Mercurial > hg > octave-max
changeset 10944:a7655cdba470
add the cquad integrator, by Pedro Gonnet
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Fri, 03 Sep 2010 13:18:59 +0200 |
| parents | 90450db218e4 |
| children | aa40bdbfa478 |
| files | src/ChangeLog src/DLD-FUNCTIONS/cquad.cc src/DLD-FUNCTIONS/module-files |
| diffstat | 3 files changed, 2255 insertions(+), 0 deletions(-) [+] |
line wrap: on
line diff
--- a/src/ChangeLog +++ b/src/ChangeLog @@ -1,3 +1,8 @@ +2010-09-03 Jaroslav Hajek <highegg@gmail.com> + + * DLD-FUNCTIONS/cquad.cc: New source, courtesy by Pedro Gonnet. + * DLD-FUNCTIONS/module-files: Update. + 2010-09-01 Marco Caliari <marco.caliari@univr.it> * graphics.cc: Fix axes limits underflow to zero for logscale.
new file mode 100644 --- /dev/null +++ b/src/DLD-FUNCTIONS/cquad.cc @@ -0,0 +1,2249 @@ +/* + +Copyright (C) 2010 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 <stdlib.h> +#include "lo-math.h" +#include "lo-ieee.h" +#include "oct.h" +#include "parse.h" +#include "ov-fcn-handle.h" + +/* Define the size of the interval heap. */ +#define 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, + .38576221976287138036e-1, .32712713318999268739e-1, + .26583214101668429944e-1, .20122785860913684493e-1, + .13600665164323186111e-1, .66419939766331555194e-2, + .69120897476690862600e-3, -.11972090629438798134e-2, + -.11448874821643225573e-1, -.23104401104002905904e-1, + -.33352899418646530133e-1, -.42538626424075425908e-1, + -.49969730733911825941e-1, -.55555454015360728353e-1, + -.58955533624852604918e-1, -.60126044219122513907e-1, + 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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 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_DLD (cquad, args, nargout, +"-*- texinfo -*-\n\ +@deftypefn {Function File} {[@var{int}, @var{err}, @var{nr_points}] =} cquad (@var{f}, @var{a}, @var{b}, @var{tol})\n\ +@deftypefnx {Function File} {[@var{int}, @var{err}, @var{nr_points}] =} cquad (@var{f}, @var{a}, @var{b}, @var{tol}, @var{sing})\n\ +Numerically evaluates an integral using the doubly-adaptive\n\ +quadrature described by P. Gonnet in @cite{\"Increasing the\n\ +Reliability of Adaptive Quadrature Using Explicit Interpolants\",\n\ +ACM Transactions on Mathematical Software, in Press, 2010}.\n\ +The algorithm uses Clenshaw-Curtis quadrature rules of increasing\n\ +degree in each interval and bisects the interval if either the\n\ +function does not appear to be smooth or a rule of maximum\n\ +degree has been reached. The error estimate is computed from the\n\ +L2-norm of the difference between two successive interpolations\n\ +of the integrand over the nodes of the respective quadrature rules.\n\ +\n\ +For example,\n\ +\n\ +@example\n\ + int = cquad ( f , a , b , 1.0e-6 );\n\ +@end example\n\ +\n\ +@noindent computes the integral of a function @var{f} in the interval\n\ +[@var{a},@var{b}] to the relative precision of six\n\ +decimal digits.\n\ +The integrand @var{f} should accept a vector argument and return a vector\n\ +result containing the integrand evaluated at each element of the\n\ +argument, for example\n\ +\n\ +@example\n\ + f = @@(x) x .* sin ( 1 ./ x ) .* sqrt ( abs ( 1 - x ) );\n\ +@end example\n\ +\n\ +If the integrand has known singularieites or discontinuities\n\ +in any of its derivatives inside the interval,\n\ +as does the above example at x=1, these can be specified in\n\ +the additional argument @var{sing} as follows\n\ +\n\ +@example\n\ + int = cquad ( f , a , b , 1.0e-6 , [ 1 ] );\n\ +@end example\n\ +\n\ +The two additional output variables @var{err} and @var{nr_points}\n\ +return an estimate of the absolute integration error and\n\ +the number of points at which the integrand was evaluated\n\ +respectively.\n\ +If the adaptive integration did not converge, the value of\n\ +@var{err} will be larger than the requested tolerance. It is\n\ +therefore recommended to verify this value for difficult\n\ +integrands.\n\ +\n\ +If either @var{a} or @var{b} are @code{+/-Inf}, @code{cquad}\n\ +integrates @var{f} by substituting the variable of integration\n\ +with @code{x=tan(pi/2*u)}.\n\ +\n\ +@code{cquad} is capable of dealing with non-numerical\n\ +values of the integrand such as @code{NaN}, @code{Inf}\n\ +or @code{-Inf}, as the above example at x=0.\n\ +If the integral diverges and @code{cquad} detects this, \n\ +a warning is issued and @code{Inf} or @code{-Inf} is returned.\n\ +\n\ +Note that @code{cquad} is a general purpose quadrature algorithm\n\ +and as such may be less efficient for smooth or otherwise\n\ +well-behaved integrand than other methods such as\n\ +@code{quadgk} or @code{trapz}.