This is the mail archive of the gsl-discuss@sourceware.org mailing list for the GSL project.

Index Nav: [Date Index] [Subject Index] [Author Index] [Thread Index]
Message Nav: [Date Prev] [Date Next] [Thread Prev] [Thread Next]
Other format: [Raw text]

Re: [PATCH] Adding fixed-order Gauss-Legendre integration routines

> 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

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, 0.3062450888108541472266190, 0.3091627729648165073212094, 0.3120775500006891993287636, 0.3149893925102530283167230, 0.3178982731128827248285835, 0.3208041644558044102645582, 0.3237070392143528003701590, 0.3266068700922281444141618, 0.3295036298217528976399056, 0.3323972911641281245763845, 0.3352878269096896307981228, 0.3381752098781638207253743, 0.3410594129189232790587667, 0.3439404089112420734451077, 0.3468181707645507759736923, 0.3496926714186912011050938, 0.3525638838441708576370887, 0.3554317810424171123150528, 0.3582963360460310626968790, 0.3611575219190411168852009, 0.3640153117571562777424605, 0.3668696786880191292071420, 0.3697205958714585223322883, 0.3725680364997419586702471, 0.3754119737978276686304337, 0.3782523810236163824397703, 0.3810892314682027913383487, 0.3839224984561266966457784, 0.3867521553456238443366159, 0.3895781755288764427662286, 0.3924005324322633611914264, 0.3952191995166100067331951, 0.3980341502774378774318886, 0.4008453582452137890482864, 0.4036527969855987732669841, 0.4064564400996966449616823, 0.4092562612243022361850445, 0.4120522340321492945489319, 0.4148443322321580436639788, 0.4176325295696824033106488, 0.4204167998267568670171117, 0.4231971168223430347225035, 0.4259734544125757982073747, 0.4287457864910091769763965, 0.4315140869888618022816824, 0.4342783298752620469783905, 0.4370384891574927989076034, 0.4397945388812358755048319, 0.4425464531308160773358662, 0.4452942060294448782650898, 0.4480377717394637499647905, 0.4507771244625871184774399, 0.4535122384401449505463744, 0.4562430879533249674337895, 0.4589696473234144839484647, 0.4616918909120418704091584, 0.4644097931214176352731591, 0.4671233283945751261630457, 0.4698324712156108470282980, 0.4725371961099243891820077, 0.4752374776444579739565725, 0.4779332904279356047259052, 0.4806246091111018260453658, 0.4833114083869600876643171, 0.4859936629910107111699206, 0.4886713477014884570245255, 0.4913444373395996897627612, 0.4940129067697591391182235, 0.4966767308998262548534419, 0.4993358846813411530706387, 0.5019903431097601517846292, 0.5046400812246908935430768, 0.5072850741101270528831987, 0.5099252968946826264179220, 0.5125607247518258033484145, 0.5151913329001124142038603, 0.5178170966034189556133159, 0.5204379911711751889184691, 0.5230539919585963104401304, 0.5256650743669146912153147, 0.5282712138436111840258187, 0.5308723858826459955432696, 0.5334685660246891214197081, 0.5360597298573503421568799, 0.5386458530154087775915395, 0.5412269111810419978382210, 0.5438028800840546885350993, 0.5463737355021068682427603, 0.5489394532609416558499039, 0.5515000092346125858442412, 0.5540553793457104693110943, 0.5566055395655897985264809, 0.5591504659145946930157566, 0.5616901344622843849532002, 0.5642245213276582417822586, 0.5667536026793803239405196, 0.5692773547360034755778519, 0.5717957537661929461605442, 0.5743087760889495408586850, 0.5768163980738322976184566, 0.5793185961411806888254667, 0.5818153467623363454697137, 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, 0.98765432098765427000, 0.98780487804878048000, 0.98795180722891562000, 