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[Axiom-developer] 20080107.02.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080107.02.tpd.patch |
|
Date: |
Sun, 6 Jan 2008 23:53:17 -0600 |
This patch adds two files missing from prior updates. -- Tim
========================================================================
diff --git a/changelog b/changelog
index 9c87000..41ba008 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080107 tpd src/input/gamma.input missing file added
+20080107 tpd src/input/pfaffian.input missing file added
20080107 tpd src/input/Makefile add free to test bug 7089/343
20080107 tpd src/input/free.input added to test bug 7089/343
20080107 mxr src/input/free.spad fix FreeAbelianGroup bug 7089/343
diff --git a/src/input/gamma.input.pamphlet b/src/input/gamma.input.pamphlet
new file mode 100644
index 0000000..608cf75
--- /dev/null
+++ b/src/input/gamma.input.pamphlet
@@ -0,0 +1,1596 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input gamma.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a set of tests for the Gamma and polygamma functions
+as well as a set of graphics that illustrate these curves.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of Gamma, ranging from 1.000 to 1.995. The
+second column is the reference value of Gamma from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+
+Notice that certain entries have reasonably large differences:
+\begin{verbatim}
+[1.215, 0.9143261399876792, 0.91432614997006778, 9.9823885824079639E-9]
+[1.455, 0.88562206999631599, 0.88562207999314335, 9.9968273659101214E-9]
+\end{verbatim}
+
+<<*>>=
+)spool gamma.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1 of 3
+[[1.000,1.0000000000,Gamma(1.000),Gamma(1.000)-1.0000000000],_
+ [1.005,0.9971385354,Gamma(1.005),Gamma(1.005)-0.9971385354],_
+ [1.010,0.9943258512,Gamma(1.010),Gamma(1.010)-0.9943258512],_
+ [1.015,0.9915612888,Gamma(1.015),Gamma(1.015)-0.9915612888],_
+ [1.020,0.9888442033,Gamma(1.020),Gamma(1.020)-0.9888442033],_
+ [1.025,0.9861739633,Gamma(1.025),Gamma(1.025)-0.9861739633],_
+ [1.030,0.9835499506,Gamma(1.030),Gamma(1.030)-0.9835499506],_
+ [1.035,0.9809715606,Gamma(1.035),Gamma(1.035)-0.9809715606],_
+ [1.040,0.9784382009,Gamma(1.040),Gamma(1.040)-0.9784382009],_
+ [1.045,0.9759492919,Gamma(1.045),Gamma(1.045)-0.9759492919],_
+ [1.050,0.9735042656,Gamma(1.050),Gamma(1.050)-0.9735042656],_
+ [1.055,0.9711025663,Gamma(1.055),Gamma(1.055)-0.9711025663],_
+ [1.060,0.9687436495,Gamma(1.060),Gamma(1.060)-0.9687436495],_
+ [1.065,0.9664269823,Gamma(1.065),Gamma(1.065)-0.9664269823],_
+ [1.070,0.9641520425,Gamma(1.070),Gamma(1.070)-0.9641520425],_
+ [1.075,0.9619183189,Gamma(1.075),Gamma(1.075)-0.9619183189],_
+ [1.080,0.9597253107,Gamma(1.080),Gamma(1.080)-0.9597253107],_
+ [1.085,0.9575725273,Gamma(1.085),Gamma(1.085)-0.9575725273],_
+ [1.090,0.9554594882,Gamma(1.090),Gamma(1.090)-0.9554594882],_
+ [1.095,0.9533857227,Gamma(1.095),Gamma(1.095)-0.9533857227],_
+ [1.100,0.9513507699,Gamma(1.100),Gamma(1.100)-0.9513507699],_
+ [1.105,0.9493541778,Gamma(1.105),Gamma(1.105)-0.9493541778],_
+ [1.110,0.9473955040,Gamma(1.110),Gamma(1.110)-0.9473955040],_
+ [1.115,0.9454743149,Gamma(1.115),Gamma(1.115)-0.9454743149],_
+ [1.120,0.9435901856,Gamma(1.120),Gamma(1.120)-0.9435901856],_
+ [1.125,0.9417426997,Gamma(1.125),Gamma(1.125)-0.9417426997],_
+ [1.130,0.9399314497,Gamma(1.130),Gamma(1.130)-0.9399314497],_
+ [1.135,0.9381560356,Gamma(1.135),Gamma(1.135)-0.9381560356],_
+ [1.140,0.9364160657,Gamma(1.140),Gamma(1.140)-0.9364160657],_
+ [1.145,0.9347111562,Gamma(1.145),Gamma(1.145)-0.9347111562],_
+ [1.150,0.9330409311,Gamma(1.150),Gamma(1.150)-0.9330409311],_
+ [1.155,0.9314050217,Gamma(1.155),Gamma(1.155)-0.9314050217],_
+ [1.160,0.9298030666,Gamma(1.160),Gamma(1.160)-0.9298030666],_
+ [1.165,0.9282347120,Gamma(1.165),Gamma(1.165)-0.9282347120],_
+ [1.170,0.9266996106,Gamma(1.170),Gamma(1.170)-0.9266996106],_
+ [1.175,0.9251974225,Gamma(1.175),Gamma(1.175)-0.9251974225],_
+ [1.180,0.9237278143,Gamma(1.180),Gamma(1.180)-0.9237278143],_
+ [1.185,0.9222904591,Gamma(1.185),Gamma(1.185)-0.9222904591],_
+ [1.190,0.9208850371,Gamma(1.190),Gamma(1.190)-0.9208850371],_
+ [1.195,0.9195112341,Gamma(1.195),Gamma(1.195)-0.9195112341],_
+ [1.200,0.9181687424,Gamma(1.200),Gamma(1.200)-0.9181687424],_
+ [1.205,0.9168572606,Gamma(1.205),Gamma(1.205)-0.9168572606],_
+ [1.210,0.9155764930,Gamma(1.210),Gamma(1.210)-0.9155764930],_
+ [1.215,0.9143261400,Gamma(1.215),Gamma(1.215)-0.9143261400],_
+ [1.220,0.9131059475,Gamma(1.220),Gamma(1.220)-0.9131059475],_
+ [1.225,0.9119156071,Gamma(1.225),Gamma(1.225)-0.9119156071],_
+ [1.230,0.9107548564,Gamma(1.230),Gamma(1.230)-0.9107548564],_
+ [1.235,0.9096234274,Gamma(1.235),Gamma(1.235)-0.9096234274],_
+ [1.240,0.9085210583,Gamma(1.240),Gamma(1.240)-0.9085210583],_
+ [1.245,0.9074474922,Gamma(1.245),Gamma(1.245)-0.9074474922],_
+ [1.250,0.9064024771,Gamma(1.250),Gamma(1.250)-0.9064024771],_
+ [1.255,0.9053857663,Gamma(1.255),Gamma(1.255)-0.9053857663],_
+ [1.260,0.9043971178,Gamma(1.260),Gamma(1.260)-0.9043971178],_
+ [1.265,0.9034362946,Gamma(1.265),Gamma(1.265)-0.9034362946],_
+ [1.270,0.9025030645,Gamma(1.270),Gamma(1.270)-0.9025030645],_
+ [1.275,0.9015971994,Gamma(1.275),Gamma(1.275)-0.9015971994],_
+ [1.280,0.9007184765,Gamma(1.280),Gamma(1.280)-0.9007184765],_
+ [1.285,0.8998666769,Gamma(1.285),Gamma(1.285)-0.8998666769],_
+ [1.290,0.8990415863,Gamma(1.290),Gamma(1.290)-0.8990415863],_
+ [1.295,0.8982429947,Gamma(1.295),Gamma(1.295)-0.8982429947],_
+ [1.300,0.8974706963,Gamma(1.300),Gamma(1.300)-0.8974706963],_
+ [1.305,0.8967244895,Gamma(1.305),Gamma(1.305)-0.8967244895],_
+ [1.310,0.8960041767,Gamma(1.310),Gamma(1.310)-0.8960041767],_
+ [1.315,0.8953095644,Gamma(1.315),Gamma(1.315)-0.8953095644],_
+ [1.320,0.8946404630,Gamma(1.320),Gamma(1.320)-0.8946404630],_
+ [1.325,0.8939966866,Gamma(1.325),Gamma(1.325)-0.8939966866],_
+ [1.330,0.8933780535,Gamma(1.330),Gamma(1.330)-0.8933780535],_
+ [1.335,0.8927843850,Gamma(1.335),Gamma(1.335)-0.8927843850],_
+ [1.340,0.8922155072,Gamma(1.340),Gamma(1.340)-0.8922155072],_
+ [1.345,0.8916712485,Gamma(1.345),Gamma(1.345)-0.8916712485],_
+ [1.350,0.8911514420,Gamma(1.350),Gamma(1.350)-0.8911514420],_
+ [1.355,0.8906559235,Gamma(1.355),Gamma(1.355)-0.8906559235],_
+ [1.360,0.8901845324,Gamma(1.360),Gamma(1.360)-0.8901845324],_
+ [1.365,0.8897371116,Gamma(1.365),Gamma(1.365)-0.8897371116],_
+ [1.370,0.8893135074,Gamma(1.370),Gamma(1.370)-0.8893135074],_
+ [1.375,0.8889135692,Gamma(1.375),Gamma(1.375)-0.8889135692],_
+ [1.380,0.8885371494,Gamma(1.380),Gamma(1.380)-0.8885371494],_
+ [1.385,0.8881841041,Gamma(1.385),Gamma(1.385)-0.8881841041],_
+ [1.390,0.8878542918,Gamma(1.390),Gamma(1.390)-0.8878542918],_
+ [1.395,0.8875475748,Gamma(1.395),Gamma(1.395)-0.8875475748],_
+ [1.400,0.8872638175,Gamma(1.400),Gamma(1.400)-0.8872638175],_
+ [1.405,0.8870028884,Gamma(1.405),Gamma(1.405)-0.8870028884],_
+ [1.410,0.8867646576,Gamma(1.410),Gamma(1.410)-0.8867646576],_
+ [1.415,0.8865489993,Gamma(1.415),Gamma(1.415)-0.8865489993],_
+ [1.420,0.8863557896,Gamma(1.420),Gamma(1.420)-0.8863557896],_
+ [1.425,0.8861849081,Gamma(1.425),Gamma(1.425)-0.8861849081],_
+ [1.430,0.8860362361,Gamma(1.430),Gamma(1.430)-0.8860362361],_
+ [1.435,0.8859096587,Gamma(1.435),Gamma(1.435)-0.8859096587],_
+ [1.440,0.8858050635,Gamma(1.440),Gamma(1.440)-0.8858050635],_
+ [1.445,0.8857223397,Gamma(1.445),Gamma(1.445)-0.8857223397],_
+ [1.450,0.8856613803,Gamma(1.450),Gamma(1.450)-0.8856613803],_
+ [1.455,0.8856220700,Gamma(1.455),Gamma(1.455)-0.8856220700],_
+ [1.460,0.8856043364,Gamma(1.460),Gamma(1.460)-0.8856043364],_
+ [1.465,0.8856080495,Gamma(1.465),Gamma(1.465)-0.8856080495],_
+ [1.470,0.8856331217,Gamma(1.470),Gamma(1.470)-0.8856331217],_
+ [1.475,0.8856794575,Gamma(1.475),Gamma(1.475)-0.8856794575],_
+ [1.480,0.8857469646,Gamma(1.480),Gamma(1.480)-0.8857469646],_