\n\ +\n\ +@seealso{triplequad, dblquad, quadgk, quadl, quadv, trapz}\n\ +@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; + + /* The interval heap. */ + cquad_ival ivals[cquad_heapsize]; + int heap[cquad_heapsize]; + + /* Arguments left and right */ + int nargin = args.length (); + octave_function *fcn; + double a, b, tol, iivals[cquad_heapsize], *sing; + + /* Variables needed for transforming the integrand. */ + int wrap = 0; + double xw; + + /* Stuff we will need to call the integrand. */ + octave_value_list fargs, retval; + + /* Actual variables (as opposed to constants above). */ + double m, h, ml, hl, mr, hr, temp; + double igral, err, igral_final, err_final, err_excess; + 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 < 1) + { + error + ("cquad: first argument (integrand) of type function handle required."); + return octave_value_list (); + } + else + { + if (args (0).is_function_handle () || args (0).is_inline_function ()) + fcn = args (0).function_value (); + else { + error + ("cquad: first argument (integrand) must be a function handle or an inline function."); + return octave_value_list(); + } + } + + if (nargin < 2 || !args (1).is_real_scalar ()) + { + error + ("cquad: second argument (left interval edge) must be a single real scalar."); + return octave_value_list (); + } + else + a = args (1).double_value (); + + if (nargin < 3 || !args (2).is_real_scalar ()) + { + error + ("cquad: third argument (right interval edge) must be a single real scalar."); + return octave_value_list (); + } + else + b = args (2).double_value (); + + if (nargin < 4) + tol = 1.0e-6; + else if (!args (3).is_real_scalar ()) + { + error + ("cquad: fourth argument (tolerance) must be a single real scalar."); + return octave_value_list (); + } + else + tol = args (3).double_value (); + + if (nargin < 5) + { + nivals = 1; + iivals[0] = a; + iivals[1] = b; + } + else if (!(args (4).is_real_scalar () || args (4).is_real_matrix ())) + { + error + ("cquad: fifth argument (singularities) must be a vector of real values."); + return octave_value_list (); + } + else + { + nivals = 1 + args (4).length (); + if ( nivals > cquad_heapsize ) { + error("cquad: maximum number of singular points is limited to %i.",cquad_heapsize-1); + return octave_value_list(); + } + sing = args (4).array_value ().fortran_vec (); + iivals[0] = a; + for (i = 0; i < nivals - 2; i++) + iivals[i + 1] = sing[i]; + iivals[nivals] = b; + } + + /* If a or b are +/-Inf, transform the integral. */ + if (xisinf (a) || xisinf (b)) + { + wrap = 1; + for (i = 0; i <= nivals; i++) + if (xisinf (iivals[i])) + iivals[i] = 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; + retval = feval (fcn, fargs, 1); + if (retval.length () != 1 || !retval (0).is_real_matrix ()) + { + error + ("cquad: integrand must return a single, real-valued vector."); + return octave_value_list (); + } + Matrix effex = retval (0).matrix_value (); + if (effex.length () != ex.length ()) + { + error + ("cquad: integrand must return a single, real-valued vector of the same size as the input"); + return octave_value_list (); + } + 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[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; + err_excess = 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; + +/* printf + ("cquad: 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); +*/ + /* 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; + retval = feval (fcn, fargs, 1); + if (retval.length () != 1 || !retval (0).is_real_matrix ()) + { + error + ("cquad: integrand must return a single, real-valued vector."); + return octave_value_list (); + } + Matrix effex = retval (0).matrix_value (); + if (effex.length () != ex.length ()) + { + error + ("cquad: integrand must return a single, real-valued vector of the same size as the input."); + return octave_value_list (); + } + 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[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) * DBL_EPSILON * 10) + { + +/* printf + ("cquad: 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); +*/ + /* 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; + retval = feval (fcn, fargs, 1); + if (retval.length () != 1 || !retval (0).is_real_matrix ()) + { + error + ("cquad: integrand must return a single, real-valued vector."); + return octave_value_list (); + } + Matrix effex = retval (0).matrix_value (); + if (effex.length () != ex.length ()) + { + error + ("cquad: integrand must return a single, real-valued vector of the same size as the input."); + return octave_value_list (); + } + 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[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 = copysign (INFINITY, igral); + warning ("cquad: 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; + retval = feval (fcn, fargs, 1); + if (retval.length () != 1 || !retval (0).is_real_matrix ()) + { + error + ("cquad: integrand must return a single, real-valued vector."); + return octave_value_list (); + } + Matrix effex = retval (0).matrix_value (); + if (effex.length () != ex.length ()) + { + error + ("cquad: integrand must return a single, real-valued vector of the same size as the input."); + return octave_value_list (); + } + 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[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 = copysign (INFINITY, igral); + warning ("cquad: 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]]); +/* printf + ("cquad: 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); +*/ + 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. */ +/* for (i = 0; i < nivals; i++) + { + iv = &(ivals[heap[i]]); + printf + ("cquad: 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); + } +*/ + /* Clean up and present the results. */ + retval (0) = igral; + if (nargout > 1) + retval (1) = err; + if (nargout > 2) + retval (2) = neval; + /* All is well that ends well. */ + return retval; +}