0.98809523809523814000, 0.98823529411764710000, 0.98837209302325579000, 0.98850574712643680000, 0.98863636363636365000, 0.98876404494382020000, 0.98888888888888893000, 0.98901098901098905000, 0.98913043478260865000, 0.98924731182795700000, 0.98936170212765961000, 0.98947368421052628000, 0.98958333333333337000, 0.98969072164948457000, 0.98979591836734693000, 0.98989898989898994000, 0.98999999999999999000, 0.99009900990099009000, 0.99019607843137258000, 0.99029126213592233000, 0.99038461538461542000, 0.99047619047619051000, 0.99056603773584906000, 0.99065420560747663000, 0.99074074074074070000, 0.99082568807339455000, 0.99090909090909096000, 0.99099099099099097000, 0.99107142857142860000, 0.99115044247787609000, 0.99122807017543857000, 0.99130434782608701000, 0.99137931034482762000, 0.99145299145299148000, 0.99152542372881358000, 0.99159663865546221000, 0.99166666666666670000, 0.99173553719008267000, 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, 0.99386503067484666000, 0.99390243902439024000, 0.99393939393939390000, 0.99397590361445787000, 0.99401197604790414000, 0.99404761904761907000, 0.99408284023668636000, 0.99411764705882355000, 0.99415204678362579000, 0.99418604651162790000, 0.99421965317919070000, 0.99425287356321834000, 0.99428571428571433000, 0.99431818181818177000, 0.99435028248587576000, 0.99438202247191010000, 0.99441340782122900000, 0.99444444444444446000, 0.99447513812154698000, 0.99450549450549453000, 0.99453551912568305000, 0.99456521739130432000, 0.99459459459459465000, 0.99462365591397850000, 0.99465240641711228000, 0.99468085106382975000, 0.99470899470899465000, 0.99473684210526314000, 0.99476439790575921000, 0.99479166666666663000, 0.99481865284974091000, 0.99484536082474229000, 0.99487179487179489000, 0.99489795918367352000, 0.99492385786802029000, 0.99494949494949492000, 0.99497487437185927000, 0.99500000000000000000, 0.99502487562189057000, 0.99504950495049505000, 0.99507389162561577000, 0.99509803921568629000, 0.99512195121951219000, 0.99514563106796117000, 0.99516908212560384000, 0.99519230769230771000, 0.99521531100478466000, 0.99523809523809526000, 0.99526066350710896000, 0.99528301886792447000, 0.99530516431924887000, 0.99532710280373837000, 0.99534883720930234000, 0.99537037037037035000, 0.99539170506912444000, 0.99541284403669728000, 0.99543378995433796000, 0.99545454545454548000, 0.99547511312217196000, 0.99549549549549554000, 0.99551569506726456000, 0.99553571428571430000, 0.99555555555555553000, 0.99557522123893805000, 0.99559471365638763000, 0.99561403508771928000, 0.99563318777292575000, 0.99565217391304350000, 0.99567099567099571000, 0.99568965517241381000, 0.99570815450643779000, 0.99572649572649574000, 0.99574468085106382000, 0.99576271186440679000, 0.99578059071729963000, 0.99579831932773111000, 0.99581589958159000000, 0.99583333333333335000, 0.99585062240663902000, 0.99586776859504134000, 0.99588477366255146000, 0.99590163934426235000, 0.99591836734693873000, 0.99593495934959353000, 0.99595141700404854000, 0.99596774193548387000, 0.99598393574297184000, 0.99600000000000000000, 0.99601593625498008000, 0.99603174603174605000, 0.99604743083003955000, 0.99606299212598426000, 0.99607843137254903000, 0.99609375000000000000, 0.99610894941634243000, 0.99612403100775193000, 0.99613899613899615000, 0.99615384615384617000, 0.99616858237547889000, 0.99618320610687028000, 0.99619771863117867000, 0.99621212121212122000, 0.99622641509433962000, 0.99624060150375937000, 0.99625468164794007000, 0.99626865671641796000, 