Mercurial > octave
view libinterp/corefcn/quadcc.cc @ 27918:b442ec6dda5c
use centralized file for copyright info for individual contributors
* COPYRIGHT.md: New file.
* In most other files, use "Copyright (C) YYYY-YYYY The Octave Project
Developers" instead of tracking individual names in separate source
files. The motivation is to reduce the effort required to update the
notices each year.
Until now, the Octave source files contained copyright notices that
list individual contributors. I adopted these file-scope copyright
notices because that is what everyone was doing 30 years ago in the
days before distributed version control systems. But now, with many
contributors and modern version control systems, having these
file-scope copyright notices causes trouble when we update copyright
years or refactor code.
Over time, the file-scope copyright notices may become outdated as new
contributions are made or code is moved from one file to
another. Sometimes people contribute significant patches but do not
add a line claiming copyright. Other times, people add a copyright
notice for their contribution but then a later refactoring moves part
or all of their contribution to another file and the notice is not
moved with the code. As a practical matter, moving such notices is
difficult -- determining what parts are due to a particular
contributor requires a time-consuming search through the project
history. Even managing the yearly update of copyright years is
problematic. We have some contributors who are no longer
living. Should we update the copyright dates for their contributions
when we release new versions? Probably not, but we do still want to
claim copyright for the project as a whole.
To minimize the difficulty of maintaining the copyright notices, I
would like to change Octave's sources to use what is described here:
https://softwarefreedom.org/resources/2012/ManagingCopyrightInformation.html
in the section "Maintaining centralized copyright notices":
The centralized notice approach consolidates all copyright
notices in a single location, usually a top-level file.
This file should contain all of the copyright notices
provided project contributors, unless the contribution was
clearly insignificant. It may also credit -- without a copyright
notice -- anyone who helped with the project but did not
contribute code or other copyrighted material.
This approach captures less information about contributions
within individual files, recognizing that the DVCS is better
equipped to record those details. As we mentioned before, it
does have one disadvantage as compared to the file-scope
approach: if a single file is separated from the distribution,
the recipient won't see the contributors' copyright notices.
But this can be easily remedied by including a single
copyright notice in each file's header, pointing to the
top-level file:
Copyright YYYY-YYYY The Octave Project Developers
See the COPYRIGHT file at the top-level directory
of this distribution or at https://octave.org/COPYRIGHT.html.
followed by the usual GPL copyright statement.
For more background, see the discussion here:
https://lists.gnu.org/archive/html/octave-maintainers/2020-01/msg00009.html
Most files in the following directories have been skipped intentinally
in this changeset:
doc
libgui/qterminal
liboctave/external
m4
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Mon, 06 Jan 2020 15:38:17 -0500 |
parents | f426dd9750a3 |
children | 1891570abac8 |
line wrap: on
line source
/* Copyright (C) 2010-2019 The Octave Project Developers See the file COPYRIGHT.md in the top-level directory of this distribution or <https://octave.org/COPYRIGHT.html/>. 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, 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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 ])) */