+ [1.485,0.8858355520,Gamma(1.485),Gamma(1.485)-0.8858355520],_
+ [1.490,0.8859451316,Gamma(1.490),Gamma(1.490)-0.8859451316],_
+ [1.495,0.8860756174,Gamma(1.495),Gamma(1.495)-0.8860756174],_
+ [1.500,0.8862269255,Gamma(1.500),Gamma(1.500)-0.8862269255],_
+ [1.505,0.8863989744,Gamma(1.505),Gamma(1.505)-0.8863989744],_
+ [1.510,0.8865916850,Gamma(1.510),Gamma(1.510)-0.8865916850],_
+ [1.515,0.8868049797,Gamma(1.515),Gamma(1.515)-0.8868049797],_
+ [1.520,0.8870387833,Gamma(1.520),Gamma(1.520)-0.8870387833],_
+ [1.525,0.8872930231,Gamma(1.525),Gamma(1.525)-0.8872930231],_
+ [1.530,0.8875676278,Gamma(1.530),Gamma(1.530)-0.8875676278],_
+ [1.535,0.8878625287,Gamma(1.535),Gamma(1.535)-0.8878625287],_
+ [1.540,0.8881776586,Gamma(1.540),Gamma(1.540)-0.8881776586],_
+ [1.545,0.8885129527,Gamma(1.545),Gamma(1.545)-0.8885129527],_
+ [1.550,0.8888683478,Gamma(1.550),Gamma(1.550)-0.8888683478],_
+ [1.555,0.8892437830,Gamma(1.555),Gamma(1.555)-0.8892437830],_
+ [1.560,0.8896391990,Gamma(1.560),Gamma(1.560)-0.8896391990],_
+ [1.565,0.8900545387,Gamma(1.565),Gamma(1.565)-0.8900545387],_
+ [1.570,0.8904897463,Gamma(1.570),Gamma(1.570)-0.8904897463],_
+ [1.575,0.8909447686,Gamma(1.575),Gamma(1.575)-0.8909447686],_
+ [1.580,0.8914195537,Gamma(1.580),Gamma(1.580)-0.8914195537],_
+ [1.585,0.8919140515,Gamma(1.585),Gamma(1.585)-0.8919140515],_
+ [1.590,0.8924282141,Gamma(1.590),Gamma(1.590)-0.8924282141],_
+ [1.595,0.8929619949,Gamma(1.595),Gamma(1.595)-0.8929619949],_
+ [1.600,0.8935153493,Gamma(1.600),Gamma(1.600)-0.8935153493],_
+ [1.605,0.8940882342,Gamma(1.605),Gamma(1.605)-0.8940882342],_
+ [1.610,0.8946806085,Gamma(1.610),Gamma(1.610)-0.8946806085],_
+ [1.615,0.8952924327,Gamma(1.615),Gamma(1.615)-0.8952924327],_
+ [1.620,0.8959236685,Gamma(1.620),Gamma(1.620)-0.8959236685],_
+ [1.625,0.8965742800,Gamma(1.625),Gamma(1.625)-0.8965742800],_
+ [1.630,0.8972442326,Gamma(1.630),Gamma(1.630)-0.8972442326],_
+ [1.635,0.8979334930,Gamma(1.635),Gamma(1.635)-0.8979334930],_
+ [1.640,0.8986420302,Gamma(1.640),Gamma(1.640)-0.8986420302],_
+ [1.645,0.8993698138,Gamma(1.645),Gamma(1.645)-0.8993698138],_
+ [1.650,0.9001168163,Gamma(1.650),Gamma(1.650)-0.9001168163],_
+ [1.655,0.9008830104,Gamma(1.655),Gamma(1.655)-0.9008830104],_
+ [1.660,0.9016683712,Gamma(1.660),Gamma(1.660)-0.9016683712],_
+ [1.665,0.9024728748,Gamma(1.665),Gamma(1.665)-0.9024728748],_
+ [1.670,0.9032964995,Gamma(1.670),Gamma(1.670)-0.9032964995],_
+ [1.675,0.9041392243,Gamma(1.675),Gamma(1.675)-0.9041392243],_
+ [1.680,0.9050010302,Gamma(1.680),Gamma(1.680)-0.9050010302],_
+ [1.685,0.9058818996,Gamma(1.685),Gamma(1.685)-0.9058818996],_
+ [1.690,0.9067818160,Gamma(1.690),Gamma(1.690)-0.9067818160],_
+ [1.695,0.9077007650,Gamma(1.695),Gamma(1.695)-0.9077007650],_
+ [1.700,0.9086387329,Gamma(1.700),Gamma(1.700)-0.9086387329],_
+ [1.705,0.9095957079,Gamma(1.705),Gamma(1.705)-0.9095957079],_
+ [1.710,0.9105716796,Gamma(1.710),Gamma(1.710)-0.9105716796],_
+ [1.715,0.9115666390,Gamma(1.715),Gamma(1.715)-0.9115666390],_
+ [1.720,0.9125805779,Gamma(1.720),Gamma(1.720)-0.9125805779],_
+ [1.725,0.9136134904,Gamma(1.725),Gamma(1.725)-0.9136134904],_
+ [1.730,0.9146653712,Gamma(1.730),Gamma(1.730)-0.9146653712],_
+ [1.735,0.9157362171,Gamma(1.735),Gamma(1.735)-0.9157362171],_
+ [1.740,0.9168260252,Gamma(1.740),Gamma(1.740)-0.9168260252],_
+ [1.745,0.9179347950,Gamma(1.745),Gamma(1.745)-0.9179347950],_
+ [1.750,0.9190625268,Gamma(1.750),Gamma(1.750)-0.9190625268],_
+ [1.755,0.9202092224,Gamma(1.755),Gamma(1.755)-0.9202092224],_
+ [1.760,0.9213748846,Gamma(1.760),Gamma(1.760)-0.9213748846],_
+ [1.765,0.9225595178,Gamma(1.765),Gamma(1.765)-0.9225595178],_
+ [1.770,0.9237631277,Gamma(1.770),Gamma(1.770)-0.9237631277],_
+ [1.775,0.9249857211,Gamma(1.775),Gamma(1.775)-0.9249857211],_
+ [1.780,0.9262273062,Gamma(1.780),Gamma(1.780)-0.9262273062],_
+ [1.785,0.9274878926,Gamma(1.785),Gamma(1.785)-0.9274878926],_
+ [1.790,0.9287674904,Gamma(1.790),Gamma(1.790)-0.9287674904],_
+ [1.795,0.9300661123,Gamma(1.795),Gamma(1.795)-0.9300661123],_
+ [1.800,0.9313837710,Gamma(1.800),Gamma(1.800)-0.9313837710],_
+ [1.805,0.9327204811,Gamma(1.805),Gamma(1.805)-0.9327204811],_
+ [1.810,0.9340762585,Gamma(1.810),Gamma(1.810)-0.9340762585],_
+ [1.815,0.9354511198,Gamma(1.815),Gamma(1.815)-0.9354511198],_
+ [1.820,0.9368450832,Gamma(1.820),Gamma(1.820)-0.9368450832],_
+ [1.825,0.9382581682,Gamma(1.825),Gamma(1.825)-0.9382581682],_
+ [1.830,0.9396903951,Gamma(1.830),Gamma(1.830)-0.9396903951],_
+ [1.835,0.9411417859,Gamma(1.835),Gamma(1.835)-0.9411417859],_
+ [1.840,0.9426123634,Gamma(1.840),Gamma(1.840)-0.9426123634],_
+ [1.845,0.9441021519,Gamma(1.845),Gamma(1.845)-0.9441021519],_
+ [1.850,0.9456111764,Gamma(1.850),Gamma(1.850)-0.9456111764],_
+ [1.855,0.9471394637,Gamma(1.855),Gamma(1.855)-0.9471394637],_
+ [1.860,0.9486870417,Gamma(1.860),Gamma(1.860)-0.9486870417],_
+ [1.865,0.9502539389,Gamma(1.865),Gamma(1.865)-0.9502539389],_
+ [1.870,0.9518401855,Gamma(1.870),Gamma(1.870)-0.9518401855],_
+ [1.875,0.9534458127,Gamma(1.875),Gamma(1.875)-0.9534458127],_
+ [1.880,0.9550708530,Gamma(1.880),Gamma(1.880)-0.9550708530],_
+ [1.885,0.9567153398,Gamma(1.885),Gamma(1.885)-0.9567153398],_
+ [1.890,0.9583793077,Gamma(1.890),Gamma(1.890)-0.9583793077],_
+ [1.895,0.9600627927,Gamma(1.895),Gamma(1.895)-0.9600627927],_
+ [1.900,0.9617658319,Gamma(1.900),Gamma(1.900)-0.9617658319],_
+ [1.905,0.9634884632,Gamma(1.905),Gamma(1.905)-0.9634884632],_
+ [1.910,0.9652307261,Gamma(1.910),Gamma(1.910)-0.9652307261],_
+ [1.915,0.9669926608,Gamma(1.915),Gamma(1.915)-0.9669926608],_
+ [1.920,0.9687743090,Gamma(1.920),Gamma(1.920)-0.9687743090],_
+ [1.925,0.9705757134,Gamma(1.925),Gamma(1.925)-0.9705757134],_
+ [1.930,0.9723969178,Gamma(1.930),Gamma(1.930)-0.9723969178],_
+ [1.935,0.9742379672,Gamma(1.935),Gamma(1.935)-0.9742379672],_
+ [1.940,0.9760989075,Gamma(1.940),Gamma(1.940)-0.9760989075],_
+ [1.945,0.9779797861,Gamma(1.945),Gamma(1.945)-0.9779797861],_
+ [1.950,0.9798806513,Gamma(1.950),Gamma(1.950)-0.9798806513],_
+ [1.955,0.9818015524,Gamma(1.955),Gamma(1.955)-0.9818015524],_
+ [1.960,0.9837425404,Gamma(1.960),Gamma(1.960)-0.9837425404],_
+ [1.965,0.9857036664,Gamma(1.965),Gamma(1.965)-0.9857036664],_
+ [1.970,0.9876849838,Gamma(1.970),Gamma(1.970)-0.9876849838],_
+ [1.975,0.9896865462,Gamma(1.975),Gamma(1.975)-0.9896865462],_
+ [1.980,0.9917084087,Gamma(1.980),Gamma(1.980)-0.9917084087],_
+ [1.985,0.9937506274,Gamma(1.985),Gamma(1.985)-0.9937506274],_
+ [1.990,0.9958132598,Gamma(1.990),Gamma(1.990)-0.9958132598],_
+ [1.995,0.9978963643,Gamma(1.995),Gamma(1.995)-0.9978963643]]
+--R
+--R
+--R (1)
+--R [[1.,1.,1.,0.],
+--R
+--R [1.0049999999999999, 0.99713853539999997, 0.99713853525101781,
+--R - 1.4898215994207931E-10]
+--R ,
+--R [1.01,0.99432585120000005,0.99432585119150585,- 8.4942053391046102E-12],
+--R
+--R [1.0149999999999999, 0.99156128880000005, 0.99156128884897066,
+--R 4.8970605348586105E-11]
+--R ,
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+--R [1.7849999999999999, 0.92748789259999997, 0.92748789238727514,
+--R - 2.1272483774481543E-10]
+--R ,
+--R [1.79,0.92876749039999995,0.92876749040598172,5.9817706343778809E-12],
+--R
+--R [1.7949999999999999, 0.93006611230000003, 0.93006611220278679,
+--R - 9.7213237459925494E-11]
+--R ,
+--R [1.8,0.93138377100000003,0.93138377098024239,- 1.9757639968531748E-11],
+--R
+--R [1.8049999999999999, 0.93272048110000005, 0.93272048118181339,
+--R 8.1813333885349948E-11]
+--R ,
+--R
+--R [1.8100000000000001, 0.93407625849999998, 0.93407625848531362,
+--R - 1.4686363236648958E-11]
+--R ,
+--R
+--R [1.8149999999999999, 0.93545111979999995, 0.93545111979654905,
+--R - 3.4509062274423741E-12]
+--R ,
+--R
+--R [1.8200000000000001, 0.93684508320000004, 0.93684508324316496,
+--R 4.3164916085913774E-11]
+--R ,
+--R [1.825,0.93825816819999996,0.93825816816869001,-
3.1309954628966352E-11],
+--R
+--R [1.8300000000000001, 0.93969039509999996, 0.93969039512678099,