0.99628252788104088000, 0.99629629629629635000, 0.99630996309963105000, 0.99632352941176472000, 0.99633699633699635000, 0.99635036496350360000, 0.99636363636363634000, 0.99637681159420288000, 0.99638989169675085000, 0.99640287769784175000, 0.99641577060931896000, 0.99642857142857144000, 0.99644128113879005000, 0.99645390070921991000, 0.99646643109540634000, 0.99647887323943662000, 0.99649122807017543000, 0.99650349650349646000, 0.99651567944250874000, 0.99652777777777779000, 0.99653979238754320000, 0.99655172413793103000, 0.99656357388316152000, 0.99657534246575341000, 0.99658703071672350000, 0.99659863945578231000, 0.99661016949152548000, 0.99662162162162160000, 0.99663299663299665000, 0.99664429530201337000, 0.99665551839464883000, 0.99666666666666670000, 0.99667774086378735000, 0.99668874172185429000, 0.99669966996699666000, 0.99671052631578949000, 0.99672131147540988000, 0.99673202614379086000, 0.99674267100977199000, 0.99675324675324672000, 0.99676375404530748000, 0.99677419354838714000, 0.99678456591639875000, 0.99679487179487181000, 0.99680511182108622000, 0.99681528662420382000, 0.99682539682539684000, 0.99683544303797467000, 0.99684542586750791000, 0.99685534591194969000, 0.99686520376175547000, 0.99687499999999996000, 0.99688473520249221000, 0.99689440993788825000, 0.99690402476780182000, 0.99691358024691357000, 0.99692307692307691000, 0.99693251533742333000, 0.99694189602446481000, 0.99695121951219512000, 0.99696048632218848000, 0.99696969696969695000, 0.99697885196374625000, 0.99698795180722888000, 0.99699699699699695000, 0.99700598802395213000, 0.99701492537313430000, 0.99702380952380953000, 0.99703264094955490000, 0.99704142011834318000, 0.99705014749262533000, 0.99705882352941178000, 0.99706744868035191000, 0.99707602339181289000, 0.99708454810495628000, 0.99709302325581395000, 0.99710144927536237000, 0.99710982658959535000, 0.99711815561959649000, 0.99712643678160917000, 0.99713467048710602000, 0.99714285714285711000, 0.99715099715099720000, 0.99715909090909094000, 0.99716713881019825000, 0.99717514124293782000, 0.99718309859154930000, 0.99719101123595510000, 0.99719887955182074000, 0.99720670391061450000, 0.99721448467966578000, 0.99722222222222223000, 0.99722991689750695000, 0.99723756906077343000, 0.99724517906336085000, 0.99725274725274726000, 0.99726027397260275000, 0.99726775956284153000, 0.99727520435967298000, 0.99728260869565222000, 0.99728997289972898000, 0.99729729729729732000, 0.99730458221024254000, 0.99731182795698925000, 0.99731903485254692000, 0.99732620320855614000, 0.99733333333333329000, 0.99734042553191493000, 0.99734748010610075000, 0.99735449735449733000, 0.99736147757255933000, 0.99736842105263157000, 0.99737532808398954000, 0.99738219895287961000, 0.99738903394255873000, 0.99739583333333337000, 0.99740259740259740000, 0.99740932642487046000, 0.99741602067183466000, 0.99742268041237114000, 0.99742930591259638000, 0.99743589743589745000, 0.99744245524296671000, 0.99744897959183676000, 0.99745547073791352000, 0.99746192893401020000, 0.99746835443037973000, 0.99747474747474751000, 0.99748110831234260000, 0.99748743718592969000, 0.99749373433583965000, 0.99750000000000005000, 0.99750623441396513000, 0.99751243781094523000, 0.99751861042183620000, 0.99752475247524752000, 0.99753086419753090000, 0.99753694581280783000, 0.99754299754299758000, 0.99754901960784315000, 0.99755501222493892000, 0.99756097560975610000, 0.99756690997566910000, 0.99757281553398058000, 0.99757869249394671000, 0.99758454106280192000, 0.99759036144578317000, 