* Re: [PATCH] Adding fixed-order Gauss-Legendre integration routines
2009-06-23 16:52 ` Brian Gough
@ 2009-06-23 16:58 ` Rhys Ulerich
2009-06-24 16:49 ` Rhys Ulerich
2009-06-28 20:41 ` Brian Gough
0 siblings, 2 replies; 6+ messages in thread
From: Rhys Ulerich @ 2009-06-23 16:58 UTC (permalink / raw)
To: Brian Gough; +Cc: gsl-discuss
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> Could you redo the patch without the whitespace changes to make it
> easier to see the actual changes, thanks.
Definitely (attached) Didn't know about that git format-patch
argument until just now...
- Rhys
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From 5b39872f96668df87c912448aa8600bb142cef9c Mon Sep 17 00:00:00 2001
From: Rhys Ulerich <rhys.ulerich@gmail.com>
Date: Sun, 21 Jun 2009 14:47:06 -0500
Subject: [PATCH] Added gsl_integration_glfixed routines
Added fixed order Gauss-Legendre quadrature with
high precision weights based on the work of
Pavel Holoborodko (http://www.holoborodko.com/pavel/?page_id=679).
Includes implementation, documentation, and unit tests.
---
AUTHORS | 1 +
NEWS | 4 +
doc/integration.texi | 35 ++++-
integration/Makefile.am | 2 +-
integration/glfixed.c | 435 +++++++++++++++++++++++++++++++++++++++++
integration/gsl_integration.h | 26 +++
integration/test.c | 116 +++++++++++-
integration/tests.c | 30 +++
integration/tests.h | 10 +
9 files changed, 655 insertions(+), 4 deletions(-)
create mode 100644 integration/glfixed.c
diff --git a/AUTHORS b/AUTHORS
index fdbf33a..13e416b 100644
--- a/AUTHORS
+++ b/AUTHORS
@@ -25,3 +25,4 @@ Ivo Alxneit (ivo.alxneit@psi.ch) - multidimensional minimization, wavelet
transforms
Jason H. Stover (jason@sakla.net) - cumulative distribution functions
Patrick Alken <patrick.alken@colorado.edu> - nonsymmetric and generalized eigensystems, B-splines
+Pavel Holoborodko <pavel@holoborodko.com> - fixed order Gauss-Legendre quadrature
diff --git a/NEWS b/NEWS
index 9ffa0b2..bec0a85 100644
--- a/NEWS
+++ b/NEWS
@@ -1,5 +1,9 @@
* What is new in gsl-1.12+:
+** Added gsl_integrate_glfixed routines for performing fixed order
+ Gauss-Legendre integration
+ (Pavel Holoborodko, Konstantin Holoborodko, Rhys Ulerich)
+
** Improved the implementation of gsl_ran_discrete_preproc to avoid
internal errors due to inconsistencies from excess precision on
some platforms.
diff --git a/doc/integration.texi b/doc/integration.texi
index 33c9776..d576c5a 100644
--- a/doc/integration.texi
+++ b/doc/integration.texi
@@ -12,7 +12,9 @@ logarithmic singularities, computation of Cauchy principal values and
oscillatory integrals. The library reimplements the algorithms used in
@sc{quadpack}, a numerical integration package written by Piessens,
Doncker-Kapenga, Uberhuber and Kahaner. Fortran code for @sc{quadpack} is
-available on Netlib.
+available on Netlib. Also included are non-adaptive, fixed-order
+Gauss-Legendre integration routines with high precision coefficients
+by Pavel Holoborodko.
The functions described in this chapter are declared in the header file
@file{gsl_integration.h}.
@@ -28,6 +30,7 @@ The functions described in this chapter are declared in the header file
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
+* Fixed order Gauss-Legendre integration::
* Numerical integration error codes::
* Numerical integration examples::
* Numerical integration References and Further Reading::
@@ -776,6 +779,36 @@ The integration over each subinterval uses the memory provided by
@end deftypefun
+@node Fixed order Gauss-Legendre integration
+@section Gauss-Legendre integration
+
+The fixed-order Gauss-Legendre integration routines are provided for fast
+integration of smooth functions with known polynomial order. The
+@math{n}-point Gauss-Legendre rule is exact for polynomials of order
+@math{2*n-1} or less. For example, these rules are useful when integrating
+basis functions to form mass matrices for the Galerkin method. Unlike other
+numerical integration routines within the library, these routines do not accept
+absolute or relative error bounds.
+
+@deftypefun {gsl_integration_glfixed_table *} gsl_integration_glfixed_table_alloc (size_t @var{n})
+@tpindex gsl_integration_glfixed_table
+This function determines the Gauss-Legendre abscissae and weights necessary for
+an @math{n}-point fixed order integration scheme. If possible, high precision
+precomputed coefficients are used. If precomputed weights are not available,
+lower precision coefficients are computed on the fly.
+@end deftypefun
+
+@deftypefun double gsl_integration_glfixed (const gsl_function * @var{f}, double @var{a}, double @var{b}, {gsl_integration_glfixed_table *} @var{t})
+This function applies the Gauss-Legendre integration rule
+contained in table @var{t} and returns the result.
+@end deftypefun
+
+
+@deftypefun void gsl_integration_glfixed_table_alloc (gsl_integration_glfixed_table * @var{t})
+@tpindex gsl_integration_glfixed_table
+This function frees the memory associated with the table @var{t}.
+@end deftypefun
+
@node Numerical integration error codes
@section Error codes
diff --git a/integration/Makefile.am b/integration/Makefile.am
index 12c5b6f..f5d6da7 100644
--- a/integration/Makefile.am
+++ b/integration/Makefile.am
@@ -2,7 +2,7 @@ noinst_LTLIBRARIES = libgslintegration.la
INCLUDES = -I$(top_srcdir)
-libgslintegration_la_SOURCES = qk15.c qk21.c qk31.c qk41.c qk51.c qk61.c qk.c qng.c qng.h qag.c qags.c qagp.c workspace.c qcheb.c qawc.c qmomo.c qaws.c qmomof.c qawo.c qawf.c
+libgslintegration_la_SOURCES = qk15.c qk21.c qk31.c qk41.c qk51.c qk61.c qk.c qng.c qng.h qag.c qags.c qagp.c workspace.c qcheb.c qawc.c qmomo.c qaws.c qmomof.c qawo.c qawf.c glfixed.c
pkginclude_HEADERS = gsl_integration.h
noinst_HEADERS = qpsrt.c qpsrt2.c qelg.c qc25c.c qc25s.c qc25f.c ptsort.c util.c err.c positivity.c append.c initialise.c set_initial.c reset.c
diff --git a/integration/glfixed.c b/integration/glfixed.c
new file mode 100644
index 0000000..827ba42
--- /dev/null
+++ b/integration/glfixed.c
@@ -0,0 +1,435 @@
+/*
+ * Numerical Integration by Gauss-Legendre Quadrature Formulas of high orders.
+ * High-precision abscissas and weights are used.
+ *
+ * Original project homepage: http://www.holoborodko.com/pavel/?page_id=679
+ * Original contact e-mail: pavel@holoborodko.com
+ *
+ * Copyright (c)2007-2008 Pavel Holoborodko
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU Lesser General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ *
+ * Contributors:
+ * Konstantin Holoborodko - Optimization of Legendre polynomial computing.
+ * Rhys Ulerich - Inclusion within GNU Scientific Library.
+ *
+ */
+
+#include <config.h>
+#include <math.h>
+#include <float.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_integration.h>
+
+void gauss_legendre_tbl(int n, double* x, double* w, double eps);
+
+/* n = 2 */
+static double x2[1] = {0.5773502691896257645091488};
+static double w2[1] = {1.0000000000000000000000000};
+
+/* n = 4 */
+static double x4[2] = {0.3399810435848562648026658, 0.8611363115940525752239465};
+static double w4[2] = {0.6521451548625461426269361, 0.3478548451374538573730639};
+
+/* n = 6 */
+static double x6[3] = {0.2386191860831969086305017, 0.6612093864662645136613996, 0.9324695142031520278123016};
+static double w6[3] = {0.4679139345726910473898703, 0.3607615730481386075698335, 0.1713244923791703450402961};
+
+/* n = 8 */
+static double x8[4] = {0.1834346424956498049394761, 0.5255324099163289858177390, 0.7966664774136267395915539, 0.9602898564975362316835609};
+static double w8[4] = {0.3626837833783619829651504, 0.3137066458778872873379622, 0.2223810344533744705443560, 0.1012285362903762591525314};
+
+/* n = 10 */
+static double x10[5] = {0.1488743389816312108848260, 0.4333953941292471907992659, 0.6794095682990244062343274, 0.8650633666889845107320967, 0.9739065285171717200779640};
+static double w10[5] = {0.2955242247147528701738930, 0.2692667193099963550912269, 0.2190863625159820439955349, 0.1494513491505805931457763, 0.0666713443086881375935688};
+
+/* n = 12 */
+static double x12[6] = {0.1252334085114689154724414, 0.3678314989981801937526915, 0.5873179542866174472967024, 0.7699026741943046870368938, 0.9041172563704748566784659, 0.9815606342467192506905491};
+static double w12[6] = {0.2491470458134027850005624, 0.2334925365383548087608499, 0.2031674267230659217490645, 0.1600783285433462263346525, 0.1069393259953184309602547, 0.0471753363865118271946160};
+
+/* n = 14 */
+static double x14[7] = {0.1080549487073436620662447, 0.3191123689278897604356718, 0.5152486363581540919652907, 0.6872929048116854701480198, 0.8272013150697649931897947, 0.9284348836635735173363911, 0.9862838086968123388415973};
+static double w14[7] = {0.2152638534631577901958764, 0.2051984637212956039659241, 0.1855383974779378137417166, 0.1572031671581935345696019, 0.1215185706879031846894148, 0.0801580871597602098056333, 0.0351194603317518630318329};
+
+/* n = 16 */
+static double x16[8] = {0.0950125098376374401853193, 0.2816035507792589132304605, 0.4580167776572273863424194, 0.6178762444026437484466718, 0.7554044083550030338951012, 0.8656312023878317438804679, 0.9445750230732325760779884, 0.9894009349916499325961542};
+static double w16[8] = {0.1894506104550684962853967, 0.1826034150449235888667637, 0.1691565193950025381893121, 0.1495959888165767320815017, 0.1246289712555338720524763, 0.0951585116824927848099251, 0.0622535239386478928628438, 0.0271524594117540948517806};
+
+/* n = 18 */
+static double x18[9] = {0.0847750130417353012422619, 0.2518862256915055095889729, 0.4117511614628426460359318, 0.5597708310739475346078715, 0.6916870430603532078748911, 0.8037049589725231156824175, 0.8926024664975557392060606, 0.9558239495713977551811959, 0.9915651684209309467300160};
+static double w18[9] = {0.1691423829631435918406565, 0.1642764837458327229860538, 0.1546846751262652449254180, 0.1406429146706506512047313, 0.1225552067114784601845191, 0.1009420441062871655628140, 0.0764257302548890565291297, 0.0497145488949697964533349, 0.0216160135264833103133427};
+
+/* n = 20 */
+static double x20[10] = {0.0765265211334973337546404, 0.2277858511416450780804962, 0.3737060887154195606725482, 0.5108670019508270980043641, 0.6360536807265150254528367, 0.7463319064601507926143051, 0.8391169718222188233945291, 0.9122344282513259058677524, 0.9639719272779137912676661, 0.9931285991850949247861224};
+static double w20[10] = {0.1527533871307258506980843, 0.1491729864726037467878287, 0.1420961093183820513292983, 0.1316886384491766268984945, 0.1181945319615184173123774, 0.1019301198172404350367501, 0.0832767415767047487247581, 0.0626720483341090635695065, 0.0406014298003869413310400, 0.0176140071391521183118620};
+
+/* n = 32 */
+static double x32[16] = {0.0483076656877383162348126, 0.1444719615827964934851864, 0.2392873622521370745446032, 0.3318686022821276497799168, 0.4213512761306353453641194, 0.5068999089322293900237475, 0.5877157572407623290407455, 0.6630442669302152009751152, 0.7321821187402896803874267, 0.7944837959679424069630973, 0.8493676137325699701336930, 0.8963211557660521239653072, 0.9349060759377396891709191, 0.9647622555875064307738119, 0.9856115115452683354001750, 0.9972638618494815635449811};
+static double w32[16] = {0.0965400885147278005667648, 0.0956387200792748594190820, 0.0938443990808045656391802, 0.0911738786957638847128686, 0.0876520930044038111427715, 0.0833119242269467552221991, 0.0781938957870703064717409, 0.0723457941088485062253994, 0.0658222227763618468376501, 0.0586840934785355471452836, 0.0509980592623761761961632, 0.0428358980222266806568786, 0.0342738629130214331026877, 0.0253920653092620594557526, 0.0162743947309056706051706, 0.0070186100094700966004071};