+--R 2.6781021844612951E-11]
+--R ,
+--R [1.835,0.94114178589999997,0.941141785875659,- 2.4340973681091782E-11],
+--R
+--R [1.8400000000000001, 0.94261236339999999, 0.94261236337273746,
+--R - 2.7262525570392881E-11]
+--R ,
+--R [1.845,0.94410215190000002,0.94410215176943868,-
1.3056133951749871E-10],
+--R
+--R [1.8500000000000001, 0.94561117640000003, 0.94561117640619552,
+--R 6.1954885666182236E-12]
+--R ,
+--R [1.855,0.94713946370000002,0.94713946380763558,1.0763556712589661E-10],
+--R
+--R [1.8600000000000001, 0.94868704169999996, 0.94868704167794815,
+--R - 2.2051804826617172E-11]
+--R ,
+--R [1.865,0.95025393889999998,0.95025393889642473,-
3.5752512062003916E-12],
+--R
+--R [1.8700000000000001, 0.95184018550000005, 0.95184018551317828,
+--R 1.3178236279998146E-11]
+--R ,
+--R [1.875,0.95344581269999995,0.95344581274503493,4.5034975748592387E-11],
+--R
+--R [1.8799999999999999, 0.95507085300000005, 0.95507085297159322,
+--R - 2.840683244187403E-11]
+--R ,
+--R [1.885,0.95671533980000001,0.95671533973145539,-
6.8544614428844852E-11],
+--R
+--R [1.8899999999999999, 0.95837930770000002, 0.95837930771862279,
+--R 1.8622769992759913E-11]
+--R ,
+--R [1.895,0.96006279269999995,0.96006279277905426,7.9054318646853972E-11],
+--R
+--R [1.8999999999999999, 0.96176583189999998, 0.96176583190738751,
+--R 7.3875350281582541E-12]
+--R ,
+--R [1.905,0.96348846320000003,0.96348846324382054,4.3820502781954929E-11],
+--R
+--R [1.9099999999999999, 0.96523072610000005, 0.96523072607114846,
+--R - 2.8851587785538868E-11]
+--R ,
+--R [1.915,0.96699266080000001,0.96699266081195756,1.1957546064422786E-11],
+--R
+--R [1.9199999999999999, 0.96877430899999994, 0.96877430902597383,
+--R 2.5973889705710462E-11]
+--R ,
+--R [1.925,0.97057571340000004,0.97057571340755988,7.5598416415800784E-12],
+--R
+--R [1.9299999999999999, 0.97239691780000004, 0.97239691778336623,
+--R - 1.6633805444143945E-11]
+--R ,
+--R
+--R [1.9350000000000001, 0.9742379672, 0.97423796711012756,
+--R - 8.9872442821103959E-11]
+--R ,
+--R
+--R [1.9399999999999999, 0.97609890749999995, 0.97609890747260597,
+--R - 2.73939759765085E-11]
+--R ,
+--R
+--R [1.9450000000000001, 0.97797978610000003, 0.97797978608168112,
+--R - 1.8318901950920008E-11]
+--R ,
+--R [1.95,0.97988065130000002,0.97988065127258051,- 2.7419511106074879E-11],
+--R
[1.9550000000000001,0.9818015524,0.98180155250325418,1.032541829815159E-10],
+--R [1.96,0.98374254039999998,0.98374254035288666,- 4.711331325069068E-11],
+--R
+--R [1.9650000000000001, 0.98570366639999996, 0.98570366652054964,
+--R 1.2054968134833643E-10]
+--R ,
+--R [1.97,0.98768498380000003,0.98768498382399139,2.399136445063732E-11],
+--R
+--R [1.9750000000000001, 0.98968654619999996, 0.98968654619855956,
+--R - 1.4404033521486781E-12]
+--R ,
+--R [1.98,0.99170840869999999,0.99170840869626087,- 3.7391201246350647E-12],
+--R
+--R [1.9850000000000001, 0.99375062739999998, 0.99375062748495313,
+--R 8.4953155621292353E-11]
+--R ,
+--R [1.99,0.99581325980000002,0.995813259847667,4.7666981473071246E-11],
+--R
+--R [1.9950000000000001, 0.99789636429999995, 0.99789636418206007,
+--R - 1.1793988008435008E-10]
+--R ]
+--R Type: List List
DoubleFloat
+--E 1
+
+--S 2 of 3
+Psi(x:DFLOAT):DFLOAT==polygamma(0,x)
+--R
+--R Function declaration Psi : DoubleFloat -> DoubleFloat has been added
+--R to workspace.
+--R Type:
Void
+--E 2
+@
+In the following table there are 4 columns. The first column
+is the argument of Psi, ranging from 1.000 to 1.995. The
+second column is the reference value of Psi from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+
+<<*>>=
+
+--S 3 of 3
+[[1.000, -0.5772156649, Psi(1.000), Psi(1.000)- -0.5772156649],_
+ [1.005, -0.5690209113, Psi(1.005), Psi(1.005)- -0.5690209113],_
+ [1.010, -0.5608854579, Psi(1.010), Psi(1.010)- -0.5608854579],_
+ [1.015, -0.5528085156, Psi(1.015), Psi(1.015)- -0.5528085156],_
+ [1.020, -0.5447893105, Psi(1.020), Psi(1.020)- -0.5447893105],_
+ [1.025, -0.5368270828, Psi(1.025), Psi(1.025)- -0.5368270828],_
+ [1.030, -0.5289210873, Psi(1.030), Psi(1.030)- -0.5289210873],_
+ [1.035, -0.5210705921, Psi(1.035), Psi(1.035)- -0.5210705921],_
+ [1.040, -0.5132748789, Psi(1.040), Psi(1.040)- -0.5132748789],_
+ [1.045, -0.5055332428, Psi(1.045), Psi(1.045)- -0.5055332428],_
+ [1.050, -0.4978449913, Psi(1.050), Psi(1.050)- -0.4978449913],_
+ [1.055, -0.4902094448, Psi(1.055), Psi(1.055)- -0.4902094448],_
+ [1.060, -0.4826259358, Psi(1.060), Psi(1.060)- -0.4826259358],_
+ [1.065, -0.4750938088, Psi(1.065), Psi(1.065)- -0.4750938088],_
+ [1.070, -0.4676124199, Psi(1.070), Psi(1.070)- -0.4676124199],_
+ [1.075, -0.4601811367, Psi(1.075), Psi(1.075)- -0.4601811367],_
+ [1.080, -0.4527993380, Psi(1.080), Psi(1.080)- -0.4527993380],_
+ [1.085, -0.4454664135, Psi(1.085), Psi(1.085)- -0.4454664135],_
+ [1.090, -0.4381817635, Psi(1.090), Psi(1.090)- -0.4381817635],_
+ [1.095, -0.4309447988, Psi(1.095), Psi(1.095)- -0.4309447988],_
+ [1.100, -0.4237549404, Psi(1.100), Psi(1.100)- -0.4237549404],_
+ [1.105, -0.4166116193, Psi(1.105), Psi(1.105)- -0.4166116193],_
+ [1.110, -0.4095142761, Psi(1.110), Psi(1.110)- -0.4095142761],_
+ [1.115, -0.4024623611, Psi(1.115), Psi(1.115)- -0.4024623611],_
+ [1.120, -0.3954553339, Psi(1.120), Psi(1.120)- -0.3954553339],_
+ [1.125, -0.3884926633, Psi(1.125), Psi(1.125)- -0.3884926633],_
+ [1.130, -0.3815738268, Psi(1.130), Psi(1.130)- -0.3815738268],_
+ [1.135, -0.3746983110, Psi(1.135), Psi(1.135)- -0.3746983110],_
+ [1.140, -0.3678656106, Psi(1.140), Psi(1.140)- -0.3678656106],_
+ [1.145, -0.3610752291, Psi(1.145), Psi(1.145)- -0.3610752291],_
+ [1.150, -0.3543266780, Psi(1.150), Psi(1.150)- -0.3543266780],_
+ [1.155, -0.3476194768, Psi(1.155), Psi(1.155)- -0.3476194768],_
+ [1.160, -0.3409531528, Psi(1.160), Psi(1.160)- -0.3409531528],_
+ [1.165, -0.3343272413, Psi(1.165), Psi(1.165)- -0.3343272413],_
+ [1.170, -0.3277412847, Psi(1.170), Psi(1.170)- -0.3277412847],_
+ [1.175, -0.3211948332, Psi(1.175), Psi(1.175)- -0.3211948332],_
+ [1.180, -0.3146874438, Psi(1.180), Psi(1.180)- -0.3146874438],_
+ [1.185, -0.3082186809, Psi(1.185), Psi(1.185)- -0.3082186809],_
+ [1.190, -0.3017881156, Psi(1.190), Psi(1.190)- -0.3017881156],_
+ [1.195, -0.2953953259, Psi(1.195), Psi(1.195)- -0.2953953259],_
+ [1.200, -0.2890398966, Psi(1.200), Psi(1.200)- -0.2890398966],_
+ [1.205, -0.2827214187, Psi(1.205), Psi(1.205)- -0.2827214187],_
+ [1.210, -0.2764394897, Psi(1.210), Psi(1.210)- -0.2764394897],_
+ [1.215, -0.2701937135, Psi(1.215), Psi(1.215)- -0.2701937135],_
+ [1.220, -0.2639837000, Psi(1.220), Psi(1.220)- -0.2639837000],_
+ [1.225, -0.2578090652, Psi(1.225), Psi(1.225)- -0.2578090652],_
+ [1.230, -0.2516694307, Psi(1.230), Psi(1.230)- -0.2516694307],_
+ [1.235, -0.2455644243, Psi(1.235), Psi(1.235)- -0.2455644243],_
+ [1.240, -0.2394936791, Psi(1.240), Psi(1.240)- -0.2394936791],_
+ [1.245, -0.2334568341, Psi(1.245), Psi(1.245)- -0.2334568341],_
+ [1.250, -0.2274535334, Psi(1.250), Psi(1.250)- -0.2274535334],_
+ [1.255, -0.2214834266, Psi(1.255), Psi(1.255)- -0.2214834266],_
+ [1.260, -0.2155461686, Psi(1.260), Psi(1.260)- -0.2155461686],_
+ [1.265, -0.2096414193, Psi(1.265), Psi(1.265)- -0.2096414193],_
+ [1.270, -0.2037688437, Psi(1.270), Psi(1.270)- -0.2037688437],_
+ [1.275, -0.1979281118, Psi(1.275), Psi(1.275)- -0.1979281118],_
+ [1.280, -0.1921188983, Psi(1.280), Psi(1.280)- -0.1921188983],_
+ [1.285, -0.1863408828, Psi(1.285), Psi(1.285)- -0.1863408828],_
+ [1.290, -0.1805937494, Psi(1.290), Psi(1.290)- -0.1805937494],_
+ [1.295, -0.1748771870, Psi(1.295), Psi(1.295)- -0.1748771870],_
+ [1.300, -0.1691908889, Psi(1.300), Psi(1.300)- -0.1691908889],_
+ [1.305, -0.1635345526, Psi(1.305), Psi(1.305)- -0.1635345526],_
+ [1.310, -0.1579078803, Psi(1.310), Psi(1.310)- -0.1579078803],_
+ [1.315, -0.1523105782, Psi(1.315), Psi(1.315)- -0.1523105782],_
+ [1.320, -0.1467423568, Psi(1.320), Psi(1.320)- -0.1467423568],_
