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

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

[Rhys Ulerich]

[-- Attachment #1: Type: text/plain, Size: 1161 bytes --]

Hi all,

Attached is a gzipped patch that adds fixed-order Gauss-Legendre quadrature
to the numerical integration routines.  This is work by Pavel Holoborodko
(http://www.holoborodko.com/pavel/?page_id=679) which I have modified
to fit GSL conventions.  The routines include precomputed full double
precision points and weights for n =
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
32, 64, 96, 100, 128, 256, 512, and 1024 Gauss points.  If the user
selects any other number, the necessary points and weights are
computed on the fly and remain quite good.  Documentation and unit
tests are provided.  Functions/structs live under the name
gsl_integration_glfixed*

One might justly ask why include this code when gsl_integration_qng
already provides non-adaptive Gauss-Kronrod rules.  For integrands
with known polynomial order, it is possible to cut down the number of
evaluations required when using these routines.

Would love to hear feedback,
Rhys

P.S. Sorry about the patch size.  It's a bit hefty from coefficients
and trailing whitespace elimination in the documentation and
integration unit tests files.

[-- Attachment #2: 0001-Added-gsl_integration_glfixed-routines.patch.gz --]
[-- Type: application/x-gzip, Size: 57402 bytes --]

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

[Rhys Ulerich]

> The routines include precomputed full double precision points and
> weights for n = 2, ..., 512, and 1024 Gauss points.  If the user selects
> any other number, the necessary points and weights are computed
> on the fly and remain quite good.

A quick comment from Pavel about the numerical precision in the
patch's coefficients...

> ... fixed tables included in [the patch] are way beyond full double precision.
> Full double precision is about 1e-16, fixed tables are of 1e-25.
> This doesn't hurt the double calculations and allows future extensions
> e.g. for 'long double' and multi-precision libraries (like gmp or mpfr).
>
> Tables generated on the fly are of 1e-10, which is pretty much the
> maximum precision we can get using double to calculate weights.
> So, on-the-fly tables are of full double precision, fixed have much
> higher precision.

- Rhys

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

[Brian Gough]

At Sun, 21 Jun 2009 15:31:24 -0500,
Rhys Ulerich wrote:
> P.S. Sorry about the patch size.  It's a bit hefty from coefficients
> and trailing whitespace elimination in the documentation and
> integration unit tests files.

Could you redo the patch without the whitespace changes to make it
easier to see the actual changes, thanks.

-- 
Brian Gough

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

[Rhys Ulerich]

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> Could you redo the patch without the whitespace changes to make it
> easier to see the actual changes, thanks.

Definitely (attached)  Didn't know about that git format-patch
argument until just now...

- Rhys

[-- Attachment #2: 0001-Added-gsl_integration_glfixed-routines.patch --]
[-- Type: text/x-diff, Size: 118487 bytes --]

From 5b39872f96668df87c912448aa8600bb142cef9c Mon Sep 17 00:00:00 2001
From: Rhys Ulerich <rhys.ulerich@gmail.com>
Date: Sun, 21 Jun 2009 14:47:06 -0500
Subject: [PATCH] Added gsl_integration_glfixed routines

Added fixed order Gauss-Legendre quadrature with
high precision weights based on the work of
Pavel Holoborodko (http://www.holoborodko.com/pavel/?page_id=679).
Includes implementation, documentation, and unit tests.
---
 AUTHORS                       |    1 +
 NEWS                          |    4 +
 doc/integration.texi          |   35 ++++-
 integration/Makefile.am       |    2 +-
 integration/glfixed.c         |  435 +++++++++++++++++++++++++++++++++++++++++
 integration/gsl_integration.h |   26 +++
 integration/test.c            |  116 +++++++++++-
 integration/tests.c           |   30 +++
 integration/tests.h           |   10 +
 9 files changed, 655 insertions(+), 4 deletions(-)
 create mode 100644 integration/glfixed.c

diff --git a/AUTHORS b/AUTHORS
index fdbf33a..13e416b 100644
--- a/AUTHORS
+++ b/AUTHORS
@@ -25,3 +25,4 @@ Ivo Alxneit (ivo.alxneit@psi.ch) - multidimensional minimization, wavelet
   transforms
 Jason H. Stover (jason@sakla.net) - cumulative distribution functions
 Patrick Alken <patrick.alken@colorado.edu> - nonsymmetric and generalized eigensystems, B-splines
+Pavel Holoborodko <pavel@holoborodko.com> - fixed order Gauss-Legendre quadrature
diff --git a/NEWS b/NEWS
index 9ffa0b2..bec0a85 100644
--- a/NEWS
+++ b/NEWS
@@ -1,5 +1,9 @@
 * What is new in gsl-1.12+:
 
+** Added gsl_integrate_glfixed routines for performing fixed order
+   Gauss-Legendre integration 
+   (Pavel Holoborodko, Konstantin Holoborodko, Rhys Ulerich)
+
 ** Improved the implementation of gsl_ran_discrete_preproc to avoid
    internal errors due to inconsistencies from excess precision on
    some platforms.
diff --git a/doc/integration.texi b/doc/integration.texi
index 33c9776..d576c5a 100644
--- a/doc/integration.texi
+++ b/doc/integration.texi
@@ -12,7 +12,9 @@ logarithmic singularities, computation of Cauchy principal values and
 oscillatory integrals.  The library reimplements the algorithms used in
 @sc{quadpack}, a numerical integration package written by Piessens,
 Doncker-Kapenga, Uberhuber and Kahaner.  Fortran code for @sc{quadpack} is
-available on Netlib.
+available on Netlib.  Also included are non-adaptive, fixed-order
+Gauss-Legendre integration routines with high precision coefficients
+by Pavel Holoborodko.
 
 The functions described in this chapter are declared in the header file
 @file{gsl_integration.h}.
@@ -28,6 +30,7 @@ The functions described in this chapter are declared in the header file
 * QAWS adaptive integration for singular functions::  
 * QAWO adaptive integration for oscillatory functions::  
 * QAWF adaptive integration for Fourier integrals::  
+* Fixed order Gauss-Legendre integration::
 * Numerical integration error codes::  
 * Numerical integration examples::  
 * Numerical integration References and Further Reading::  
@@ -776,6 +779,36 @@ The integration over each subinterval uses the memory provided by
 
 @end deftypefun
 
+@node Fixed order Gauss-Legendre integration
+@section Gauss-Legendre integration
+
+The fixed-order Gauss-Legendre integration routines are provided for fast
+integration of smooth functions with known polynomial order.  The
+@math{n}-point Gauss-Legendre rule is exact for polynomials of order
+@math{2*n-1} or less.  For example, these rules are useful when integrating
+basis functions to form mass matrices for the Galerkin method.  Unlike other
+numerical integration routines within the library, these routines do not accept
+absolute or relative error bounds.
+
+@deftypefun {gsl_integration_glfixed_table *} gsl_integration_glfixed_table_alloc (size_t @var{n})
+@tpindex gsl_integration_glfixed_table
+This function determines the Gauss-Legendre abscissae and weights necessary for
+an @math{n}-point fixed order integration scheme.  If possible, high precision
+precomputed coefficients are used.  If precomputed weights are not available,
+lower precision coefficients are computed on the fly.
+@end deftypefun
+
+@deftypefun double gsl_integration_glfixed (const gsl_function * @var{f}, double @var{a}, double @var{b}, {gsl_integration_glfixed_table *} @var{t})
+This function applies the Gauss-Legendre integration rule
+contained in table @var{t} and returns the result.
+@end deftypefun
+
+
+@deftypefun void gsl_integration_glfixed_table_alloc (gsl_integration_glfixed_table * @var{t})
+@tpindex gsl_integration_glfixed_table
+This function frees the memory associated with the table @var{t}.
+@end deftypefun
+
 @node Numerical integration error codes
 @section Error codes
 
diff --git a/integration/Makefile.am b/integration/Makefile.am
index 12c5b6f..f5d6da7 100644
--- a/integration/Makefile.am
+++ b/integration/Makefile.am
@@ -2,7 +2,7 @@ noinst_LTLIBRARIES = libgslintegration.la
 
 INCLUDES = -I$(top_srcdir)
 
-libgslintegration_la_SOURCES = qk15.c qk21.c qk31.c qk41.c qk51.c qk61.c qk.c qng.c qng.h qag.c	qags.c qagp.c workspace.c qcheb.c qawc.c qmomo.c qaws.c	qmomof.c qawo.c	qawf.c 
+libgslintegration_la_SOURCES = qk15.c qk21.c qk31.c qk41.c qk51.c qk61.c qk.c qng.c qng.h qag.c	qags.c qagp.c workspace.c qcheb.c qawc.c qmomo.c qaws.c	qmomof.c qawo.c	qawf.c glfixed.c
 