0.99759615384615385000, 0.99760191846522783000, 0.99760765550239239000, 0.99761336515513122000, 0.99761904761904763000, 0.99762470308788598000, 0.99763033175355453000, 0.99763593380614657000, 0.99764150943396224000, 0.99764705882352944000, 0.99765258215962438000, 0.99765807962529274000, 0.99766355140186913000, 0.99766899766899764000, 0.99767441860465111000, 0.99767981438515085000, 0.99768518518518523000, 0.99769053117782913000, 0.99769585253456217000, 0.99770114942528731000, 0.99770642201834858000, 0.99771167048054921000, 0.99771689497716898000, 0.99772209567198178000, 0.99772727272727268000, 0.99773242630385484000, 0.99773755656108598000, 0.99774266365688491000, 0.99774774774774777000, 0.99775280898876406000, 0.99775784753363228000, 0.99776286353467558000, 0.99776785714285710000, 0.99777282850779514000, 0.99777777777777776000, 0.99778270509977829000, 0.99778761061946908000, 0.99779249448123619000, 0.99779735682819382000, 0.99780219780219781000, 0.99780701754385970000, 0.99781181619256021000, 0.99781659388646293000, 0.99782135076252720000, 0.99782608695652175000, 0.99783080260303691000, 0.99783549783549785000, 0.99784017278617709000, 0.99784482758620685000, 0.99784946236559136000, 0.99785407725321884000, 0.99785867237687365000, 0.99786324786324787000, 0.99786780383795304000, 0.99787234042553197000, 0.99787685774946921000, 0.99788135593220340000, 0.99788583509513740000, 0.99789029535864981000, 0.99789473684210528000, 0.99789915966386555000, 0.99790356394129975000, 0.99790794979079500000, 0.99791231732776620000, 0.99791666666666667000, 0.99792099792099798000, 0.99792531120331951000, 0.99792960662525876000, 0.99793388429752061000, 0.99793814432989691000, 0.99794238683127567000, 0.99794661190965095000, 0.99795081967213117000, 0.99795501022494892000, 0.99795918367346936000, 0.99796334012219956000, 0.99796747967479671000, 0.99797160243407712000, 0.99797570850202433000, 0.99797979797979797000, 0.99798387096774188000, 0.99798792756539234000, 0.99799196787148592000, 0.99799599198396793000, 0.99800000000000000000, 0.99800399201596801000, 0.99800796812749004000, 0.99801192842942343000, 0.99801587301587302000, 0.99801980198019802000, 0.99802371541501977000, 0.99802761341222879000, 0.99803149606299213000, 0.99803536345776034000, 0.99803921568627452000, 0.99804305283757344000, 0.99804687500000000000, 0.99805068226120852000, 0.99805447470817121000, 0.99805825242718449000, 0.99806201550387597000, 0.99806576402321079000, 0.99806949806949807000, 0.99807321772639690000, 0.99807692307692308000, 0.99808061420345484000, 0.99808429118773945000, 0.99808795411089868000, 0.99809160305343514000, 0.99809523809523815000, 0.99809885931558939000, 0.99810246679316883000, 0.99810606060606055000, 0.99810964083175802000, 0.99811320754716981000, 0.99811676082862522000, 0.99812030075187974000, 0.99812382739212002000, 0.99812734082397003000, 0.99813084112149530000, 0.99813432835820892000, 0.99813780260707630000, 0.99814126394052050000, 0.99814471243042668000, 0.99814814814814812000, 0.99815157116451014000, 0.99815498154981552000, 0.99815837937384899000, 0.99816176470588236000, 0.99816513761467895000, 0.99816849816849818000, 0.99817184643510060000, 0.99817518248175185000, 0.99817850637522765000, 0.99818181818181817000, 0.99818511796733211000, 0.99818840579710144000, 0.99819168173598549000, 0.99819494584837543000, 0.99819819819819822000, 0.99820143884892087000, 0.99820466786355477000, 0.99820788530465954000, 0.99821109123434704000, 0.99821428571428572000, 0.99821746880570406000, 0.99822064056939497000, 