+ [1.325, -0.1412029305, Psi(1.325), Psi(1.325)- -0.1412029305],_
+ [1.330, -0.1356920180, Psi(1.330), Psi(1.330)- -0.1356920180],_
+ [1.335, -0.1302093416, Psi(1.335), Psi(1.335)- -0.1302093416],_
+ [1.340, -0.1247546279, Psi(1.340), Psi(1.340)- -0.1247546279],_
+ [1.345, -0.1193276069, Psi(1.345), Psi(1.345)- -0.1193276069],_
+ [1.350, -0.1139280127, Psi(1.350), Psi(1.350)- -0.1139280127],_
+ [1.355, -0.1085555827, Psi(1.355), Psi(1.355)- -0.1085555827],_
+ [1.360, -0.1032100582, Psi(1.360), Psi(1.360)- -0.1032100582],_
+ [1.365, -0.0978911840, Psi(1.365), Psi(1.365)- -0.0978911840],_
+ [1.370, -0.0925987082, Psi(1.370), Psi(1.370)- -0.0925987082],_
+ [1.375, -0.0873323825, Psi(1.375), Psi(1.375)- -0.0873323825],_
+ [1.380, -0.0820919619, Psi(1.380), Psi(1.380)- -0.0820919619],_
+ [1.385, -0.0768772046, Psi(1.385), Psi(1.385)- -0.0768772046],_
+ [1.390, -0.0716878723, Psi(1.390), Psi(1.390)- -0.0716878723],_
+ [1.395, -0.0665237297, Psi(1.395), Psi(1.395)- -0.0665237297],_
+ [1.400, -0.0613845446, Psi(1.400), Psi(1.400)- -0.0613845446],_
+ [1.405, -0.0562700879, Psi(1.405), Psi(1.405)- -0.0562700879],_
+ [1.410, -0.0511801337, Psi(1.410), Psi(1.410)- -0.0511801337],_
+ [1.415, -0.0461144589, Psi(1.415), Psi(1.415)- -0.0461144589],_
+ [1.420, -0.0410728433, Psi(1.420), Psi(1.420)- -0.0410728433],_
+ [1.425, -0.0360550697, Psi(1.425), Psi(1.425)- -0.0360550697],_
+ [1.430, -0.0310609237, Psi(1.430), Psi(1.430)- -0.0310609237],_
+ [1.435, -0.0260901935, Psi(1.435), Psi(1.435)- -0.0260901935],_
+ [1.440, -0.0211426703, Psi(1.440), Psi(1.440)- -0.0211426703],_
+ [1.445, -0.0162181479, Psi(1.445), Psi(1.445)- -0.0162181479],_
+ [1.450, -0.0113164226, Psi(1.450), Psi(1.450)- -0.0113164226],_
+ [1.455, -0.0064372934, Psi(1.455), Psi(1.455)- -0.0064372934],_
+ [1.460, -0.0015805620, Psi(1.460), Psi(1.460)- -0.0015805620],_
+ [1.465, 0.0032539677, Psi(1.465), Psi(1.465)- 0.0032539677],_
+ [1.470, 0.0080664890, Psi(1.470), Psi(1.470)- 0.0080664890],_
+ [1.475, 0.0128571930, Psi(1.475), Psi(1.475)- 0.0128571930],_
+ [1.480, 0.0176262684, Psi(1.480), Psi(1.480)- 0.0176262684],_
+ [1.485, 0.0223739013, Psi(1.485), Psi(1.485)- 0.0223739013],_
+ [1.490, 0.0271002758, Psi(1.490), Psi(1.490)- 0.0271002758],_
+ [1.495, 0.0318055736, Psi(1.495), Psi(1.495)- 0.0318055736],_
+ [1.500, 0.0364899740, Psi(1.500), Psi(1.500)- 0.0364899740],_
+ [1.505, 0.0411536543, Psi(1.505), Psi(1.505)- 0.0411536543],_
+ [1.510, 0.0457967896, Psi(1.510), Psi(1.510)- 0.0457967896],_
+ [1.515, 0.0504195527, Psi(1.515), Psi(1.515)- 0.0504195527],_
+ [1.520, 0.0550221146, Psi(1.520), Psi(1.520)- 0.0550221146],_
+ [1.525, 0.0596046439, Psi(1.525), Psi(1.525)- 0.0596046439],_
+ [1.530, 0.0641673074, Psi(1.530), Psi(1.530)- 0.0641673074],_
+ [1.535, 0.0687102697, Psi(1.535), Psi(1.535)- 0.0687102697],_
+ [1.540, 0.0732336936, Psi(1.540), Psi(1.540)- 0.0732336936],_
+ [1.545, 0.0777377300, Psi(1.545), Psi(1.545)- 0.0777377300],_
+ [1.550, 0.0822225675, Psi(1.550), Psi(1.550)- 0.0822225675],_
+ [1.555, 0.0866883334, Psi(1.555), Psi(1.555)- 0.0866883334],_
+ [1.560, 0.0911351925, Psi(1.560), Psi(1.560)- 0.0911351925],_
+ [1.565, 0.0955632984, Psi(1.565), Psi(1.565)- 0.0955632984],_
+ [1.570, 0.0999728024, Psi(1.570), Psi(1.570)- 0.0999728024],_
+ [1.575, 0.1043638544, Psi(1.575), Psi(1.575)- 0.1043638544],_
+ [1.580, 0.1087366023, Psi(1.580), Psi(1.580)- 0.1087366023],_
+ [1.585, 0.1130911923, Psi(1.585), Psi(1.585)- 0.1130911923],_
+ [1.590, 0.1174277690, Psi(1.590), Psi(1.590)- 0.1174277690],_
+ [1.595, 0.1217464754, Psi(1.595), Psi(1.595)- 0.1217464754],_
+ [1.600, 0.1260474528, Psi(1.600), Psi(1.600)- 0.1260474528],_
+ [1.605, 0.1303308407, Psi(1.605), Psi(1.605)- 0.1303308407],_
+ [1.610, 0.1345967772, Psi(1.610), Psi(1.610)- 0.1345967772],_
+ [1.615, 0.1388453988, Psi(1.615), Psi(1.615)- 0.1388453988],_
+ [1.620, 0.1430768404, Psi(1.620), Psi(1.620)- 0.1430768404],_
+ [1.625, 0.1472912354, Psi(1.625), Psi(1.625)- 0.1472912354],_
+ [1.630, 0.1514887158, Psi(1.630), Psi(1.630)- 0.1514887158],_
+ [1.635, 0.1556694120, Psi(1.635), Psi(1.635)- 0.1556694120],_
+ [1.640, 0.1598334529, Psi(1.640), Psi(1.640)- 0.1598334529],_
+ [1.645, 0.1639809660, Psi(1.645), Psi(1.645)- 0.1639809660],_
+ [1.650, 0.1681120776, Psi(1.650), Psi(1.650)- 0.1681120776],_
+ [1.655, 0.1722269122, Psi(1.655), Psi(1.655)- 0.1722269122],_
+ [1.660, 0.1763255933, Psi(1.660), Psi(1.660)- 0.1763255933],_
+ [1.665, 0.1804082427, Psi(1.665), Psi(1.665)- 0.1804082427],_
+ [1.670, 0.1844749813, Psi(1.670), Psi(1.670)- 0.1844749813],_
+ [1.675, 0.1885259282, Psi(1.675), Psi(1.675)- 0.1885259282],_
+ [1.680, 0.1925612015, Psi(1.680), Psi(1.680)- 0.1925612015],_
+ [1.685, 0.1965809180, Psi(1.685), Psi(1.685)- 0.1965809180],_
+ [1.690, 0.2005851931, Psi(1.690), Psi(1.690)- 0.2005851931],_
+ [1.695, 0.2045741410, Psi(1.695), Psi(1.695)- 0.2045741410],_
+ [1.700, 0.2085478749, Psi(1.700), Psi(1.700)- 0.2085478749],_
+ [1.705, 0.2125065064, Psi(1.705), Psi(1.705)- 0.2125065064],_
+ [1.710, 0.2164501462, Psi(1.710), Psi(1.710)- 0.2164501462],_
+ [1.715, 0.2203789037, Psi(1.715), Psi(1.715)- 0.2203789037],_
+ [1.720, 0.2242928871, Psi(1.720), Psi(1.720)- 0.2242928871],_
+ [1.725, 0.2281922037, Psi(1.725), Psi(1.725)- 0.2281922037],_
+ [1.730, 0.2320769593, Psi(1.730), Psi(1.730)- 0.2320769593],_
+ [1.735, 0.2359472589, Psi(1.735), Psi(1.735)- 0.2359472589],_
+ [1.740, 0.2398032061, Psi(1.740), Psi(1.740)- 0.2398032061],_
+ [1.745, 0.2436449038, Psi(1.745), Psi(1.745)- 0.2436449038],_
+ [1.750, 0.2474724535, Psi(1.750), Psi(1.750)- 0.2474724535],_
+ [1.755, 0.2512859559, Psi(1.755), Psi(1.755)- 0.2512859559],_
+ [1.760, 0.2550855103, Psi(1.760), Psi(1.760)- 0.2550855103],_
+ [1.765, 0.2588712154, Psi(1.765), Psi(1.765)- 0.2588712154],_
+ [1.770, 0.2626431686, Psi(1.770), Psi(1.770)- 0.2626431686],_
+ [1.775, 0.2664014664, Psi(1.775), Psi(1.775)- 0.2664014664],_
+ [1.780, 0.2701462043, Psi(1.780), Psi(1.780)- 0.2701462043],_
+ [1.785, 0.2738774769, Psi(1.785), Psi(1.785)- 0.2738774769],_
+ [1.790, 0.2775953776, Psi(1.790), Psi(1.790)- 0.2775953776],_
+ [1.795, 0.2812999992, Psi(1.795), Psi(1.795)- 0.2812999992],_
+ [1.800, 0.2849914333, Psi(1.800), Psi(1.800)- 0.2849914333],_
+ [1.805, 0.2886697707, Psi(1.805), Psi(1.805)- 0.2886697707],_
+ [1.810, 0.2923351012, Psi(1.810), Psi(1.810)- 0.2923351012],_
+ [1.815, 0.2959875138, Psi(1.815), Psi(1.815)- 0.2959875138],_
+ [1.820, 0.2996270966, Psi(1.820), Psi(1.820)- 0.2996270966],_
+ [1.825, 0.3032539367, Psi(1.825), Psi(1.825)- 0.3032539367],_
+ [1.830, 0.3068681205, Psi(1.830), Psi(1.830)- 0.3068681205],_
+ [1.835, 0.3104697335, Psi(1.835), Psi(1.835)- 0.3104697335],_
+ [1.840, 0.3140588602, Psi(1.840), Psi(1.840)- 0.3140588602],_
+ [1.845, 0.3176355846, Psi(1.845), Psi(1.845)- 0.3176355846],_
+ [1.850, 0.3211999895, Psi(1.850), Psi(1.850)- 0.3211999895],_
+ [1.855, 0.3247521572, Psi(1.855), Psi(1.855)- 0.3247521572],_
+ [1.860, 0.3282921691, Psi(1.860), Psi(1.860)- 0.3282921691],_
+ [1.865, 0.3318201056, Psi(1.865), Psi(1.865)- 0.3318201056],_
+ [1.870, 0.3353360467, Psi(1.870), Psi(1.870)- 0.3353360467],_
+ [1.875, 0.3388400713, Psi(1.875), Psi(1.875)- 0.3388400713],_
+ [1.880, 0.3423322577, Psi(1.880), Psi(1.880)- 0.3423322577],_
+ [1.885, 0.3458126835, Psi(1.885), Psi(1.885)- 0.3458126835],_
+ [1.890, 0.3492814255, Psi(1.890), Psi(1.890)- 0.3492814255],_
+ [1.895, 0.3527385596, Psi(1.895), Psi(1.895)- 0.3527385596],_
+ [1.900, 0.3561841612, Psi(1.900), Psi(1.900)- 0.3561841612],_
+ [1.905, 0.3596183049, Psi(1.905), Psi(1.905)- 0.3596183049],_
+ [1.910, 0.3630410646, Psi(1.910), Psi(1.910)- 0.3630410646],_
+ [1.915, 0.3664525136, Psi(1.915), Psi(1.915)- 0.3664525136],_
+ [1.920, 0.3698527244, Psi(1.920), Psi(1.920)- 0.3698527244],_
+ [1.925, 0.3732417688, Psi(1.925), Psi(1.925)- 0.3732417688],_
+ [1.930, 0.3766197179, Psi(1.930), Psi(1.930)- 0.3766197179],_
+ [1.935, 0.3799866424, Psi(1.935), Psi(1.935)- 0.3799866424],_
