| from sympy.core import S, Rational | |
| from sympy.integrals.quadrature import (gauss_legendre, gauss_laguerre, | |
| gauss_hermite, gauss_gen_laguerre, | |
| gauss_chebyshev_t, gauss_chebyshev_u, | |
| gauss_jacobi, gauss_lobatto) | |
| def test_legendre(): | |
| x, w = gauss_legendre(1, 17) | |
| assert [str(r) for r in x] == ['0'] | |
| assert [str(r) for r in w] == ['2.0000000000000000'] | |
| x, w = gauss_legendre(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.57735026918962576', | |
| '0.57735026918962576'] | |
| assert [str(r) for r in w] == [ | |
| '1.0000000000000000', | |
| '1.0000000000000000'] | |
| x, w = gauss_legendre(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.77459666924148338', | |
| '0', | |
| '0.77459666924148338'] | |
| assert [str(r) for r in w] == [ | |
| '0.55555555555555556', | |
| '0.88888888888888889', | |
| '0.55555555555555556'] | |
| x, w = gauss_legendre(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.86113631159405258', | |
| '-0.33998104358485626', | |
| '0.33998104358485626', | |
| '0.86113631159405258'] | |
| assert [str(r) for r in w] == [ | |
| '0.34785484513745386', | |
| '0.65214515486254614', | |
| '0.65214515486254614', | |
| '0.34785484513745386'] | |
| def test_legendre_precise(): | |
| x, w = gauss_legendre(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '-0.7745966692414833770358530799564799221666', | |
| '0', | |
| '0.7745966692414833770358530799564799221666'] | |
| assert [str(r) for r in w] == [ | |
| '0.5555555555555555555555555555555555555556', | |
| '0.8888888888888888888888888888888888888889', | |
| '0.5555555555555555555555555555555555555556'] | |
| def test_laguerre(): | |
| x, w = gauss_laguerre(1, 17) | |
| assert [str(r) for r in x] == ['1.0000000000000000'] | |
| assert [str(r) for r in w] == ['1.0000000000000000'] | |
| x, w = gauss_laguerre(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.58578643762690495', | |
| '3.4142135623730950'] | |
| assert [str(r) for r in w] == [ | |
| '0.85355339059327376', | |
| '0.14644660940672624'] | |
| x, w = gauss_laguerre(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.41577455678347908', | |
| '2.2942803602790417', | |
| '6.2899450829374792', | |
| ] | |
| assert [str(r) for r in w] == [ | |
| '0.71109300992917302', | |
| '0.27851773356924085', | |
| '0.010389256501586136', | |
| ] | |
| x, w = gauss_laguerre(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.32254768961939231', | |
| '1.7457611011583466', | |
| '4.5366202969211280', | |
| '9.3950709123011331'] | |
| assert [str(r) for r in w] == [ | |
| '0.60315410434163360', | |
| '0.35741869243779969', | |
| '0.038887908515005384', | |
| '0.00053929470556132745'] | |
| x, w = gauss_laguerre(5, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.26356031971814091', | |
| '1.4134030591065168', | |
| '3.5964257710407221', | |
| '7.0858100058588376', | |
| '12.640800844275783'] | |
| assert [str(r) for r in w] == [ | |
| '0.52175561058280865', | |
| '0.39866681108317593', | |
| '0.075942449681707595', | |
| '0.0036117586799220485', | |
| '2.3369972385776228e-5'] | |
| def test_laguerre_precise(): | |
| x, w = gauss_laguerre(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '0.4157745567834790833115338731282744735466', | |
| '2.294280360279041719822050361359593868960', | |
| '6.289945082937479196866415765512131657493'] | |
| assert [str(r) for r in w] == [ | |
| '0.7110930099291730154495901911425944313094', | |
| '0.2785177335692408488014448884567264810349', | |
| '0.01038925650158613574896492040067908765572'] | |
| def test_hermite(): | |
| x, w = gauss_hermite(1, 17) | |