0.99822380106571940000, 0.99822695035460995000, 0.99823008849557526000, 0.99823321554770317000, 0.99823633156966496000, 0.99823943661971826000, 0.99824253075571179000, 0.99824561403508771000, 0.99824868651488619000, 0.99825174825174823000, 0.99825479930191974000, 0.99825783972125437000, 0.99826086956521742000, 0.99826388888888884000, 0.99826689774696709000, 0.99826989619377160000, 0.99827288428324701000, 0.99827586206896557000, 0.99827882960413084000, 0.99828178694158076000, 0.99828473413379071000, 0.99828767123287676000, 0.99829059829059830000, 0.99829351535836175000, 0.99829642248722317000, 0.99829931972789121000, 0.99830220713073003000, 0.99830508474576274000, 0.99830795262267347000, 0.99831081081081086000, 0.99831365935919059000, 0.99831649831649827000, 0.99831932773109244000, 0.99832214765100669000, 0.99832495812395305000, 0.99832775919732442000, 0.99833055091819700000, 0.99833333333333329000, 0.99833610648918469000, 0.99833887043189373000, 0.99834162520729686000, 0.99834437086092720000, 0.99834710743801658000, 0.99834983498349839000, 0.99835255354200991000, 0.99835526315789469000, 0.99835796387520526000, 0.99836065573770494000, 0.99836333878887074000, 0.99836601307189543000, 0.99836867862969003000, 0.99837133550488599000, 0.99837398373983743000, 0.99837662337662336000, 0.99837925445705022000, 0.99838187702265369000, 0.99838449111470118000, 0.99838709677419357000, 0.99838969404186795000, 0.99839228295819937000, 0.99839486356340290000, 0.99839743589743590000, 0.99839999999999995000, 0.99840255591054317000, 0.99840510366826152000, 0.99840764331210186000, 0.99841017488076311000, 0.99841269841269842000, 0.99841521394611732000, 0.99841772151898733000, 0.99842022116903628000, 0.99842271293375395000, 0.99842519685039366000, 0.99842767295597479000, 0.99843014128728413000, 0.99843260188087779000, 0.99843505477308292000, 0.99843749999999998000, 0.99843993759750393000, 0.99844236760124616000, 0.99844479004665632000, 0.99844720496894412000, 0.99844961240310082000, 0.99845201238390091000, 0.99845440494590421000, 0.99845679012345678000, 0.99845916795069334000, 0.99846153846153851000, 0.99846390168970811000, 0.99846625766871167000, 0.99846860643185298000, 0.99847094801223246000, 0.99847328244274813000, 0.99847560975609762000, 0.99847792998477924000, 0.99848024316109418000, 0.99848254931714719000, 0.99848484848484853000, 0.99848714069591527000, 0.99848942598187307000, 0.99849170437405732000, 0.99849397590361444000, 0.99849624060150377000, 0.99849849849849848000, 0.99850074962518742000, 0.99850299401197606000, 0.99850523168908822000, 0.99850746268656720000, 0.99850968703427723000, 0.99851190476190477000, 0.99851411589895989000, 0.99851632047477745000, 0.99851851851851847000, 0.99852071005917165000, 0.99852289512555392000, 0.99852507374631272000, 0.99852724594992637000, 0.99852941176470589000, 0.99853157121879588000, 0.99853372434017595000, 0.99853587115666176000, 0.99853801169590639000, 0.99854014598540142000, 0.99854227405247808000, 0.99854439592430855000, 0.99854651162790697000, 0.99854862119013066000, 0.99855072463768113000, 0.99855282199710560000, 0.99855491329479773000, 0.99855699855699853000, 0.99855907780979825000, 0.99856115107913668000, 0.99856321839080464000, 0.99856527977044474000, 0.99856733524355301000, 0.99856938483547930000, 0.99857142857142855000, 0.99857346647646217000, 0.99857549857549854000, 0.99857752489331442000, 0.99857954545454541000, 0.99858156028368794000, 0.99858356940509918000, 0.99858557284299854000, 0.99858757062146897000, 0.99858956276445698000, 