+ [1.940, 0.3833426119, Psi(1.940), Psi(1.940)- 0.3833426119],_
+ [1.945, 0.3866876959, Psi(1.945), Psi(1.945)- 0.3866876959],_
+ [1.950, 0.3900219627, Psi(1.950), Psi(1.950)- 0.3900219627],_
+ [1.955, 0.3933454805, Psi(1.955), Psi(1.955)- 0.3933454805],_
+ [1.960, 0.3966583163, Psi(1.960), Psi(1.960)- 0.3966583163],_
+ [1.965, 0.3999605371, Psi(1.965), Psi(1.965)- 0.3999605371],_
+ [1.970, 0.4032522088, Psi(1.970), Psi(1.970)- 0.4032522088],_
+ [1.975, 0.4065333970, Psi(1.975), Psi(1.975)- 0.4065333970],_
+ [1.980, 0.4098041664, Psi(1.980), Psi(1.980)- 0.4098041664],_
+ [1.985, 0.4130645816, Psi(1.985), Psi(1.985)- 0.4130645816],_
+ [1.990, 0.4163147060, Psi(1.990), Psi(1.990)- 0.4163147060],_
+ [1.995, 0.4195546030, Psi(1.995), Psi(1.995)- 0.4195546030],_
+ [2.000, 0.4227843351, Psi(2.000), Psi(2.000)- 0.4227843351]]
+--R
+--R Compiling function Psi with type DoubleFloat -> DoubleFloat
+--R
+--R (3)
+--R [[1.,- 0.57721566489999998,- 0.57721566490153275,-
1.5327739077974911E-12],
+--R
+--R [1.0049999999999999, - 0.56902091129999999, - 0.56902091134438304,
+--R - 4.4383052788532495E-11]
+--R ,
+--R [1.01,- 0.5608854579,- 0.56088545786867472,3.1325275706706179E-11],
+--R
+--R [1.0149999999999999, - 0.55280851559999999, - 0.55280851559434629,
+--R 5.6536997306011472E-12]
+--R ,
+--R [1.02,- 0.54478931050000001,-
0.54478931045617984,4.3820169715047541E-11],
+--R
+--R [1.0249999999999999, - 0.53682708280000002, - 0.53682708284938863,
+--R - 4.9388604317357476E-11]
+--R ,
+--R [1.03,- 0.52892108729999998,-
0.5289210872854303,1.4569678796760854E-11],
+--R
+--R [1.0349999999999999, - 0.52107059209999995, - 0.52107059205771,
+--R 4.2289949320206688E-11]
+--R ,
+--R [1.04,- 0.51327487890000001,- 0.51327487891683021,-
1.6830203897200136E-11],
+--R
+--R [1.0449999999999999, - 0.5055332428, - 0.50553324275508449,
+--R 4.4915515751142721E-11]
+--R ,
+--R [1.05,- 0.49784499129999998,-
0.49784499129987031,1.2967404927621828E-13],
+--R
+--R [1.0549999999999999, - 0.4902094448, - 0.49020944481574569,
+--R - 1.5745682535595051E-11]
+--R ,
+--R
+--R [1.0600000000000001, - 0.48262593580000002, - 0.48262593581482538,
+--R - 1.4825363159332028E-11]
+--R ,
+--R
+--R [1.0649999999999999, - 0.47509380880000002, - 0.47509380877526647,
+--R 2.4733548542599237E-11]
+--R ,
+--R
+--R [1.0700000000000001, - 0.46761241990000002, - 0.46761241986755342,
+--R 3.2446600961577587E-11]
+--R ,
+--R [1.075,- 0.46018113669999999,-
0.4601811366883593,1.1640688413194766E-11],
+--R
+--R [1.0800000000000001, - 0.452799338, - 0.45279933800171246,
+--R - 1.7124635043330727E-12]
+--R ,
+--R [1.085,- 0.44546641349999999,-
0.44546641348725191,1.2748080369107129E-11],
+--R
+--R [1.0900000000000001, - 0.43818176349999999, - 0.43818176349533489,
+--R 4.6651016383236765E-12]
+--R ,
+--R [1.095,- 0.43094479879999997,- 0.43094479880878706,-
8.7870821730007265E-12]
+--R ,
+--R
+--R [1.1000000000000001, - 0.4237549404, - 0.42375494041107653,
+--R - 1.1076528583231493E-11]
+--R ,
+--R [1.105,- 0.41661161930000001,-
0.41661161926071655,3.9283465369521764E-11],
+--R
+--R [1.1100000000000001, - 0.40951427610000002, - 0.40951427607169383,
+--R 2.830619072469176E-11]
+--R ,
+--R [1.115,- 0.4024623611,- 0.40246236109974648,2.535194276731545E-13],
+--R
+--R [1.1200000000000001, - 0.39545533389999998, - 0.39545533393429283,
+--R - 3.4292846340377992E-11]
+--R ,
+--R [1.125,- 0.38849266329999999,-
0.38849266329585463,4.1453507293454095E-12],
+--R
+--R [1.1299999999999999, - 0.38157382680000002, - 0.38157382683879215,
+--R - 3.8792136169973901E-11]
+--R ,
+--R [1.135,- 0.37469831100000001,-
0.37469831095919082,4.0809189361112885E-11],
+--R
+--R [1.1399999999999999, - 0.3678656106, - 0.36786561060774969,
+--R - 7.7496897787909802E-12]
+--R ,
+--R [1.145,- 0.36107522910000001,- 0.361075229107509,-
7.5089934270522463E-12],
+--R
+--R [1.1499999999999999, - 0.35432667800000001, - 0.35432667797627904,
+--R 2.3720969632989863E-11]
+--R ,
+--R [1.155,- 0.34761947679999999,-
0.34761947675362337,4.6376624762700658E-11],
+--R
+--R [1.1599999999999999, - 0.34095315279999999, - 0.34095315283226135,
+--R - 3.226136024991888E-11]
+--R ,
+--R [1.165,- 0.33432724130000002,-
0.3343272412937619,6.2381211307638296E-12],
+--R
+--R [1.1699999999999999, - 0.32774128470000002, - 0.3277412847483927,
+--R - 4.839267875311748E-11]
+--R ,
+--R [1.175,- 0.32119483319999997,-
0.3211948331790081,2.0991874905007535E-11],
+--R
+--R [1.1799999999999999, - 0.31468744380000002, - 0.31468744378886082,
+--R 1.1139200672971583E-11]
+--R ,
+--R
+--R [1.1850000000000001, - 0.30821868089999999, - 0.30821868085320625,
+--R 4.6793735553052329E-11]
+--R ,
+--R
+--R [1.1899999999999999, - 0.30178811560000002, - 0.30178811557461016,
+--R 2.5389856883606399E-11]
+--R ,
+--R
+--R [1.1950000000000001, - 0.29539532590000001, - 0.2953953259418296,
+--R - 4.1829595343045867E-11]
+--R ,
+--R [1.2,- 0.28903989660000001,-
0.28903989659218843,7.8115847124138327E-12],
+--R
+--R [1.2050000000000001, - 0.2827214187, - 0.28272141867731704,
+--R 2.2682966616116573E-11]
+--R ,
+--R [1.21,- 0.27643948969999999,- 0.2764394897321919,-
3.2191915799728577E-11],
+--R
+--R [1.2150000000000001, - 0.27019371349999999, - 0.27019371354735244,
+--R - 4.7352455290194939E-11]
+--R ,
+--R [1.22,- 0.26398369999999999,- 0.26398370004422023,-
4.4220238581971216E-11],
+--R
+--R [1.2250000000000001, - 0.25780906520000002, - 0.25780906515343338,
+--R 4.6566639433365253E-11]
+--R ,
+--R [1.23,- 0.25166943069999997,-
0.25166943069609982,3.9001579743569437E-12],
+--R
+--R [1.2350000000000001, - 0.2455644243, - 0.24556442426789726,
+--R 3.2102737135275561E-11]
+--R ,
+--R [1.24,- 0.2394936791,- 0.23949367912593666,- 2.5936669478809904E-11],
+--R
+--R [1.2450000000000001, - 0.23345683410000001, - 0.23345683407831253,
+--R 2.1687485141086427E-11]
+--R ,
+--R [1.25,- 0.22745353339999999,- 0.22745353337626528,2.37347086429196E-11],
+--R
+--R [1.2549999999999999, - 0.22148342660000001, - 0.22148342660888165,
+--R - 8.8816454191231742E-12]
+--R ,
+--R [1.26,- 0.2155461686,- 0.21554616860026521,- 2.6520452500733427E-13],
+--R
+--R [1.2649999999999999, - 0.20964141929999999, - 0.20964141930911384,
+--R - 9.1138485647235257E-12]
+--R ,
+--R [1.27,- 0.20376884370000001,- 0.20376884373062343,-
3.0623420466113771E-11],
+--R
+--R [1.2749999999999999, - 0.1979281118, - 0.19792811180067393,
+--R - 6.7393313152308565E-13]
+--R ,
+--R [1.28,- 0.19211889830000001,- 0.19211889830222173,-
2.221722805728632E-12],
+--R
+--R [1.2849999999999999, - 0.1863408828, - 0.18634088277384209,
+--R 2.6157909172042082E-11]
+--R ,
+--R [1.29,- 0.1805937494,- 0.1805937494203691,- 2.0369095299344053E-11],
+--R
+--R [1.2949999999999999, - 0.17487718699999999, - 0.17487718702556942,
+--R - 2.5569435457839518E-11]
+--R ,
+--R [1.3,- 0.1691908889,- 0.16919088886679934,3.3200664439902994E-11],
+--R
+--R [1.3049999999999999, - 0.16353455259999999, - 0.163534552631597,
+--R - 3.1597002791983186E-11]
+--R ,
+--R
+--R [1.3100000000000001, - 0.1579078803, - 0.15790788033614178,
+--R - 3.6141784010013112E-11]
+--R ,
+--R
+--R [1.3149999999999999, - 0.15231057819999999, - 0.15231057824555994,
+--R - 4.5559944705786393E-11]
+--R ,
+--R
+--R [1.3200000000000001, - 0.1467423568, - 0.1467423567959959,
+--R 4.0041026050374739E-12]
+--R ,
+--R [1.325,- 0.14120293049999999,- 0.14120293051842803,-
1.8428036874240661E-11]
+--R ,
+--R
+--R [1.3300000000000001, - 0.135692018, - 0.13569201796416941,
+--R 3.583058849621068E-11]
+--R ,
+--R [1.335,- 0.1302093416,- 0.13020934163201769,- 3.2017694051589274E-11],
+--R
+--R [1.3400000000000001, - 0.1247546279, - 0.12475462789700376,
+--R 2.9962421432827568E-12]
+--R ,
+--R [1.345,- 0.11932760689999999,- 0.11932760694070754,-
4.0707548443208452E-11]
+--R ,
+--R
+--R [1.3500000000000001, - 0.1139280127, - 0.11392801268308839,
+--R 1.6911611000480775E-11]
+--R ,
+--R [1.355,- 0.1085555827,- 0.10855558271580501,- 1.5805010078473458E-11],
+--R