| assert [str(r) for r in x] == ['0'] | |
| assert [str(r) for r in w] == ['1.7724538509055160'] | |
| x, w = gauss_hermite(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.70710678118654752', | |
| '0.70710678118654752'] | |
| assert [str(r) for r in w] == [ | |
| '0.88622692545275801', | |
| '0.88622692545275801'] | |
| x, w = gauss_hermite(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1.2247448713915890', | |
| '0', | |
| '1.2247448713915890'] | |
| assert [str(r) for r in w] == [ | |
| '0.29540897515091934', | |
| '1.1816359006036774', | |
| '0.29540897515091934'] | |
| x, w = gauss_hermite(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1.6506801238857846', | |
| '-0.52464762327529032', | |
| '0.52464762327529032', | |
| '1.6506801238857846'] | |
| assert [str(r) for r in w] == [ | |
| '0.081312835447245177', | |
| '0.80491409000551284', | |
| '0.80491409000551284', | |
| '0.081312835447245177'] | |
| x, w = gauss_hermite(5, 17) | |
| assert [str(r) for r in x] == [ | |
| '-2.0201828704560856', | |
| '-0.95857246461381851', | |
| '0', | |
| '0.95857246461381851', | |
| '2.0201828704560856'] | |
| assert [str(r) for r in w] == [ | |
| '0.019953242059045913', | |
| '0.39361932315224116', | |
| '0.94530872048294188', | |
| '0.39361932315224116', | |
| '0.019953242059045913'] | |
| def test_hermite_precise(): | |
| x, w = gauss_hermite(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '-1.224744871391589049098642037352945695983', | |
| '0', | |
| '1.224744871391589049098642037352945695983'] | |
| assert [str(r) for r in w] == [ | |
| '0.2954089751509193378830279138901908637996', | |
| '1.181635900603677351532111655560763455198', | |
| '0.2954089751509193378830279138901908637996'] | |
| def test_gen_laguerre(): | |
| x, w = gauss_gen_laguerre(1, Rational(-1, 2), 17) | |
| assert [str(r) for r in x] == ['0.50000000000000000'] | |
| assert [str(r) for r in w] == ['1.7724538509055160'] | |
| x, w = gauss_gen_laguerre(2, Rational(-1, 2), 17) | |
| assert [str(r) for r in x] == [ | |
| '0.27525512860841095', | |
| '2.7247448713915890'] | |
| assert [str(r) for r in w] == [ | |
| '1.6098281800110257', | |
| '0.16262567089449035'] | |
| x, w = gauss_gen_laguerre(3, Rational(-1, 2), 17) | |
| assert [str(r) for r in x] == [ | |
| '0.19016350919348813', | |
| '1.7844927485432516', | |
| '5.5253437422632603'] | |
| assert [str(r) for r in w] == [ | |
| '1.4492591904487850', | |
| '0.31413464064571329', | |
| '0.0090600198110176913'] | |
| x, w = gauss_gen_laguerre(4, Rational(-1, 2), 17) | |
| assert [str(r) for r in x] == [ | |
| '0.14530352150331709', | |
| '1.3390972881263614', | |
| '3.9269635013582872', | |
| '8.5886356890120343'] | |
| assert [str(r) for r in w] == [ | |
| '1.3222940251164826', | |
| '0.41560465162978376', | |
| '0.034155966014826951', | |
| '0.00039920814442273524'] | |
| x, w = gauss_gen_laguerre(5, Rational(-1, 2), 17) | |
| assert [str(r) for r in x] == [ | |
| '0.11758132021177814', | |
| '1.0745620124369040', | |
| '3.0859374437175500', | |
| '6.4147297336620305', | |
| '11.807189489971737'] | |
| assert [str(r) for r in w] == [ | |
| '1.2217252674706516', | |
| '0.48027722216462937', | |
| '0.067748788910962126', | |
| '0.0026872914935624654', | |
| '1.5280865710465241e-5'] | |
| x, w = gauss_gen_laguerre(1, 2, 17) | |
| assert [str(r) for r in x] == ['3.0000000000000000'] | |
| assert [str(r) for r in w] == ['2.0000000000000000'] | |
| x, w = gauss_gen_laguerre(2, 2, 17) | |
| assert [str(r) for r in x] == [ | |
| '2.0000000000000000', | |
| '6.0000000000000000'] | |
| assert [str(r) for r in w] == [ | |
| '1.5000000000000000', | |
| '0.50000000000000000'] | |