+
+/* n = 64 */
+static double x64[32] = {0.0243502926634244325089558, 0.0729931217877990394495429, 0.1214628192961205544703765, 0.1696444204239928180373136, 0.2174236437400070841496487, 0.2646871622087674163739642, 0.3113228719902109561575127, 0.3572201583376681159504426, 0.4022701579639916036957668, 0.4463660172534640879849477, 0.4894031457070529574785263, 0.5312794640198945456580139, 0.5718956462026340342838781, 0.6111553551723932502488530, 0.6489654712546573398577612, 0.6852363130542332425635584, 0.7198818501716108268489402, 0.7528199072605318966118638, 0.7839723589433414076102205, 0.8132653151227975597419233, 0.8406292962525803627516915, 0.8659993981540928197607834, 0.8893154459951141058534040, 0.9105221370785028057563807, 0.9295691721319395758214902, 0.9464113748584028160624815, 0.9610087996520537189186141, 0.9733268277899109637418535, 0.9833362538846259569312993, 0.9910133714767443207393824, 0.9963401167719552793469245, 0.9993050417357721394569056};
+static double w64[32] = {0.0486909570091397203833654, 0.0485754674415034269347991, 0.0483447622348029571697695, 0.0479993885964583077281262, 0.0475401657148303086622822, 0.0469681828162100173253263, 0.0462847965813144172959532, 0.0454916279274181444797710, 0.0445905581637565630601347, 0.0435837245293234533768279, 0.0424735151236535890073398, 0.0412625632426235286101563, 0.0399537411327203413866569, 0.0385501531786156291289625, 0.0370551285402400460404151, 0.0354722132568823838106931, 0.0338051618371416093915655, 0.0320579283548515535854675, 0.0302346570724024788679741, 0.0283396726142594832275113, 0.0263774697150546586716918, 0.0243527025687108733381776, 0.0222701738083832541592983, 0.0201348231535302093723403, 0.0179517157756973430850453, 0.0157260304760247193219660, 0.0134630478967186425980608, 0.0111681394601311288185905, 0.0088467598263639477230309, 0.0065044579689783628561174, 0.0041470332605624676352875, 0.0017832807216964329472961};
+
+/* n = 96 */
+static double x96[48] = {0.0162767448496029695791346, 0.0488129851360497311119582, 0.0812974954644255589944713, 0.1136958501106659209112081, 0.1459737146548969419891073, 0.1780968823676186027594026, 0.2100313104605672036028472, 0.2417431561638400123279319, 0.2731988125910491414872722, 0.3043649443544963530239298, 0.3352085228926254226163256, 0.3656968614723136350308956, 0.3957976498289086032850002, 0.4254789884073005453648192, 0.4547094221677430086356761, 0.4834579739205963597684056, 0.5116941771546676735855097, 0.5393881083243574362268026, 0.5665104185613971684042502, 0.5930323647775720806835558, 0.6189258401254685703863693, 0.6441634037849671067984124, 0.6687183100439161539525572, 0.6925645366421715613442458, 0.7156768123489676262251441, 0.7380306437444001328511657, 0.7596023411766474987029704, 0.7803690438674332176036045, 0.8003087441391408172287961, 0.8194003107379316755389996, 0.8376235112281871214943028, 0.8549590334346014554627870, 0.8713885059092965028737748, 0.8868945174024204160568774, 0.9014606353158523413192327, 0.9150714231208980742058845, 0.9277124567223086909646905, 0.9393703397527552169318574, 0.9500327177844376357560989, 0.9596882914487425393000680, 0.9683268284632642121736594, 0.9759391745851364664526010, 0.9825172635630146774470458, 0.9880541263296237994807628, 0.9925439003237626245718923, 0.9959818429872092906503991, 0.9983643758631816777241494, 0.9996895038832307668276901};
+static double w96[48] = {0.0325506144923631662419614, 0.0325161187138688359872055, 0.0324471637140642693640128, 0.0323438225685759284287748, 0.0322062047940302506686671, 0.0320344562319926632181390, 0.0318287588944110065347537, 0.0315893307707271685580207, 0.0313164255968613558127843, 0.0310103325863138374232498, 0.0306713761236691490142288, 0.0302999154208275937940888, 0.0298963441363283859843881, 0.0294610899581679059704363, 0.0289946141505552365426788, 0.0284974110650853856455995, 0.0279700076168483344398186, 0.0274129627260292428234211, 0.0268268667255917621980567, 0.0262123407356724139134580, 0.0255700360053493614987972, 0.0249006332224836102883822, 0.0242048417923646912822673, 0.0234833990859262198422359, 0.0227370696583293740013478, 0.0219666444387443491947564, 0.0211729398921912989876739, 0.0203567971543333245952452, 0.0195190811401450224100852, 0.0186606796274114673851568, 0.0177825023160452608376142, 0.0168854798642451724504775, 0.0159705629025622913806165, 0.0150387210269949380058763, 0.0140909417723148609158616, 0.0131282295669615726370637, 0.0121516046710883196351814, 0.0111621020998384985912133, 0.0101607705350084157575876, 0.0091486712307833866325846, 0.0081268769256987592173824, 0.0070964707911538652691442, 0.0060585455042359616833167, 0.0050142027429275176924702, 0.0039645543384446866737334, 0.0029107318179349464084106, 0.0018539607889469217323359, 0.0007967920655520124294381};
+
+/* n = 100 */
+static double x100[50] = {0.0156289844215430828722167, 0.0468716824215916316149239, 0.0780685828134366366948174, 0.1091892035800611150034260, 0.1402031372361139732075146, 0.1710800805386032748875324, 0.2017898640957359972360489, 0.2323024818449739696495100, 0.2625881203715034791689293, 0.2926171880384719647375559, 0.3223603439005291517224766, 0.3517885263724217209723438, 0.3808729816246299567633625, 0.4095852916783015425288684, 0.4378974021720315131089780, 0.4657816497733580422492166, 0.4932107892081909335693088, 0.5201580198817630566468157, 0.5465970120650941674679943, 0.5725019326213811913168704, 0.5978474702471787212648065, 0.6226088602037077716041908, 0.6467619085141292798326303, 0.6702830156031410158025870, 0.6931491993558019659486479, 0.7153381175730564464599671, 0.7368280898020207055124277, 0.7575981185197071760356680, 0.7776279096494954756275514, 0.7968978923903144763895729, 0.8153892383391762543939888, 0.8330838798884008235429158, 0.8499645278795912842933626, 0.8660146884971646234107400, 0.8812186793850184155733168, 0.8955616449707269866985210, 0.9090295709825296904671263, 0.9216092981453339526669513, 0.9332885350430795459243337, 0.9440558701362559779627747, 0.9539007829254917428493369, 0.9628136542558155272936593, 0.9707857757637063319308979, 0.9778093584869182885537811, 0.9838775407060570154961002, 0.9889843952429917480044187, 0.9931249370374434596520099, 0.9962951347331251491861317, 0.9984919506395958184001634, 0.9997137267734412336782285};
+static double w100[50] = {0.0312554234538633569476425, 0.0312248842548493577323765, 0.0311638356962099067838183, 0.0310723374275665165878102, 0.0309504788504909882340635, 0.0307983790311525904277139, 0.0306161865839804484964594, 0.0304040795264548200165079, 0.0301622651051691449190687, 0.0298909795933328309168368, 0.0295904880599126425117545, 0.0292610841106382766201190, 0.0289030896011252031348762, 0.0285168543223950979909368, 0.0281027556591011733176483, 0.0276611982207923882942042, 0.0271926134465768801364916, 0.0266974591835709626603847, 0.0261762192395456763423087, 0.0256294029102081160756420, 0.0250575444815795897037642, 0.0244612027079570527199750, 0.0238409602659682059625604, 0.0231974231852541216224889, 0.0225312202563362727017970, 0.0218430024162473863139537, 0.0211334421125276415426723, 0.0204032326462094327668389, 0.0196530874944353058653815, 0.0188837396133749045529412, 0.0180959407221281166643908, 0.0172904605683235824393442, 0.0164680861761452126431050, 0.0156296210775460027239369, 0.0147758845274413017688800, 0.0139077107037187726879541, 0.0130259478929715422855586, 0.0121314576629794974077448, 0.0112251140231859771172216, 0.0103078025748689695857821, 0.0093804196536944579514182, 0.0084438714696689714026208, 0.0074990732554647115788287, 0.0065469484508453227641521, 0.0055884280038655151572119, 0.0046244500634221193510958, 0.0036559612013263751823425, 0.0026839253715534824194396, 0.0017093926535181052395294, 0.0007346344905056717304063};
+
+/* n = 128 */
+static double x128[64] = {0.0122236989606157641980521, 0.0366637909687334933302153, 0.0610819696041395681037870, 0.0854636405045154986364980, 0.1097942311276437466729747, 0.1340591994611877851175753, 0.1582440427142249339974755, 0.1823343059853371824103826, 0.2063155909020792171540580, 0.2301735642266599864109866, 0.2538939664226943208556180, 0.2774626201779044028062316, 0.3008654388776772026671541, 0.3240884350244133751832523, 0.3471177285976355084261628, 0.3699395553498590266165917, 0.3925402750332674427356482, 0.4149063795522750154922739, 0.4370245010371041629370429, 0.4588814198335521954490891, 0.4804640724041720258582757, 0.5017595591361444642896063, 0.5227551520511754784539479, 0.5434383024128103634441936, 0.5637966482266180839144308, 0.5838180216287630895500389, 0.6034904561585486242035732, 0.6228021939105849107615396, 0.6417416925623075571535249, 0.6602976322726460521059468, 0.6784589224477192593677557, 0.6962147083695143323850866, 0.7135543776835874133438599, 0.7304675667419088064717369, 0.7469441667970619811698824, 0.7629743300440947227797691, 0.7785484755064119668504941, 0.7936572947621932902433329, 0.8082917575079136601196422, 0.8224431169556438424645942, 0.8361029150609068471168753, 0.8492629875779689691636001, 0.8619154689395484605906323, 0.8740527969580317986954180, 0.8856677173453972174082924, 0.8967532880491581843864474, 0.9073028834017568139214859, 0.9173101980809605370364836, 0.9267692508789478433346245, 0.9356743882779163757831268, 0.9440202878302201821211114, 0.9518019613412643862177963, 0.9590147578536999280989185, 0.9656543664319652686458290, 0.9717168187471365809043384, 0.9771984914639073871653744, 0.9820961084357185360247656, 0.9864067427245862088712355, 0.9901278184917343833379303, 0.9932571129002129353034372, 0.9957927585349811868641612, 0.9977332486255140198821574, 0.9990774599773758950119878, 0.9998248879471319144736081};
+static double w128[64] = {0.0244461801962625182113259, 0.0244315690978500450548486, 0.0244023556338495820932980, 0.0243585572646906258532685, 0.0243002001679718653234426, 0.0242273192228152481200933, 0.0241399579890192849977167, 0.0240381686810240526375873, 0.0239220121367034556724504, 0.0237915577810034006387807, 0.0236468835844476151436514, 0.0234880760165359131530253, 0.0233152299940627601224157, 0.0231284488243870278792979, 0.0229278441436868469204110, 0.0227135358502364613097126, 0.0224856520327449668718246, 0.0222443288937997651046291, 0.0219897106684604914341221, 0.0217219495380520753752610, 0.0214412055392084601371119, 0.0211476464682213485370195, 0.0208414477807511491135839, 0.0205227924869600694322850, 0.0201918710421300411806732, 0.0198488812328308622199444, 0.0194940280587066028230219, 0.0191275236099509454865185, 0.0187495869405447086509195, 0.0183604439373313432212893, 0.0179603271850086859401969, 0.0175494758271177046487069, 0.0171281354231113768306810, 0.0166965578015892045890915, 0.0162550009097851870516575, 0.0158037286593993468589656, 0.0153430107688651440859909, 0.0148731226021473142523855, 0.0143943450041668461768239, 0.0139069641329519852442880, 0.0134112712886163323144890, 0.0129075627392673472204428, 0.0123961395439509229688217, 0.0118773073727402795758911, 0.0113513763240804166932817, 0.0108186607395030762476596, 0.0102794790158321571332153, 0.0097341534150068058635483, 0.0091830098716608743344787, 0.0086263777986167497049788, 0.0080645898904860579729286, 0.0074979819256347286876720, 0.0069268925668988135634267, 0.0063516631617071887872143, 0.0057726375428656985893346, 0.0051901618326763302050708, 0.0046045842567029551182905, 0.0040162549837386423131943, 0.0034255260409102157743378, 0.0028327514714579910952857, 0.0022382884309626187436221, 0.0016425030186690295387909, 0.0010458126793403487793129, 0.0004493809602920903763943};