+--R [1.3600000000000001, - 0.1032100582, - 0.10321005823697738,
+--R - 3.6977379491709428E-11]
+--R ,
+--R
+--R [1.365, - 9.7891184000000006E-2, - 9.7891183987354968E-2,
+--R 1.2645037794634106E-11]
+--R ,
+--R
+--R [1.3700000000000001, - 9.2598708200000004E-2, - 9.2598708187860979E-2,
+--R 1.2139025895585576E-11]
+--R ,
+--R [1.375,- 8.73323825E-2,- 8.7332382478473081E-2,2.1526919136150013E-11],
+--R
+--R [1.3799999999999999, - 8.2091961899999996E-2, - 8.2091961858406615E-2,
+--R 4.1593381516769057E-11]
+--R ,
+--R
+--R [1.385, - 7.6877204599999999E-2, - 7.6877204627574525E-2,
+--R - 2.7574525995888166E-11]
+--R ,
+--R
+--R [1.3899999999999999, - 7.1687872299999997E-2, - 7.1687872329281643E-2,
+--R - 2.928164655191523E-11]
+--R ,
+--R
+--R [1.395, - 6.6523729700000006E-2, - 6.6523729694132228E-2,
+--R 5.867778485324493E-12]
+--R ,
+--R
+--R [1.3999999999999999, - 6.13845446E-2, - 6.1384544585116108E-2,
+--R 1.4883892729411485E-11]
+--R ,
+--R
+--R [1.405, - 5.6270087900000001E-2, - 5.6270087943841696E-2,
+--R - 4.3841694163937461E-11]
+--R ,
+--R
+--R [1.4099999999999999, - 5.1180133699999998E-2, - 5.1180133737897426E-2,
+--R - 3.7897428251110199E-11]
+--R ,
+--R
+--R [1.415, - 4.6114458900000002E-2, - 4.6114458909301992E-2,
+--R - 9.3019897340340663E-12]
+--R ,
+--R
+--R [1.4199999999999999, - 4.1072843300000002E-2, - 4.1072843324024277E-2,
+--R - 2.4024275624423552E-11]
+--R ,
+--R
+--R [1.425, - 3.6055069699999998E-2, - 3.6055069722547906E-2,
+--R - 2.2547907985170923E-11]
+--R ,
+--R
+--R [1.4299999999999999, - 3.10609237E-2, - 3.1060923671447194E-2,
+--R 2.8552805952930527E-11]
+--R ,
+--R
+--R [1.4350000000000001, - 2.6090193500000001E-2, - 2.609019351596098E-2,
+--R - 1.5960979066198533E-11]
+--R ,
+--R
+--R [1.4399999999999999, - 2.1142670299999999E-2, - 2.1142670333530678E-2,
+--R - 3.3530678233972822E-11]
+--R ,
+--R
+--R [1.4450000000000001, - 1.62181479E-2, - 1.6218147888283685E-2,
+--R 1.1716315417853451E-11]
+--R ,
+--R
+--R [1.45, - 1.1316422600000001E-2, - 1.1316422586445718E-2,
+--R 1.3554282696226494E-11]
+--R ,
+--R
+--R [1.4550000000000001, - 6.4372934000000003E-3, - 6.4372934326406561E-3,
+--R - 3.2640655803217733E-11]
+--R ,
+--R [1.46,- 1.580562E-3,- 1.5805619870833398E-3,1.2916660145981251E-11],
+--R
+--R [1.4650000000000001, 3.2539677000000002E-3, 3.2539676763745362E-3,
+--R - 2.3625463998339091E-11]
+--R ,
+--R
[1.47,8.0664889999999996E-3,8.0664890113649745E-3,1.1364974933369965E-11],
+--R
+--R [1.4750000000000001, 1.2857192999999999E-2, 1.2857193039295334E-2,
+--R 3.9295334347544397E-11]
+--R ,
+--R [1.48,1.7626268399999999E-2,1.7626268388849287E-2,-
1.1150712297958165E-11],
+--R
+--R [1.4850000000000001, 2.2373901299999999E-2, 2.2373901334705404E-2,
+--R 3.47054052163287E-11]
+--R ,
+--R [1.49,2.71002758E-2,2.7100275835486465E-2,3.5486464461387257E-11],
+--R
+--R [1.4950000000000001, 3.1805573599999998E-2, 3.1805573570971468E-2,
+--R - 2.9028529580088502E-11]
+--R ,
+--R [1.5,3.6489974000000001E-2,3.6489973978576673E-2,-
2.1423328389058582E-11],
+--R
+--R [1.5049999999999999, 4.1153654300000002E-2, 4.1153654289123542E-2,
+--R - 1.0876459455300136E-11]
+--R ,
+--R [1.51,4.5796789599999999E-2,4.5796789561914686E-2,-
3.8085312681346295E-11],
+--R
+--R [1.5149999999999999, 5.0419552700000002E-2, 5.0419552719128236E-2,
+--R 1.9128233719190035E-11]
+--R ,
+--R [1.52,5.5022114599999998E-2,5.5022114579551307E-2,-
2.0448691351315773E-11],
+--R
+--R [1.5249999999999999, 5.9604643899999997E-2, 5.960464389166209E-2,
+--R - 8.3379067539190999E-12]
+--R ,
+--R [1.53,6.4167307399999998E-2,6.4167307366077231E-2,-
3.3922767372907003E-11],
+--R
+--R [1.5349999999999999, 6.8710269700000007E-2, 6.8710269707385141E-2,
+--R 7.3851341708675022E-12]
+--R ,
+--R
[1.54,7.3233693599999997E-2,7.3233693645366138E-2,4.5366141399050264E-11],
+--R
+--R [1.5449999999999999, 7.7737730000000005E-2, 7.7737739965624497E-2,
+--R 9.9656244922918802E-9]
+--R ,
+--R
[1.55,8.2222567499999996E-2,8.2222567539644631E-2,3.9644634797220135E-11],
+--R
+--R [1.5549999999999999, 8.6688333399999998E-2, 8.6688333354268288E-2,
+--R - 4.5731710085483712E-11]
+--R ,
+--R
+--R [1.5600000000000001, 9.1135192500000003E-2, 9.1135192540635401E-2,
+--R 4.0635397824395625E-11]
+--R ,
+--R
+--R [1.5649999999999999, 9.5563298399999996E-2, 9.5563298402570163E-2,
+--R 2.5701663020072374E-12]
+--R ,
+--R
+--R [1.5700000000000001, 9.9972802400000005E-2, 9.9972802444444731E-2,
+--R 4.4444725677550423E-11]
+--R ,
+--R [1.575,0.1043638544,0.10436385439851947,- 1.4805240367010697E-12],
+--R
+--R [1.5800000000000001, 0.1087366023, 0.10873660225178161,
+--R - 4.8218387616039138E-11]
+--R ,
+--R [1.585,0.1130911923,0.11309119227228603,- 2.7713970007781086E-11],
+--R
[1.5900000000000001,0.117427769,0.11742776903501095,3.5010952470493351E-11],
+--R [1.595,0.1217464754,0.12174647544723916,4.7239157030531942E-11],
+--R
+--R [1.6000000000000001, 0.12604745279999999, 0.12604745277347584,
+--R - 2.6524143992290306E-11]
+--R ,
+--R [1.605,0.13033084070000001,0.13033084065991318,-
4.0086822750140527E-11],
+--R
+--R [1.6100000000000001, 0.13459677719999999, 0.13459677715844587,
+--R - 4.155412125506075E-11]
+--R ,
+--R [1.615,0.13884539879999999,0.13884539875025736,-
4.9742626684334823E-11],
+--R
+--R [1.6200000000000001, 0.14307684039999999, 0.14307684036898005,
+--R - 3.1019936619358646E-11]
+--R ,
+--R [1.625,0.1472912354,0.14729123542343325,2.3433255336158254E-11],
+--R
+--R [1.6299999999999999, 0.15148871580000001, 0.15148871581995815,
+--R 1.9958146246779052E-11]
+--R ,
+--R [1.635,0.15566941200000001,0.15566941198435302,-
1.5646983708705875E-11],
+--R
+--R [1.6399999999999999, 0.15983345290000001, 0.15983345288341522,
+--R - 1.6584789097606745E-11]
+--R ,
+--R [1.645,0.16398096600000001,0.16398096604610457,4.6104564610516263E-11],
+--R
+--R [1.6499999999999999, 0.16811207759999999, 0.16811207758432767,
+--R - 1.5672324549242944E-11]
+--R ,
+--R [1.655,0.1722269122,0.17222691221335784,1.335784260980688E-11],
+--R
+--R [1.6599999999999999, 0.1763255933, 0.17632559327189457,
+--R - 2.8105434646263916E-11]
+--R ,
+--R [1.665,0.1804082427,0.18040824274177392,4.1773917658360915E-11],
+--R
+--R [1.6699999999999999, 0.18447498130000001, 0.1844749812673292,
+--R - 3.2670810501400638E-11]
+--R ,
+--R [1.675,0.1885259282,0.18852592817442249,- 2.5577512330343666E-11],
+--R
+--R [1.6799999999999999, 0.1925612015, 0.19256120148913258,
+--R - 1.0867418076543345E-11]
+--R ,
+--R
+--R [1.6850000000000001, 0.19658091799999999, 0.19658091795613342,
+--R - 4.3866577037476873E-11]
+--R ,
+--R
+--R [1.6899999999999999, 0.2005851931, 0.20058519305674649,
+--R - 4.3253511883278861E-11]
+--R ,
+--R
+--R [1.6950000000000001, 0.20457414099999999, 0.20457414102668603,
+--R 2.6686042264856269E-11]
+--R ,
+--R [1.7,0.20854787490000001,0.20854787487349435,- 2.6505658778930297E-11],
+--R
+--R [1.7050000000000001, 0.21250650639999999, 0.21250650639368796,
+--R - 6.3120342286282494E-12]
+--R ,
+--R [1.71,0.2164501462,0.21645014618960501,- 1.0394990423989725E-11],
+--R
+--R [1.7150000000000001, 0.22037890369999999, 0.2203789036859658,
+--R - 1.4034190476408526E-11]
+--R ,
+--R [1.72,0.2242928871,0.22429288714615725,4.6157244693034727E-11],
+--R
+--R [1.7250000000000001, 0.22819220370000001, 0.22819220368823745,
+--R - 1.1762563145722993E-11]
+--R ,
+--R [1.73,0.2320769593,0.23207695930067274,6.7273964177161361E-13],
+--R
+--R [1.7350000000000001, 0.23594725890000001, 0.23594725885781176,
+--R - 4.2188252891151024E-11]
+--R ,
+--R [1.74,0.2398032061,0.23980320613509676,3.5096758832509067E-11],
+--R
+--R [1.7450000000000001, 0.24364490380000001, 0.24364490382402559,
+--R 2.4025587075371391E-11]
+--R ,
+--R [1.75,0.24747245349999999,0.2474724535468612,4.6861209357373923E-11],
+--R
+--R [1.7549999999999999, 0.25128595590000002, 0.25128595587109781,
+--R - 2.8902213955461775E-11]
+--R ,
+--R [1.76,0.25508551029999998,0.25508551032368809,2.3688107031460959E-11],