| x, w = gauss_gen_laguerre(3, 2, 17) | |
| assert [str(r) for r in x] == [ | |
| '1.5173870806774125', | |
| '4.3115831337195203', | |
| '9.1710297856030672'] | |
| assert [str(r) for r in w] == [ | |
| '1.0374949614904253', | |
| '0.90575000470306537', | |
| '0.056755033806509347'] | |
| x, w = gauss_gen_laguerre(4, 2, 17) | |
| assert [str(r) for r in x] == [ | |
| '1.2267632635003021', | |
| '3.4125073586969460', | |
| '6.9026926058516134', | |
| '12.458036771951139'] | |
| assert [str(r) for r in w] == [ | |
| '0.72552499769865438', | |
| '1.0634242919791946', | |
| '0.20669613102835355', | |
| '0.0043545792937974889'] | |
| x, w = gauss_gen_laguerre(5, 2, 17) | |
| assert [str(r) for r in x] == [ | |
| '1.0311091440933816', | |
| '2.8372128239538217', | |
| '5.6202942725987079', | |
| '9.6829098376640271', | |
| '15.828473921690062'] | |
| assert [str(r) for r in w] == [ | |
| '0.52091739683509184', | |
| '1.0667059331592211', | |
| '0.38354972366693113', | |
| '0.028564233532974658', | |
| '0.00026271280578124935'] | |
| def test_gen_laguerre_precise(): | |
| x, w = gauss_gen_laguerre(3, Rational(-1, 2), 40) | |
| assert [str(r) for r in x] == [ | |
| '0.1901635091934881328718554276203028970878', | |
| '1.784492748543251591186722461957367638500', | |
| '5.525343742263260275941422110422329464413'] | |
| assert [str(r) for r in w] == [ | |
| '1.449259190448785048183829411195134343108', | |
| '0.3141346406457132878326231270167565378246', | |
| '0.009060019811017691281714945129254301865020'] | |
| x, w = gauss_gen_laguerre(3, 2, 40) | |
| assert [str(r) for r in x] == [ | |
| '1.517387080677412495020323111016672547482', | |
| '4.311583133719520302881184669723530562299', | |
| '9.171029785603067202098492219259796890218'] | |
| assert [str(r) for r in w] == [ | |
| '1.037494961490425285817554606541269153041', | |
| '0.9057500047030653669269785048806009945254', | |
| '0.05675503380650934725546688857812985243312'] | |
| def test_chebyshev_t(): | |
| x, w = gauss_chebyshev_t(1, 17) | |
| assert [str(r) for r in x] == ['0'] | |
| assert [str(r) for r in w] == ['3.1415926535897932'] | |
| x, w = gauss_chebyshev_t(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.70710678118654752', | |
| '-0.70710678118654752'] | |
| assert [str(r) for r in w] == [ | |
| '1.5707963267948966', | |
| '1.5707963267948966'] | |
| x, w = gauss_chebyshev_t(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.86602540378443865', | |
| '0', | |
| '-0.86602540378443865'] | |
| assert [str(r) for r in w] == [ | |
| '1.0471975511965977', | |
| '1.0471975511965977', | |
| '1.0471975511965977'] | |
| x, w = gauss_chebyshev_t(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.92387953251128676', | |
| '0.38268343236508977', | |
| '-0.38268343236508977', | |
| '-0.92387953251128676'] | |
| assert [str(r) for r in w] == [ | |
| '0.78539816339744831', | |
| '0.78539816339744831', | |
| '0.78539816339744831', | |
| '0.78539816339744831'] | |
| x, w = gauss_chebyshev_t(5, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.95105651629515357', | |
| '0.58778525229247313', | |
| '0', | |
| '-0.58778525229247313', | |
| '-0.95105651629515357'] | |
| assert [str(r) for r in w] == [ | |
| '0.62831853071795865', | |
| '0.62831853071795865', | |
| '0.62831853071795865', | |
| '0.62831853071795865', | |
| '0.62831853071795865'] | |
| def test_chebyshev_t_precise(): | |
| x, w = gauss_chebyshev_t(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '0.8660254037844386467637231707529361834714', | |
| '0', | |
| '-0.8660254037844386467637231707529361834714'] | |
| assert [str(r) for r in w] == [ | |
| '1.047197551196597746154214461093167628066', | |