 pkginclude_HEADERS = gsl_integration.h
 noinst_HEADERS = qpsrt.c qpsrt2.c qelg.c qc25c.c qc25s.c qc25f.c ptsort.c util.c err.c positivity.c append.c initialise.c set_initial.c reset.c
diff --git a/integration/glfixed.c b/integration/glfixed.c
new file mode 100644
index 0000000..827ba42
--- /dev/null
+++ b/integration/glfixed.c
@@ -0,0 +1,435 @@
+/*
+ * Numerical Integration by Gauss-Legendre Quadrature Formulas of high orders.
+ * High-precision abscissas and weights are used.
+ *
+ * Original project homepage: http://www.holoborodko.com/pavel/?page_id=679
+ * Original contact e-mail:   pavel@holoborodko.com
+ *
+ * Copyright (c)2007-2008 Pavel Holoborodko
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ * GNU Lesser General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ *
+ * Contributors:
+ * Konstantin Holoborodko - Optimization of Legendre polynomial computing.
+ * Rhys Ulerich - Inclusion within GNU Scientific Library.
+ *
+ */
+
+#include <config.h>
+#include <math.h>
+#include <float.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_integration.h>
+
+void gauss_legendre_tbl(int n, double* x, double* w, double eps);
+
+/* n = 2 */
+static double x2[1] = {0.5773502691896257645091488};
+static double w2[1] = {1.0000000000000000000000000};
+
+/* n = 4 */
+static double x4[2] = {0.3399810435848562648026658, 0.8611363115940525752239465};
+static double w4[2] = {0.6521451548625461426269361, 0.3478548451374538573730639};
+
+/* n = 6 */
+static double x6[3] = {0.2386191860831969086305017, 0.6612093864662645136613996, 0.9324695142031520278123016};
+static double w6[3] = {0.4679139345726910473898703, 0.3607615730481386075698335, 0.1713244923791703450402961};
+
+/* n = 8 */
+static double x8[4] = {0.1834346424956498049394761, 0.5255324099163289858177390, 0.7966664774136267395915539, 0.9602898564975362316835609};
+static double w8[4] = {0.3626837833783619829651504, 0.3137066458778872873379622, 0.2223810344533744705443560, 0.1012285362903762591525314};
+
+/* n = 10 */
+static double x10[5] = {0.1488743389816312108848260, 0.4333953941292471907992659, 0.6794095682990244062343274, 0.8650633666889845107320967, 0.9739065285171717200779640};
+static double w10[5] = {0.2955242247147528701738930, 0.2692667193099963550912269, 0.2190863625159820439955349, 0.1494513491505805931457763, 0.0666713443086881375935688};
+
+/* n = 12 */
+static double x12[6] = {0.1252334085114689154724414, 0.3678314989981801937526915, 0.5873179542866174472967024, 0.7699026741943046870368938, 0.9041172563704748566784659, 0.9815606342467192506905491};
+static double w12[6] = {0.2491470458134027850005624, 0.2334925365383548087608499, 0.2031674267230659217490645, 0.1600783285433462263346525, 0.1069393259953184309602547, 0.0471753363865118271946160};
+
+/* n = 14 */
+static double x14[7] = {0.1080549487073436620662447, 0.3191123689278897604356718, 0.5152486363581540919652907, 0.6872929048116854701480198, 0.8272013150697649931897947, 0.9284348836635735173363911, 0.9862838086968123388415973};
+static double w14[7] = {0.2152638534631577901958764, 0.2051984637212956039659241, 0.1855383974779378137417166, 0.1572031671581935345696019, 0.1215185706879031846894148, 0.0801580871597602098056333, 0.0351194603317518630318329};
+
+/* n = 16 */
+static double x16[8] = {0.0950125098376374401853193, 0.2816035507792589132304605, 0.4580167776572273863424194, 0.6178762444026437484466718, 0.7554044083550030338951012, 0.8656312023878317438804679, 0.9445750230732325760779884, 0.9894009349916499325961542};
+static double w16[8] = {0.1894506104550684962853967, 0.1826034150449235888667637, 0.1691565193950025381893121, 0.1495959888165767320815017, 0.1246289712555338720524763, 0.0951585116824927848099251, 0.0622535239386478928628438, 0.0271524594117540948517806};
+
+/* n = 18 */
+static double x18[9] = {0.0847750130417353012422619, 0.2518862256915055095889729, 0.4117511614628426460359318, 0.5597708310739475346078715, 0.6916870430603532078748911, 0.8037049589725231156824175, 0.8926024664975557392060606, 0.9558239495713977551811959, 0.9915651684209309467300160};
+static double w18[9] = {0.1691423829631435918406565, 0.1642764837458327229860538, 0.1546846751262652449254180, 0.1406429146706506512047313, 0.1225552067114784601845191, 0.1009420441062871655628140, 0.0764257302548890565291297, 0.0497145488949697964533349, 0.0216160135264833103133427};
+
+/* n = 20 */
+static double x20[10] = {0.0765265211334973337546404, 0.2277858511416450780804962, 0.3737060887154195606725482, 0.5108670019508270980043641, 0.6360536807265150254528367, 0.7463319064601507926143051, 0.8391169718222188233945291, 0.9122344282513259058677524, 0.9639719272779137912676661, 0.9931285991850949247861224};
+static double w20[10] = {0.1527533871307258506980843, 0.1491729864726037467878287, 0.1420961093183820513292983, 0.1316886384491766268984945, 0.1181945319615184173123774, 0.1019301198172404350367501, 0.0832767415767047487247581, 0.0626720483341090635695065, 0.0406014298003869413310400, 0.0176140071391521183118620};
+
+/* n = 32 */
+static double x32[16] = {0.0483076656877383162348126, 0.1444719615827964934851864, 0.2392873622521370745446032, 0.3318686022821276497799168, 0.4213512761306353453641194, 0.5068999089322293900237475, 0.5877157572407623290407455, 0.6630442669302152009751152, 0.7321821187402896803874267, 0.7944837959679424069630973, 0.8493676137325699701336930, 0.8963211557660521239653072, 0.9349060759377396891709191, 0.9647622555875064307738119, 0.9856115115452683354001750, 0.9972638618494815635449811};
+static double w32[16] = {0.0965400885147278005667648, 0.0956387200792748594190820, 0.0938443990808045656391802, 0.0911738786957638847128686, 0.0876520930044038111427715, 0.0833119242269467552221991, 0.0781938957870703064717409, 0.0723457941088485062253994, 0.0658222227763618468376501, 0.0586840934785355471452836, 0.0509980592623761761961632, 0.0428358980222266806568786, 0.0342738629130214331026877, 0.0253920653092620594557526, 0.0162743947309056706051706, 0.0070186100094700966004071};
+
+/* n = 64 */
+static double x64[32] = {0.0243502926634244325089558, 0.0729931217877990394495429, 0.1214628192961205544703765, 0.1696444204239928180373136, 0.2174236437400070841496487, 0.2646871622087674163739642, 0.3113228719902109561575127, 0.3572201583376681159504426, 0.4022701579639916036957668, 0.4463660172534640879849477, 0.4894031457070529574785263, 0.5312794640198945456580139, 0.5718956462026340342838781, 0.6111553551723932502488530, 0.6489654712546573398577612, 0.6852363130542332425635584, 0.7198818501716108268489402, 0.7528199072605318966118638, 0.7839723589433414076102205, 0.8132653151227975597419233, 0.8406292962525803627516915, 0.8659993981540928197607834, 0.8893154459951141058534040, 0.9105221370785028057563807, 0.9295691721319395758214902, 0.9464113748584028160624815, 0.9610087996520537189186141, 0.9733268277899109637418535, 0.9833362538846259569312993, 0.9910133714767443207393824, 0.9963401167719552793469245, 0.9993050417357721394569056};
+static double w64[32] = {0.0486909570091397203833654, 0.0485754674415034269347991, 0.0483447622348029571697695, 0.0479993885964583077281262, 0.0475401657148303086622822, 0.0469681828162100173253263, 0.0462847965813144172959532, 0.0454916279274181444797710, 0.0445905581637565630601347, 0.0435837245293234533768279, 0.0424735151236535890073398, 0.0412625632426235286101563, 0.0399537411327203413866569, 0.0385501531786156291289625, 0.0370551285402400460404151, 0.0354722132568823838106931, 0.0338051618371416093915655, 0.0320579283548515535854675, 0.0302346570724024788679741, 0.0283396726142594832275113, 0.0263774697150546586716918, 0.0243527025687108733381776, 0.0222701738083832541592983, 0.0201348231535302093723403, 0.0179517157756973430850453, 0.0157260304760247193219660, 0.0134630478967186425980608, 0.0111681394601311288185905, 0.0088467598263639477230309, 0.0065044579689783628561174, 0.0041470332605624676352875, 0.0017832807216964329472961};
+
+/* n = 96 */
+static double x96[48] = {0.0162767448496029695791346, 0.0488129851360497311119582, 0.0812974954644255589944713, 0.1136958501106659209112081, 0.1459737146548969419891073, 0.1780968823676186027594026, 0.2100313104605672036028472, 0.2417431561638400123279319, 0.2731988125910491414872722, 0.3043649443544963530239298, 0.3352085228926254226163256, 0.3656968614723136350308956, 0.3957976498289086032850002, 0.4254789884073005453648192, 0.4547094221677430086356761, 0.4834579739205963597684056, 0.5116941771546676735855097, 0.5393881083243574362268026, 0.5665104185613971684042502, 0.5930323647775720806835558, 0.6189258401254685703863693, 0.6441634037849671067984124, 0.6687183100439161539525572, 0.6925645366421715613442458, 0.7156768123489676262251441, 0.7380306437444001328511657, 0.7596023411766474987029704, 0.7803690438674332176036045, 0.8003087441391408172287961, 0.8194003107379316755389996, 0.8376235112281871214943028, 0.8549590334346014554627870, 0.8713885059092965028737748, 0.8868945174024204160568774, 0.9014606353158523413192327, 0.9150714231208980742058845, 0.9277124567223086909646905, 0.9393703397527552169318574, 0.9500327177844376357560989, 0.9596882914487425393000680, 0.9683268284632642121736594, 0.9759391745851364664526010, 0.9825172635630146774470458, 0.9880541263296237994807628, 0.9925439003237626245718923, 0.9959818429872092906503991, 0.9983643758631816777241494, 0.9996895038832307668276901};
+static double w96[48] = {0.0325506144923631662419614, 0.0325161187138688359872055, 0.0324471637140642693640128, 0.0323438225685759284287748, 0.0322062047940302506686671, 0.0320344562319926632181390, 0.0318287588944110065347537, 0.0315893307707271685580207, 0.0313164255968613558127843, 0.0310103325863138374232498, 0.0306713761236691490142288, 0.0302999154208275937940888, 0.0298963441363283859843881, 0.0294610899581679059704363, 0.0289946141505552365426788, 0.0284974110650853856455995, 0.0279700076168483344398186, 0.0274129627260292428234211, 0.0268268667255917621980567, 0.0262123407356724139134580, 0.0255700360053493614987972, 0.0249006332224836102883822, 0.0242048417923646912822673, 0.0234833990859262198422359, 0.0227370696583293740013478, 0.0219666444387443491947564, 0.0211729398921912989876739, 0.0203567971543333245952452, 0.0195190811401450224100852, 0.0186606796274114673851568, 0.0177825023160452608376142, 0.0168854798642451724504775, 0.0159705629025622913806165, 0.0150387210269949380058763, 0.0140909417723148609158616, 0.0131282295669615726370637, 0.0121516046710883196351814, 0.0111621020998384985912133, 0.0101607705350084157575876, 0.0091486712307833866325846, 0.0081268769256987592173824, 0.0070964707911538652691442, 0.0060585455042359616833167, 0.0050142027429275176924702, 0.0039645543384446866737334, 0.0029107318179349464084106, 0.0018539607889469217323359, 0.0007967920655520124294381};
+
+/* n = 100 */