+
+/* n = 256 */
+static double x256[128] = {0.0061239123751895295011702, 0.0183708184788136651179263, 0.0306149687799790293662786, 0.0428545265363790983812423, 0.0550876556946339841045614, 0.0673125211657164002422903, 0.0795272891002329659032271, 0.0917301271635195520311456, 0.1039192048105094036391969, 0.1160926935603328049407349, 0.1282487672706070947420496, 0.1403856024113758859130249, 0.1525013783386563953746068, 0.1645942775675538498292845, 0.1766624860449019974037218, 0.1887041934213888264615036, 0.2007175933231266700680007, 0.2127008836226259579370402, 0.2246522667091319671478783, 0.2365699497582840184775084, 0.2484521450010566668332427, 0.2602970699919425419785609, 0.2721029478763366095052447, 0.2838680076570817417997658, 0.2955904844601356145637868, 0.3072686197993190762586103, 0.3189006618401062756316834, 0.3304848656624169762291870, 0.3420194935223716364807297, 0.3535028151129699895377902, 0.3649331078236540185334649, 0.3763086569987163902830557, 0.3876277561945155836379846, 0.3988887074354591277134632, 0.4100898214687165500064336, 0.4212294180176238249768124, 0.4323058260337413099534411, 0.4433173839475273572169258, 0.4542624399175899987744552, 0.4651393520784793136455705, 0.4759464887869833063907375, 0.4866822288668903501036214, 0.4973449618521814771195124, 0.5079330882286160362319249, 0.5184450196736744762216617, 0.5288791792948222619514764, 0.5392340018660591811279362, 0.5495079340627185570424269, 0.5596994346944811451369074, 0.5698069749365687590576675, 0.5798290385590829449218317, 0.5897641221544543007857861, 0.5996107353629683217303882, 0.6093674010963339395223108, 0.6190326557592612194309676, 0.6286050494690149754322099, 0.6380831462729113686686886, 0.6474655243637248626170162, 0.6567507762929732218875002, 0.6659375091820485599064084, 0.6750243449311627638559187, 0.6840099204260759531248771, 0.6928928877425769601053416, 0.7016719143486851594060835, 0.7103456833045433133945663, 0.7189128934599714483726399, 0.7273722596496521265868944, 0.7357225128859178346203729, 0.7439624005491115684556831, 0.7520906865754920595875297, 0.7601061516426554549419068, 0.7680075933524456359758906, 0.7757938264113257391320526, 0.7834636828081838207506702, 0.7910160119895459945467075, 0.7984496810321707587825429, 0.8057635748129986232573891, 0.8129565961764315431364104, 0.8200276660989170674034781, 0.8269757238508125142890929, 0.8337997271555048943484439, 0.8404986523457627138950680, 0.8470714945172962071870724, 0.8535172676795029650730355, 0.8598350049033763506961731, 0.8660237584665545192975154, 0.8720825999954882891300459, 0.8780106206047065439864349, 0.8838069310331582848598262, 0.8894706617776108888286766, 0.8950009632230845774412228, 0.9003970057703035447716200, 0.9056579799601446470826819, 0.9107830965950650118909072, 0.9157715868574903845266696, 0.9206227024251464955050471, 0.9253357155833162028727303, 0.9299099193340056411802456, 0.9343446275020030942924765, 0.9386391748378148049819261, 0.9427929171174624431830761, 0.9468052312391274813720517, 0.9506755153166282763638521, 0.9544031887697162417644479, 0.9579876924111781293657904, 0.9614284885307321440064075, 0.9647250609757064309326123, 0.9678769152284894549090038, 0.9708835784807430293209233, 0.9737445997043704052660786, 0.9764595497192341556210107, 0.9790280212576220388242380, 0.9814496290254644057693031, 0.9837240097603154961666861, 0.9858508222861259564792451, 0.9878297475648606089164877, 0.9896604887450652183192437, 0.9913427712075830869221885, 0.9928763426088221171435338, 0.9942609729224096649628775, 0.9954964544810963565926471, 0.9965826020233815404305044, 0.9975192527567208275634088, 0.9983062664730064440555005, 0.9989435258434088565550263, 0.9994309374662614082408542, 0.9997684374092631861048786, 0.9999560500189922307348012};
+static double w256[128] = {0.0122476716402897559040703, 0.0122458343697479201424639, 0.0122421601042728007697281, 0.0122366493950401581092426, 0.0122293030687102789041463, 0.0122201222273039691917087, 0.0122091082480372404075141, 0.0121962627831147135181810, 0.0121815877594817721740476, 0.0121650853785355020613073, 0.0121467581157944598155598, 0.0121266087205273210347185, 0.0121046402153404630977578, 0.0120808558957245446559752, 0.0120552593295601498143471, 0.0120278543565825711612675, 0.0119986450878058119345367, 0.0119676359049058937290073, 0.0119348314595635622558732, 0.0119002366727664897542872, 0.0118638567340710787319046, 0.0118256971008239777711607, 0.0117857634973434261816901, 0.0117440619140605503053767, 0.0117005986066207402881898, 0.0116553800949452421212989, 0.0116084131622531057220847, 0.0115597048540436357726687, 0.0115092624770394979585864, 0.0114570935980906391523344, 0.0114032060430391859648471, 0.0113476078955454919416257, 0.0112903074958755095083676, 0.0112313134396496685726568, 0.0111706345765534494627109, 0.0111082800090098436304608, 0.0110442590908139012635176, 0.0109785814257295706379882, 0.0109112568660490397007968, 0.0108422955111147959952935, 0.0107717077058046266366536, 0.0106995040389797856030482, 0.0106256953418965611339617, 0.0105502926865814815175336, 0.0104733073841704030035696, 0.0103947509832117289971017, 0.0103146352679340150682607, 0.0102329722564782196569549, 0.0101497741990948656546341, 0.0100650535763063833094610, 0.0099788230970349101247339, 0.0098910956966958286026307, 0.0098018845352573278254988, 0.0097112029952662799642497, 0.0096190646798407278571622, 0.0095254834106292848118297, 0.0094304732257377527473528, 0.0093340483776232697124660, 0.0092362233309563026873787, 0.0091370127604508064020005, 0.0090364315486628736802278, 0.0089344947837582075484084, 0.0088312177572487500253183, 0.0087266159616988071403366, 0.0086207050884010143053688, 0.0085135010250224906938384, 0.0084050198532215357561803, 0.0082952778462352254251714, 0.0081842914664382699356198, 0.0080720773628734995009470, 0.0079586523687543483536132, 0.0078440334989397118668103, 0.0077282379473815556311102, 0.0076112830845456594616187, 0.0074931864548058833585998, 0.0073739657738123464375724, 0.0072536389258339137838291, 0.0071322239610753900716724, 0.0070097390929698226212344, 0.0068862026954463203467133, 0.0067616333001737987809279, 0.0066360495937810650445900, 0.0065094704150536602678099, 0.0063819147521078805703752, 0.0062534017395424012720636, 0.0061239506555679325423891, 0.0059935809191153382211277, 0.0058623120869226530606616, 0.0057301638506014371773844, 0.0055971560336829100775514, 0.0054633085886443102775705, 0.0053286415939159303170811, 0.0051931752508692809303288, 0.0050569298807868423875578, 0.0049199259218138656695588, 0.0047821839258926913729317, 0.0046437245556800603139791, 0.0045045685814478970686418, 0.0043647368779680566815684, 0.0042242504213815362723565, 0.0040831302860526684085998, 0.0039413976414088336277290, 0.0037990737487662579981170, 0.0036561799581425021693892, 0.0035127377050563073309711, 0.0033687685073155510120191, 0.0032242939617941981570107, 0.0030793357411993375832054, 0.0029339155908297166460123, 0.0027880553253277068805748, 0.0026417768254274905641208, 0.0024951020347037068508395, 0.0023480529563273120170065, 0.0022006516498399104996849, 0.0020529202279661431745488, 0.0019048808534997184044191, 0.0017565557363307299936069, 0.0016079671307493272424499, 0.0014591373333107332010884, 0.0013100886819025044578317, 0.0011608435575677247239706, 0.0010114243932084404526058, 0.0008618537014200890378141, 0.0007121541634733206669090, 0.0005623489540314098028152, 0.0004124632544261763284322, 0.0002625349442964459062875, 0.0001127890178222721755125};
+
+/* n = 512 */
+static double x512[256] = {0.0030649621851593961529232, 0.0091947713864329108047442, 0.0153242350848981855249677, 0.0214531229597748745137841, 0.0275812047119197840615246, 0.0337082500724805951232271, 0.0398340288115484476830396, 0.0459583107468090617788760, 0.0520808657521920701127271, 0.0582014637665182372392330, 0.0643198748021442404045319, 0.0704358689536046871990309, 0.0765492164062510452915674, 0.0826596874448871596284651, 0.0887670524624010326092165, 0.0948710819683925428909483, 0.1009715465977967786264323, 0.1070682171195026611052004, 0.1131608644449665349442888, 0.1192492596368204011642726, 0.1253331739174744696875513, 0.1314123786777137080093018, 0.1374866454852880630171099, 0.1435557460934960331730353, 0.1496194524497612685217272, 0.1556775367042018762501969, 0.1617297712181921097989489, 0.1677759285729161198103670, 0.1738157815779134454985394, 0.1798491032796159253350647, 0.1858756669698757062678115, 0.1918952461944840310240859, 0.1979076147616804833961808, 0.2039125467506523717658375, 0.2099098165200239314947094, 0.2158991987163350271904893, 0.2218804682825090362529109, 0.2278534004663095955103621, 0.2338177708287858931763260, 0.2397733552527061887852891, 0.2457199299509792442100997, 0.2516572714750633493170137, 0.2575851567233626262808095, 0.2635033629496102970603704, 0.2694116677712385990250046, 0.2753098491777350342234845, 0.2811976855389846383013106, 0.2870749556135979555970354, 0.2929414385572244074855835, 0.2987969139308507415853707, 0.3046411617090842500066247, 0.3104739622884204453906292, 0.3162950964954948840736281, 0.3221043455953188263048133, 0.3279014912994984240551598, 0.3336863157744371275728377, 0.3394586016495210024715049, 0.3452181320252866497799379, 0.3509646904815714220351686, 0.3566980610856456291665404, 0.3624180284003264285948478, 0.3681243774920730946589582, 0.3738168939390633631820054, 0.3794953638392505477003659, 0.3851595738184011246011504, 0.3908093110381124851478484, 0.3964443632038105531190080, 0.4020645185727269675414064, 0.4076695659618555307670286, 0.4132592947558876229222955, 0.4188334949151262845483445, 0.4243919569833786700527309, 0.4299344720958265754056529, 0.4354608319868747443376920, 0.4409708289979766581310498, 0.4464642560854375149423431, 0.4519409068281941054521446, 0.4574005754355712925046003, 0.4628430567550148032795831, 0.4682681462798000434299255, 0.4736756401567166435172692, 0.4790653351937284489919577, 0.4844370288676086658851277, 0.4897905193315498753147078, 0.4951256054227486308513615, 0.5004420866699643537454866, 0.5057397633010522419821678, 0.5110184362504699101074361, 0.5162779071667574777562819, 0.5215179784199908258105606, 0.5267384531092077401231844, 0.5319391350698066637637706, 0.5371198288809177797701793, 0.5422803398727461474300859, 0.5474204741338866161668468, 0.5525400385186102421644070, 0.5576388406541219339368088, 0.5627166889477890541289656, 0.5677733925943407059267120, 0.5728087615830374335557009, 0.5778226067048110674604360, 0.5828147395593744458765762, 0.5877849725623007456415722, 0.5927331189520721562306608, 0.5976589927970976321572046, 0.6025624090026994600382737, 0.6074431833180683777981926, 0.6123011323431869846644595, 0.6171360735357211818019505, 0.6219478252178793846326095, 0.6267362065832392490988318, 0.6315010377035416553494506, 0.6362421395354516935575740, 0.6409593339272863978194482, 0.6456524436257089753330001, 0.6503212922823892793136899, 0.6549657044606302753737317, 0.6595855056419602523685720, 0.6641805222326905300017078, 0.6687505815704384167754210, 0.6732955119306151731807642, 0.6778151425328787363350998, 0.6823093035475509635996236, 0.6867778261019991540425409, 0.6912205422869816079558685, 0.6956372851629569859851427, 0.7000278887663572307915895, 0.7043921881158238155354902, 0.7087300192184070848475163, 0.7130412190757284553416507, 0.7173256256901052441189100, 0.7215830780706378951153816, 0.7258134162392593745610389, 0.7300164812367465082373380, 0.7341921151286930346516885, 0.7383401610114441496854630, 