| '1.047197551196597746154214461093167628066', | |
| '1.047197551196597746154214461093167628066'] | |
| def test_chebyshev_u(): | |
| x, w = gauss_chebyshev_u(1, 17) | |
| assert [str(r) for r in x] == ['0'] | |
| assert [str(r) for r in w] == ['1.5707963267948966'] | |
| x, w = gauss_chebyshev_u(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.50000000000000000', | |
| '-0.50000000000000000'] | |
| assert [str(r) for r in w] == [ | |
| '0.78539816339744831', | |
| '0.78539816339744831'] | |
| x, w = gauss_chebyshev_u(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.70710678118654752', | |
| '0', | |
| '-0.70710678118654752'] | |
| assert [str(r) for r in w] == [ | |
| '0.39269908169872415', | |
| '0.78539816339744831', | |
| '0.39269908169872415'] | |
| x, w = gauss_chebyshev_u(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.80901699437494742', | |
| '0.30901699437494742', | |
| '-0.30901699437494742', | |
| '-0.80901699437494742'] | |
| assert [str(r) for r in w] == [ | |
| '0.21707871342270599', | |
| '0.56831944997474231', | |
| '0.56831944997474231', | |
| '0.21707871342270599'] | |
| x, w = gauss_chebyshev_u(5, 17) | |
| assert [str(r) for r in x] == [ | |
| '0.86602540378443865', | |
| '0.50000000000000000', | |
| '0', | |
| '-0.50000000000000000', | |
| '-0.86602540378443865'] | |
| assert [str(r) for r in w] == [ | |
| '0.13089969389957472', | |
| '0.39269908169872415', | |
| '0.52359877559829887', | |
| '0.39269908169872415', | |
| '0.13089969389957472'] | |
| def test_chebyshev_u_precise(): | |
| x, w = gauss_chebyshev_u(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '0.7071067811865475244008443621048490392848', | |
| '0', | |
| '-0.7071067811865475244008443621048490392848'] | |
| assert [str(r) for r in w] == [ | |
| '0.3926990816987241548078304229099378605246', | |
| '0.7853981633974483096156608458198757210493', | |
| '0.3926990816987241548078304229099378605246'] | |
| def test_jacobi(): | |
| x, w = gauss_jacobi(1, Rational(-1, 2), S.Half, 17) | |
| assert [str(r) for r in x] == ['0.50000000000000000'] | |
| assert [str(r) for r in w] == ['3.1415926535897932'] | |
| x, w = gauss_jacobi(2, Rational(-1, 2), S.Half, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.30901699437494742', | |
| '0.80901699437494742'] | |
| assert [str(r) for r in w] == [ | |
| '0.86831485369082398', | |
| '2.2732777998989693'] | |
| x, w = gauss_jacobi(3, Rational(-1, 2), S.Half, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.62348980185873353', | |
| '0.22252093395631440', | |
| '0.90096886790241913'] | |
| assert [str(r) for r in w] == [ | |
| '0.33795476356635433', | |
| '1.0973322242791115', | |
| '1.7063056657443274'] | |
| x, w = gauss_jacobi(4, Rational(-1, 2), S.Half, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.76604444311897804', | |
| '-0.17364817766693035', | |
| '0.50000000000000000', | |
| '0.93969262078590838'] | |
| assert [str(r) for r in w] == [ | |
| '0.16333179083642836', | |
| '0.57690240318269103', | |
| '1.0471975511965977', | |
| '1.3541609083740761'] | |
| x, w = gauss_jacobi(5, Rational(-1, 2), S.Half, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.84125353283118117', | |
| '-0.41541501300188643', | |
| '0.14231483827328514', | |
| '0.65486073394528506', | |
| '0.95949297361449739'] | |
| assert [str(r) for r in w] == [ | |
| '0.090675770007435372', | |
| '0.33391416373675607', | |
| '0.65248870981926643', | |
| '0.94525424081394926', | |
| '1.1192597692123861'] | |
| x, w = gauss_jacobi(1, 2, 3, 17) | |
| assert [str(r) for r in x] == ['0.14285714285714286'] | |
| assert [str(r) for r in w] == ['1.0666666666666667'] | |