+static double x100[50] = {0.0156289844215430828722167, 0.0468716824215916316149239, 0.0780685828134366366948174, 0.1091892035800611150034260, 0.1402031372361139732075146, 0.1710800805386032748875324, 0.2017898640957359972360489, 0.2323024818449739696495100, 0.2625881203715034791689293, 0.2926171880384719647375559, 0.3223603439005291517224766, 0.3517885263724217209723438, 0.3808729816246299567633625, 0.4095852916783015425288684, 0.4378974021720315131089780, 0.4657816497733580422492166, 0.4932107892081909335693088, 0.5201580198817630566468157, 0.5465970120650941674679943, 0.5725019326213811913168704, 0.5978474702471787212648065, 0.6226088602037077716041908, 0.6467619085141292798326303, 0.6702830156031410158025870, 0.6931491993558019659486479, 0.7153381175730564464599671, 0.7368280898020207055124277, 0.7575981185197071760356680, 0.7776279096494954756275514, 0.7968978923903144763895729, 0.8153892383391762543939888, 0.8330838798884008235429158, 0.8499645278795912842933626, 0.8660146884971646234107400, 0.8812186793850184155733168, 0.8955616449707269866985210, 0.9090295709825296904671263, 0.9216092981453339526669513, 0.9332885350430795459243337, 0.9440558701362559779627747, 0.9539007829254917428493369, 0.9628136542558155272936593, 0.9707857757637063319308979, 0.9778093584869182885537811, 0.9838775407060570154961002, 0.9889843952429917480044187, 0.9931249370374434596520099, 0.9962951347331251491861317, 0.9984919506395958184001634, 0.9997137267734412336782285};
+static double w100[50] = {0.0312554234538633569476425, 0.0312248842548493577323765, 0.0311638356962099067838183, 0.0310723374275665165878102, 0.0309504788504909882340635, 0.0307983790311525904277139, 0.0306161865839804484964594, 0.0304040795264548200165079, 0.0301622651051691449190687, 0.0298909795933328309168368, 0.0295904880599126425117545, 0.0292610841106382766201190, 0.0289030896011252031348762, 0.0285168543223950979909368, 0.0281027556591011733176483, 0.0276611982207923882942042, 0.0271926134465768801364916, 0.0266974591835709626603847, 0.0261762192395456763423087, 0.0256294029102081160756420, 0.0250575444815795897037642, 0.0244612027079570527199750, 0.0238409602659682059625604, 0.0231974231852541216224889, 0.0225312202563362727017970, 0.0218430024162473863139537, 0.0211334421125276415426723, 0.0204032326462094327668389, 0.0196530874944353058653815, 0.0188837396133749045529412, 0.0180959407221281166643908, 0.0172904605683235824393442, 0.0164680861761452126431050, 0.0156296210775460027239369, 0.0147758845274413017688800, 0.0139077107037187726879541, 0.0130259478929715422855586, 0.0121314576629794974077448, 0.0112251140231859771172216, 0.0103078025748689695857821, 0.0093804196536944579514182, 0.0084438714696689714026208, 0.0074990732554647115788287, 0.0065469484508453227641521, 0.0055884280038655151572119, 0.0046244500634221193510958, 0.0036559612013263751823425, 0.0026839253715534824194396, 0.0017093926535181052395294, 0.0007346344905056717304063};
+
+/* n = 128 */
+static double x128[64] = {0.0122236989606157641980521, 0.0366637909687334933302153, 0.0610819696041395681037870, 0.0854636405045154986364980, 0.1097942311276437466729747, 0.1340591994611877851175753, 0.1582440427142249339974755, 0.1823343059853371824103826, 0.2063155909020792171540580, 0.2301735642266599864109866, 0.2538939664226943208556180, 0.2774626201779044028062316, 0.3008654388776772026671541, 0.3240884350244133751832523, 0.3471177285976355084261628, 0.3699395553498590266165917, 0.3925402750332674427356482, 0.4149063795522750154922739, 0.4370245010371041629370429, 0.4588814198335521954490891, 0.4804640724041720258582757, 0.5017595591361444642896063, 0.5227551520511754784539479, 0.5434383024128103634441936, 0.5637966482266180839144308, 0.5838180216287630895500389, 0.6034904561585486242035732, 0.6228021939105849107615396, 0.6417416925623075571535249, 0.6602976322726460521059468, 0.6784589224477192593677557, 0.6962147083695143323850866, 0.7135543776835874133438599, 0.7304675667419088064717369, 0.7469441667970619811698824, 0.7629743300440947227797691, 0.7785484755064119668504941, 0.7936572947621932902433329, 0.8082917575079136601196422, 0.8224431169556438424645942, 0.8361029150609068471168753, 0.8492629875779689691636001, 0.8619154689395484605906323, 0.8740527969580317986954180, 0.8856677173453972174082924, 0.8967532880491581843864474, 0.9073028834017568139214859, 0.9173101980809605370364836, 0.9267692508789478433346245, 0.9356743882779163757831268, 0.9440202878302201821211114, 0.9518019613412643862177963, 0.9590147578536999280989185, 0.9656543664319652686458290, 0.9717168187471365809043384, 0.9771984914639073871653744, 0.9820961084357185360247656, 0.9864067427245862088712355, 0.9901278184917343833379303, 0.9932571129002129353034372, 0.9957927585349811868641612, 0.9977332486255140198821574, 0.9990774599773758950119878, 0.9998248879471319144736081};
+static double w128[64] = {0.0244461801962625182113259, 0.0244315690978500450548486, 0.0244023556338495820932980, 0.0243585572646906258532685, 0.0243002001679718653234426, 0.0242273192228152481200933, 0.0241399579890192849977167, 0.0240381686810240526375873, 0.0239220121367034556724504, 0.0237915577810034006387807, 0.0236468835844476151436514, 0.0234880760165359131530253, 0.0233152299940627601224157, 0.0231284488243870278792979, 0.0229278441436868469204110, 0.0227135358502364613097126, 0.0224856520327449668718246, 0.0222443288937997651046291, 0.0219897106684604914341221, 0.0217219495380520753752610, 0.0214412055392084601371119, 0.0211476464682213485370195, 0.0208414477807511491135839, 0.0205227924869600694322850, 0.0201918710421300411806732, 0.0198488812328308622199444, 0.0194940280587066028230219, 0.0191275236099509454865185, 0.0187495869405447086509195, 0.0183604439373313432212893, 0.0179603271850086859401969, 0.0175494758271177046487069, 0.0171281354231113768306810, 0.0166965578015892045890915, 0.0162550009097851870516575, 0.0158037286593993468589656, 0.0153430107688651440859909, 0.0148731226021473142523855, 0.0143943450041668461768239, 0.0139069641329519852442880, 0.0134112712886163323144890, 0.0129075627392673472204428, 0.0123961395439509229688217, 0.0118773073727402795758911, 0.0113513763240804166932817, 0.0108186607395030762476596, 0.0102794790158321571332153, 0.0097341534150068058635483, 0.0091830098716608743344787, 0.0086263777986167497049788, 0.0080645898904860579729286, 0.0074979819256347286876720, 0.0069268925668988135634267, 0.0063516631617071887872143, 0.0057726375428656985893346, 0.0051901618326763302050708, 0.0046045842567029551182905, 0.0040162549837386423131943, 0.0034255260409102157743378, 0.0028327514714579910952857, 0.0022382884309626187436221, 0.0016425030186690295387909, 0.0010458126793403487793129, 0.0004493809602920903763943};
+
+/* n = 256 */
+static double x256[128] = {0.0061239123751895295011702, 0.0183708184788136651179263, 0.0306149687799790293662786, 0.0428545265363790983812423, 0.0550876556946339841045614, 0.0673125211657164002422903, 0.0795272891002329659032271, 0.0917301271635195520311456, 0.1039192048105094036391969, 0.1160926935603328049407349, 0.1282487672706070947420496, 0.1403856024113758859130249, 0.1525013783386563953746068, 0.1645942775675538498292845, 0.1766624860449019974037218, 0.1887041934213888264615036, 0.2007175933231266700680007, 0.2127008836226259579370402, 0.2246522667091319671478783, 0.2365699497582840184775084, 0.2484521450010566668332427, 0.2602970699919425419785609, 0.2721029478763366095052447, 0.2838680076570817417997658, 0.2955904844601356145637868, 0.3072686197993190762586103, 0.3189006618401062756316834, 0.3304848656624169762291870, 0.3420194935223716364807297, 0.3535028151129699895377902, 0.3649331078236540185334649, 0.3763086569987163902830557, 0.3876277561945155836379846, 0.3988887074354591277134632, 0.4100898214687165500064336, 0.4212294180176238249768124, 0.4323058260337413099534411, 0.4433173839475273572169258, 0.4542624399175899987744552, 0.4651393520784793136455705, 0.4759464887869833063907375, 0.4866822288668903501036214, 0.4973449618521814771195124, 0.5079330882286160362319249, 0.5184450196736744762216617, 0.5288791792948222619514764, 0.5392340018660591811279362, 0.5495079340627185570424269, 0.5596994346944811451369074, 0.5698069749365687590576675, 0.5798290385590829449218317, 0.5897641221544543007857861, 0.5996107353629683217303882, 0.6093674010963339395223108, 0.6190326557592612194309676, 0.6286050494690149754322099, 0.6380831462729113686686886, 0.6474655243637248626170162, 0.6567507762929732218875002, 0.6659375091820485599064084, 0.6750243449311627638559187, 0.6840099204260759531248771, 0.6928928877425769601053416, 0.7016719143486851594060835, 0.7103456833045433133945663, 0.7189128934599714483726399, 0.7273722596496521265868944, 0.7357225128859178346203729, 0.7439624005491115684556831, 0.7520906865754920595875297, 0.7601061516426554549419068, 0.7680075933524456359758906, 0.7757938264113257391320526, 0.7834636828081838207506702, 0.7910160119895459945467075, 0.7984496810321707587825429, 0.8057635748129986232573891, 0.8129565961764315431364104, 0.8200276660989170674034781, 0.8269757238508125142890929, 0.8337997271555048943484439, 0.8404986523457627138950680, 0.8470714945172962071870724, 0.8535172676795029650730355, 0.8598350049033763506961731, 0.8660237584665545192975154, 0.8720825999954882891300459, 0.8780106206047065439864349, 0.8838069310331582848598262, 0.8894706617776108888286766, 0.8950009632230845774412228, 0.9003970057703035447716200, 0.9056579799601446470826819, 0.9107830965950650118909072, 0.9157715868574903845266696, 0.9206227024251464955050471, 0.9253357155833162028727303, 0.9299099193340056411802456, 0.9343446275020030942924765, 0.9386391748378148049819261, 0.9427929171174624431830761, 0.9468052312391274813720517, 0.9506755153166282763638521, 0.9544031887697162417644479, 0.9579876924111781293657904, 0.9614284885307321440064075, 0.9647250609757064309326123, 0.9678769152284894549090038, 0.9708835784807430293209233, 0.9737445997043704052660786, 0.9764595497192341556210107, 0.9790280212576220388242380, 0.9814496290254644057693031, 0.9837240097603154961666861, 0.9858508222861259564792451, 0.9878297475648606089164877, 0.9896604887450652183192437, 0.9913427712075830869221885, 0.9928763426088221171435338, 0.9942609729224096649628775, 0.9954964544810963565926471, 0.9965826020233815404305044, 0.9975192527567208275634088, 0.9983062664730064440555005, 0.9989435258434088565550263, 0.9994309374662614082408542, 0.9997684374092631861048786, 0.9999560500189922307348012};