0.7424604630179923197192207, 0.7465528663238341416942072, 0.7506172171527880300329109, 0.7546533627827725118134392, 0.7586611515515449130726824, 0.7626404328624002206015913, 0.7665910571898299050923647, 0.7705128760851404930018538, 0.7744057421820316760079998, 0.7782695092021337484565606, 0.7821040319605041647237048, 0.7859091663710830099561901, 0.7896847694521071791947507, 0.7934306993314830614379285, 0.7971468152521175267628422, 0.8008329775772070161862372, 0.8044890477954845355235412, 0.8081148885264243560855026, 0.8117103635254042266412553, 0.8152753376888249026732770, 0.8188096770591868005536242, 0.8223132488301235858819787, 0.8257859213513925068443721, 0.8292275641338212850768968, 0.8326380478542113781512150, 0.8360172443601974294381733, 0.8393650266750627227522641, 0.8426812690025104608329811, 0.8459658467313906883792422, 0.8492186364403826820199251, 0.8524395159026326312771384, 0.8556283640903464362590494, 0.8587850611793374495058711, 0.8619094885535289911058997, 0.8650015288094114678982387, 0.8680610657604539292849800, 0.8710879844414698938880857, 0.8740821711129372830049576, 0.8770435132652722985416439, 0.8799718996230570848337538, 0.8828672201492210155023745, 0.8857293660491754482355527, 0.8885582297749017921351663, 0.8913537050289927340242104, 0.8941156867686464718706125, 0.8968440712096138052506156, 0.8995387558300979345474886, 0.9021996393746068223597927, 0.9048266218577579723776075, 0.9074196045680354827749729, 0.9099784900714992329623006, 0.9125031822154460643436214, 0.9149935861320228175302595, 0.9174496082417910902748409, 0.9198711562572435822074657, 0.9222581391862718942794141, 0.9246104673355856526489486, 0.9269280523140828285786768, 0.9292108070361711277546193, 0.9314586457250403242837002, 0.9336714839158854164789745, 0.9358492384590804834007204, 0.9379918275233031229867813, 0.9400991705986093544775539, 0.9421711884994588697201555, 0.9442078033676905198230562, 0.9462089386754479255274304, 0.9481745192280551015654245, 0.9501044711668419871894147, 0.9519987219719197769813274, 0.9538572004649059479887372, 0.9556798368115988811866200, 0.9574665625246019772327448, 0.9592173104658971684737507, 0.9609320148493677311718534, 0.9626106112432703039637754, 0.9642530365726560206402068, 0.9658592291217406674540047, 0.9674291285362237773389233, 0.9689626758255565756615864, 0.9704598133651586944555050, 0.9719204848985835745206522, 0.9733446355396324773464471, 0.9747322117744170315712560, 0.9760831614633702416830300, 0.9773974338432058899681861, 0.9786749795288262664309572, 0.9799157505151781656726285, 0.9811197001790570947322311, 0.9822867832808596419166429, 0.9834169559662839640681455, 0.9845101757679783590716126, 0.9855664016071379024692622, 0.9865855937950491429603684, 0.9875677140345828729848910, 0.9885127254216350200148487, 0.9894205924465157453777048, 0.9902912809952868962106899, 0.9911247583510480415528399, 0.9919209931951714500244370, 0.9926799556084865573546763, 0.9934016170724147657859271, 0.9940859504700558793702825, 0.9947329300872282225027936, 0.9953425316134657151476031, 0.9959147321429772566997088, 0.9964495101755774022837600, 0.9969468456176038804367370, 0.9974067197828498321611183, 0.9978291153935628466036470, 0.9982140165816127953876923, 0.9985614088900397275573677, 0.9988712792754494246541769, 0.9991436161123782382453400, 0.9993784092025992514480161, 0.9995756497983108555936109, 0.9997353306710426625827368, 0.9998574463699794385446275, 0.9999419946068456536361287, 0.9999889909843818679872841};
+static double w512[256] = {0.0061299051754057857591564, 0.0061296748380364986664278, 0.0061292141719530834395471, 0.0061285231944655327693402, 0.0061276019315380226384508, 0.0061264504177879366912426, 0.0061250686964845654506976, 0.0061234568195474804311878, 0.0061216148475445832082156, 0.0061195428496898295184288, 0.0061172409038406284754329, 0.0061147090964949169991245, 0.0061119475227879095684795, 0.0061089562864885234199252, 0.0061057354999954793256260, 0.0061022852843330780981965, 0.0060986057691466529805468, 0.0060946970926976980917399, 0.0060905594018586731119147, 0.0060861928521074844014940, 0.0060815976075216427620556, 0.0060767738407720980583934, 0.0060717217331167509334394, 0.0060664414743936418598512, 0.0060609332630138177841916, 0.0060551973059538766317450, 0.0060492338187481899521175, 0.0060430430254808039978627, 0.0060366251587770195404584, 0.0060299804597946507400317, 0.0060231091782149633972884, 0.0060160115722332929281516, 0.0060086879085493424136484, 0.0060011384623571610896056, 0.0059933635173348036527221, 0.0059853633656336707715812, 0.0059771383078675312031423, 0.0059686886531012259272183, 0.0059600147188390547233923, 0.0059511168310128456267588, 0.0059419953239697077107922, 0.0059326505404594676575446, 0.0059230828316217905872556, 0.0059132925569729856313229, 0.0059032800843924967444267, 0.0058930457901090792634301, 0.0058825900586866627324847, 0.0058719132830099005255609, 0.0058610158642694068093892, 0.0058498982119466814015496, 0.0058385607437987230901727, 0.0058270038858423319934219, 0.0058152280723381015486124, 0.0058032337457741007324836, 0.0057910213568492471257818, 0.0057785913644563714469284, 0.0057659442356649741911390, 0.0057530804457036750229319, 0.0057400004779423555815070, 0.0057267048238739963699973, 0.0057131939830962084110906, 0.0056994684632924603629882, 0.0056855287802130018011102, 0.0056713754576554833823756, 0.0056570090274452746202723, 0.0056424300294154800102991, 0.0056276390113866542566918, 0.0056126365291462173626557, 0.0055974231464275703576030, 0.0055819994348889124461425, 0.0055663659740917603747899, 0.0055505233514791708235538, 0.0055344721623536666407146, 0.0055182130098548677502395, 0.0055017465049368275723757, 0.0054850732663450758090285, 0.0054681939205933684565648, 0.0054511091019401459196852, 0.0054338194523647001109732, 0.0054163256215430514316688, 0.0053986282668235365401123, 0.0053807280532021078251738, 0.0053626256532973455128155, 0.0053443217473251833447318, 0.0053258170230733487787774, 0.0053071121758755186716175, 0.0052882079085851914147269, 0.0052691049315492765055207, 0.0052498039625814025460136, 0.0052303057269349446719890, 0.0052106109572757724261988, 0.0051907203936547190996206, 0.0051706347834797735752665, 0.0051503548814879957194620, 0.0051298814497171563759039, 0.0051092152574771030281542, 0.0050883570813208522065339, 0.0050673077050154097256505, 0.0050460679195123198490183, 0.0050246385229179444874178, 0.0050030203204634735477834, 0.0049812141244746675595135, 0.0049592207543413337151533, 0.0049370410364865364724225, 0.0049146758043355438745290, 0.0048921258982845107556462, 0.0048693921656689000083132, 0.0048464754607316430993636, 0.0048233766445910410307843, 0.0048000965852084069516609, 0.0047766361573554516370718, 0.0047529962425814130594576, 0.0047291777291799312876071, 0.0047051815121556699579709, 0.0046810084931906855725376, 0.0046566595806105458869828, 0.0046321356893501986622283, 0.0046074377409195920619320, 0.0045825666633690479877601, 0.0045575233912543896535753, 0.0045323088656018247089130, 0.0045069240338725852313010, 0.0044813698499273259161146, 0.0044556472739902818017469, 0.0044297572726131868769073, 0.0044037008186389549258496, 0.0043774788911651239762643, 0.0043510924755070657234522, 0.0043245425631609613132305, 0.0042978301517665448748000, 0.0042709562450696162035304, 0.0042439218528843240022977, 0.0042167279910552210986262, 0.0041893756814190930634598, 0.0041618659517665616659011, 0.0041341998358034646067195, 0.0041063783731120129818357, 0.0040784026091117279353449, 0.0040502735950201579699371, 0.0040219923878133783908191, 0.0039935600501862743674273, 0.0039649776505126091053562, 0.0039362462628048786290012, 0.0039073669666739546834366, 0.0038783408472885172720108, 0.0038491689953342783540510, 0.0038198525069729982349166, 0.0037903924838012961884344, 0.0037607900328092568594835, 0.0037310462663388340021755, 0.0037011623020420531166926, 0.0036711392628390145554094, 0.0036409782768756986764252, 0.0036106804774815746300758, 0.0035802470031270143713799, 0.0035496789973805134987000, 0.0035189776088657205261605, 0.0034881439912182762045767, 0.0034571793030424645127888, 0.0034260847078676769483860, 0.0033948613741046917538288, 0.0033635104750017697209450, 0.0033320331886005682236783, 0.0033004306976918751358177, 0.0032687041897711642972145, 0.0032368548569939741987234, 0.0032048838961311115627642, 0.0031727925085236815030060, 0.0031405819000379459532169, 0.0031082532810200120618074, 0.0030758078662503522550163, 0.0030432468748981576780527, 0.0030105715304755267298129, 0.0029777830607914904130339, 0.0029448826979058762279357, 0.0029118716780830123435331, 0.0028787512417452737868732, 0.0028455226334264723964728, 0.0028121871017250922921949, 0.0027787458992573726197173, 0.0027452002826102393336092, 0.0027115515122940877888456, 0.0026778008526954179163600, 0.0026439495720293237639656, 0.0026099989422918391896635, 0.0025759502392121415000167, 0.0025418047422046148318992, 0.0025075637343207750815413, 0.0024732285022010581903898, 0.0024388003360264736029032, 0.0024042805294701247170072, 0.0023696703796485981535706, 0.0023349711870732236769383, 0.0023001842556012066042973, 0.0022653108923866345474810, 0.0022303524078313603367724, 0.0021953101155357629823745, 0.0021601853322493885355395, 0.0021249793778214727179358, 0.0020896935751513471947536, 0.0020543292501387313744068, 0.0020188877316339116255770, 0.0019833703513878098109153, 0.0019477784440019430461334, 0.0019121133468782766036998, 0.0018763764001689718921795, 0.0018405689467260314557679, 0.0018046923320508429542037, 0.0017687479042436241015783, 0.0017327370139527705642995, 0.0016966610143241088445575, 0.0016605212609500562072903, 0.0016243191118186897474239, 0.0015880559272627267421479, 0.0015517330699084184928942, 0.0015153519046243599371387, 0.0014789137984702174059640, 0.0014424201206453770259886, 0.0014058722424375164225552, 0.0013692715371711025869345, 0.0013326193801558190401403, 0.0012959171486349257824991, 0.0012591662217335559930561, 0.0012223679804069540808915, 0.0011855238073886605549070, 0.0011486350871386503607080, 0.0011117032057914329649653, 0.0010747295511041247428251, 0.0010377155124045074300544, 0.0010006624805390909706032, 0.0009635718478212056798501, 0.0009264450079791582697455, 0.0008892833561045005372012, 0.0008520882886004809402792, 0.0008148612031307819965602, 0.0007776034985686972438014, 0.0007403165749469818962867, 0.0007030018334087411433900, 0.0006656606761599343409382, 0.0006282945064244358390880, 0.0005909047284032230162400, 0.0005534927472403894647847, 0.0005160599690007674370993, 0.0004786078006679509066920, 0.0004411376501795405636493, 0.0004036509265333198797447, 0.0003661490400356268530141, 0.0003286334028523334162522, 0.0002911054302514885125319, 0.0002535665435705865135866, 0.0002160181779769908583388, 0.0001784618055459532946077, 0.0001408990173881984930124, 0.0001033319034969132362968, 0.0000657657316592401958310, 0.0000282526373739346920387};
+
+/* n = 1024 */