| x, w = gauss_jacobi(2, 2, 3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.24025307335204215', | |
| '0.46247529557426437'] | |
| assert [str(r) for r in w] == [ | |
| '0.48514624517838660', | |
| '0.58152042148828007'] | |
| x, w = gauss_jacobi(3, 2, 3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.46115870378089762', | |
| '0.10438533038323902', | |
| '0.62950064612493132'] | |
| assert [str(r) for r in w] == [ | |
| '0.17937613502213266', | |
| '0.61595640991147154', | |
| '0.27133412173306246'] | |
| x, w = gauss_jacobi(4, 2, 3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.59903470850824782', | |
| '-0.14761105199952565', | |
| '0.32554377081188859', | |
| '0.72879429738819258'] | |
| assert [str(r) for r in w] == [ | |
| '0.067809641836772187', | |
| '0.38956404952032481', | |
| '0.47995970868024150', | |
| '0.12933326662932816'] | |
| x, w = gauss_jacobi(5, 2, 3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-0.69045775012676106', | |
| '-0.32651993134900065', | |
| '0.082337849552034905', | |
| '0.47517887061283164', | |
| '0.79279429464422850'] | |
| assert [str(r) for r in w] == [ | |
| '0.027410178066337099', | |
| '0.21291786060364828', | |
| '0.43908437944395081', | |
| '0.32220656547221822', | |
| '0.065047683080512268'] | |
| def test_jacobi_precise(): | |
| x, w = gauss_jacobi(3, Rational(-1, 2), S.Half, 40) | |
| assert [str(r) for r in x] == [ | |
| '-0.6234898018587335305250048840042398106323', | |
| '0.2225209339563144042889025644967947594664', | |
| '0.9009688679024191262361023195074450511659'] | |
| assert [str(r) for r in w] == [ | |
| '0.3379547635663543330553835737094171534907', | |
| '1.097332224279111467485302294320899710461', | |
| '1.706305665744327437921957515249186020246'] | |
| x, w = gauss_jacobi(3, 2, 3, 40) | |
| assert [str(r) for r in x] == [ | |
| '-0.4611587037808976179121958105554375981274', | |
| '0.1043853303832390210914918407615869143233', | |
| '0.6295006461249313240934312425211234110769'] | |
| assert [str(r) for r in w] == [ | |
| '0.1793761350221326596137764371503859752628', | |
| '0.6159564099114715430909548532229749439714', | |
| '0.2713341217330624639619353762933057474325'] | |
| def test_lobatto(): | |
| x, w = gauss_lobatto(2, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1', | |
| '1'] | |
| assert [str(r) for r in w] == [ | |
| '1.0000000000000000', | |
| '1.0000000000000000'] | |
| x, w = gauss_lobatto(3, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1', | |
| '0', | |
| '1'] | |
| assert [str(r) for r in w] == [ | |
| '0.33333333333333333', | |
| '1.3333333333333333', | |
| '0.33333333333333333'] | |
| x, w = gauss_lobatto(4, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1', | |
| '-0.44721359549995794', | |
| '0.44721359549995794', | |
| '1'] | |
| assert [str(r) for r in w] == [ | |
| '0.16666666666666667', | |
| '0.83333333333333333', | |
| '0.83333333333333333', | |
| '0.16666666666666667'] | |
| x, w = gauss_lobatto(5, 17) | |
| assert [str(r) for r in x] == [ | |
| '-1', | |
| '-0.65465367070797714', | |
| '0', | |
| '0.65465367070797714', | |
| '1'] | |
| assert [str(r) for r in w] == [ | |
| '0.10000000000000000', | |
| '0.54444444444444444', | |
| '0.71111111111111111', | |
| '0.54444444444444444', | |
| '0.10000000000000000'] | |
| def test_lobatto_precise(): | |
| x, w = gauss_lobatto(3, 40) | |
| assert [str(r) for r in x] == [ | |
| '-1', | |
| '0', | |
| '1'] | |
| assert [str(r) for r in w] == [ | |
| '0.3333333333333333333333333333333333333333', | |
| '1.333333333333333333333333333333333333333', | |
| '0.3333333333333333333333333333333333333333'] | |