+static double w256[128] = {0.0122476716402897559040703, 0.0122458343697479201424639, 0.0122421601042728007697281, 0.0122366493950401581092426, 0.0122293030687102789041463, 0.0122201222273039691917087, 0.0122091082480372404075141, 0.0121962627831147135181810, 0.0121815877594817721740476, 0.0121650853785355020613073, 0.0121467581157944598155598, 0.0121266087205273210347185, 0.0121046402153404630977578, 0.0120808558957245446559752, 0.0120552593295601498143471, 0.0120278543565825711612675, 0.0119986450878058119345367, 0.0119676359049058937290073, 0.0119348314595635622558732, 0.0119002366727664897542872, 0.0118638567340710787319046, 0.0118256971008239777711607, 0.0117857634973434261816901, 0.0117440619140605503053767, 0.0117005986066207402881898, 0.0116553800949452421212989, 0.0116084131622531057220847, 0.0115597048540436357726687, 0.0115092624770394979585864, 0.0114570935980906391523344, 0.0114032060430391859648471, 0.0113476078955454919416257, 0.0112903074958755095083676, 0.0112313134396496685726568, 0.0111706345765534494627109, 0.0111082800090098436304608, 0.0110442590908139012635176, 0.0109785814257295706379882, 0.0109112568660490397007968, 0.0108422955111147959952935, 0.0107717077058046266366536, 0.0106995040389797856030482, 0.0106256953418965611339617, 0.0105502926865814815175336, 0.0104733073841704030035696, 0.0103947509832117289971017, 0.0103146352679340150682607, 0.0102329722564782196569549, 0.0101497741990948656546341, 0.0100650535763063833094610, 0.0099788230970349101247339, 0.0098910956966958286026307, 0.0098018845352573278254988, 0.0097112029952662799642497, 0.0096190646798407278571622, 0.0095254834106292848118297, 0.0094304732257377527473528, 0.0093340483776232697124660, 0.0092362233309563026873787, 0.0091370127604508064020005, 0.0090364315486628736802278, 0.0089344947837582075484084, 0.0088312177572487500253183, 0.0087266159616988071403366, 0.0086207050884010143053688, 0.0085135010250224906938384, 0.0084050198532215357561803, 0.0082952778462352254251714, 0.0081842914664382699356198, 0.0080720773628734995009470, 0.0079586523687543483536132, 0.0078440334989397118668103, 0.0077282379473815556311102, 0.0076112830845456594616187, 0.0074931864548058833585998, 0.0073739657738123464375724, 0.0072536389258339137838291, 0.0071322239610753900716724, 0.0070097390929698226212344, 0.0068862026954463203467133, 0.0067616333001737987809279, 0.0066360495937810650445900, 0.0065094704150536602678099, 0.0063819147521078805703752, 0.0062534017395424012720636, 0.0061239506555679325423891, 0.0059935809191153382211277, 0.0058623120869226530606616, 0.0057301638506014371773844, 0.0055971560336829100775514, 0.0054633085886443102775705, 0.0053286415939159303170811, 0.0051931752508692809303288, 0.0050569298807868423875578, 0.0049199259218138656695588, 0.0047821839258926913729317, 0.0046437245556800603139791, 0.0045045685814478970686418, 0.0043647368779680566815684, 0.0042242504213815362723565, 0.0040831302860526684085998, 0.0039413976414088336277290, 0.0037990737487662579981170, 0.0036561799581425021693892, 0.0035127377050563073309711, 0.0033687685073155510120191, 0.0032242939617941981570107, 0.0030793357411993375832054, 0.0029339155908297166460123, 0.0027880553253277068805748, 0.0026417768254274905641208, 0.0024951020347037068508395, 0.0023480529563273120170065, 0.0022006516498399104996849, 0.0020529202279661431745488, 0.0019048808534997184044191, 0.0017565557363307299936069, 0.0016079671307493272424499, 0.0014591373333107332010884, 0.0013100886819025044578317, 0.0011608435575677247239706, 0.0010114243932084404526058, 0.0008618537014200890378141, 0.0007121541634733206669090, 0.0005623489540314098028152, 0.0004124632544261763284322, 0.0002625349442964459062875, 0.0001127890178222721755125};
+
+/* n = 512 */
+static double x512[256] = {0.0030649621851593961529232, 0.0091947713864329108047442, 0.0153242350848981855249677, 0.0214531229597748745137841, 0.0275812047119197840615246, 0.0337082500724805951232271, 0.0398340288115484476830396, 0.0459583107468090617788760, 0.0520808657521920701127271, 0.0582014637665182372392330, 0.0643198748021442404045319, 0.0704358689536046871990309, 0.0765492164062510452915674, 0.0826596874448871596284651, 0.0887670524624010326092165, 0.0948710819683925428909483, 0.1009715465977967786264323, 0.1070682171195026611052004, 0.1131608644449665349442888, 0.1192492596368204011642726, 0.1253331739174744696875513, 0.1314123786777137080093018, 0.1374866454852880630171099, 0.1435557460934960331730353, 0.1496194524497612685217272, 0.1556775367042018762501969, 0.1617297712181921097989489, 0.1677759285729161198103670, 0.1738157815779134454985394, 0.1798491032796159253350647, 0.1858756669698757062678115, 0.1918952461944840310240859, 0.1979076147616804833961808, 0.2039125467506523717658375, 0.2099098165200239314947094, 0.2158991987163350271904893, 0.2218804682825090362529109, 0.2278534004663095955103621, 0.2338177708287858931763260, 0.2397733552527061887852891, 0.2457199299509792442100997, 0.2516572714750633493170137, 0.2575851567233626262808095, 0.2635033629496102970603704, 0.2694116677712385990250046, 0.2753098491777350342234845, 0.2811976855389846383013106, 0.2870749556135979555970354, 0.2929414385572244074855835, 0.2987969139308507415853707, 0.3046411617090842500066247, 0.3104739622884204453906292, 0.3162950964954948840736281, 0.3221043455953188263048133, 0.3279014912994984240551598, 0.3336863157744371275728377, 0.3394586016495210024715049, 0.3452181320252866497799379, 0.3509646904815714220351686, 0.3566980610856456291665404, 0.3624180284003264285948478, 0.3681243774920730946589582, 0.3738168939390633631820054, 0.3794953638392505477003659, 0.3851595738184011246011504, 0.3908093110381124851478484, 0.3964443632038105531190080, 0.4020645185727269675414064, 0.4076695659618555307670286, 0.4132592947558876229222955, 0.4188334949151262845483445, 0.4243919569833786700527309, 0.4299344720958265754056529, 0.4354608319868747443376920, 0.4409708289979766581310498, 0.4464642560854375149423431, 0.4519409068281941054521446, 0.4574005754355712925046003, 0.4628430567550148032795831, 0.4682681462798000434299255, 0.4736756401567166435172692, 0.4790653351937284489919577, 0.4844370288676086658851277, 0.4897905193315498753147078, 0.4951256054227486308513615, 0.5004420866699643537454866, 0.5057397633010522419821678, 0.5110184362504699101074361, 0.5162779071667574777562819, 0.5215179784199908258105606, 0.5267384531092077401231844, 0.5319391350698066637637706, 0.5371198288809177797701793, 0.5422803398727461474300859, 0.5474204741338866161668468, 0.5525400385186102421644070, 0.5576388406541219339368088, 0.5627166889477890541289656, 0.5677733925943407059267120, 0.5728087615830374335557009, 0.5778226067048110674604360, 0.5828147395593744458765762, 0.5877849725623007456415722, 0.5927331189520721562306608, 0.5976589927970976321572046, 0.6025624090026994600382737, 0.6074431833180683777981926, 0.6123011323431869846644595, 0.6171360735357211818019505, 0.6219478252178793846326095, 0.6267362065832392490988318, 0.6315010377035416553494506, 0.6362421395354516935575740, 0.6409593339272863978194482, 0.6456524436257089753330001, 0.6503212922823892793136899, 0.6549657044606302753737317, 0.6595855056419602523685720, 0.6641805222326905300017078, 0.6687505815704384167754210, 0.6732955119306151731807642, 0.6778151425328787363350998, 0.6823093035475509635996236, 0.6867778261019991540425409, 0.6912205422869816079558685, 0.6956372851629569859851427, 0.7000278887663572307915895, 0.7043921881158238155354902, 0.7087300192184070848475163, 0.7130412190757284553416507, 0.7173256256901052441189100, 0.7215830780706378951153816, 0.7258134162392593745610389, 0.7300164812367465082373380, 0.7341921151286930346516885, 0.7383401610114441496854630, 0.7424604630179923197192207, 0.7465528663238341416942072, 0.7506172171527880300329109, 0.7546533627827725118134392, 0.7586611515515449130726824, 0.7626404328624002206015913, 0.7665910571898299050923647, 0.7705128760851404930018538, 0.7744057421820316760079998, 0.7782695092021337484565606, 0.7821040319605041647237048, 0.7859091663710830099561901, 0.7896847694521071791947507, 0.7934306993314830614379285, 0.7971468152521175267628422, 0.8008329775772070161862372, 0.8044890477954845355235412, 0.8081148885264243560855026, 0.8117103635254042266412553, 0.8152753376888249026732770, 0.8188096770591868005536242, 0.8223132488301235858819787, 0.8257859213513925068443721, 0.8292275641338212850768968, 0.8326380478542113781512150, 0.8360172443601974294381733, 0.8393650266750627227522641, 0.8426812690025104608329811, 0.8459658467313906883792422, 0.8492186364403826820199251, 0.8524395159026326312771384, 0.8556283640903464362590494, 0.8587850611793374495058711, 0.8619094885535289911058997, 0.8650015288094114678982387, 0.8680610657604539292849800, 0.8710879844414698938880857, 0.8740821711129372830049576, 0.8770435132652722985416439, 0.8799718996230570848337538, 0.8828672201492210155023745, 0.8857293660491754482355527, 0.8885582297749017921351663, 0.8913537050289927340242104, 0.8941156867686464718706125, 0.8968440712096138052506156, 0.8995387558300979345474886, 0.9021996393746068223597927, 0.9048266218577579723776075, 0.9074196045680354827749729, 0.9099784900714992329623006, 0.9125031822154460643436214, 0.9149935861320228175302595, 0.9174496082417910902748409, 0.9198711562572435822074657, 0.9222581391862718942794141, 0.9246104673355856526489486, 0.9269280523140828285786768, 0.9292108070361711277546193, 0.9314586457250403242837002, 0.9336714839158854164789745, 0.9358492384590804834007204, 0.9379918275233031229867813, 0.9400991705986093544775539, 0.9421711884994588697201555, 0.9442078033676905198230562, 0.9462089386754479255274304, 0.9481745192280551015654245, 0.9501044711668419871894147, 0.9519987219719197769813274, 0.9538572004649059479887372, 0.9556798368115988811866200, 0.9574665625246019772327448, 0.9592173104658971684737507, 0.9609320148493677311718534, 0.9626106112432703039637754, 0.9642530365726560206402068, 0.9658592291217406674540047, 0.9674291285362237773389233, 0.9689626758255565756615864, 0.9704598133651586944555050, 0.9719204848985835745206522, 0.9733446355396324773464471, 0.9747322117744170315712560, 0.9760831614633702416830300, 0.9773974338432058899681861, 0.9786749795288262664309572, 0.9799157505151781656726285, 0.9811197001790570947322311, 0.9822867832808596419166429, 0.9834169559662839640681455, 0.9845101757679783590716126, 0.9855664016071379024692622, 0.9865855937950491429603684, 0.9875677140345828729848910, 0.9885127254216350200148487, 0.9894205924465157453777048, 0.9902912809952868962106899, 0.9911247583510480415528399, 0.9919209931951714500244370, 0.9926799556084865573546763, 0.9934016170724147657859271, 0.9940859504700558793702825, 0.9947329300872282225027936, 0.9953425316134657151476031, 0.9959147321429772566997088, 0.9964495101755774022837600, 0.9969468456176038804367370, 0.9974067197828498321611183, 0.9978291153935628466036470, 0.9982140165816127953876923, 0.9985614088900397275573677, 0.9988712792754494246541769, 0.9991436161123782382453400, 0.9993784092025992514480161, 0.9995756497983108555936109, 0.9997353306710426625827368, 0.9998574463699794385446275, 0.9999419946068456536361287, 0.9999889909843818679872841};