+static double x1024[512] = {0.0015332313560626384065387, 0.0045996796509132604743248, 0.0076660846940754867627839, 0.0107324176515422803327458, 0.0137986496899844401539048, 0.0168647519770217265449962, 0.0199306956814939776907024, 0.0229964519737322146859283, 0.0260619920258297325581921, 0.0291272870119131747190088, 0.0321923081084135882953009, 0.0352570264943374577920498, 0.0383214133515377145376052, 0.0413854398649847193632977, 0.0444490772230372159692514, 0.0475122966177132524285687, 0.0505750692449610682823599, 0.0536373663049299446784129, 0.0566991590022410150066456, 0.0597604185462580334848567, 0.0628211161513580991486838, 0.0658812230372023327000985, 0.0689407104290065036692117, 0.0719995495578116053446277, 0.0750577116607543749280791, 0.0781151679813377563695878, 0.0811718897697013033399379, 0.0842278482828915197978074, 0.0872830147851321356094940, 0.0903373605480943146797811, 0.0933908568511667930531222, 0.0964434749817259444449839, 0.0994951862354057706638682, 0.1025459619163678143852404, 0.1055957733375709917393206, 0.1086445918210413421754502, 0.1116923886981416930665228, 0.1147391353098412365177689, 0.1177848030069850158450139, 0.1208293631505633191883714, 0.1238727871119809777282145, 0.1269150462733265659711591, 0.1299561120276415015747167, 0.1329959557791890421802183, 0.1360345489437231767245806, 0.1390718629487574087024745, 0.1421078692338334288514767, 0.1451425392507896747338214, 0.1481758444640297746894331, 0.1512077563507908736360111, 0.1542382464014118381930443, 0.1572672861196013386077717, 0.1602948470227058049622614, 0.1633209006419772551419632, 0.1663454185228409920472972, 0.1693683722251631675310675, 0.1723897333235182105457458, 0.1754094734074561169859457, 0.1784275640817695987127083, 0.1814439769667610892475458, 0.1844586836985096036255346, 0.1874716559291374498981239, 0.1904828653270767897777182, 0.1934922835773360459175133, 0.1964998823817661533215037, 0.1995056334593266523810493, 0.2025095085463516210358758, 0.2055114793968154435588961, 0.2085115177825984134657778, 0.2115095954937521680517391, 0.2145056843387649520596422, 0.2174997561448267079850562, 0.2204917827580939905255947, 0.2234817360439547026834844, 0.2264695878872926510320010, 0.2294553101927519176581055, 0.2324388748850010462953415, 0.2354202539089970401627982, 0.2383994192302491690277166, 0.2413763428350825830111093, 0.2443509967309017306575811, 0.2473233529464535787923793, 0.2502933835320906316905658, 0.2532610605600337470850902, 0.2562263561246347465424530, 0.2591892423426388177365829, 0.2621496913534467061535080, 0.2651076753193766937613805, 0.2680631664259263621824189, 0.2710161368820341379053566, 0.2739665589203406170790369, 0.2769144047974496674298651, 0.2798596467941893048479266, 0.2828022572158723421886958, 0.2857422083925568078394062, 0.2886794726793061316013119, 0.2916140224564490954412652, 0.2945458301298395466682397, 0.2974748681311158710926665, 0.3004011089179602237287060, 0.3033245249743575146018584, 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0.5843066264598643017272666, 0.5867924118077737578782574, 0.5892726794317383594853053, 0.5917474060093159907610475, 0.5942165682701680800580147, 0.5966801429962784154186793, 0.5991381070221714681281111, 0.6015904372351302222163013, 0.6040371105754135078618616, 0.6064781040364728366534687, 0.6089133946651687366701116, 0.6113429595619865853458987, 0.6137667758812519380899084, 0.6161848208313453506363029, 0.6185970716749166931046915, 0.6210035057290989537555048, 0.6234041003657215304299416, 0.6257988330115230076688675, 0.6281876811483634175098794, 0.6305706223134359819666081, 0.6329476340994783351992008, 0.6353186941549832233898213, 0.6376837801844086803419153, 0.6400428699483876768269192, 0.6423959412639372417070377, 0.6447429720046670528676835, 0.6470839401009874959981582, 0.6494188235403171892641570, 0.6517476003672899719207013, 0.6540702486839613549191454, 0.6563867466500144315669620, 0.6586970724829652463040876, 0.6610012044583676196647058, 0.6632991209100174274984589, 0.6655908002301563325302097, 0.6678762208696749663426270, 0.6701553613383155598710345, 0.6724282002048740205051479, 0.6746947160974014538975312, 0.6769548877034051285838219, 0.6792086937700488815250166, 0.6814561131043529626873631, 0.6836971245733933167806834, 0.6859317071045003002812397, 0.6881598396854568318705713, 0.6903815013646959744270519, 0.6925966712514979467122689, 0.6948053285161865628996815, 0.6970074523903250980984011, 0.6992030221669115780303307, 0.7013920172005734910243170, 0.7035744169077619204963997, 0.7057502007669450960906928, 0.7079193483188013616608982, 0.7100818391664115582779368, 0.7122376529754508204546805, 0.7143867694743797837842896, 0.7165291684546352021941915, 0.7186648297708199730232898, 0.7207937333408925681355609, 0.7229158591463558692887801, 0.7250311872324454059827217, 0.7271396977083169940167956, 0.7292413707472337729927181, 0.7313361865867526410034676, 0.7334241255289100847554419, 0.7355051679404074033764222, 0.7375792942527953241676460, 0.7396464849626580085640129, 0.7417067206317964465721772, 0.7437599818874112379620360, 0.7458062494222847584928838, 0.7478455039949627094612890, 0.7498777264299350488635483, 0.7519028976178163024713854, 0.7539209985155252531253957, 0.7559320101464640065565832, 0.7579359136006964320521972, 0.7599326900351259762879594, 0.7619223206736728486546595, 0.7639047868074505764130149, 0.7658800697949419280166093, 0.7678481510621742029486694, 0.7698090121028938864243967, 0.7717626344787406673165402, 0.7737089998194208176678866, 0.7756480898228799321603470, 0.7775798862554750259163361, 0.7795043709521459890141759, 0.7814215258165863961053031, 0.7833313328214136695271245, 0.7852337740083385943114429, 0.7871288314883341834944720, 0.7890164874418038921405657, 0.7908967241187491784979139, 0.7927695238389364107105941, 0.7946348689920631175175217, 0.7964927420379235813750136, 0.7983431255065737724458586, 0.8001860019984956219039900, 0.8020213541847606330100649, 0.8038491648071928284194859, 0.8056694166785310321906380, 0.8074820926825904849673728, 0.8092871757744237908160400, 0.8110846489804811942036542, 0.8128744953987701856100790, 0.8146566981990144342734272, 0.8164312406228120465742028, 0.8181981059837931485700490, 0.8199572776677767911993239, 0.8217087391329271766780945, 0.8234524739099092046215225, 0.8251884656020433364270094, 0.8269166978854597764628854, 0.8286371545092519686128428, 0.8303498192956294067327593, 0.8320546761400697575830038, 0.8337517090114702948057846, 0.8354409019522986425235764, 0.8371222390787428271411563, 0.8387957045808606359402829, 0.8404612827227282810625704, 0.8421189578425883674826439, 0.8437687143529971635802028, 0.8454105367409711729261812, 0.8470444095681330059047621, 0.8486703174708565497995875, 0.8502882451604114359791023, 0.8518981774231068028225812, 0.8535000991204343530350070, 0.8550939951892107040056078, 0.8566798506417190298715048, 0.8582576505658499939545848, 0.8598273801252419702463831, 0.8613890245594205526224495, 0.8629425691839373504743648, 0.8644879993905080694542896, 0.8660253006471498760336444, 0.8675544584983180445842596, 0.8690754585650418856970762, 0.8705882865450599544602407, 0.8720929282129545374252050, 0.8735893694202854169962281, 0.8750775960957229119854680, 0.8765575942451801930826613, 0.8780293499519448719952049, 0.8794928493768098630212838, 0.8809480787582035158255322, 0.8823950244123190181935674, 0.8838336727332430675485994, 0.8852640101930838100201983, 0.8866860233420980458621863, 0.8880996988088177000235219, 0.8895050233001755566829532, 0.8909019836016302565651375, 0.8922905665772905558628607, 0.8936707591700388455969280, 0.8950425484016539302522575, 0.8964059213729330645356690, 0.8977608652638132471078410, 0.8991073673334917701488930, 0.9004454149205460236240486, 0.9017749954430525531228459, 0.9030960963987053701523781, 0.9044087053649335137720782, 0.9057128099990178624646022, 0.9070083980382071951444166, 0.9082954572998335002127549, 0.9095739756814265315746820, 0.9108439411608276105410847, 0.9121053417963026725455006, 0.9133581657266545576127977, 0.9146024011713345435238301, 0.9158380364305531206273175, 0.9170650598853900072573273, 0.9182834599979034047218800, 0.9194932253112384908353520, 0.9206943444497351509745089, 0.9218868061190349456451742, 0.9230705991061873135537215, 0.9242457122797550091847637, 0.9254121345899187738936182, 0.9265698550685812395293315, 0.9277188628294700636112689, 0.9288591470682402950895005, 0.9299906970625759697264543, 0.9311135021722909341445515, 0.9322275518394288975917975, 0.9333328355883627104845635, 0.9344293430258928687940732, 0.9355170638413452433503852, 0.9365959878066680331449597, 0.9376661047765279417201973, 0.9387274046884055757416456, 0.9397798775626900648558921, 0.9408235135027729019444869, 0.9418583026951410028915762, 0.9428842354094689849902736, 0.9439013019987106631201510, 0.9449094928991897628355911, 0.9459087986306898495121205, 0.9468992097965434727052183, 0.9478807170837205248834878, 0.9488533112629158137054760, 0.9498169831886358470168335, 0.9507717237992848297519245, 0.9517175241172498719314184, 0.9526543752489854069548347, 0.9535822683850968193944507, 0.9545011948004232815044368, 0.9554111458541197976665483, 0.9563121129897384560011695, 0.9572040877353088863799924, 0.9580870617034179240840996, 0.9589610265912884783587268, 0.9598259741808576051234879, 0.9606818963388537831043733, 0.9615287850168733926613630, 0.9623666322514563965930439, 0.9631954301641612222071790, 0.9640151709616388439537466, 0.9648258469357060659245549, 0.9656274504634180035311332, 0.9664199740071397636802195, 0.9672034101146173227737943, 0.9679777514190476018682591, 0.9687429906391477383350273, 0.9694991205792235533724866, 0.9702461341292372147270016, 0.9709840242648740939883669, 0.9717127840476088178328839, 0.9724324066247705125950353, 0.9731428852296072415565604, 0.9738442131813496343496072, 0.9745363838852737078785517, 0.9752193908327628781730396, 0.9758932276013691625928266, 0.9765578878548735718130775, 0.9772133653433456910269459, 0.9778596539032024498104955, 0.9784967474572660801033674, 0.9791246400148212617670490, 0.9797433256716714551911835, 0.9803527986101944204270933, 0.9809530530993969223366037, 0.9815440834949686212533729, 0.9821258842393351486632952, 0.9826984498617103674201996, 0.9832617749781478160230522, 0.9838158542915913364912672, 0.9843606825919248853856025, 0.9848962547560215275335618, 0.9854225657477916120303537, 0.9859396106182301300994116, 0.9864473845054632544104222, 0.9869458826347940594679517, 0.9874351003187474227003598, 0.9879150329571141058970610, 0.9883856760369940166627304, 0.9888470251328386495802522, 0.9892990759064927068006818, 0.9897418241072348978090276, 0.9901752655718179181502248, 0.9905993962245076069415402, 0.9910142120771212830473891, 0.9914197092290652598522332, 0.9918158838673715386394944, 0.9922027322667336806727008, 0.9925802507895418581838653, 0.9929484358859170846092543, 0.9933072840937446245820355, 0.9936567920387065844051246, 0.9939969564343136839997662, 0.9943277740819362116746914, 0.9946492418708341635125525, 0.9949613567781865697596566, 0.9952641158691200113800912, 0.9955575162967363309635588, 0.9958415553021395435525955, 0.9961162302144619548145649, 0.9963815384508894965215124, 0.9966374775166862927999356, 0.9968840450052184754903082, 0.9971212385979772738362093, 0.9973490560646014135491635, 0.9975674952628988745188845, 0.9977765541388680773265018, 0.9979762307267185998745420, 0.9981665231488915727109186, 0.9983474296160799746514418, 0.9985189484272491654281575, 0.9986810779696581776171579, 0.9988338167188825964389443, 0.9989771632388403756649803, 0.9991111161818228462260355, 0.9992356742885348165163858, 0.9993508363881507486653971, 0.9994566013984000492749057, 0.9995529683257070064969677, 0.9996399362654382464576482, 0.9997175044023747284307007, 0.9997856720116889628341744, 0.9998444384611711916084367, 0.9998938032169419878731474, 0.9999337658606177711221103, 0.9999643261538894550943330, 0.9999854843850284447675914, 0.9999972450545584403516182};