+--R
+--R [1.7649999999999999, 0.25887121540000002, 0.25887121540503744,
+--R 5.0374149296317228E-12]
+--R ,
+--R [1.77,0.26264316859999998,0.26264316860276249,2.7625124410235458E-12],
+--R
+--R [1.7749999999999999, 0.26640146640000001, 0.2664014664052331,
+--R 5.2330917377219066E-12]
+--R ,
+--R [1.78,0.27014620430000003,0.27014620431488368,1.4883649868124849E-11],
+--R
+--R [1.7849999999999999, 0.27387747690000003, 0.27387747686131236,
+--R - 3.8687664183356674E-11]
+--R ,
+--R [1.79,0.27759537760000003,0.27759537761416786,1.4167833572997779E-11],
+--R
+--R [1.7949999999999999, 0.28129999919999998, 0.2812999991958266,
+--R - 4.1733838607171947E-12]
+--R ,
+--R [1.8,0.28499143329999999,0.2849914332938619,- 6.138090036245103E-12],
+--R
+--R [1.8049999999999999, 0.28866977069999999, 0.28866977067331689,
+--R - 2.6683100173841012E-11]
+--R ,
+--R
+--R [1.8100000000000001, 0.2923351012, 0.29233510118877948,
+--R - 1.1220524509525376E-11]
+--R ,
+--R
+--R [1.8149999999999999, 0.29598751379999999, 0.29598751379626109,
+--R - 3.7388980800301397E-12]
+--R ,
+--R
+--R [1.8200000000000001, 0.29962709659999998, 0.29962709656488773,
+--R - 3.5112246443702588E-11]
+--R ,
+--R [1.825,0.3032539367,0.30325393668840539,- 1.1594614157672822E-11],
+--R
+--R [1.8300000000000001, 0.30686812050000001, 0.30686812049650136,
+--R - 3.4986458175012558E-12]
+--R ,
+--R [1.835,0.31046973350000001,0.31046973346594764,-
3.4052372033244183E-11],
+--R
+--R [1.8400000000000001, 0.31405886020000001, 0.31405886023156859,
+--R 3.1568581082552782E-11]
+--R ,
+--R [1.845,0.31763558460000002,0.31763558459703256,-
2.9674596113693497E-12],
+--R
[1.8500000000000001,0.3211999895,0.32119998954547946,4.5479453536501069E-11]
+--R ,
+--R [1.855,0.3247521572,0.32475215724997797,4.9977966209979741E-11],
+--R
+--R [1.8600000000000001, 0.32829216909999998, 0.32829216908382075,
+--R - 1.6179224626711175E-11]
+--R ,
+--R [1.865,0.33182010560000003,0.33182010563065989,3.0659863536897092E-11],
+--R
+--R [1.8700000000000001, 0.3353360467, 0.33533604669448569,
+--R - 5.5143112298594588E-12]
+--R ,
+--R [1.875,0.33884007129999999,0.33884007130944738,9.4473873168965383E-12],
+--R
+--R [1.8799999999999999, 0.34233225769999998, 0.34233225774952925,
+--R 4.9529269574577484E-11]
+--R ,
+--R [1.885,0.3458126835,0.34581268353806771,3.8067715646405986E-11],
+--R
+--R [1.8899999999999999, 0.34928142550000002, 0.34928142545713492,
+--R - 4.286510035811375E-11]
+--R ,
+--R [1.895,0.3527385596,0.35273855955676792,- 4.3232084578903596E-11],
+--R
+--R [1.8999999999999999, 0.35618416120000002, 0.35618416116406026,
+--R - 3.5939751175106949E-11]
+--R ,
+--R [1.905,0.35961830490000002,0.35961830489211799,-
7.8820283633262989E-12],
+--R
+--R [1.9099999999999999, 0.36304106460000002, 0.36304106464888108,
+--R 4.8881065861650086E-11]
+--R ,
+--R [1.915,0.36645251359999997,0.36645251364580167,4.580169576939852E-11],
+--R
[1.9199999999999999,0.3698527244,0.36985272440640171,6.4017124934423464E-12]
+--R ,
+--R [1.925,0.37324176879999998,0.37324176877469795,-
2.5302038242358549E-11],
+--R
+--R [1.9299999999999999, 0.37661971789999998, 0.37661971792349891,
+--R 2.3498925028064832E-11]
+--R ,
+--R
+--R [1.9350000000000001, 0.37998664240000002, 0.37998664236258128,
+--R - 3.7418734777361351E-11]
+--R ,
+--R
+--R [1.9399999999999999, 0.38334261190000002, 0.38334261194674013,
+--R 4.6740111780962934E-11]
+--R ,
+--R
+--R [1.9450000000000001, 0.38668769590000002, 0.38668769588372298,
+--R - 1.6277035275180651E-11]
+--R ,
+--R [1.95,0.3900219627,0.39002196274204304,4.2043035719530053E-11],
+--R
+--R [1.9550000000000001, 0.39334548050000001, 0.39334548045868012,
+--R - 4.1319891952440457E-11]
+--R ,
+--R [1.96,0.39665831629999998,0.39665831634666171,4.6661730035424398E-11],
+--R
+--R [1.9650000000000001, 0.39996053710000001, 0.39996053710254509,
+--R 2.5450752616507089E-12]
+--R ,
+--R [1.97,0.40325220880000001,0.40325220881377177,1.3771761508962754E-11],
+--R
+--R [1.9750000000000001, 0.40653339700000002, 0.40653339696592627,
+--R - 3.4073743826468217E-11]
+--R ,
+--R [1.98,0.4098041664,0.40980416644989071,4.9890702680244203E-11],
+--R
+--R [1.9850000000000001, 0.41306458159999998, 0.41306458156888626,
+--R - 3.1113722709363856E-11]
+--R ,
+--R [1.99,0.41631470599999998,0.41631470604541487,4.5414894067619116E-11],
+--R
+--R [1.9950000000000001, 0.41955460300000003, 0.41955460302810832,
+--R 2.8108293470552326E-11]
+--R ,
+--R [2.,0.42278433510000002,0.42278433509846725,- 1.5327739077974911E-12]]
+--R Type: List List
DoubleFloat
+--E 3
+
+)spool
+)lisp (bye)
+
+@
+
+\subsection{Gamma}
+Because we use this file for regression testing and regression tests
+do not yet allow graphics the rest of this file is commented out.
+
+This is the graph of the gamma and 1/gamma functions.
+We are going to combine two graphs in a single viewport so we enable
+color differences for the two graphs.
+\begin{verbatim}
+)clear all
+lineColorDefault(green())
+viewport1:=draw(Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+lineColorDefault(blue())
+viewport2:=draw(1/Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+putGraph(viewport2,getGraph(viewport1,1),2)
+points(viewport2,1,"off")
+points(viewport2,2,"off")
+makeViewport2D(viewport2)
+close(viewport1)
+close(viewport2)
+\end{verbatim}
+
+This is a graph of the log(gamma) function.
+\begin{verbatim}
+)clear all
+viewport:=draw(log Gamma(i), i=0..8, adaptive==true, unit==[1.0,1.0])
+points(viewport1,1,"off")
+close(viewport)
+\end{verbatim}
+
+This is a graph of the Psi function
+\begin{verbatim}
+)clear all
+Psi(x)==polygamma(0,x)
+viewport:=draw(Psi(y),y=-3.5..4,adaptive==true)
+scale(viewport,1,0.9,22.5)
+translate(viewport,1,0,-0.02)
+points(viewport,1,"off")
+close(viewport)
+\end{verbatim}
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf Digital Library of Mathematical Functions},
+http://dlmf.nist.gov
+\bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+\end{thebibliography}
+\end{document}
diff --git a/src/input/pfaffian.input.pamphlet
b/src/input/pfaffian.input.pamphlet
new file mode 100644
index 0000000..0e2f33b
--- /dev/null
+++ b/src/input/pfaffian.input.pamphlet
@@ -0,0 +1,524 @@
+\documentclass{article}
+\usepackage{amsfonts}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input pfaffian}
+\author{Timothy Daly, Gunter Rote, Martin Rubey}
+\maketitle
+\begin{abstract}
+The Pfaffian is a function defined on skew-symmetric $2n\times 2n$ matrices.
+It is the square root of the determinant.\cite{1}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{Examples}
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cc}
+0 & a\\
+-a & 0
+\end{array}
+\right] = a
+$$
+
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cccc}
+0 & a & b & c\\
+-a & 0 & d & e\\
+-b & -d & 0 & f\\
+-c & -e & -f & 0
+\end{array}
+\right] = af -be + dc
+$$
+
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cccc}
+\begin{array}{cc}
+0 & \lambda_1\\
+-\lambda_1 & 0
+\end{array} & 0 & \cdots & 0\\
+0 &
+\begin{array}{cc}
+0 & \lambda_2\\
+-\lambda_2 & 0
+\end{array} & & 0\\
+\vdots & & \ddots & \vdots\\
+0 & 0 & \cdots &
+\begin{array}{cc}
+0 & \lambda_n\\
+-\lambda_n & 0
+\end{array}
+\end{array}
+\right] = \lambda_1\lambda_2\cdots\lambda_n
+$$
+\section{Formal definition}
+
+A matrix is skew-symmetric if $A^T = -A$
+
+Let $A = \{a_{i,j}\}$ be a $2n \times 2n$ skew-symmetric matrix.
+The Pfaffian of A is defined by the equation
+
+$$Pf(A) = \frac{1}{s^n n!}\sum_{\sigma\in{}S_{2n}}sgn(\sigma)
+\prod_{i=1}^n{a_{\sigma(2i-1),\sigma(2i)}}$$
+
+where $S_{2n}$ is the symmetric group and $sgn(\sigma)$
+is the signature of $\sigma$.
+
+The Pfaffian of a $n\times n$ skew-symmetric matrix for n odd is
+defined to be zero.
+
+\section{Axiom code}
+<<*>>=
+)spool pfaffian.output
+)set message test on
+)set message auto off
+)clear all
+
+@
+The function B0 has an input range of PositiveInteger. It computes
+a square matrix.