0.99859154929577465000, 0.99859353023909991000, 0.99859550561797750000, 0.99859747545582045000, 0.99859943977591037000, 0.99860139860139863000, 0.99860335195530725000, 0.99860529986053004000, 0.99860724233983289000, 0.99860917941585536000, 0.99861111111111112000, 0.99861303744798890000, 0.99861495844875348000, 0.99861687413554634000, 0.99861878453038677000, 0.99862068965517237000, 0.99862258953168048000, 0.99862448418156813000, 0.99862637362637363000, 0.99862825788751719000, 0.99863013698630132000, 0.99863201094391241000, 0.99863387978142082000, 0.99863574351978168000, 0.99863760217983655000, 0.99863945578231295000, 0.99864130434782605000, 0.99864314789687925000, 0.99864498644986455000, 0.99864682002706362000, 0.99864864864864866000, 0.99865047233468285000, 0.99865229110512133000, 0.99865410497981155000, 0.99865591397849462000, 0.99865771812080539000, 0.99865951742627346000, 0.99866131191432395000, 0.99866310160427807000, 0.99866488651535379000, 0.99866666666666670000, 0.99866844207723038000, 0.99867021276595747000, 0.99867197875166003000, 0.99867374005305043000, 0.99867549668874167000, 0.99867724867724872000, 0.99867899603698806000, 0.99868073878627972000, 0.99868247694334655000, 0.99868421052631584000, 0.99868593955321949000, 0.99868766404199472000, 0.99868938401048490000, 0.99869109947643975000, 0.99869281045751634000, 0.99869451697127942000, 0.99869621903520212000, 0.99869791666666663000, 0.99869960988296491000, 0.99870129870129876000, 0.99870298313878081000, 0.99870466321243523000, 0.99870633893919791000, 0.99870801033591727000, 0.99870967741935479000, 0.99871134020618557000, 0.99871299871299868000, 0.99871465295629824000, 0.99871630295250324000, 0.99871794871794872000, 0.99871959026888601000, 0.99872122762148341000, 0.99872286079182626000, 0.99872448979591832000, 0.99872611464968153000, 0.99872773536895676000, 0.99872935196950441000, 0.99873096446700504000, 0.99873257287705952000, 0.99873417721518987000, 0.99873577749683939000, 0.99873737373737370000, 0.99873896595208067000, 0.99874055415617125000, 0.99874213836477987000, 0.99874371859296485000, 0.99874529485570895000, 0.99874686716791983000, 0.99874843554443049000, 0.99875000000000003000, 0.99875156054931336000, 0.99875311720698257000, 0.99875466998754669000, 0.99875621890547261000, 0.99875776397515525000, 0.99875930521091816000, 0.99876084262701359000, 0.99876237623762376000, 0.99876390605686027000, 0.99876543209876545000, 0.99876695437731200000, 0.99876847290640391000, 0.99876998769987702000, 0.99877149877149873000, 0.99877300613496933000, 0.99877450980392157000, 0.99877600979192172000, 0.99877750611246940000, 0.99877899877899878000, 0.99878048780487805000, 0.99878197320341044000, 0.99878345498783450000, 0.99878493317132444000, 0.99878640776699024000, 0.99878787878787878000, 0.99878934624697335000, 0.99879081015719473000, 0.99879227053140096000, 0.99879372738238847000, 0.99879518072289153000, 0.99879663056558365000, 0.99879807692307687000, 0.99879951980792314000, 0.99880095923261392000, 0.99880239520958081000, 0.99880382775119614000, 0.99880525686977295000, 0.99880668257756566000, 0.99880810488676997000, 0.99880952380952381000, 0.99881093935790721000, 0.99881235154394299000, 0.99881376037959668000, 0.99881516587677721000, 0.99881656804733732000, 0.99881796690307334000, 0.99881936245572611000, 0.99882075471698117000, 0.99882214369846878000, 0.99882352941176467000, 0.99882491186839018000, 0.99882629107981225000, 0.99882766705744430000, 0.99882903981264637000, 0.99883040935672518000, 