+static double w512[256] = {0.0061299051754057857591564, 0.0061296748380364986664278, 0.0061292141719530834395471, 0.0061285231944655327693402, 0.0061276019315380226384508, 0.0061264504177879366912426, 0.0061250686964845654506976, 0.0061234568195474804311878, 0.0061216148475445832082156, 0.0061195428496898295184288, 0.0061172409038406284754329, 0.0061147090964949169991245, 0.0061119475227879095684795, 0.0061089562864885234199252, 0.0061057354999954793256260, 0.0061022852843330780981965, 0.0060986057691466529805468, 0.0060946970926976980917399, 0.0060905594018586731119147, 0.0060861928521074844014940, 0.0060815976075216427620556, 0.0060767738407720980583934, 0.0060717217331167509334394, 0.0060664414743936418598512, 0.0060609332630138177841916, 0.0060551973059538766317450, 0.0060492338187481899521175, 0.0060430430254808039978627, 0.0060366251587770195404584, 0.0060299804597946507400317, 0.0060231091782149633972884, 0.0060160115722332929281516, 0.0060086879085493424136484, 0.0060011384623571610896056, 0.0059933635173348036527221, 0.0059853633656336707715812, 0.0059771383078675312031423, 0.0059686886531012259272183, 0.0059600147188390547233923, 0.0059511168310128456267588, 0.0059419953239697077107922, 0.0059326505404594676575446, 0.0059230828316217905872556, 0.0059132925569729856313229, 0.0059032800843924967444267, 0.0058930457901090792634301, 0.0058825900586866627324847, 0.0058719132830099005255609, 0.0058610158642694068093892, 0.0058498982119466814015496, 0.0058385607437987230901727, 0.0058270038858423319934219, 0.0058152280723381015486124, 0.0058032337457741007324836, 0.0057910213568492471257818, 0.0057785913644563714469284, 0.0057659442356649741911390, 0.0057530804457036750229319, 0.0057400004779423555815070, 0.0057267048238739963699973, 0.0057131939830962084110906, 0.0056994684632924603629882, 0.0056855287802130018011102, 0.0056713754576554833823756, 0.0056570090274452746202723, 0.0056424300294154800102991, 0.0056276390113866542566918, 0.0056126365291462173626557, 0.0055974231464275703576030, 0.0055819994348889124461425, 0.0055663659740917603747899, 0.0055505233514791708235538, 0.0055344721623536666407146, 0.0055182130098548677502395, 0.0055017465049368275723757, 0.0054850732663450758090285, 0.0054681939205933684565648, 0.0054511091019401459196852, 0.0054338194523647001109732, 0.0054163256215430514316688, 0.0053986282668235365401123, 0.0053807280532021078251738, 0.0053626256532973455128155, 0.0053443217473251833447318, 0.0053258170230733487787774, 0.0053071121758755186716175, 0.0052882079085851914147269, 0.0052691049315492765055207, 0.0052498039625814025460136, 0.0052303057269349446719890, 0.0052106109572757724261988, 0.0051907203936547190996206, 0.0051706347834797735752665, 0.0051503548814879957194620, 0.0051298814497171563759039, 0.0051092152574771030281542, 0.0050883570813208522065339, 0.0050673077050154097256505, 0.0050460679195123198490183, 0.0050246385229179444874178, 0.0050030203204634735477834, 0.0049812141244746675595135, 0.0049592207543413337151533, 0.0049370410364865364724225, 0.0049146758043355438745290, 0.0048921258982845107556462, 0.0048693921656689000083132, 0.0048464754607316430993636, 0.0048233766445910410307843, 0.0048000965852084069516609, 0.0047766361573554516370718, 0.0047529962425814130594576, 0.0047291777291799312876071, 0.0047051815121556699579709, 0.0046810084931906855725376, 0.0046566595806105458869828, 0.0046321356893501986622283, 0.0046074377409195920619320, 0.0045825666633690479877601, 0.0045575233912543896535753, 0.0045323088656018247089130, 0.0045069240338725852313010, 0.0044813698499273259161146, 0.0044556472739902818017469, 0.0044297572726131868769073, 0.0044037008186389549258496, 0.0043774788911651239762643, 0.0043510924755070657234522, 0.0043245425631609613132305, 0.0042978301517665448748000, 0.0042709562450696162035304, 0.0042439218528843240022977, 0.0042167279910552210986262, 0.0041893756814190930634598, 0.0041618659517665616659011, 0.0041341998358034646067195, 0.0041063783731120129818357, 0.0040784026091117279353449, 0.0040502735950201579699371, 0.0040219923878133783908191, 0.0039935600501862743674273, 0.0039649776505126091053562, 0.0039362462628048786290012, 0.0039073669666739546834366, 0.0038783408472885172720108, 0.0038491689953342783540510, 0.0038198525069729982349166, 0.0037903924838012961884344, 0.0037607900328092568594835, 0.0037310462663388340021755, 0.0037011623020420531166926, 0.0036711392628390145554094, 0.0036409782768756986764252, 0.0036106804774815746300758, 0.0035802470031270143713799, 0.0035496789973805134987000, 0.0035189776088657205261605, 0.0034881439912182762045767, 0.0034571793030424645127888, 0.0034260847078676769483860, 0.0033948613741046917538288, 0.0033635104750017697209450, 0.0033320331886005682236783, 0.0033004306976918751358177, 0.0032687041897711642972145, 0.0032368548569939741987234, 0.0032048838961311115627642, 0.0031727925085236815030060, 0.0031405819000379459532169, 0.0031082532810200120618074, 0.0030758078662503522550163, 0.0030432468748981576780527, 0.0030105715304755267298129, 0.0029777830607914904130339, 0.0029448826979058762279357, 0.0029118716780830123435331, 0.0028787512417452737868732, 0.0028455226334264723964728, 0.0028121871017250922921949, 0.0027787458992573726197173, 0.0027452002826102393336092, 0.0027115515122940877888456, 0.0026778008526954179163600, 0.0026439495720293237639656, 0.0026099989422918391896635, 0.0025759502392121415000167, 0.0025418047422046148318992, 0.0025075637343207750815413, 0.0024732285022010581903898, 0.0024388003360264736029032, 0.0024042805294701247170072, 0.0023696703796485981535706, 0.0023349711870732236769383, 0.0023001842556012066042973, 0.0022653108923866345474810, 0.0022303524078313603367724, 0.0021953101155357629823745, 0.0021601853322493885355395, 0.0021249793778214727179358, 0.0020896935751513471947536, 0.0020543292501387313744068, 0.0020188877316339116255770, 0.0019833703513878098109153, 0.0019477784440019430461334, 0.0019121133468782766036998, 0.0018763764001689718921795, 0.0018405689467260314557679, 0.0018046923320508429542037, 0.0017687479042436241015783, 0.0017327370139527705642995, 0.0016966610143241088445575, 0.0016605212609500562072903, 0.0016243191118186897474239, 0.0015880559272627267421479, 0.0015517330699084184928942, 0.0015153519046243599371387, 0.0014789137984702174059640, 0.0014424201206453770259886, 0.0014058722424375164225552, 0.0013692715371711025869345, 0.0013326193801558190401403, 0.0012959171486349257824991, 0.0012591662217335559930561, 0.0012223679804069540808915, 0.0011855238073886605549070, 0.0011486350871386503607080, 0.0011117032057914329649653, 0.0010747295511041247428251, 0.0010377155124045074300544, 0.0010006624805390909706032, 0.0009635718478212056798501, 0.0009264450079791582697455, 0.0008892833561045005372012, 0.0008520882886004809402792, 0.0008148612031307819965602, 0.0007776034985686972438014, 0.0007403165749469818962867, 0.0007030018334087411433900, 0.0006656606761599343409382, 0.0006282945064244358390880, 0.0005909047284032230162400, 0.0005534927472403894647847, 0.0005160599690007674370993, 0.0004786078006679509066920, 0.0004411376501795405636493, 0.0004036509265333198797447, 0.0003661490400356268530141, 0.0003286334028523334162522, 0.0002911054302514885125319, 0.0002535665435705865135866, 0.0002160181779769908583388, 0.0001784618055459532946077, 0.0001408990173881984930124, 0.0001033319034969132362968, 0.0000657657316592401958310, 0.0000282526373739346920387};
+
+/* n = 1024 */
+static double x1024[512] = {0.0015332313560626384065387, 0.0045996796509132604743248, 0.0076660846940754867627839, 0.0107324176515422803327458, 0.0137986496899844401539048, 0.0168647519770217265449962, 0.0199306956814939776907024, 0.0229964519737322146859283, 0.0260619920258297325581921, 0.0291272870119131747190088, 0.0321923081084135882953009, 0.0352570264943374577920498, 0.0383214133515377145376052, 0.0413854398649847193632977, 0.0444490772230372159692514, 0.0475122966177132524285687, 0.0505750692449610682823599, 0.0536373663049299446784129, 0.0566991590022410150066456, 0.0597604185462580334848567, 0.0628211161513580991486838, 0.0658812230372023327000985, 0.0689407104290065036692117, 0.0719995495578116053446277, 0.0750577116607543749280791, 0.0781151679813377563695878, 0.0811718897697013033399379, 0.0842278482828915197978074, 0.0872830147851321356094940, 0.0903373605480943146797811, 0.0933908568511667930531222, 0.0964434749817259444449839, 0.0994951862354057706638682, 0.1025459619163678143852404, 0.1055957733375709917393206, 0.1086445918210413421754502, 0.1116923886981416930665228, 0.1147391353098412365177689, 0.1177848030069850158450139, 0.1208293631505633191883714, 0.1238727871119809777282145, 0.1269150462733265659711591, 0.1299561120276415015747167, 0.1329959557791890421802183, 0.1360345489437231767245806, 0.1390718629487574087024745, 0.1421078692338334288514767, 0.1451425392507896747338214, 0.1481758444640297746894331, 0.1512077563507908736360111, 0.1542382464014118381930443, 0.1572672861196013386077717, 0.1602948470227058049622614, 0.1633209006419772551419632, 0.1663454185228409920472972, 0.1693683722251631675310675, 0.1723897333235182105457458, 0.1754094734074561169859457, 0.1784275640817695987127083, 0.1814439769667610892475458, 0.1844586836985096036255346, 0.1874716559291374498981239, 0.1904828653270767897777182, 0.1934922835773360459175133, 0.1964998823817661533215037, 0.1995056334593266523810493, 0.2025095085463516210358758, 0.2055114793968154435588961, 0.2085115177825984134657778, 0.2115095954937521680517391, 0.2145056843387649520596422, 0.2174997561448267079850562, 0.2204917827580939905255947, 0.2234817360439547026834844, 0.2264695878872926510320010, 0.2294553101927519176581055, 0.2324388748850010462953415, 0.2354202539089970401627982, 0.2383994192302491690277166, 0.2413763428350825830111093, 0.2443509967309017306575811, 0.2473233529464535787923793, 0.2502933835320906316905658, 0.2532610605600337470850902, 0.2562263561246347465424530, 0.2591892423426388177365829, 0.2621496913534467061535080, 0.2651076753193766937613805, 0.2680631664259263621824189, 0.2710161368820341379053566, 0.2739665589203406170790369, 0.2769144047974496674298651, 0.2798596467941893048479266, 0.2828022572158723421886958, 0.2857422083925568078394062, 0.2886794726793061316013119, 0.2916140224564490954412652, 0.2945458301298395466682397, 0.2974748681311158710926665, 0.3004011089179602237287060, 0.3033245249743575146018584, 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0.5843066264598643017272666, 0.5867924118077737578782574, 0.5892726794317383594853053, 0.5917474060093159907610475, 0.5942165682701680800580147, 0.5966801429962784154186793, 0.5991381070221714681281111, 0.6015904372351302222163013, 0.6040371105754135078618616, 0.6064781040364728366534687, 0.6089133946651687366701116, 0.6113429595619865853458987, 0.6137667758812519380899084, 0.6161848208313453506363029, 0.6185970716749166931046915, 0.6210035057290989537555048, 0.6234041003657215304299416, 0.6257988330115230076688675, 0.6281876811483634175098794, 0.6305706223134359819666081, 0.6329476340994783351992008, 0.6353186941549832233898213, 0.6376837801844086803419153, 0.6400428699483876768269192, 0.6423959412639372417070377, 0.6447429720046670528676835, 0.6470839401009874959981582, 0.6494188235403171892641570, 0.6517476003672899719207013, 0.6540702486839613549191454, 0.6563867466500144315669620, 0.6586970724829652463040876, 0.6610012044583676196647058, 0.6632991209100174274984589, 0.6655908002301563325302097, 0.6678762208696749663426270, 0.6701553613383155598710345, 0.6724282002048740205051479, 0.6746947160974014538975312, 0.6769548877034051285838219, 0.6792086937700488815250166, 0.6814561131043529626873631, 0.6836971245733933167806834, 0.6859317071045003002812397, 0.6881598396854568318705713, 0.6903815013646959744270519, 0.6925966712514979467122689, 0.6948053285161865628996815, 0.6970074523903250980984011, 0.6992030221669115780303307, 0.7013920172005734910243170, 0.7035744169077619204963997, 0.7057502007669450960906928, 0.7079193483188013616608982, 0.7100818391664115582779368, 0.7122376529754508204546805, 0.7143867694743797837842896, 0.7165291684546352021941915, 0.7186648297708199730232898, 0.7207937333408925681355609, 0.7229158591463558692887801, 0.7250311872324454059827217, 0.7271396977083169940167956, 0.7292413707472337729927181, 0.7313361865867526410034676, 0.7334241255289100847554419, 0.7355051679404074033764222, 0.7375792942527953241676460, 0.7396464849626580085640129, 0.7417067206317964465721772, 0.7437599818874112379620360, 0.7458062494222847584928838, 0.7478455039949627094612890, 0.7498777264299350488635483, 0.7519028976178163024713854, 0.7539209985155252531253957, 0.7559320101464640065565832, 0.7579359136006964320521972, 0.7599326900351259762879594, 0.7619223206736728486546595, 0.7639047868074505764130149, 0.7658800697949419280166093, 0.7678481510621742029486694, 