! (err_final > std::max (abstol, fabs (igral) * reltol)
               && err - err_final < std::max (abstol, fabs (igral) * reltol)))
    {
      // 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 = octave::feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              error ("quadcc: integrand F must return a single, real-valued vector");
 
            Matrix effex = fvals(0).matrix_value ();
            if (effex.numel () != ex.numel ())
              error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
 
            neval += effex.numel ();
            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 (! octave::math::isfinite (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::numeric_limits<double>::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;
              temp = iv->c[idx[d] + i];
              nc += temp * temp;
            }
          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);
        }
      else
        {
          // Maximum degree reached, just split.
          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) * DROP_RELTOL)
        {
#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.
          std::swap (heap[nivals - 1], heap[0]);
          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
                {
                  std::swap (heap[j], heap[i]);
                  i = j;
                }
            }
        }
      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 = octave::feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              error ("quadcc: integrand F must return a single, real-valued vector");
 
            Matrix effex = fvals(0).matrix_value ();
            if (effex.numel () != ex.numel ())
              error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
 
            neval += effex.numel ();
            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 (! octave::math::isfinite (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::numeric_limits<double>::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 = std::copysign (octave::numeric_limits<double>::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 = octave::feval (fcn, fargs, 1);
            if (fvals.length () != 1 || ! fvals(0).is_real_matrix ())
              error ("quadcc: integrand F must return a single, real-valued vector");
 
            Matrix effex = fvals(0).matrix_value ();
            if (effex.numel () != ex.numel ())
              error ("quadcc: integrand F must return a single, real-valued vector of the same size as the input");
 
            neval += effex.numel ();
            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 (! octave::math::isfinite (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::numeric_limits<double>::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 = std::copysign (octave::numeric_limits<double>::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.
          std::swap (heap[nivals - 1], heap[0]);
          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
                {
                  std::swap (heap[j], heap[i]);
                  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)
                {
                  std::swap (heap[j], heap[i]);
                  i = j;
                }
              else
                break;
            }
        }
      else
        {
          // Otherwise, just fix-up the heap.
          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
                {
                  std::swap (heap[j], heap[i]);
                  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;
        }
    }
 
#if (DEBUG_QUADCC)
  // Dump the contents of the heap.
  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
 
  if (issingle)
    return ovl (static_cast<float> (igral), err, neval);
  else
    return ovl (igral, err, neval);
}
 
/*
%!assert (quadcc (@sin, -pi, pi), 0, 1e-10)
%!assert (quadcc (inline ("sin"), -pi, pi), 0, 1e-10)
%!assert (quadcc ("sin", -pi, pi), 0, 1e-10)
 
%!assert (quadcc (@sin, -pi, 0), -2, 1e-10)
%!assert (quadcc (@sin, 0, pi), 2, 1e-10)
%!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) 1./(sqrt (x).*(x+1)), 0, Inf, [0, 1e-8]), pi, -1e-8)
 
%!assert (quadcc (@(x) exp (-x .^ 2), -Inf, Inf), sqrt (pi), 1e-10)
%!assert (quadcc (@(x) exp (-x .^ 2), -Inf, 0), sqrt (pi)/2, 1e-10)
 
## 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, [0, 1e-6]);
%! assert (q, 0, -1e-6);
%! assert (err, 0, 15*eps);
 
%!test
%! assert (class (quadcc (@sin, 0, 1)), "double");
%! assert (class (quadcc (@sin, single (0), 1)), "single");
%! assert (class (quadcc (@sin, 0, single (1))), "single");
%! assert (class (quadcc (@sin, single (0), single (1))), "single");
 
## Test input validation
%!error (quadcc ())
%!error (quadcc (@sin))
%!error (quadcc (@sin, 0))
%!error <lower limit .* must be a .* scalar> (quadcc (@sin, ones (2), pi))
%!error <lower limit .* must be a real scalar> (quadcc (@sin, -i, pi))
%!error <upper limit .* must be a .* scalar> (quadcc (@sin, 0, ones (2)))
%!error <upper limit .* must be a real scalar> (quadcc (@sin, 0, i))
%!error <TOL must be a 1- or 2-element vector> (quadcc (@sin, 0, pi, {1}))
%!error <TOL must be a 1- or 2-element vector> (quadcc (@sin, 0, pi, [1,2,3]))
%!error <absolute tolerance must be .=0> (quadcc (@sin, 0, pi, -1))
%!error <relative tolerance must be .=0> (quadcc (@sin, 0, pi, [1, -1]))
%!error <SING.* must be .* real values> (quadcc (@sin, 0, pi, 1e-6, [ i ]))
*/