0.99883177570093462000, 0.99883313885647607000, 0.99883449883449882000, 0.99883585564610011000, 0.99883720930232556000, 0.99883855981416958000, 0.99883990719257543000, 0.99884125144843572000, 0.99884259259259256000, 0.99884393063583810000, 0.99884526558891451000, 0.99884659746251436000, 0.99884792626728114000, 0.99884925201380903000, 0.99885057471264371000, 0.99885189437428246000, 0.99885321100917435000, 0.99885452462772051000, 0.99885583524027455000, 0.99885714285714289000, 0.99885844748858443000, 0.99885974914481190000, 0.99886104783599083000, 0.99886234357224113000, 0.99886363636363640000, 0.99886492622020429000, 0.99886621315192747000, 0.99886749716874290000, 0.99886877828054299000, 0.99887005649717520000, 0.99887133182844245000, 0.99887260428410374000, 0.99887387387387383000, 0.99887514060742411000, 0.99887640449438198000, 0.99887766554433222000, 0.99887892376681620000, 0.99888017917133254000, 0.99888143176733779000, 0.99888268156424576000, 0.99888392857142860000, 0.99888517279821631000, 0.99888641425389757000, 0.99888765294771964000, 0.99888888888888894000, 0.99889012208657046000, 0.99889135254988914000, 0.99889258028792915000, 0.99889380530973448000, 0.99889502762430937000, 0.99889624724061810000, 0.99889746416758540000, 0.99889867841409696000, 0.99889988998899892000, 0.99890109890109891000, 0.99890230515916578000, 0.99890350877192979000, 0.99890470974808321000, 0.99890590809628010000, 0.99890710382513659000, 0.99890829694323147000, 0.99890948745910579000, 0.99891067538126366000, 0.99891186071817195000, 0.99891304347826082000, 0.99891422366992400000, 0.99891540130151846000, 0.99891657638136511000, 0.99891774891774887000, 0.99891891891891893000, 0.99892008639308860000, 0.99892125134843579000, 0.99892241379310343000, 0.99892357373519913000, 0.99892473118279568000, 0.99892588614393130000, 0.99892703862660948000, 0.99892818863879962000, 0.99892933618843682000, 0.99893048128342243000, 0.99893162393162394000, 0.99893276414087517000, 0.99893390191897657000, 0.99893503727369537000, 0.99893617021276593000, 0.99893730074388953000, 0.99893842887473461000, 0.99893955461293749000, 0.99894067796610164000, 0.99894179894179891000, 0.99894291754756870000, 0.99894403379091867000, 0.99894514767932485000, 0.99894625922023184000, 0.99894736842105258000, 0.99894847528916930000, 0.99894957983193278000, 0.99895068205666315000, 0.99895178197064993000, 0.99895287958115186000, 0.99895397489539750000, 0.99895506792058519000, 0.99895615866388310000, 0.99895724713242962000, 0.99895833333333328000, 0.99895941727367321000, 0.99896049896049899000, 0.99896157840083077000, 0.99896265560165975000, 0.99896373056994814000, 0.99896480331262938000, 0.99896587383660806000, 0.99896694214876036000, 0.99896800825593390000, 0.99896907216494846000, 0.99897013388259526000, 0.99897119341563789000, 0.99897225077081198000, 0.99897330595482547000, 0.99897435897435893000, 0.99897540983606559000, 0.99897645854657113000, 0.99897750511247441000, 0.99897854954034726000, 0.99897959183673468000, 0.99898063200815490000, 0.99898167006109984000, 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, &params };
+
+    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, &params),
+        (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, &params };
+    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, &params);
+        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
Index Nav: [Date Index] [Subject Index] [Author Index] [Thread Index]
Message Nav: [Date Prev] [Date Next] [Thread Prev] [Thread Next]