0.7698090121028938864243967, 0.7717626344787406673165402, 0.7737089998194208176678866, 0.7756480898228799321603470, 0.7775798862554750259163361, 0.7795043709521459890141759, 0.7814215258165863961053031, 0.7833313328214136695271245, 0.7852337740083385943114429, 0.7871288314883341834944720, 0.7890164874418038921405657, 0.7908967241187491784979139, 0.7927695238389364107105941, 0.7946348689920631175175217, 0.7964927420379235813750136, 0.7983431255065737724458586, 0.8001860019984956219039900, 0.8020213541847606330100649, 0.8038491648071928284194859, 0.8056694166785310321906380, 0.8074820926825904849673728, 0.8092871757744237908160400, 0.8110846489804811942036542, 0.8128744953987701856100790, 0.8146566981990144342734272, 0.8164312406228120465742028, 0.8181981059837931485700490, 0.8199572776677767911993239, 0.8217087391329271766780945, 0.8234524739099092046215225, 0.8251884656020433364270094, 0.8269166978854597764628854, 0.8286371545092519686128428, 0.8303498192956294067327593, 0.8320546761400697575830038, 0.8337517090114702948057846, 0.8354409019522986425235764, 0.8371222390787428271411563, 0.8387957045808606359402829, 0.8404612827227282810625704, 0.8421189578425883674826439, 0.8437687143529971635802028, 0.8454105367409711729261812, 0.8470444095681330059047621, 0.8486703174708565497995875, 0.8502882451604114359791023, 0.8518981774231068028225812, 0.8535000991204343530350070, 0.8550939951892107040056078, 0.8566798506417190298715048, 0.8582576505658499939545848, 0.8598273801252419702463831, 0.8613890245594205526224495, 0.8629425691839373504743648, 0.8644879993905080694542896, 0.8660253006471498760336444, 0.8675544584983180445842596, 0.8690754585650418856970762, 0.8705882865450599544602407, 0.8720929282129545374252050, 0.8735893694202854169962281, 0.8750775960957229119854680, 0.8765575942451801930826613, 0.8780293499519448719952049, 0.8794928493768098630212838, 0.8809480787582035158255322, 0.8823950244123190181935674, 0.8838336727332430675485994, 0.8852640101930838100201983, 0.8866860233420980458621863, 0.8880996988088177000235219, 0.8895050233001755566829532, 0.8909019836016302565651375, 0.8922905665772905558628607, 0.8936707591700388455969280, 0.8950425484016539302522575, 0.8964059213729330645356690, 0.8977608652638132471078410, 0.8991073673334917701488930, 0.9004454149205460236240486, 0.9017749954430525531228459, 0.9030960963987053701523781, 0.9044087053649335137720782, 0.9057128099990178624646022, 0.9070083980382071951444166, 0.9082954572998335002127549, 0.9095739756814265315746820, 0.9108439411608276105410847, 0.9121053417963026725455006, 0.9133581657266545576127977, 0.9146024011713345435238301, 0.9158380364305531206273175, 0.9170650598853900072573273, 0.9182834599979034047218800, 0.9194932253112384908353520, 0.9206943444497351509745089, 0.9218868061190349456451742, 0.9230705991061873135537215, 0.9242457122797550091847637, 0.9254121345899187738936182, 0.9265698550685812395293315, 0.9277188628294700636112689, 0.9288591470682402950895005, 0.9299906970625759697264543, 0.9311135021722909341445515, 0.9322275518394288975917975, 0.9333328355883627104845635, 0.9344293430258928687940732, 0.9355170638413452433503852, 0.9365959878066680331449597, 0.9376661047765279417201973, 0.9387274046884055757416456, 0.9397798775626900648558921, 0.9408235135027729019444869, 0.9418583026951410028915762, 0.9428842354094689849902736, 0.9439013019987106631201510, 0.9449094928991897628355911, 0.9459087986306898495121205, 0.9468992097965434727052183, 0.9478807170837205248834878, 0.9488533112629158137054760, 0.9498169831886358470168335, 0.9507717237992848297519245, 0.9517175241172498719314184, 0.9526543752489854069548347, 0.9535822683850968193944507, 0.9545011948004232815044368, 0.9554111458541197976665483, 0.9563121129897384560011695, 0.9572040877353088863799924, 0.9580870617034179240840996, 0.9589610265912884783587268, 0.9598259741808576051234879, 0.9606818963388537831043733, 0.9615287850168733926613630, 0.9623666322514563965930439, 0.9631954301641612222071790, 0.9640151709616388439537466, 0.9648258469357060659245549, 0.9656274504634180035311332, 0.9664199740071397636802195, 0.9672034101146173227737943, 0.9679777514190476018682591, 0.9687429906391477383350273, 0.9694991205792235533724866, 0.9702461341292372147270016, 0.9709840242648740939883669, 0.9717127840476088178328839, 0.9724324066247705125950353, 0.9731428852296072415565604, 0.9738442131813496343496072, 0.9745363838852737078785517, 0.9752193908327628781730396, 0.9758932276013691625928266, 0.9765578878548735718130775, 0.9772133653433456910269459, 0.9778596539032024498104955, 0.9784967474572660801033674, 0.9791246400148212617670490, 0.9797433256716714551911835, 0.9803527986101944204270933, 0.9809530530993969223366037, 0.9815440834949686212533729, 0.9821258842393351486632952, 0.9826984498617103674201996, 0.9832617749781478160230522, 0.9838158542915913364912672, 0.9843606825919248853856025, 0.9848962547560215275335618, 0.9854225657477916120303537, 0.9859396106182301300994116, 0.9864473845054632544104222, 0.9869458826347940594679517, 0.9874351003187474227003598, 0.9879150329571141058970610, 0.9883856760369940166627304, 0.9888470251328386495802522, 0.9892990759064927068006818, 0.9897418241072348978090276, 0.9901752655718179181502248, 0.9905993962245076069415402, 0.9910142120771212830473891, 0.9914197092290652598522332, 0.9918158838673715386394944, 0.9922027322667336806727008, 0.9925802507895418581838653, 0.9929484358859170846092543, 0.9933072840937446245820355, 0.9936567920387065844051246, 0.9939969564343136839997662, 0.9943277740819362116746914, 0.9946492418708341635125525, 0.9949613567781865697596566, 0.9952641158691200113800912, 0.9955575162967363309635588, 0.9958415553021395435525955, 0.9961162302144619548145649, 0.9963815384508894965215124, 0.9966374775166862927999356, 0.9968840450052184754903082, 0.9971212385979772738362093, 0.9973490560646014135491635, 0.9975674952628988745188845, 0.9977765541388680773265018, 0.9979762307267185998745420, 0.9981665231488915727109186, 0.9983474296160799746514418, 0.9985189484272491654281575, 0.9986810779696581776171579, 0.9988338167188825964389443, 0.9989771632388403756649803, 0.9991111161818228462260355, 0.9992356742885348165163858, 0.9993508363881507486653971, 0.9994566013984000492749057, 0.9995529683257070064969677, 0.9996399362654382464576482, 0.9997175044023747284307007, 0.9997856720116889628341744, 0.9998444384611711916084367, 0.9998938032169419878731474, 0.9999337658606177711221103, 0.9999643261538894550943330, 0.9999854843850284447675914, 0.9999972450545584403516182};
+static double w1024[512] = {0.0030664603092439082115513, 0.0030664314747171934849726, 0.0030663738059349007324470, 0.0030662873034393008056861, 0.0030661719680437936084028, 0.0030660278008329004477528, 0.0030658548031622538363679, 0.0030656529766585847450783, 0.0030654223232197073064431, 0.0030651628450145009692318, 0.0030648745444828901040266, 0.0030645574243358210601357, 0.0030642114875552366740338, 0.0030638367373940482295700, 0.0030634331773761048702058, 0.0030630008112961604635720, 0.0030625396432198379186545, 0.0030620496774835909559465, 0.0030615309186946633309249, 0.0030609833717310455112352, 0.0030604070417414288079918, 0.0030598019341451569616257, 0.0030591680546321751827342, 0.0030585054091629766484119, 0.0030578140039685464545661, 0.0030570938455503030247440, 0.0030563449406800369760227, 0.0030555672963998474425352, 0.0030547609200220758572342, 0.0030539258191292371925135, 0.0030530620015739486603347, 0.0030521694754788558725307, 0.0030512482492365564619779, 0.0030502983315095211653578, 0.0030493197312300123682482, 0.0030483124576000001133114, 0.0030472765200910755723677, 0.0030462119284443619831693, 0.0030451186926704230517109, 0.0030439968230491688209395, 0.0030428463301297590067471, 0.0030416672247305038021562, 0.0030404595179387621506312, 0.0030392232211108374894710, 0.0030379583458718709642643, 0.0030366649041157321154111, 0.0030353429080049070377385, 0.0030339923699703840142628, 0.0030326133027115366251721, 0.0030312057191960043331307, 0.0030297696326595705460252, 0.0030283050566060381583022, 0.0030268120048071025720655, 0.0030252904913022221991274, 0.0030237405303984864452325, 0.0030221621366704811776946, 0.0030205553249601516777118, 0.0030189201103766630786495, 0.0030172565082962582916016, 0.0030155645343621134195681, 0.0030138442044841906616068, 0.0030120955348390887083441, 0.0030103185418698906302495, 0.0030085132422860092601062, 0.0030066796530630300711306, 0.0030048177914425515522176, 0.0030029276749320230818149, 0.0030010093213045803019478, 0.0029990627485988779939449, 0.0029970879751189204574353, 0.0029950850194338893942123, 0.0029930539003779692985814, 0.0029909946370501703558363, 0.0029889072488141488505262, 0.0029867917552980250862041, 0.0029846481763941988183689, 0.0029824765322591622023349, 0.0029802768433133102577897, 0.0029780491302407488518214, 0.0029757934139891002022209, 0.0029735097157693059028890, 0.0029711980570554274731990, 0.0029688584595844444331918, 0.0029664909453560499065010, 0.0029640955366324437529314, 0.0029616722559381232326340, 0.0029592211260596712038487, 0.0029567421700455418562030, 0.0029542354112058439815854, 0.0029517008731121217846274, 0.0029491385795971332348581, 0.0029465485547546259626151, 0.0029439308229391107008170, 0.0029412854087656322747309, 0.0029386123371095381418860, 0.0029359116331062444843108, 0.0029331833221509998552933, 0.0029304274298986463828860, 0.0029276439822633785324025, 0.0029248330054184994301727, 0.0029219945257961747508486, 0.0029191285700871841705750, 0.0029162351652406703883623, 0.0029133143384638857180205, 0.0029103661172219362530391, 0.0029073905292375236068160, 0.0029043876024906842306667, 0.0029013573652185263120627, 0.0028982998459149642555740, 0.0028952150733304507490135, 0.0028921030764717064173001, 0.0028889638846014470665859, 0.0028857975272381085212091, 0.0028826040341555690560623, 0.0028793834353828694269858, 0.0028761357612039305018167, 0.0028728610421572684947521, 0.0028695593090357078067012, 0.0028662305928860914743281, 0.0028628749250089892305081, 0.0028594923369584031789413, 0.0028560828605414710856927, 0.0028526465278181672904478, 0.0028491833711010012402964, 0.0028456934229547136488796, 0.0028421767161959702837564, 0.0028386332838930533848701, 0.0028350631593655507170153, 0.0028314663761840422592303, 0.0028278429681697845340603, 0.0028241929693943925796601, 0.0028205164141795195677262, 0.0028168133370965340702726, 0.0028130837729661949782821, 0.0028093277568583240752928, 0.0028055453240914762689974, 0.0028017365102326074839556, 0.0027979013510967402185435, 0.0027940398827466267692845, 0.0027901521414924101257281, 0.0027862381638912825390663, 0.0027822979867471417676962, 0.0027783316471102450029635, 0.0027743391822768604783394, 0.0027703206297889167653083, 0.0027662760274336497592617, 0.0027622054132432473587211, 0.0027581088254944918412282, 0.0027539863027083999392661, 0.0027498378836498606195970, 0.0027456636073272705694208, 0.0027414635129921673927833, 0.0027372376401388605206822, 0.0027329860285040598383428, 0.0027287087180665020331547, 0.0027244057490465746667821, 0.0027200771619059379749851, 0.0027157229973471443987056, 0.0027113432963132558499974, 0.0027069380999874587163979, 0.0027025074497926766073634, 0.0026980513873911808464073, 0.0026935699546841987126055, 0.0026890631938115194351518, 0.0026845311471510979446691, 