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 , +--R [1.02,0.9888442033,0.98884420326391309,- 3.6086911237021013E-11], +--R +--R [1.0249999999999999, 0.98617396329999996, 0.98617396314825367, +--R - 1.5174628220648856E-10] +--R , +--R [1.03,0.98354995059999994,0.98354995055382399,- 4.6175951950999661E-11], +--R +--R [1.0349999999999999, 0.98097156060000001, 0.98097156055058576, +--R - 4.9414250469226317E-11] +--R , +--R [1.04,0.9784382009,0.97843820091424472,1.4244716517453071E-11], +--R +--R [1.0449999999999999, 0.97594929190000002, 0.97594929182295154, +--R - 7.7048478708263701E-11] +--R , +--R [1.05,0.97350426560000003,0.97350426556277558,- 3.7224445748051949E-11], +--R +--R [1.0549999999999999, 0.97110256630000003, 0.97110256624166991, +--R - 5.8330118513083562E-11] +--R , +--R +--R [1.0600000000000001, 0.96874364950000003, 0.9687436495116386, +--R 1.1638578989447979E-11] +--R , +--R +--R [1.0649999999999999, 0.96642698230000001, 0.96642698229884005, +--R - 1.1599610161283636E-12] +--R , +--R +--R [1.0700000000000001, 0.96415204249999997, 0.96415204254136644, +--R 4.1366465808323483E-11] +--R , +--R [1.075,0.96191831890000001,0.96191831893444502,3.4445002405902869E-11], +--R +--R [1.0800000000000001, 0.9597253107, 0.95972531068282241, +--R - 1.7177592681605347E-11] +--R , +--R [1.085,0.95757252729999998,0.957572527260101,- 3.9898973014373951E-11], +--R +--R [1.0900000000000001, 0.95545948820000004, 0.95545948817480109, +--R - 2.5198954034522103E-11] +--R , +--R [1.095,0.95338572269999999,0.95338572274293298,4.2932990496069579E-11], +--R +--R [1.1000000000000001, 0.9513507699, 0.95135076986687306, +--R - 3.3126945631067883E-11] +--R , +--R [1.105,0.94935417779999998,0.94935417782034381,2.0343837725533831E-11], +--R +--R [1.1100000000000001, 0.94739550400000005, 0.94739550403930195, +--R 3.9301895071730542E-11] +--R , +--R [1.115,0.9454743149,0.94547431491855183,1.8551826741486366E-11], +--R +--R [1.1200000000000001, 0.94359018559999996, 0.9435901856139034, +--R 1.3903433959683298E-11] +--R , +--R [1.125,0.94174269970000002,0.94174269984970138,1.4970136241743148E-10], +--R +--R [1.1299999999999999, 0.93993144969999998, 0.93993144973155829, +--R 3.1558311519575E-11] +--R , +--R [1.135,0.93815603560000005,0.93815603556413329,- 3.5866754011237845E-11], +--R +--R [1.1399999999999999, 0.93641606570000002, 0.9364160656737992, +--R - 2.6200819291943844E-11] +--R , +--R [1.145,0.9347111562,0.93471115623604595,3.6045944007412345E-11], +--R +--R [1.1499999999999999, 0.93304093109999997, 0.93304093110748143, +--R 7.4814598960415424E-12] +--R , +--R [1.155,0.93140502169999995,0.93140502166228489,- 3.7715053302633805E-11], +--R +--R [1.1599999999999999, 0.92980306660000001, 0.9298030666329804, +--R 3.2980396191817363E-11] +--R , +--R [1.165,0.92823471199999996,0.92823471195540086,- 4.4599102189124551E-11], +--R +--R [1.1699999999999999, 0.92669961059999995, 0.92669961061771577, +--R 1.7715828803943623E-11] +--R , +--R [1.175,0.92519742250000003,0.9251974225134012,1.3401169063342877E-11], +--R +--R [1.1799999999999999, 0.92372781429999995, 0.92372781429803652, +--R - 1.9634294190495893E-12] +--R , +--R +--R [1.1850000000000001, 0.92229045909999996, 0.9222904592498119, +--R 1.4981194063068415E-10] +--R , +--R +--R [1.1899999999999999, 0.92088503710000003, 0.92088503713363956, +--R 3.3639535601537318E-11] +--R , +--R +--R [1.1950000000000001, 0.91951123410000002, 0.91951123406876334, +--R - 3.1236679909341092E-11] +--R , +--R [1.2,0.91816874240000002,0.91816874239976076,- 2.3925306180672123E-13], +--R +--R [1.2050000000000001, 0.91685726059999995, 0.91685726057084371, +--R - 2.9156232983496011E-11] +--R , +--R [1.21,0.91557649299999999,0.91557649300335908,3.3590907833058736E-12], +--R +--R [1.2150000000000001, 0.91432614000000001, 0.91432614997639472, +--R 9.9763947103426176E-9] +--R , +--R [1.22,0.91310594749999996,0.9131059475104113,1.041133845802733E-11], +--R +--R [1.2250000000000001, 0.91191560709999997, 0.91191560725379928, +--R 1.537993066236254E-10] +--R , +--R [1.23,0.91075485639999998,0.91075485637228959,- 2.771038953852667E-11], +--R +--R [1.2350000000000001, 0.90962342740000002, 0.90962342744113123, +--R 4.1131209549405412E-11] +--R , +--R [1.24,0.90852105829999996,0.90852105833995966,3.9959702213820947E-11], +--R +--R [1.2450000000000001, 0.90744749219999998, 0.90744749215027698, +--R - 4.9723003492374573E-11] +--R , +--R [1.25,0.90640247709999999,0.90640247705547716,- 4.4522829867332803E-11], +--R +--R [1.2549999999999999, 0.90538576630000001, 0.90538576624333822, +--R - 5.6661786373979339E-11] +--R , +--R [1.26,0.90439711779999998,0.90439711781091747,1.0917489134953939E-11], +--R +--R [1.2649999999999999, 0.90343629459999997, 0.90343629467178155, +--R 7.1781580679441959E-11] +--R , +--R [1.27,0.90250306449999995,0.9025030644655061,- 3.4493852218986376E-11], +--R +--R [1.2749999999999999, 0.90159719940000005, 0.90159719946938677, +--R 6.9386718593023033E-11] +--R , +--R [1.28,0.90071847650000003,0.90071847651229453,1.2294498752396521E-11], +--R +--R [1.2849999999999999, 0.8998666769, 0.89986667689062316, +--R - 9.3768326436816096E-12] +--R , +--R [1.29,0.89904158629999997,0.89904158628626873,- 1.3731238368563936E-11], +--R +--R [1.2949999999999999, 0.89824299470000002, 0.89824299468658741, +--R - 1.3412604360496516E-11] +--R , +--R [1.3,0.89747069629999998,0.89747069630627696,6.27697893662571E-12], +--R +--R [1.3049999999999999, 0.89672448950000005, 0.89672448951113382, +--R 1.113376058015092E-11] +--R , +--R +--R [1.3100000000000001, 0.89600417669999999, 0.89600417674363142, +--R 4.3631431800861264E-11] +--R , +--R +--R [1.3149999999999999, 0.89530956439999998, 0.89530956445027288, +--R 5.0272896956471413E-11] +--R , +--R [1.3200000000000001,0.894640463,0.8946404630106749,1.0674905404073343E-11], +--R [1.325,0.89399668659999998,0.89399668666833076,6.8330785474302047E-11], +--R +--R [1.3300000000000001, 0.89337805349999999, 0.89337805346301302, +--R - 3.6986969043084628E-11] +--R , +--R [1.335,0.89278438500000001,0.89278438516477132,1.6477130770908843E-10], +--R +--R [1.3400000000000001, 0.89221550719999998, 0.89221550720948417, +--R 9.4841912101628623E-12] +--R , +--R [1.345,0.89167124850000001,0.89167124863592484,1.3592482694946284E-10], +--R +--R [1.3500000000000001, 0.89115144199999996, 0.89115144202430063, +--R 2.4300672585297889E-11] +--R , +--R [1.355,0.8906559235,0.89065592343622957,- 6.3770433378351754E-11], +--R +--R [1.3600000000000001, 0.89018453239999995, 0.8901845323561175, +--R - 4.3882453226729012E-11] +--R , +--R [1.365,0.88973711160000002,0.88973711163389768,3.3897662454762667E-11], +--R +--R [1.3700000000000001, 0.88931350740000004, 0.88931350742910176, +--R 2.9101721032986916E-11] +--R , +--R [1.375,0.88891356919999998,0.88891356915622532,- 4.377465057103791E-11], +--R +--R [1.3799999999999999, 0.88853714939999995, 0.88853714943135709, +--R 3.1357139107512921E-11] +--R , +--R [1.385,0.88818410410000004,0.88818410402003933,- 7.9960704724157949E-11], +--R +--R [1.3899999999999999, 0.88785429179999997, 0.88785429178632957, +--R - 1.3670398146814478E-11] +--R , +--R [1.395,0.88754757480000002,0.88754757464303291,- 1.5696710597978836E-10], +--R +--R [1.3999999999999999, 0.88726381750000005, 0.88726381750307515, +--R 3.0750957336067586E-12] +--R , +--R [1.405,0.88700288839999997,0.88700288823199203,- 1.6800794089277815E-10], +--R +--R [1.4099999999999999, 0.88676465760000001, 0.88676465760150136, +--R 1.5013545962005992E-12] +--R , +--R [1.415,0.88654899929999997,0.88654899924413844,- 5.5861537617829526E-11], +--R +--R [1.4199999999999999, 0.88635578960000005, 0.88635578960892225, +--R 8.922196315097608E-12] +--R , +--R [1.425,0.88618490809999995,0.88618490791803228,- 1.8196766315980994E-10], +--R +--R [1.4299999999999999, 0.88603623610000004, 0.88603623612446902, +--R 2.4468982395831063E-11] +--R , +--R +--R [1.4350000000000001, 0.88590965870000005, 0.88590965887067441, +--R 1.7067436353102039E-10] +--R , +--R +--R [1.4399999999999999, 0.88580506349999999, 0.88580506344809162, +--R - 5.1908366494046732E-11] +--R , +--R +--R [1.4450000000000001, 0.88572233970000003, 0.88572233975763781, +--R 5.7637783434927314E-11] +--R , +--R [1.45,0.88566138029999997,0.88566138027107244,- 2.8927527040423229E-11], +--R +--R [1.4550000000000001, 0.88562207000000004, 0.88562207999323628, +--R 9.9932362385146689E-9] +--R , +--R [1.46,0.88560433640000003,0.88560433642514391,2.5143886972500695E-11], +--R +--R [1.4650000000000001, 0.88560804950000005, 0.88560804952790473, +--R 2.7904678567836072E-11] +--R , +--R [1.47,0.88563312169999997,0.88563312168746056,- 1.2539413951628831E-11], +--R +--R [1.4750000000000001, 0.88567945749999999, 0.88567945768011203, +--R 1.8011203639645146E-10] +--R , +--R [1.48,0.88574696460000002,0.88574696463882385,3.8823833037326949E-11], +--R +--R [1.4850000000000001, 0.88583555199999997, 0.88583555202028463, +--R 2.028466283832131E-11] +--R , +--R [1.49,0.88594513159999999,0.88594513157270771,- 2.7292279547452836E-11], +--R +--R [1.4950000000000001, 0.88607561739999996, 0.88607561730435302, +--R - 9.5646934816784324E-11] +--R , +--R [1.5,0.88622692550000004,0.88622692545275816,- 4.7241877076942274E-11], +--R +--R [1.5049999999999999, 0.88639897440000004, 0.8863989744546561, +--R 5.4656057457691531E-11] +--R , +--R [1.51,0.88659168499999996,0.88659168491657048,- 8.3429485542296788E-11], +--R +--R [1.5149999999999999, 0.88680497970000005, 0.88680497958606908, +--R - 1.139309757647311E-10] +--R , +--R [1.52,0.88703878329999997,0.88703878332366115,2.3661184123113799E-11], +--R +--R [1.5249999999999999, 0.88729302310000002, 0.88729302307532742, +--R - 2.4672597298547316E-11] +--R , +--R [1.53,0.88756762779999998,0.8875676278456639,4.5663917092042539E-11], +--R +--R [1.5349999999999999, 0.88786252870000004, 0.88786252867163051, +--R - 2.8369528948246625E-11] +--R , +--R [1.54,0.88817765859999998,0.88817765859688746,- 3.1125102495366264E-12], +--R +--R [1.5449999999999999, 0.88851295269999997, 0.88851295264671037, +--R - 5.3289594958982889E-11] +--R , +--R [1.55,0.88886834780000001,0.88886834780346613,3.4661162828797387E-12], +--R +--R [1.5549999999999999, 0.88924378299999995, 0.88924378298264395, +--R - 1.735600552166261E-11] +--R , +--R +--R [1.5600000000000001, 0.88963919899999999, 0.88963919900942479, +--R 9.4247942783454164E-12] +--R , +--R +--R [1.5649999999999999, 0.89005453869999995, 0.89005453859577743, +--R - 1.0422251950359396E-10] +--R , +--R [1.5700000000000001,0.8904897463,0.89048974631807609,1.8076096175434486E-11] +--R , +--R [1.575,0.89094476860000005,0.89094476859521765,- 4.7823967008753243E-12], +--R +--R [1.5800000000000001, 0.89141955370000003, 0.89141955366724079, +--R - 3.2759239765312032E-11] +--R , +--R [1.585,0.89191405150000003,0.89191405157442527,7.4425243745679381E-11], +--R +--R [1.5900000000000001, 0.89242821409999995, 0.89242821413686702, +--R 3.6867064956425111E-11] +--R , +--R [1.595,0.89296199489999994,0.89296199493452022,3.4520275526972455E-11], +--R +--R [1.6000000000000001, 0.89351534929999998, 0.89351534928769027, +--R - 1.2309708807833886E-11] +--R , +--R [1.605,0.89408823420000005,0.89408823423797623,3.7976177758025642E-11], +--R +--R [1.6100000000000001, 0.89468060849999997, 0.89468060852964915, +--R 2.9649172006429581E-11] +--R , +--R [1.615,0.89529243270000003,0.89529243259145597,- 1.0854406262694738E-10], +--R +--R [1.6200000000000001, 0.89592366850000005, 0.89592366851884431, +--R 1.8844259486172632E-11] +--R , +--R [1.625,0.89657427999999995,0.89657428005659789,5.6597948550063393E-11], +--R +--R [1.6299999999999999, 0.89724423259999997, 0.89724423258187269, +--R - 1.8127277456869706E-11] +--R , +--R [1.635,0.89793349300000003,0.89793349308762749,8.762746084300943E-11], +--R +--R [1.6399999999999999, 0.8986420302, 0.89864203016644328, +--R - 3.3556712963900281E-11] +--R , +--R [1.645,0.89936981380000003,0.89936981399471749,1.9471746437460524E-10], +--R +--R [1.6499999999999999, 0.90011681629999996, 0.90011681631723162, +--R 1.7231660542904592E-11] +--R , +--R [1.655,0.90088301039999996,0.90088301043208341,3.2083447010222699E-11], +--R +--R [1.6599999999999999, 0.90166837119999999, 0.90166837117597354, +--R - 2.4026447498215475E-11] +--R , +--R [1.665,0.90247287480000005,0.90247287490984562,1.0984557707871545E-10], +--R +--R [1.6699999999999999, 0.90329649950000002, 0.90329649950486623, +--R 4.8662185392345236E-12] +--R , +--R [1.675,0.90413922430000004,0.90413922432874227,2.874223081761329E-11], +--R [1.6799999999999999,0.9050010302,0.90500103023237111,3.2371105795903077E-11] +--R , +--R +--R [1.6850000000000001, 0.90588189959999998, 0.90588189953681197, +--R - 6.3188010379633397E-11] +--R , +--R +--R [1.6899999999999999, 0.90678181599999996, 0.90678181602057484, +--R 2.0574875136958326E-11] +--R , +--R +--R [1.6950000000000001, 0.90770076499999996, 0.90770076490722529, +--R - 9.2774676829776581E-11] +--R , +--R [1.7,0.90863873289999997,0.90863873285329078,- 4.6709192069727123E-11], +--R +--R [1.7050000000000001, 0.90959570789999999, 0.9095957079364706, +--R 3.6470604314331467E-11] +--R , +--R [1.71,0.91057167959999996,0.91057167964414076,4.4140802124559286E-11], +--R +--R [1.7150000000000001, 0.91156663900000001, 0.91156663886214651, +--R - 1.3785350638784166E-10] +--R , +--R [1.72,0.91258057790000002,0.91258057786388114,- 3.6118885660130218E-11], +--R +--R [1.7250000000000001, 0.91361349039999995, 0.91361349029964423, +--R - 1.00355723731127E-10] +--R , +--R [1.73,0.91466537120000002,0.91466537118627245,- 1.3727574632582673E-11], +--R +--R [1.7350000000000001, 0.91573621709999997, 0.9157362168970431, +--R - 2.0295687352955838E-10] +--R , +--R [1.74,0.91682602520000001,0.91682602515183886,- 4.8161141741331903E-11], +--R +--R [1.7450000000000001, 0.91793479499999997, 0.91793479500757513, +--R 7.5751627193199056E-12] +--R , +--R [1.75,0.9190625268,0.91906252684888312,4.8883119774245642E-11], +--R +--R [1.7549999999999999, 0.92020922240000003, 0.92020922237904224, +--R - 2.0957791058151543E-11] +--R , +--R [1.76,0.92137488460000005,0.92137488461116102,1.1160961044254236E-11], +--R +--R [1.7649999999999999, 0.92255951780000001, 0.92255951785960111, +--R 5.9601101831674441E-11] +--R , +--R [1.77,0.92376312770000002,0.92376312773163893,3.1638913711162786E-11], +--R +--R [1.7749999999999999, 0.92498572109999999, 0.92498572111936284, +--R 1.9362844660975043E-11] +--R , +--R [1.78,0.92622730620000004,0.92622730619180083,- 8.1992190814617061E-12], +--R +--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 ij 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}