+static double w1024[512] = {0.0030664603092439082115513, 0.0030664314747171934849726, 0.0030663738059349007324470, 0.0030662873034393008056861, 0.0030661719680437936084028, 0.0030660278008329004477528, 0.0030658548031622538363679, 0.0030656529766585847450783, 0.0030654223232197073064431, 0.0030651628450145009692318, 0.0030648745444828901040266, 0.0030645574243358210601357, 0.0030642114875552366740338, 0.0030638367373940482295700, 0.0030634331773761048702058, 0.0030630008112961604635720, 0.0030625396432198379186545, 0.0030620496774835909559465, 0.0030615309186946633309249, 0.0030609833717310455112352, 0.0030604070417414288079918, 0.0030598019341451569616257, 0.0030591680546321751827342, 0.0030585054091629766484119, 0.0030578140039685464545661, 0.0030570938455503030247440, 0.0030563449406800369760227, 0.0030555672963998474425352, 0.0030547609200220758572342, 0.0030539258191292371925135, 0.0030530620015739486603347, 0.0030521694754788558725307, 0.0030512482492365564619779, 0.0030502983315095211653578, 0.0030493197312300123682482, 0.0030483124576000001133114, 0.0030472765200910755723677, 0.0030462119284443619831693, 0.0030451186926704230517109, 0.0030439968230491688209395, 0.0030428463301297590067471, 0.0030416672247305038021562, 0.0030404595179387621506312, 0.0030392232211108374894710, 0.0030379583458718709642643, 0.0030366649041157321154111, 0.0030353429080049070377385, 0.0030339923699703840142628, 0.0030326133027115366251721, 0.0030312057191960043331307, 0.0030297696326595705460252, 0.0030283050566060381583022, 0.0030268120048071025720655, 0.0030252904913022221991274, 0.0030237405303984864452325, 0.0030221621366704811776946, 0.0030205553249601516777118, 0.0030189201103766630786495, 0.0030172565082962582916016, 0.0030155645343621134195681, 0.0030138442044841906616068, 0.0030120955348390887083441, 0.0030103185418698906302495, 0.0030085132422860092601062, 0.0030066796530630300711306, 0.0030048177914425515522176, 0.0030029276749320230818149, 0.0030010093213045803019478, 0.0029990627485988779939449, 0.0029970879751189204574353, 0.0029950850194338893942123, 0.0029930539003779692985814, 0.0029909946370501703558363, 0.0029889072488141488505262, 0.0029867917552980250862041, 0.0029846481763941988183689, 0.0029824765322591622023349, 0.0029802768433133102577897, 0.0029780491302407488518214, 0.0029757934139891002022209, 0.0029735097157693059028890, 0.0029711980570554274731990, 0.0029688584595844444331918, 0.0029664909453560499065010, 0.0029640955366324437529314, 0.0029616722559381232326340, 0.0029592211260596712038487, 0.0029567421700455418562030, 0.0029542354112058439815854, 0.0029517008731121217846274, 0.0029491385795971332348581, 0.0029465485547546259626151, 0.0029439308229391107008170, 0.0029412854087656322747309, 0.0029386123371095381418860, 0.0029359116331062444843108, 0.0029331833221509998552933, 0.0029304274298986463828860, 0.0029276439822633785324025, 0.0029248330054184994301727, 0.0029219945257961747508486, 0.0029191285700871841705750, 0.0029162351652406703883623, 0.0029133143384638857180205, 0.0029103661172219362530391, 0.0029073905292375236068160, 0.0029043876024906842306667, 0.0029013573652185263120627, 0.0028982998459149642555740, 0.0028952150733304507490135, 0.0028921030764717064173001, 0.0028889638846014470665859, 0.0028857975272381085212091, 0.0028826040341555690560623, 0.0028793834353828694269858, 0.0028761357612039305018167, 0.0028728610421572684947521, 0.0028695593090357078067012, 0.0028662305928860914743281, 0.0028628749250089892305081, 0.0028594923369584031789413, 0.0028560828605414710856927, 0.0028526465278181672904478, 0.0028491833711010012402964, 0.0028456934229547136488796, 0.0028421767161959702837564, 0.0028386332838930533848701, 0.0028350631593655507170153, 0.0028314663761840422592303, 0.0028278429681697845340603, 0.0028241929693943925796601, 0.0028205164141795195677262, 0.0028168133370965340702726, 0.0028130837729661949782821, 0.0028093277568583240752928, 0.0028055453240914762689974, 0.0028017365102326074839556, 0.0027979013510967402185435, 0.0027940398827466267692845, 0.0027901521414924101257281, 0.0027862381638912825390663, 0.0027822979867471417676962, 0.0027783316471102450029635, 0.0027743391822768604783394, 0.0027703206297889167653083, 0.0027662760274336497592617, 0.0027622054132432473587211, 0.0027581088254944918412282, 0.0027539863027083999392661, 0.0027498378836498606195970, 0.0027456636073272705694208, 0.0027414635129921673927833, 0.0027372376401388605206822, 0.0027329860285040598383428, 0.0027287087180665020331547, 0.0027244057490465746667821, 0.0027200771619059379749851, 0.0027157229973471443987056, 0.0027113432963132558499974, 0.0027069380999874587163979, 0.0027025074497926766073634, 0.0026980513873911808464073, 0.0026935699546841987126055, 0.0026890631938115194351518, 0.0026845311471510979446691, 0.0026799738573186563850015, 0.0026753913671672833892344, 0.0026707837197870311237119, 0.0026661509585045101038391, 0.0026614931268824817854798, 0.0026568102687194489357814, 0.0026521024280492437872770, 0.0026473696491406139791397, 0.0026426119764968062894804, 0.0026378294548551481626046, 0.0026330221291866270351630, 0.0026281900446954674651512, 0.0026233332468187060677353, 0.0026184517812257642618999, 0.0026135456938180188319369, 0.0026086150307283703078113, 0.0026036598383208091684657, 0.0025986801631899798721388, 0.0025936760521607427178014, 0.0025886475522877335418257, 0.0025835947108549212540321, 0.0025785175753751632172710, 0.0025734161935897584747222, 0.0025682906134679988291122, 0.0025631408832067177780710, 0.0025579670512298373098703, 0.0025527691661879125638030, 0.0025475472769576743594882, 0.0025423014326415695994010, 0.0025370316825672995489502, 0.0025317380762873559984451, 0.0025264206635785553113127, 0.0025210794944415703629476, 0.0025157146191004603745948, 0.0025103260880021986466869, 0.0025049139518161981960773, 0.0024994782614338353016280, 0.0024940190679679709626349, 0.0024885364227524702745874, 0.0024830303773417197267843, 0.0024775009835101424263432, 0.0024719482932517112531633, 0.0024663723587794599504176, 0.0024607732325249921551741, 0.0024551509671379883737605, 0.0024495056154857109065099, 0.0024438372306525067265426, 0.0024381458659393083172574, 0.0024324315748631324732279, 0.0024266944111565770692147, 0.0024209344287673158020275, 0.0024151516818575909099866, 0.0024093462248037038747545, 0.0024035181121955041103265, 0.0023976673988358756439882, 0.0023917941397402217940673, 0.0023858983901359478493246, 0.0023799802054619417548485, 0.0023740396413680528093376, 0.0023680767537145683786720, 0.0023620915985716886306938, 0.0023560842322189992961374, 0.0023500547111449424606655, 0.0023440030920462853929883, 0.0023379294318275874140606, 0.0023318337876006648123684, 0.0023257162166840538103394, 0.0023195767766024715869239, 0.0023134155250862753614165, 0.0023072325200709195436049, 0.0023010278196964109553481, 0.0022948014823067621287099, 0.0022885535664494426857857, 0.0022822841308748288053830, 0.0022759932345356507817318, 0.0022696809365864386804193, 0.0022633472963829660967620, 0.0022569923734816920218464, 0.0022506162276392008214839, 0.0022442189188116403333494, 0.0022378005071541580875846, 0.0022313610530203356561684, 0.0022249006169616211363732, 0.0022184192597267597736437, 0.0022119170422612227292520, 0.0022053940257066339981005, 0.0021988502714001954820607, 0.0021922858408741102242558, 0.0021857007958550038097087, 0.0021790951982633439377969, 0.0021724691102128581719720, 0.0021658225940099498722195, 0.0021591557121531123157498, 0.0021524685273323410114303, 0.0021457611024285442134846, 0.0021390335005129516400021, 0.0021322857848465214018174, 0.0021255180188793451473363, 0.0021187302662500514289029, 0.0021119225907852072963166, 0.0021050950564987181231273, 0.0020982477275912256713511, 0.0020913806684495044002679, 0.0020844939436458560249764, 0.0020775876179375023304007, 0.0020706617562659762464561, 0.0020637164237565111901030, 0.0020567516857174286800274, 0.0020497676076395242297101, 0.0020427642551954515246552, 0.0020357416942391048895728, 0.0020286999908050000513193, 0.0020216392111076532034194, 0.0020145594215409583780096, 0.0020074606886775631310555, 0.0020003430792682425467160, 0.0019932066602412715667394, 0.0019860514987017956507927, 0.0019788776619311997736447, 0.0019716852173864757651327, 0.0019644742326995879988655, 0.0019572447756768374356240, 0.0019499969142982240274419, 0.0019427307167168074883601, 0.0019354462512580664378677, 0.0019281435864192559230531, 0.0019208227908687633255086, 0.0019134839334454626590447, 0.0019061270831580672642844, 0.0018987523091844809062265, 0.0018913596808711472808775, 0.0018839492677323979370705, 0.0018765211394497986196010, 0.0018690753658714940398285, 0.0018616120170115510799024, 0.0018541311630493004367905, 0.0018466328743286767122991, 0.0018391172213575569552912, 0.0018315842748070976623218, 0.0018240341055110702429247, 0.0018164667844651949558009, 0.0018088823828264733221690, 0.0018012809719125190225581, 0.0017936626232008872833327, 0.0017860274083284027592567, 0.0017783753990904859184165, 0.0017707066674404779358362, 0.0017630212854889641021349, 0.0017553193255030957535871, 0.0017476008599059107299616, 0.0017398659612756523665312, 0.0017321147023450870266539, 0.0017243471560008201813452, 0.0017165633952826110422716, 0.0017087634933826857546100, 0.0017009475236450491562317, 0.0016931155595647951096823, 0.0016852676747874154134422, 0.0016774039431081072989678, 0.0016695244384710795200224, 0.0016616292349688570408253, 0.0016537184068415843295541, 0.0016457920284763272637533, 0.0016378501744063736542136, 0.0016298929193105323938983, 0.0016219203380124312385075, 0.0016139325054798132252838, 0.0016059294968238317366751, 0.0015979113872983442154825, 0.0015898782522992045381361, 0.0015818301673635540527516, 0.0015737672081691112886347, 0.0015656894505334603439125, 0.0015575969704133379579831, 0.0015494898439039192754876, 0.0015413681472381023085203, 0.0015332319567857911038062, 0.0015250813490531776215856, 0.0015169164006820223329593, 0.0015087371884489335424584, 0.0015005437892646454426166, 0.0014923362801732949073323, 0.0014841147383516970308228, 0.0014758792411086194189814, 0.0014676298658840552399621, 0.0014593666902484950408286, 0.0014510897919021973371136, 0.0014427992486744579821480, 0.0014344951385228783230315, 0.0014261775395326321501237, 0.0014178465299157314469528, 0.0014095021880102909474427, 0.0014011445922797915073771, 0.0013927738213123422970256, 0.0013843899538199418218713, 0.0013759930686377377783877, 0.0013675832447232857518263, 0.0013591605611558067629844, 0.0013507250971354436709363, 0.0013422769319825164387192, 0.0013338161451367762689788, 0.0013253428161566586165863, 0.0013168570247185350852537, 0.0013083588506159642151809, 0.0012998483737589411687807, 0.0012913256741731463215379, 0.0012827908319991927650686, 0.0012742439274918727294554, 0.0012656850410194029319476, 0.0012571142530626688591208, 0.0012485316442144679896043, 0.0012399372951787519644928, 0.0012313312867698677125706, 0.0012227136999117975374834, 0.0012140846156373981740056, 0.0012054441150876388205601, 0.0011967922795108381551550, 0.0011881291902619003419159, 0.0011794549288015500353964, 0.0011707695766955663898644, 0.0011620732156140160807669, 0.0011533659273304853455891, 0.0011446477937213110513287, 0.0011359188967648107958214, 0.0011271793185405120501566, 0.0011184291412283803494364, 0.0011096684471080465391373, 0.0011008973185580330843445, 0.0010921158380549794491381, 0.0010833240881728665534171, 0.0010745221515822403144596, 0.0010657101110494342805238, 0.0010568880494357913638046, 0.0010480560496968846800697, 0.0010392141948817375023057, 0.0010303625681320423357186, 0.0010215012526813791214350, 0.0010126303318544325762649, 0.0010037498890662086758941, 