+<<*>>=
+--S 1 of 26
+B0 n == matrix [[(if i=j+1 and odd? j then -1 else _
+ if i=j-1 and odd? i then 1 else 0) _
+ for j in 1..n] for i in 1..n]
+--R
+--R Type:
Void
+--E 1
+@
+
+The first 8 values computed are:
+<<*>>=
+--S 2 of 26
+B0 1
+--R
+--R Compiling function B0 with type PositiveInteger -> Matrix Integer
+--R
+--R (2) [0]
+--R Type: Matrix
Integer
+--E 2
+
+--S 3 of 26
+B0 2
+--R
+--R
+--R + 0 1+
+--R (3) | |
+--R +- 1 0+
+--R Type: Matrix
Integer
+--E 3
+
+--S 4 of 26
+B0 3
+--R
+--R
+--R + 0 1 0+
+--R | |
+--R (4) |- 1 0 0|
+--R | |
+--R + 0 0 0+
+--R Type: Matrix
Integer
+--E 4
+
+--S 5 of 26
+B0 4
+--R
+--R
+--R + 0 1 0 0+
+--R | |
+--R |- 1 0 0 0|
+--R (5) | |
+--R | 0 0 0 1|
+--R | |
+--R + 0 0 - 1 0+
+--R Type: Matrix
Integer
+--E 5
+
+--S 6 of 26
+B0 5
+--R
+--R
+--R + 0 1 0 0 0+
+--R | |
+--R |- 1 0 0 0 0|
+--R | |
+--R (6) | 0 0 0 1 0|
+--R | |
+--R | 0 0 - 1 0 0|
+--R | |
+--R + 0 0 0 0 0+
+--R Type: Matrix
Integer
+--E 6
+
+--S 7 of 26
+B0 6
+--R
+--R
+--R + 0 1 0 0 0 0+
+--R | |
+--R |- 1 0 0 0 0 0|
+--R | |
+--R | 0 0 0 1 0 0|
+--R (7) | |
+--R | 0 0 - 1 0 0 0|
+--R | |
+--R | 0 0 0 0 0 1|
+--R | |
+--R + 0 0 0 0 - 1 0+
+--R Type: Matrix
Integer
+--E 7
+
+--S 8 of 26
+B0 7
+--R
+--R
+--R + 0 1 0 0 0 0 0+
+--R | |
+--R |- 1 0 0 0 0 0 0|
+--R | |
+--R | 0 0 0 1 0 0 0|
+--R | |
+--R (8) | 0 0 - 1 0 0 0 0|
+--R | |
+--R | 0 0 0 0 0 1 0|
+--R | |
+--R | 0 0 0 0 - 1 0 0|
+--R | |
+--R + 0 0 0 0 0 0 0+
+--R Type: Matrix
Integer
+--E 8
+
+--S 9 of 26
+B0 8
+--R
+--R
+--R + 0 1 0 0 0 0 0 0+
+--R | |
+--R |- 1 0 0 0 0 0 0 0|
+--R | |
+--R | 0 0 0 1 0 0 0 0|
+--R | |
+--R | 0 0 - 1 0 0 0 0 0|
+--R (9) | |
+--R | 0 0 0 0 0 1 0 0|
+--R | |
+--R | 0 0 0 0 - 1 0 0 0|
+--R | |
+--R | 0 0 0 0 0 0 0 1|
+--R | |
+--R + 0 0 0 0 0 0 - 1 0+
+--R Type: Matrix
Integer
+--E 9
+
+--S 10 of 26
+PfChar(lambda, A) ==
+ n := nrows A
+ odd? n => 0
+ (n = 2) => lambda^2 + A.(1,2)
+ M := subMatrix(A, 3, n, 3, n)
+ r := subMatrix(A, 1, 1, 3, n)
+ s := subMatrix(A, 3, n, 2, 2)
+
+ p := PfChar(lambda, M)
+ d := degree(p, lambda)
+
+ B := B0(n-2)
+ C := r*B
+ g := [(C*s).(1,1), A.(1,2), 1]
+ if d >= 4 then
+ B := M*B
+ for i in 4..d by 2 repeat
+ C := C*B
+ g := cons((C*s).(1,1), g)
+ g := reverse! g
+
+ res := 0
+ for i in 0..d by 2 for j in 2..d+2 repeat
+ c := coefficient(p, lambda, i)
+ for e in first(g, j) for k in 2..-d by -2 repeat
+ res := res + c * e * lambda^(k+i)
+
+ res
+--R
+--R Type:
Void
+--E 10
+
+--S 11 of 26
+pfaffian A == eval(PfChar(l, A), l=0)
+--R
+--R Type:
Void
+--E 11
+
+--S 12 of 26
+m:Matrix(Integer):=[[0,15],[-15,0]]
+--R
+--R
+--R + 0 15+
+--R (12) | |
+--R +- 15 0 +
+--R Type: Matrix
Integer
+--E 12
+
+--S 13 of 26
+pfaffian m
+--R
+--R Compiling function B0 with type Integer -> Matrix Integer
+--R The type of the local variable res has changed in the computation.
+--R We will attempt to interpret the code.
+--R Cannot compile map: PfChar
+--R We will attempt to interpret the code.
+--R
+--R (13) 15
+--R Type: Polynomial
Integer
+--E 13
+
+--S 14 of 26
+m1:Matrix(Polynomial(Integer)):=[[0,a,b,c],[-a,0,d,e],[-b,-d,0,f],[-c,-e,-f,0]]
+--R
+--R
+--R + 0 a b c+
+--R | |
+--R |- a 0 d e|
+--R (14) | |
+--R |- b - d 0 f|
+--R | |
+--R +- c - e - f 0+
+--R Type: Matrix Polynomial
Integer
+--E 14
+
+--S 15 of 26
+pfaffian m1
+--R
+--R
+--R (15) a f - b e + c d
+--R Type: Polynomial
Integer
+--E 15
+
+--S 16 of 26
+(a,b,c,d,e,f):=(3,5,7,11,13,17)
+--R
+--R
+--R (16) 17
+--R Type:
PositiveInteger
+--E 16
+
+--S 17 of 26
+m1
+--R
+--R
+--R + 0 a b c+
+--R | |
+--R |- a 0 d e|
+--R (17) | |
+--R |- b - d 0 f|
+--R | |
+--R +- c - e - f 0+
+--R Type: Matrix Polynomial
Integer
+--E 17
+
+--S 18 of 26
+a*f-b*e+d*c
+--R
+--R
+--R (18) 63
+--R Type:
PositiveInteger
+--E 18
+
+--S 19 of 26
+n:=pfaffian m1
+--R
+--R
+--R (19) a f - b e + c d
+--R Type: Polynomial
Integer
+--E 19
+
+--S 20 of 26
+eval(n,['a,'b,'c,'d,'e,'f]::List(Symbol),[a,b,c,d,e,f])
+--R
+--R
+--R (20) 63
+--R Type: Polynomial
Integer
+--E 20
+
+@
+Here we demonstrate the pfaffian of a block matrix which is
+anti-symmetric at each level. The proper result is to return
+a single matrix in the ground field. Maybe. It's not clear.
+<<*>>=
+
+)clear properties z
+)clear properties d
+)clear properties v
+
+--S 21 of 26
+z:SQMATRIX(2,INT):=[[0,0],[0,0]]
+--R
+--R
+--R +0 0+
+--R (21) | |
+--R +0 0+
+--R Type:
SquareMatrix(2,Integer)
+--E 21
+
+--S 22 of 26
+d:SQMATRIX(2,INT):=[[0,1],[-1,0]]
+--R
+--R
+--R + 0 1+
+--R (22) | |
+--R +- 1 0+
+--R Type:
SquareMatrix(2,Integer)
+--E 22
+
+--S 23 of 26
+v:SQMATRIX(4,SQMATRIX(2,INT)):=[[z,d,d,d],[-d,z,d,d],[-d,-d,z,d],[-d,-d,-d,z]]
+--R
+--R
+--R + +0 0+ + 0 1+ + 0 1+ + 0 1++
+--R | | | | | | | | ||
+--R | +0 0+ +- 1 0+ +- 1 0+ +- 1 0+|
+--R | |
+--R |+0 - 1+ +0 0+ + 0 1+ + 0 1+|
+--R || | | | | | | ||
+--R |+1 0 + +0 0+ +- 1 0+ +- 1 0+|
+--R (23) | |
+--R |+0 - 1+ +0 - 1+ +0 0+ + 0 1+|
+--R || | | | | | | ||
+--R |+1 0 + +1 0 + +0 0+ +- 1 0+|
+--R | |
+--R |+0 - 1+ +0 - 1+ +0 - 1+ +0 0+ |
+--R || | | | | | | | |
+--R ++1 0 + +1 0 + +1 0 + +0 0+ +
+--R Type:
SquareMatrix(4,SquareMatrix(2,Integer))
+--E 23
+
+--S 24 of 26
+pfaffian v
+--R
+--R There are 1 exposed and 0 unexposed library operations named
+--R subMatrix having 5 argument(s) but none was determined to be
+--R applicable. Use HyperDoc Browse, or issue
+--R )display op subMatrix
+--R to learn more about the available operations. Perhaps
+--R package-calling the operation or using coercions on the arguments
+--R will allow you to apply the operation.
+--R
+--RDaly Bug
+--R Cannot find a definition or applicable library operation named
+--R subMatrix with argument type(s)
+--R SquareMatrix(4,SquareMatrix(2,Integer))
+--R PositiveInteger
+--R PositiveInteger
+--R PositiveInteger
+--R PositiveInteger
+--R
+--R Perhaps you should use "@" to indicate the required return type,
+--R or "$" to specify which version of the function you need.
+--E 24
+
+--S 25 of 26
+mypf(m) ==
+ nr:= nrows m
+ odd? nr => 0
+-- not zero? (nr mod 2)
+ not square? m => 0
+ not antisymmetric? m => 0
+ nr = 2 => m.(1,2)
+ nr = 4 => m.(1,2)*m.(3,4)-m.(1,3)*m.(2,4)+m.(2,3)*m.(1,4)
+ 0
+--R
+--R Type:
Void
+--E 25
+
+@
+We can construct an anti-symmetric matrix from a sequence by taking
+the first $n$ elements of the sequence:
+<<*>>=
+--S 26 of 26
+antisymmetric(seq,n) ==
+ m:= matrix [[(if i<j then (seq.(j-i)) _
+ else if i>j then -(seq.(i-j))
+ else 0) for j in 1..n] for i in 1..n]
+--R
+--R Type:
Void
+--E 26
+
+@
+According to Austin\cite{1} we can construct a map, called the
+``Pfaffian transformation'' of a sequence which maps the sequence
+$(a_0, a_1, a_2, \ldots)$ to $(Pf(A_0), Pf(A_1), Pf(A_2), \ldots)$.
+Here we construct and test the examples given in the paper:
+\begin{verbatim}
+--
+seq:=[i for i in 1..]
+--R
+--R
+--R (26) [1,2,3,4,5,6,7,8,9,10,...]
+--R Type: Stream
PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[2 for i in 1..]
+--R
+--R
+--R (27) [2,2,2,2,2,2,2,2,2,2,...]
+--R Type: Stream
PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[2^i for i in 0..]
+--R
+--R
+--R (28) [1,2,4,8,16,32,64,128,256,512,...]
+--R Type: Stream
PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[1,1,2,3,5,8,13,21,34]
+--R
+--R
+--R (29) [1,1,2,3,5,8,13,21,34]
+--R Type: List
PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+\end{verbatim}
+
+@
+<<*>>=
+)spool
+)lisp (bye)
+
+@
+The Pfaffian is the square root of the determinant. The determinant
+can be calculated by
+\begin{verbatim}
+|a b c|
+|d e f| = a|e f| - b|d f| + c|d e|
+|g h i| |h i| |g i| |g h|
+\end{verbatim}
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Austin, T., Bantilan, H., Jonas, I., Kory, P.,
+``The Pfaffian Transformation'', March 9, 2007
+{\bf ``apps.carleton.edu/curricular/math/assets/pfaffian\_comps\_06\_07.pdf''}
+\bibitem{2} {\bf ``http://en.wikipedia.org/wiki/Pfaffian''}
+\end{thebibliography}
+\end{document}
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