0.0026799738573186563850015, 0.0026753913671672833892344, 0.0026707837197870311237119, 0.0026661509585045101038391, 0.0026614931268824817854798, 0.0026568102687194489357814, 0.0026521024280492437872770, 0.0026473696491406139791397, 0.0026426119764968062894804, 0.0026378294548551481626046, 0.0026330221291866270351630, 0.0026281900446954674651512, 0.0026233332468187060677353, 0.0026184517812257642618999, 0.0026135456938180188319369, 0.0026086150307283703078113, 0.0026036598383208091684657, 0.0025986801631899798721388, 0.0025936760521607427178014, 0.0025886475522877335418257, 0.0025835947108549212540321, 0.0025785175753751632172710, 0.0025734161935897584747222, 0.0025682906134679988291122, 0.0025631408832067177780710, 0.0025579670512298373098703, 0.0025527691661879125638030, 0.0025475472769576743594882, 0.0025423014326415695994010, 0.0025370316825672995489502, 0.0025317380762873559984451, 0.0025264206635785553113127, 0.0025210794944415703629476, 0.0025157146191004603745948, 0.0025103260880021986466869, 0.0025049139518161981960773, 0.0024994782614338353016280, 0.0024940190679679709626349, 0.0024885364227524702745874, 0.0024830303773417197267843, 0.0024775009835101424263432, 0.0024719482932517112531633, 0.0024663723587794599504176, 0.0024607732325249921551741, 0.0024551509671379883737605, 0.0024495056154857109065099, 0.0024438372306525067265426, 0.0024381458659393083172574, 0.0024324315748631324732279, 0.0024266944111565770692147, 0.0024209344287673158020275, 0.0024151516818575909099866, 0.0024093462248037038747545, 0.0024035181121955041103265, 0.0023976673988358756439882, 0.0023917941397402217940673, 0.0023858983901359478493246, 0.0023799802054619417548485, 0.0023740396413680528093376, 0.0023680767537145683786720, 0.0023620915985716886306938, 0.0023560842322189992961374, 0.0023500547111449424606655, 0.0023440030920462853929883, 0.0023379294318275874140606, 0.0023318337876006648123684, 0.0023257162166840538103394, 0.0023195767766024715869239, 0.0023134155250862753614165, 0.0023072325200709195436049, 0.0023010278196964109553481, 0.0022948014823067621287099, 0.0022885535664494426857857, 0.0022822841308748288053830, 0.0022759932345356507817318, 0.0022696809365864386804193, 0.0022633472963829660967620, 0.0022569923734816920218464, 0.0022506162276392008214839, 0.0022442189188116403333494, 0.0022378005071541580875846, 0.0022313610530203356561684, 0.0022249006169616211363732, 0.0022184192597267597736437, 0.0022119170422612227292520, 0.0022053940257066339981005, 0.0021988502714001954820607, 0.0021922858408741102242558, 0.0021857007958550038097087, 0.0021790951982633439377969, 0.0021724691102128581719720, 0.0021658225940099498722195, 0.0021591557121531123157498, 0.0021524685273323410114303, 0.0021457611024285442134846, 0.0021390335005129516400021, 0.0021322857848465214018174, 0.0021255180188793451473363, 0.0021187302662500514289029, 0.0021119225907852072963166, 0.0021050950564987181231273, 0.0020982477275912256713511, 0.0020913806684495044002679, 0.0020844939436458560249764, 0.0020775876179375023304007, 0.0020706617562659762464561, 0.0020637164237565111901030, 0.0020567516857174286800274, 0.0020497676076395242297101, 0.0020427642551954515246552, 0.0020357416942391048895728, 0.0020286999908050000513193, 0.0020216392111076532034194, 0.0020145594215409583780096, 0.0020074606886775631310555, 0.0020003430792682425467160, 0.0019932066602412715667394, 0.0019860514987017956507927, 0.0019788776619311997736447, 0.0019716852173864757651327, 0.0019644742326995879988655, 0.0019572447756768374356240, 0.0019499969142982240274419, 0.0019427307167168074883601, 0.0019354462512580664378677, 0.0019281435864192559230531, 0.0019208227908687633255086, 0.0019134839334454626590447, 0.0019061270831580672642844, 0.0018987523091844809062265, 0.0018913596808711472808775, 0.0018839492677323979370705, 0.0018765211394497986196010, 0.0018690753658714940398285, 0.0018616120170115510799024, 0.0018541311630493004367905, 0.0018466328743286767122991, 0.0018391172213575569552912, 0.0018315842748070976623218, 0.0018240341055110702429247, 0.0018164667844651949558009, 0.0018088823828264733221690, 0.0018012809719125190225581, 0.0017936626232008872833327, 0.0017860274083284027592567, 0.0017783753990904859184165, 0.0017707066674404779358362, 0.0017630212854889641021349, 0.0017553193255030957535871, 0.0017476008599059107299616, 0.0017398659612756523665312, 0.0017321147023450870266539, 0.0017243471560008201813452, 0.0017165633952826110422716, 0.0017087634933826857546100, 0.0017009475236450491562317, 0.0016931155595647951096823, 0.0016852676747874154134422, 0.0016774039431081072989678, 0.0016695244384710795200224, 0.0016616292349688570408253, 0.0016537184068415843295541, 0.0016457920284763272637533, 0.0016378501744063736542136, 0.0016298929193105323938983, 0.0016219203380124312385075, 0.0016139325054798132252838, 0.0016059294968238317366751, 0.0015979113872983442154825, 0.0015898782522992045381361, 0.0015818301673635540527516, 0.0015737672081691112886347, 0.0015656894505334603439125, 0.0015575969704133379579831, 0.0015494898439039192754876, 0.0015413681472381023085203, 0.0015332319567857911038062, 0.0015250813490531776215856, 0.0015169164006820223329593, 0.0015087371884489335424584, 0.0015005437892646454426166, 0.0014923362801732949073323, 0.0014841147383516970308228, 0.0014758792411086194189814, 0.0014676298658840552399621, 0.0014593666902484950408286, 0.0014510897919021973371136, 0.0014427992486744579821480, 0.0014344951385228783230315, 0.0014261775395326321501237, 0.0014178465299157314469528, 0.0014095021880102909474427, 0.0014011445922797915073771, 0.0013927738213123422970256, 0.0013843899538199418218713, 0.0013759930686377377783877, 0.0013675832447232857518263, 0.0013591605611558067629844, 0.0013507250971354436709363, 0.0013422769319825164387192, 0.0013338161451367762689788, 0.0013253428161566586165863, 0.0013168570247185350852537, 0.0013083588506159642151809, 0.0012998483737589411687807, 0.0012913256741731463215379, 0.0012827908319991927650686, 0.0012742439274918727294554, 0.0012656850410194029319476, 0.0012571142530626688591208, 0.0012485316442144679896043, 0.0012399372951787519644928, 0.0012313312867698677125706, 0.0012227136999117975374834, 0.0012140846156373981740056, 0.0012054441150876388205601, 0.0011967922795108381551550, 0.0011881291902619003419159, 0.0011794549288015500353964, 0.0011707695766955663898644, 0.0011620732156140160807669, 0.0011533659273304853455891, 0.0011446477937213110513287, 0.0011359188967648107958214, 0.0011271793185405120501566, 0.0011184291412283803494364, 0.0011096684471080465391373, 0.0011008973185580330843445, 0.0010921158380549794491381, 0.0010833240881728665534171, 0.0010745221515822403144596, 0.0010657101110494342805238, 0.0010568880494357913638046, 0.0010480560496968846800697, 0.0010392141948817375023057, 0.0010303625681320423357186, 0.0010215012526813791214350, 0.0010126303318544325762649, 0.0010037498890662086758941, 0.0009948600078212502888805, 0.0009859607717128519688418, 0.0009770522644222739122264, 0.0009681345697179550890732, 0.0009592077714547255541688, 0.0009502719535730179460261, 0.0009413272000980781811114, 0.0009323735951391753507612, 0.0009234112228888108282347, 0.0009144401676219265933610, 0.0009054605136951127822476, 0.0008964723455458144695262, 0.0008874757476915376906225, 0.0008784708047290547115472, 0.0008694576013336085537138, 0.0008604362222581167813022, 0.0008514067523323745586954, 0.0008423692764622569855308, 0.0008333238796289207169173, 0.0008242706468880048763834, 0.0008152096633688312691343, 0.0008061410142736039032099, 0.0007970647848766078261514, 0.0007879810605234072847989, 0.0007788899266300432158601, 0.0007697914686822300749096, 0.0007606857722345520114971, 0.0007515729229096583980656, 0.0007424530063974587204051, 0.0007333261084543168373926, 0.0007241923149022446178008, 0.0007150517116280949619884, 0.0007059043845827542163241, 0.0006967504197803339882351, 0.0006875899032973623698204, 0.0006784229212719745780188, 0.0006692495599031030193850, 0.0006600699054496667875923, 0.0006508840442297606018626, 0.0006416920626198431946113, 0.0006324940470539251567018, 0.0006232900840227562488244, 0.0006140802600730121876541, 0.0006048646618064809156059, 0.0005956433758792483631993, 0.0005864164890008837132649, 0.0005771840879336241764943, 0.0005679462594915592881427, 0.0005587030905398147360662, 0.0005494546679937357307118, 0.0005402010788180699282026, 0.0005309424100261499182844, 0.0005216787486790752896494, 0.0005124101818848942860548, 0.0005031367967977850677401, 0.0004938586806172365939677, 0.0004845759205872291441124, 0.0004752886039954144966810, 0.0004659968181722957880391, 0.0004567006504904070755681, 0.0004474001883634926336095, 0.0004380955192456860150653, 0.0004287867306306889171352, 0.0004194739100509498966958, 0.0004101571450768429896514, 0.0004008365233158462997325, 0.0003915121324117206363681, 0.0003821840600436882993131, 0.0003728523939256121308821, 0.0003635172218051749865499, 0.0003541786314630598135175, 0.0003448367107121305776064, 0.0003354915473966143456333, 0.0003261432293912849189248, 0.0003167918446006485317858, 0.0003074374809581322877037, 0.0002980802264252762217455, 0.0002887201689909301727620, 0.0002793573966704570567274, 0.0002699919975049447012834, 0.0002606240595604292032823, 0.0002512536709271339139118, 0.0002418809197187298044384, 0.0002325058940716253739001, 0.0002231286821442978268308, 0.0002137493721166826096154, 0.0002043680521896465790359, 0.0001949848105845827899210, 0.0001855997355431850062940, 0.0001762129153274925249194, 0.0001668244382203495280013, 0.0001574343925265138930609, 0.0001480428665748079976500, 0.0001386499487219861751244, 0.0001292557273595155266326, 0.0001198602909254695827354, 0.0001104637279257437565603, 0.0001010661269730276014588, 0.0000916675768613669107254, 0.0000822681667164572752810, 0.0000728679863190274661367, 0.0000634671268598044229933, 0.0000540656828939400071988, 0.0000446637581285753393838, 0.0000352614859871986975067, 0.0000258591246764618586716, 0.0000164577275798968681068, 0.0000070700764101825898713};
+
+/* n = 3 */
+static double x3[2] = {0.0000000000000000000000000, 0.7745966692414833770358531};
+static double w3[2] = {0.8888888888888888888888889, 0.5555555555555555555555556};
+
+/* n = 5 */
+static double x5[3] = {0.0000000000000000000000000, 0.5384693101056830910363144, 0.9061798459386639927976269};
+static double w5[3] = {0.5688888888888888888888889, 0.4786286704993664680412915, 0.2369268850561890875142640};
+
+/* n = 7 */
+static double x7[4] = {0.0000000000000000000000000, 0.4058451513773971669066064, 0.7415311855993944398638648, 0.9491079123427585245261897};
+static double w7[4] = {0.4179591836734693877551020, 0.3818300505051189449503698, 0.2797053914892766679014678, 0.1294849661688696932706114};
+
+/* n = 9 */
+static double x9[5] = {0.0000000000000000000000000, 0.3242534234038089290385380, 0.6133714327005903973087020, 0.8360311073266357942994298, 0.9681602395076260898355762};
+static double w9[5] = {0.3302393550012597631645251, 0.3123470770400028400686304, 0.2606106964029354623187429, 0.1806481606948574040584720, 0.0812743883615744119718922};
+
+/* n = 11 */
+static double x11[6] = {0.0000000000000000000000000, 0.2695431559523449723315320, 0.5190961292068118159257257, 0.7301520055740493240934163, 0.8870625997680952990751578, 0.9782286581460569928039380};