0.0009948600078212502888805, 0.0009859607717128519688418, 0.0009770522644222739122264, 0.0009681345697179550890732, 0.0009592077714547255541688, 0.0009502719535730179460261, 0.0009413272000980781811114, 0.0009323735951391753507612, 0.0009234112228888108282347, 0.0009144401676219265933610, 0.0009054605136951127822476, 0.0008964723455458144695262, 0.0008874757476915376906225, 0.0008784708047290547115472, 0.0008694576013336085537138, 0.0008604362222581167813022, 0.0008514067523323745586954, 0.0008423692764622569855308, 0.0008333238796289207169173, 0.0008242706468880048763834, 0.0008152096633688312691343, 0.0008061410142736039032099, 0.0007970647848766078261514, 0.0007879810605234072847989, 0.0007788899266300432158601, 0.0007697914686822300749096, 0.0007606857722345520114971, 0.0007515729229096583980656, 0.0007424530063974587204051, 0.0007333261084543168373926, 0.0007241923149022446178008, 0.0007150517116280949619884, 0.0007059043845827542163241, 0.0006967504197803339882351, 0.0006875899032973623698204, 0.0006784229212719745780188, 0.0006692495599031030193850, 0.0006600699054496667875923, 0.0006508840442297606018626, 0.0006416920626198431946113, 0.0006324940470539251567018, 0.0006232900840227562488244, 0.0006140802600730121876541, 0.0006048646618064809156059, 0.0005956433758792483631993, 0.0005864164890008837132649, 0.0005771840879336241764943, 0.0005679462594915592881427, 0.0005587030905398147360662, 0.0005494546679937357307118, 0.0005402010788180699282026, 0.0005309424100261499182844, 0.0005216787486790752896494, 0.0005124101818848942860548, 0.0005031367967977850677401, 0.0004938586806172365939677, 0.0004845759205872291441124, 0.0004752886039954144966810, 0.0004659968181722957880391, 0.0004567006504904070755681, 0.0004474001883634926336095, 0.0004380955192456860150653, 0.0004287867306306889171352, 0.0004194739100509498966958, 0.0004101571450768429896514, 0.0004008365233158462997325, 0.0003915121324117206363681, 0.0003821840600436882993131, 0.0003728523939256121308821, 0.0003635172218051749865499, 0.0003541786314630598135175, 0.0003448367107121305776064, 0.0003354915473966143456333, 0.0003261432293912849189248, 0.0003167918446006485317858, 0.0003074374809581322877037, 0.0002980802264252762217455, 0.0002887201689909301727620, 0.0002793573966704570567274, 0.0002699919975049447012834, 0.0002606240595604292032823, 0.0002512536709271339139118, 0.0002418809197187298044384, 0.0002325058940716253739001, 0.0002231286821442978268308, 0.0002137493721166826096154, 0.0002043680521896465790359, 0.0001949848105845827899210, 0.0001855997355431850062940, 0.0001762129153274925249194, 0.0001668244382203495280013, 0.0001574343925265138930609, 0.0001480428665748079976500, 0.0001386499487219861751244, 0.0001292557273595155266326, 0.0001198602909254695827354, 0.0001104637279257437565603, 0.0001010661269730276014588, 0.0000916675768613669107254, 0.0000822681667164572752810, 0.0000728679863190274661367, 0.0000634671268598044229933, 0.0000540656828939400071988, 0.0000446637581285753393838, 0.0000352614859871986975067, 0.0000258591246764618586716, 0.0000164577275798968681068, 0.0000070700764101825898713};
+
+/* n = 3 */
+static double x3[2] = {0.0000000000000000000000000, 0.7745966692414833770358531};
+static double w3[2] = {0.8888888888888888888888889, 0.5555555555555555555555556};
+
+/* n = 5 */
+static double x5[3] = {0.0000000000000000000000000, 0.5384693101056830910363144, 0.9061798459386639927976269};
+static double w5[3] = {0.5688888888888888888888889, 0.4786286704993664680412915, 0.2369268850561890875142640};
+
+/* n = 7 */
+static double x7[4] = {0.0000000000000000000000000, 0.4058451513773971669066064, 0.7415311855993944398638648, 0.9491079123427585245261897};
+static double w7[4] = {0.4179591836734693877551020, 0.3818300505051189449503698, 0.2797053914892766679014678, 0.1294849661688696932706114};
+
+/* n = 9 */
+static double x9[5] = {0.0000000000000000000000000, 0.3242534234038089290385380, 0.6133714327005903973087020, 0.8360311073266357942994298, 0.9681602395076260898355762};
+static double w9[5] = {0.3302393550012597631645251, 0.3123470770400028400686304, 0.2606106964029354623187429, 0.1806481606948574040584720, 0.0812743883615744119718922};
+
+/* n = 11 */
+static double x11[6] = {0.0000000000000000000000000, 0.2695431559523449723315320, 0.5190961292068118159257257, 0.7301520055740493240934163, 0.8870625997680952990751578, 0.9782286581460569928039380};
+static double w11[6] = {0.2729250867779006307144835, 0.2628045445102466621806889, 0.2331937645919904799185237, 0.1862902109277342514260976, 0.1255803694649046246346943, 0.0556685671161736664827537};
+
+/* n = 13 */
+static double x13[7] = {0.0000000000000000000000000, 0.2304583159551347940655281, 0.4484927510364468528779129, 0.6423493394403402206439846, 0.8015780907333099127942065, 0.9175983992229779652065478, 0.9841830547185881494728294};
+static double w13[7] = {0.2325515532308739101945895, 0.2262831802628972384120902, 0.2078160475368885023125232, 0.1781459807619457382800467, 0.1388735102197872384636018, 0.0921214998377284479144218, 0.0404840047653158795200216};
+
+/* n = 15 */
+static double x15[8] = {0.0000000000000000000000000, 0.2011940939974345223006283, 0.3941513470775633698972074, 0.5709721726085388475372267, 0.7244177313601700474161861, 0.8482065834104272162006483, 0.9372733924007059043077589, 0.9879925180204854284895657};
+static double w15[8] = {0.2025782419255612728806202, 0.1984314853271115764561183, 0.1861610000155622110268006, 0.1662692058169939335532009, 0.1395706779261543144478048, 0.1071592204671719350118695, 0.0703660474881081247092674, 0.0307532419961172683546284};
+
+/* n = 17 */
+static double x17[9] = {0.0000000000000000000000000, 0.1784841814958478558506775, 0.3512317634538763152971855, 0.5126905370864769678862466, 0.6576711592166907658503022, 0.7815140038968014069252301, 0.8802391537269859021229557, 0.9506755217687677612227170, 0.9905754753144173356754340};
+static double w17[9] = {0.1794464703562065254582656, 0.1765627053669926463252710, 0.1680041021564500445099707, 0.1540457610768102880814316, 0.1351363684685254732863200, 0.1118838471934039710947884, 0.0850361483171791808835354, 0.0554595293739872011294402, 0.0241483028685479319601100};
+
+/* n = 19 */
+static double x19[10] = {0.0000000000000000000000000, 0.1603586456402253758680961, 0.3165640999636298319901173, 0.4645707413759609457172671, 0.6005453046616810234696382, 0.7209661773352293786170959, 0.8227146565371428249789225, 0.9031559036148179016426609, 0.9602081521348300308527788, 0.9924068438435844031890177};
+static double w19[10] = {0.1610544498487836959791636, 0.1589688433939543476499564, 0.1527660420658596667788554, 0.1426067021736066117757461, 0.1287539625393362276755158, 0.1115666455473339947160239, 0.0914900216224499994644621, 0.0690445427376412265807083, 0.0448142267656996003328382, 0.0194617882297264770363120};
+
+/* Merge all together */
+static gsl_integration_glfixed_table glaw[] =
+{
+ { 2, x2, w2, 1 /* precomputed */ },
+ { 3, x3, w3, 1 },
+ { 4, x4, w4, 1 },
+ { 5, x5, w5, 1 },
+ { 6, x6, w6, 1 },
+ { 7, x7, w7, 1 },
+ { 8, x8, w8, 1 },
+ { 9, x9, w9, 1 },
+ { 10, x10, w10, 1 },
+ { 11, x11, w11, 1 },
+ { 12, x12, w12, 1 },
+ { 13, x13, w13, 1 },
+ { 14, x14, w14, 1 },
+ { 15, x15, w15, 1 },
+ { 16, x16, w16, 1 },
+ { 17, x17, w17, 1 },
+ { 18, x18, w18, 1 },
+ { 19, x19, w19, 1 },
+ { 20, x20, w20, 1 },
+ { 32, x32, w32, 1 },
+ { 64, x64, w64, 1 },
+ { 96, x96, w96, 1 },
+ { 100, x100, w100, 1 },
+ { 128, x128, w128, 1 },
+ { 256, x256, w256, 1 },
+ { 512, x512, w512, 1 },
+ { 1024, x1024, w1024, 1 }
+};
+static const size_t GLAWSIZE = sizeof(glaw)/sizeof(glaw[0]);
+
+
+gsl_integration_glfixed_table *
+ gsl_integration_glfixed_table_alloc (size_t n)
+{
+ int i;
+ double *x = 0;
+ double *w = 0;
+ gsl_integration_glfixed_table * retval = 0;
+
+ /* Kludgy error check: indices in the original algorithm use ints but GSL
+ * conventions are to take a size_t parameter for n */
+ if (n > INT_MAX)
+ {
+ GSL_ERROR_NULL ("Requested n is too large", GSL_EINVAL);
+ }
+
+ /* Use a predefined table of size n if possible */
+ for (i = 0; i < GLAWSIZE;i++)
+ {
+ if (n == glaw[i].n)
+ {
+ retval = &glaw[i];
+ break;
+ }
+ }
+
+ /* No predefined table is available. Generate one on the fly. */
+ if (retval == 0)
+ {
+ const int m = (n + 1) >> 1;
+
+ x = (double *) malloc(m * sizeof(double));
+ if (x == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for abscissae",
+ GSL_ENOMEM);
+ }
+
+ w = (double *) malloc(m * sizeof(double));
+ if (w == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for weights",
+ GSL_ENOMEM);
+ free(x);
+ }
+
+ retval = malloc(sizeof(gsl_integration_glfixed_table));
+ if (retval == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for table struct",
+ GSL_ENOMEM);
+ free(x);
+ free(w);
+ }
+
+ gauss_legendre_tbl(n, x, w, 1e-10 /* precision magic number */);
+
+ retval->n = n;
+ retval->x = x;
+ retval->w = w;
+ retval->precomputed = 0;
+ }
+
+ return retval;
+}
+
+void
+ gsl_integration_glfixed_table_free (gsl_integration_glfixed_table * t)
+{
+ /* We only free the table memory if we allocated it */
+ /* Leave precomputed, static tables alone */
+ if (! t->precomputed)
+ {
+ free(t->x);
+ free(t->w);
+ free(t);
+ }
+}
+
+/*
+Gauss-Legendre n-points quadrature, exact for polynomial of degree <=2n-1
+
+1. n - even:
+
+int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1)
+ = A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=0..n/2-1)
+ A = (b-a)/2,
+ B = (a+b)/2
+
+
+2. n - odd:
+
+int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1)
+ = A*w[0]*f(B)+A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=1..(n-1)/2)
+ A = (b-a)/2,
+ B = (a+b)/2
+*/
+
+double
+ gsl_integration_glfixed (const gsl_function *f,
+ double a,
+ double b,
+ const gsl_integration_glfixed_table * t)
+{
+ const double * const x = t->x;
+ const double * const w = t->w;
+ const int n = t->n;
+ double A, B, Ax, s;
+ int i, m;
+
+ m = (n + 1) >> 1;
+ A = 0.5 * (b - a);
+ B = 0.5 * (b + a);
+
+ if (n&1) /* n - odd */
+ {
+ s = w[0] * GSL_FN_EVAL(f,B);
+
+ for (i = 1;i < m;i++)
+ {
+ Ax = A * x[i];
+ s += w[i] * (GSL_FN_EVAL(f,B+Ax) + GSL_FN_EVAL(f,B-Ax));
+ }
+
+ }
+ else /* n - even */
+ {
+
+ s = 0.0;
+
+ for (i = 0;i < m;i++)
+ {
+ Ax = A * x[i];
+ s += w[i] * (GSL_FN_EVAL(f,B+Ax) + GSL_FN_EVAL(f,B-Ax));
+ }
+ }
+
+ return A*s;
+}
+
+/* Computing of abscissas and weights for Gauss-Legendre quadrature for any(reasonable) order n
+ [in] n - order of quadrature
+ [in] eps - required precision (must be eps>=macheps(double), usually eps = 1e-10 is ok)
+ [out]x - abscisass, size = (n+1)>>1
+ [out]w - weights, size = (n+1)>>1
+*/
+
+/* Look up table for fast calculation of Legendre polynomial for n<1024 */
+/* ltbl[k] = 1.0 - 1.0/(double)k; */