+static double w11[6] = {0.2729250867779006307144835, 0.2628045445102466621806889, 0.2331937645919904799185237, 0.1862902109277342514260976, 0.1255803694649046246346943, 0.0556685671161736664827537};
+
+/* n = 13 */
+static double x13[7] = {0.0000000000000000000000000, 0.2304583159551347940655281, 0.4484927510364468528779129, 0.6423493394403402206439846, 0.8015780907333099127942065, 0.9175983992229779652065478, 0.9841830547185881494728294};
+static double w13[7] = {0.2325515532308739101945895, 0.2262831802628972384120902, 0.2078160475368885023125232, 0.1781459807619457382800467, 0.1388735102197872384636018, 0.0921214998377284479144218, 0.0404840047653158795200216};
+
+/* n = 15 */
+static double x15[8] = {0.0000000000000000000000000, 0.2011940939974345223006283, 0.3941513470775633698972074, 0.5709721726085388475372267, 0.7244177313601700474161861, 0.8482065834104272162006483, 0.9372733924007059043077589, 0.9879925180204854284895657};
+static double w15[8] = {0.2025782419255612728806202, 0.1984314853271115764561183, 0.1861610000155622110268006, 0.1662692058169939335532009, 0.1395706779261543144478048, 0.1071592204671719350118695, 0.0703660474881081247092674, 0.0307532419961172683546284};
+
+/* n = 17 */
+static double x17[9] = {0.0000000000000000000000000, 0.1784841814958478558506775, 0.3512317634538763152971855, 0.5126905370864769678862466, 0.6576711592166907658503022, 0.7815140038968014069252301, 0.8802391537269859021229557, 0.9506755217687677612227170, 0.9905754753144173356754340};
+static double w17[9] = {0.1794464703562065254582656, 0.1765627053669926463252710, 0.1680041021564500445099707, 0.1540457610768102880814316, 0.1351363684685254732863200, 0.1118838471934039710947884, 0.0850361483171791808835354, 0.0554595293739872011294402, 0.0241483028685479319601100};
+
+/* n = 19 */
+static double x19[10] = {0.0000000000000000000000000, 0.1603586456402253758680961, 0.3165640999636298319901173, 0.4645707413759609457172671, 0.6005453046616810234696382, 0.7209661773352293786170959, 0.8227146565371428249789225, 0.9031559036148179016426609, 0.9602081521348300308527788, 0.9924068438435844031890177};
+static double w19[10] = {0.1610544498487836959791636, 0.1589688433939543476499564, 0.1527660420658596667788554, 0.1426067021736066117757461, 0.1287539625393362276755158, 0.1115666455473339947160239, 0.0914900216224499994644621, 0.0690445427376412265807083, 0.0448142267656996003328382, 0.0194617882297264770363120};
+
+/* Merge all together */
+static gsl_integration_glfixed_table glaw[] =
+{
+  {   2,     x2,    w2, 1 /* precomputed */ },
+  {   3,     x3,    w3, 1 },
+  {   4,     x4,    w4, 1 },
+  {   5,     x5,    w5, 1 },
+  {   6,     x6,    w6, 1 },
+  {   7,     x7,    w7, 1 },
+  {   8,     x8,    w8, 1 },
+  {   9,     x9,    w9, 1 },
+  {  10,    x10,   w10, 1 },
+  {  11,    x11,   w11, 1 },
+  {  12,    x12,   w12, 1 },
+  {  13,    x13,   w13, 1 },
+  {  14,    x14,   w14, 1 },
+  {  15,    x15,   w15, 1 },
+  {  16,    x16,   w16, 1 },
+  {  17,    x17,   w17, 1 },
+  {  18,    x18,   w18, 1 },
+  {  19,    x19,   w19, 1 },
+  {  20,    x20,   w20, 1 },
+  {  32,    x32,   w32, 1 },
+  {  64,    x64,   w64, 1 },
+  {  96,    x96,   w96, 1 },
+  {  100,  x100,  w100, 1 },
+  {  128,  x128,  w128, 1 },
+  {  256,  x256,  w256, 1 },
+  {  512,  x512,  w512, 1 },
+  { 1024, x1024, w1024, 1 }
+};
+static const size_t GLAWSIZE = sizeof(glaw)/sizeof(glaw[0]);
+
+
+gsl_integration_glfixed_table *
+  gsl_integration_glfixed_table_alloc (size_t n)
+{
+  int i;
+  double *x = 0;
+  double *w = 0;
+  gsl_integration_glfixed_table * retval = 0;
+
+  /* Kludgy error check: indices in the original algorithm use ints but GSL
+   * conventions are to take a size_t parameter for n */
+  if (n > INT_MAX)
+    {
+      GSL_ERROR_NULL ("Requested n is too large", GSL_EINVAL);
+    }
+
+  /* Use a predefined table of size n if possible */
+  for (i = 0; i < GLAWSIZE;i++)
+    {
+      if (n == glaw[i].n)
+        {
+          retval = &glaw[i];
+          break;
+        }
+    }
+
+  /* No predefined table is available.  Generate one on the fly. */
+  if (retval == 0)
+    {
+      const int m = (n + 1) >> 1;
+
+      x = (double *) malloc(m * sizeof(double));
+      if (x == 0)
+        {
+          GSL_ERROR_NULL ("failed to allocate space for abscissae",
+                  GSL_ENOMEM);
+        }
+
+      w = (double *) malloc(m * sizeof(double));
+      if (w == 0)
+        {
+          GSL_ERROR_NULL ("failed to allocate space for weights",
+                  GSL_ENOMEM);
+          free(x);
+        }
+
+      retval = malloc(sizeof(gsl_integration_glfixed_table));
+      if (retval == 0)
+        {
+          GSL_ERROR_NULL ("failed to allocate space for table struct",
+                  GSL_ENOMEM);
+          free(x);
+          free(w);
+        }
+
+      gauss_legendre_tbl(n, x, w, 1e-10 /* precision magic number */);
+
+      retval->n           = n;
+      retval->x           = x;
+      retval->w           = w;
+      retval->precomputed = 0;
+    }
+
+    return retval;
+}
+
+void
+  gsl_integration_glfixed_table_free (gsl_integration_glfixed_table * t)
+{
+  /* We only free the table memory if we allocated it */
+  /* Leave precomputed, static tables alone */
+  if (! t->precomputed)
+    {
+      free(t->x);
+      free(t->w);
+      free(t);
+    }
+}
+
+/*
+Gauss-Legendre n-points quadrature, exact for polynomial of degree <=2n-1
+
+1. n - even:
+
+int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1)
+   = A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=0..n/2-1)
+ A = (b-a)/2,
+ B = (a+b)/2
+
+
+2. n - odd:
+
+int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1)
+   = A*w[0]*f(B)+A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=1..(n-1)/2)
+ A = (b-a)/2,
+ B = (a+b)/2
+*/
+
+double
+  gsl_integration_glfixed (const gsl_function *f,
+                           double a,
+                           double b,
+                           const gsl_integration_glfixed_table * t)
+{
+  const double * const x = t->x;
+  const double * const w = t->w;
+  const int n = t->n;
+  double A, B, Ax, s;
+  int i, m;
+
+  m = (n + 1) >> 1;
+  A = 0.5 * (b - a);
+  B = 0.5 * (b + a);
+
+  if (n&1) /* n - odd */
+    {
+      s = w[0] * GSL_FN_EVAL(f,B);
+
+      for (i = 1;i < m;i++)
+        {
+          Ax = A * x[i];
+          s += w[i] * (GSL_FN_EVAL(f,B+Ax) + GSL_FN_EVAL(f,B-Ax));
+        }
+
+    }
+  else  /* n - even */
+    {
+
+      s = 0.0;
+
+      for (i = 0;i < m;i++)
+        {
+          Ax = A * x[i];
+          s += w[i] * (GSL_FN_EVAL(f,B+Ax) + GSL_FN_EVAL(f,B-Ax));
+        }
+    }
+
+  return A*s;
+}
+
+/* Computing of abscissas and weights for Gauss-Legendre quadrature for any(reasonable) order n
+ [in] n   - order of quadrature
+ [in] eps - required precision (must be eps>=macheps(double), usually eps = 1e-10 is ok)
+ [out]x   - abscisass, size = (n+1)>>1
+ [out]w   - weights, size = (n+1)>>1
+*/
+
+/* Look up table for fast calculation of Legendre polynomial for n<1024 */
+/* ltbl[k] = 1.0 - 1.0/(double)k; */
+static double ltbl[1024] = {0.00000000000000000000, 0.00000000000000000000, 0.50000000000000000000, 0.66666666666666674000, 0.75000000000000000000, 0.80000000000000004000, 0.83333333333333337000, 0.85714285714285721000, 0.87500000000000000000, 0.88888888888888884000, 0.90000000000000002000, 0.90909090909090906000, 0.91666666666666663000, 0.92307692307692313000, 0.92857142857142860000, 0.93333333333333335000, 0.93750000000000000000, 0.94117647058823528000, 0.94444444444444442000, 0.94736842105263164000, 0.94999999999999996000, 0.95238095238095233000, 0.95454545454545459000, 0.95652173913043481000, 0.95833333333333337000, 0.95999999999999996000, 0.96153846153846156000, 0.96296296296296302000, 0.96428571428571430000, 0.96551724137931039000, 0.96666666666666667000, 0.96774193548387100000, 0.96875000000000000000, 0.96969696969696972000, 0.97058823529411764000, 0.97142857142857142000, 0.97222222222222221000, 0.97297297297297303000, 0.97368421052631582000, 0.97435897435897434000, 0.97499999999999998000, 0.97560975609756095000, 0.97619047619047616000, 0.97674418604651159000, 0.97727272727272729000, 0.97777777777777775000, 0.97826086956521741000, 0.97872340425531912000, 0.97916666666666663000, 0.97959183673469385000, 0.97999999999999998000, 0.98039215686274506000, 0.98076923076923073000, 0.98113207547169812000, 0.98148148148148151000, 0.98181818181818181000, 0.98214285714285710000, 0.98245614035087714000, 0.98275862068965514000, 0.98305084745762716000, 0.98333333333333328000, 0.98360655737704916000, 0.98387096774193550000, 0.98412698412698418000, 0.98437500000000000000, 0.98461538461538467000, 0.98484848484848486000, 0.98507462686567160000, 0.98529411764705888000, 0.98550724637681164000, 0.98571428571428577000, 0.98591549295774650000, 0.98611111111111116000, 0.98630136986301364000, 0.98648648648648651000, 0.98666666666666669000, 0.98684210526315785000, 0.98701298701298701000, 0.98717948717948723000, 0.98734177215189878000, 0.98750000000000004000, 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0.99180327868852458000, 0.99186991869918695000, 0.99193548387096775000, 0.99199999999999999000, 0.99206349206349209000, 0.99212598425196852000, 0.99218750000000000000, 0.99224806201550386000, 0.99230769230769234000, 0.99236641221374045000, 0.99242424242424243000, 0.99248120300751874000, 0.99253731343283580000, 0.99259259259259258000, 0.99264705882352944000, 0.99270072992700731000, 0.99275362318840576000, 0.99280575539568350000, 0.99285714285714288000, 0.99290780141843971000, 0.99295774647887325000, 0.99300699300699302000, 0.99305555555555558000, 0.99310344827586206000, 0.99315068493150682000, 0.99319727891156462000, 0.99324324324324320000, 0.99328859060402686000, 0.99333333333333329000, 0.99337748344370858000, 0.99342105263157898000, 0.99346405228758172000, 0.99350649350649356000, 0.99354838709677418000, 0.99358974358974361000, 0.99363057324840764000, 0.99367088607594933000, 0.99371069182389937000, 0.99375000000000002000, 0.99378881987577639000, 0.99382716049382713000, 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0.99871465295629824000, 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, 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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

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

[Rhys Ulerich]

In that patch I screwed up the order of the GSL_ERROR_NULL vs free
calls within gsl_integration_glfixed_table, e.g.

      w = (double *) malloc(m * sizeof(double));
      if (w == 0)
        {
          GSL_ERROR_NULL ("failed to allocate space for weights",
                  GSL_ENOMEM);
          free(x);
        }

should definitely have their order flipped.  The problem appears once
again just below those lines.

- Rhys


On Tue, Jun 23, 2009 at 11:58 AM, Rhys Ulerich<rhys.ulerich@gmail.com> wrote:
>> 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
>

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

[Brian Gough]

At Tue, 23 Jun 2009 11:58:00 -0500,
Rhys Ulerich wrote:
> 
> [1  <text/plain; ISO-8859-1 (7bit)>]
> > 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...

I've checked this into the branch integ-glfixed
at http://src.network-theory.com/gsl.git 

I'll merge it after the next release.

-- 
Brian Gough