+static double ltbl[1024] = {0.00000000000000000000, 0.00000000000000000000, 0.50000000000000000000, 0.66666666666666674000, 0.75000000000000000000, 0.80000000000000004000, 0.83333333333333337000, 0.85714285714285721000, 0.87500000000000000000, 0.88888888888888884000, 0.90000000000000002000, 0.90909090909090906000, 0.91666666666666663000, 0.92307692307692313000, 0.92857142857142860000, 0.93333333333333335000, 0.93750000000000000000, 0.94117647058823528000, 0.94444444444444442000, 0.94736842105263164000, 0.94999999999999996000, 0.95238095238095233000, 0.95454545454545459000, 0.95652173913043481000, 0.95833333333333337000, 0.95999999999999996000, 0.96153846153846156000, 0.96296296296296302000, 0.96428571428571430000, 0.96551724137931039000, 0.96666666666666667000, 0.96774193548387100000, 0.96875000000000000000, 0.96969696969696972000, 0.97058823529411764000, 0.97142857142857142000, 0.97222222222222221000, 0.97297297297297303000, 0.97368421052631582000, 0.97435897435897434000, 0.97499999999999998000, 0.97560975609756095000, 0.97619047619047616000, 0.97674418604651159000, 0.97727272727272729000, 0.97777777777777775000, 0.97826086956521741000, 0.97872340425531912000, 0.97916666666666663000, 0.97959183673469385000, 0.97999999999999998000, 0.98039215686274506000, 0.98076923076923073000, 0.98113207547169812000, 0.98148148148148151000, 0.98181818181818181000, 0.98214285714285710000, 0.98245614035087714000, 0.98275862068965514000, 0.98305084745762716000, 0.98333333333333328000, 0.98360655737704916000, 0.98387096774193550000, 0.98412698412698418000, 0.98437500000000000000, 0.98461538461538467000, 0.98484848484848486000, 0.98507462686567160000, 0.98529411764705888000, 0.98550724637681164000, 0.98571428571428577000, 0.98591549295774650000, 0.98611111111111116000, 0.98630136986301364000, 0.98648648648648651000, 0.98666666666666669000, 0.98684210526315785000, 0.98701298701298701000, 0.98717948717948723000, 0.98734177215189878000, 0.98750000000000004000, 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0.99180327868852458000, 0.99186991869918695000, 0.99193548387096775000, 0.99199999999999999000, 0.99206349206349209000, 0.99212598425196852000, 0.99218750000000000000, 0.99224806201550386000, 0.99230769230769234000, 0.99236641221374045000, 0.99242424242424243000, 0.99248120300751874000, 0.99253731343283580000, 0.99259259259259258000, 0.99264705882352944000, 0.99270072992700731000, 0.99275362318840576000, 0.99280575539568350000, 0.99285714285714288000, 0.99290780141843971000, 0.99295774647887325000, 0.99300699300699302000, 0.99305555555555558000, 0.99310344827586206000, 0.99315068493150682000, 0.99319727891156462000, 0.99324324324324320000, 0.99328859060402686000, 0.99333333333333329000, 0.99337748344370858000, 0.99342105263157898000, 0.99346405228758172000, 0.99350649350649356000, 0.99354838709677418000, 0.99358974358974361000, 0.99363057324840764000, 0.99367088607594933000, 0.99371069182389937000, 0.99375000000000002000, 0.99378881987577639000, 0.99382716049382713000, 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0.99898270600203454000, 0.99898373983739841000, 0.99898477157360410000, 0.99898580121703850000, 0.99898682877406286000, 0.99898785425101211000, 0.99898887765419619000, 0.99898989898989898000, 0.99899091826437947000, 0.99899193548387100000, 0.99899295065458205000, 0.99899396378269623000, 0.99899497487437183000, 0.99899598393574296000, 0.99899699097291872000, 0.99899799599198402000, 0.99899899899899902000, 0.99900000000000000000, 0.99900099900099903000, 0.99900199600798401000, 0.99900299102691925000, 0.99900398406374502000, 0.99900497512437814000, 0.99900596421471177000, 0.99900695134061568000, 0.99900793650793651000, 0.99900891972249750000, 0.99900990099009901000, 0.99901088031651830000, 0.99901185770750989000, 0.99901283316880551000, 0.99901380670611439000, 0.99901477832512320000, 0.99901574803149606000, 0.99901671583087515000, 0.99901768172888017000, 0.99901864573110888000, 0.99901960784313726000, 0.99902056807051909000, 0.99902152641878672000, 0.99902248289345064000};
+
+void gauss_legendre_tbl(int n, double* x, double* w, double eps)
+{
+ double x0, x1, dx; /* Abscissas */
+ double w0, w1, dw; /* Weights */
+ double P0, P_1, P_2; /* Legendre polynomial values */
+ double dpdx; /* Legendre polynomial derivative */
+ int i, j, k, m; /* Iterators */
+ double t0, t1, t2, t3;
+
+ m = (n + 1) >> 1;
+
+ t0 = (1.0 - (1.0 - 1.0 / (double)n) / (8.0 * (double)n * (double)n));
+ t1 = 1.0 / (4.0 * (double)n + 2.0);
+
+ for (i = 1; i <= m; i++)
+ {
+ /* Find i-th root of Legendre polynomial */
+
+ /* Initial guess */
+ const double pi_to_many_places
+ = 3.1415926535897932384626433832795028841971693993751;
+ x0 = cos(pi_to_many_places * (double)((i << 2) - 1) * t1) * t0;
+
+ /* Newton iterations, at least one */
+ j = 0;
+ dx = dw = DBL_MAX;
+
+ do
+ {
+ /* Compute Legendre polynomial value at x0 */
+ P_1 = 1.0;
+ P0 = x0;
+
+ /* Optimized version using lookup tables */
+
+ if (n < 1024)
+ {
+ /* Use fast algorithm for small n*/
+ for (k = 2; k <= n; k++)
+ {
+ P_2 = P_1;
+ P_1 = P0;
+ t2 = x0 * P_1;
+
+ P0 = t2 + ltbl[k] * (t2 - P_2);
+
+ }
+
+ }
+ else
+ {
+
+ /* Use general algorithm for other n */
+ for (k = 2; k < 1024; k++)
+ {
+ P_2 = P_1;
+ P_1 = P0;
+ t2 = x0 * P_1;
+
+ P0 = t2 + ltbl[k] * (t2 - P_2);
+ }
+
+ for (k = 1024; k <= n; k++)
+ {
+ P_2 = P_1;
+ P_1 = P0;
+ t2 = x0 * P_1;
+ t3 = (double)(k - 1) / (double)k;
+
+ P0 = t2 + t3 * (t2 - P_2);
+ }
+ }
+
+ /* Compute Legendre polynomial derivative at x0 */
+ dpdx = ((x0 * P0 - P_1) * (double)n) / (x0 * x0 - 1.0);
+
+ /* Newton step */
+ x1 = x0 - P0 / dpdx;
+
+ /* Weight computing */
+ w1 = 2.0 / ((1.0 - x1 * x1) * dpdx * dpdx);
+
+ /* Compute weight w0 on first iteration, needed for dw */
+ if (j == 0) w0 = 2.0 / ((1.0 - x0 * x0) * dpdx * dpdx);
+
+ dx = x0 - x1;
+
+ dw = w0 - w1;
+
+ x0 = x1;
+
+ w0 = w1;
+
+ j++;
+ }
+ while ((fabs(dx) > eps || fabs(dw) > eps) && j < 100);
+
+ x[(m-1)-(i-1)] = x1;
+
+ w[(m-1)-(i-1)] = w1;
+ }
+
+ return;
+}
+
diff --git a/integration/gsl_integration.h b/integration/gsl_integration.h
index dda6d0b..cb446c9 100644
--- a/integration/gsl_integration.h
+++ b/integration/gsl_integration.h
@@ -247,6 +247,32 @@ int gsl_integration_qawf (gsl_function * f,
gsl_integration_qawo_table * wf,
double *result, double *abserr);
+/* Workspace for fixed-order Gauss-Legendre integration */
+
+typedef struct
+ {
+ size_t n; /* number of points */
+ double *x; /* Gauss abscissae/points */
+ double *w; /* Gauss weights for each abscissae */
+ int precomputed; /* high precision abscissae/weights precomputed? */
+ }
+gsl_integration_glfixed_table;
+
+
+gsl_integration_glfixed_table *
+ gsl_integration_glfixed_table_alloc (size_t n);
+
+void
+ gsl_integration_glfixed_table_free (gsl_integration_glfixed_table * t);
+
+/* Routine for fixed-order Gauss-Legendre integration */
+
+double
+ gsl_integration_glfixed (const gsl_function *f,
+ double a,
+ double b,
+ const gsl_integration_glfixed_table * t);
+
__END_DECLS
#endif /* __GSL_INTEGRATION_H__ */
diff --git a/integration/test.c b/integration/test.c
index b6ec347..dff5b41 100644
--- a/integration/test.c
+++ b/integration/test.c
@@ -73,6 +73,7 @@ gsl_function make_counter (gsl_function * f, struct counter_params * p)
void my_error_handler (const char *reason, const char *file,
int line, int err);
+
int main (void)
{
gsl_ieee_env_setup ();
@@ -2093,6 +2094,119 @@ int main (void)
}
+ /* Sanity check monomial test function for fixed Gauss-Legendre rules */
+ {
+ struct monomial_params params;
+ gsl_function f = { f_monomial, ¶ms };
+
+ params.degree = 2;
+ params.constant = 1.0;
+ gsl_test_abs(GSL_FN_EVAL(&f, 2.0), 4.0, GSL_DBL_EPSILON,
+ "f_monomial sanity check 1");
+
+ params.degree = 1;
+ params.constant = 2.0;
+ gsl_test_abs(GSL_FN_EVAL(&f, 2.0), 4.0, GSL_DBL_EPSILON,
+ "f_monomial sanity check 2");
+
+ params.degree = 2;
+ params.constant = 2.0;
+ gsl_test_abs(integ_f_monomial(1.0, 2.0, ¶ms),
+ (2.0/3.0)*(2.0*2.0*2.0 - 1.0*1.0*1.0), GSL_DBL_EPSILON,
+ "integ_f_monomial sanity check");
+ }
+
+ /* Test the fixed-order Gauss-Legendre rules with a monomial. */
+ {
+ int n;
+ struct monomial_params params;
+ const gsl_function f = { f_monomial, ¶ms };
+ const double a = 0.0, b = 1.2;
+ params.constant = 1.0;
+
+ for (n = 1; n < 1025; ++n)
+ {
+ double expected, result;
+
+ gsl_integration_glfixed_table * tbl =
+ gsl_integration_glfixed_table_alloc(n);
+
+ params.degree = 2*n-1; /* n point rule exact for 2n-1 degree poly */
+ expected = integ_f_monomial(a, b, ¶ms);
+ result = gsl_integration_glfixed(&f, a, b, tbl);
+
+ if (tbl->precomputed)
+ {
+ gsl_test_rel(result, expected, 1.0e-12,
+ "glfixed %d-point: Integrating (%g*x^%d) over [%g,%g]",
+ n, params.constant, params.degree, a, b);
+ }
+ else
+ {
+ gsl_test_rel(result, expected, 1.0e-7,
+ "glfixed %d-point: Integrating (%g*x^%d) over [%g,%g]",
+ n, params.constant, params.degree, a, b);
+ }
+
+ gsl_integration_glfixed_table_free(tbl);
+ }
+ }
+
+ /* Sanity check sin(x) test function for fixed Gauss-Legendre rules */
+ {
+ gsl_function f = { f_sin, NULL };
+
+ gsl_test_abs(GSL_FN_EVAL(&f, 2.0), sin(2.0), 0.0, "f_sin sanity check 1");
+ gsl_test_abs(GSL_FN_EVAL(&f, 7.0), sin(7.0), 0.0, "f_sin sanity check 2");
+ gsl_test_abs(integ_f_sin(0.0, M_PI), 2.0, GSL_DBL_EPSILON,
+ "integ_f_sin sanity check");
+ }
+
+ /* Test the fixed-order Gauss-Legendre rules against sin(x) on [0, pi] */
+ {
+ const int n_max = 1024;
+ const gsl_function f = { f_sin, NULL };
+ const double a = 0.0, b = M_PI;
+ const double expected = integ_f_sin(a, b);
+ double result[n_max+1], abserr[n_max+1];
+ int n;
+
+ for (n = 1; n <= n_max; ++n)
+ {
+ gsl_integration_glfixed_table * const tbl =
+ gsl_integration_glfixed_table_alloc(n);
+
+ result[n] = gsl_integration_glfixed(&f, a, b, tbl);
+ abserr[n] = fabs(expected - result[n]);
+
+ if (n == 1)
+ {
+ gsl_test_abs(result[n], GSL_FN_EVAL(&f,(b+a)/2)*(b-a), 0.0,
+ "glfixed %d-point: behavior for n == 1", n);
+ }
+ else if (n < 9)
+ {
+ gsl_test(! (abserr[n] < abserr[n-1]),
+ "glfixed %d-point: observed drop in absolute error versus %d-points",
+ n, n-1);
+ }
+ else if (tbl->precomputed)
+ {
+ gsl_test_abs(result[n], expected, n/5 * GSL_DBL_EPSILON,
+ "glfixed %d-point: very low absolute error for high precision coefficients",
+ n);
+ }
+ else
+ {
+ gsl_test_abs(result[n], expected, 1.0e6 * GSL_DBL_EPSILON,
+ "glfixed %d-point: acceptable absolute error for on-the-fly coefficients",
+ n);
+ }
+
+ gsl_integration_glfixed_table_free(tbl);
+ }
+ }
+
exit (gsl_test_summary());
}
@@ -2101,5 +2215,3 @@ my_error_handler (const char *reason, const char *file, int line, int err)
{
if (0) printf ("(caught [%s:%d: %s (%d)])\n", file, line, reason, err) ;
}
-
-
diff --git a/integration/tests.c b/integration/tests.c
index 13f8824..e4ee515 100644
--- a/integration/tests.c
+++ b/integration/tests.c
@@ -218,3 +218,33 @@ double myfn2 (double x, void * params) {
double alpha = *(double *) params ;
return exp(alpha*x) ;
}
+
+
+/* f_monomial = constant * x^degree */
+double f_monomial(double x, void * params)
+{
+ struct monomial_params * p = (struct monomial_params *) params;
+
+ return p->constant * gsl_pow_int(x, p->degree);
+}
+
+/* integ(f_monomial,x,a b)=constant*(b^(degree+1)-a^(degree+1))/(degree+1) */
+double integ_f_monomial(double a, double b, struct monomial_params * p)
+{
+ const int degreep1 = p->degree + 1;
+ const double bnp1 = gsl_pow_int(b, degreep1);
+ const double anp1 = gsl_pow_int(a, degreep1);
+ return (p->constant / degreep1)*(bnp1 - anp1);
+}
+
+/* f(x) = sin(x) */
+double f_sin(double x, void * params)
+{
+ return sin(x);
+}
+
+/* integ(f_sin,x,a,b) */
+double integ_f_sin(double a, double b)
+{
+ return -cos(b) + cos(a);
+}
diff --git a/integration/tests.h b/integration/tests.h
index 2bd28c6..5145896 100644
--- a/integration/tests.h
+++ b/integration/tests.h
@@ -47,3 +47,13 @@ double f459 (double x, void * params);
double myfn1 (double x, void * params);
double myfn2 (double x, void * params);
+
+struct monomial_params {
+ int degree;
+ double constant;
+} ;
+double f_monomial(double x, void * params);
+double integ_f_monomial(double a, double b, struct monomial_params * p);
+
+double f_sin(double x, void * params);
+double integ_f_sin(double a, double b);
--
1.5.4.3
^ permalink raw reply [flat|nested] 6+ messages in thread