Rule: Q[4, 86]

Type

This rule is of Laguerre type.

Polynomials

One starts from the polynomial P1 and extends one by one with Pi until the final polynomial defining this rule is reached.
Note: Polynomials might be truncated due to size! The complete polynomials are included in the download.

Node locations in the complex plane

Plot of the node locations in the complex plane.

Magnitude of the Weights

Logarithmic magnitude of the weights
         plotted against the (real part) of the nodes.

Quadrature Rule Data

Nodes

Node (ball center) Error (ball radius)
309.47767885411761802 + 4.5468388723550336719e-4119j 3.37e-2012, 3.37e-2012j
294.39128687783059348 - 1.1988952355572968615e-4118j 1.58e-2010, 1.58e-2010j
258.8100775325624023 + 6.4760907164145993022e-4110j 8.22e-2007, 8.22e-2007j
269.56916983796216997 - 5.288927144883034765e-4112j 7.35e-2008, 7.35e-2008j
31.016134737518975915 - 6.112573447664589107e-4115j 4.38e-2008, 4.38e-2008j
53.891953037057572862 - 5.7263638034058617576e-4109j 1.02e-2002, 1.02e-2002j
103.42744082579766403 + 3.0098905284563911714e-4104j 1.23e-1998, 1.23e-1998j
155.50443735834603123 - 2.1902540450967457405e-4111j 3.12e-1999, 3.12e-1999j
129.76075552164099949 + 2.3651692417967234126e-4131j 1.57e-1998, 1.57e-1998j
239.46964934876992241 + 1.7133643885430836856e-4148j 5.85e-2005, 5.85e-2005j
17.630384347516829533 + 1.1813037351757198409e-4173j 9.51e-2014, 9.51e-2014j
281.31284774173122544 + 5.0521566302275583173e-4157j 4.32e-2009, 4.32e-2009j
39.308834796837968593 - 6.1800368823068646346e-4181j 1.1e-2005, 1.1e-2005j
43.875839770335278933 + 1.0713494787814751104e-4177j 1.36e-2004, 1.36e-2004j
27.276996508468289922 - 7.6290621614868134016e-4184j 2.16e-2009, 2.16e-2009j
120.48275140626474321 + 5.1265568806129793427e-4172j 1.84e-1998, 1.84e-1998j
19.076559248498669399 + 7.0930701840315703807e-4224j 6.01e-2013, 6.01e-2013j
68.171026506942839153 - 5.2148263469175835605e-4213j 8.08e-2001, 8.08e-2001j
125.05517930279741352 + 9.8284871137974059039e-4267j 1.82e-1998, 1.82e-1998j
328.31497468817747741 + 4.6584367675264061005e-4359j 2.23e-2014, 2.23e-2014j
185.7351116622621273 + 2.3440507375303353591e-4467j 8.35e-2001, 8.35e-2001j
32.986357056198661944 - 1.5215057282487539841e-4535j 1.88e-2007, 1.88e-2007j
22.162264314698685725 + 2.667043774358762548e-4636j 1.79e-2011, 1.79e-2011j
173.0153850367891985 + 6.9774538525872772245e-4840j 4.84e-2000, 4.84e-2000j
214.32604016551082004 - 8.5467111466684465525e-5034j 7.82e-2003, 7.82e-2003j
25.506767374609027608 - 1.0800669675475166232e-5198j 4.55e-2010, 4.55e-2010j
139.59447870123302417 - 7.2289793563405246169e-5426j 1.08e-1998, 1.08e-1998j
91.818732958514223411 - 1.5498156197316849126e-5668j 4.69e-1999, 4.69e-1999j
144.73564356067407072 + 1.8045437389877802458e-5829j 7.78e-1999, 7.78e-1999j
37.131866448051736111 + 3.4034712021748063785e-5893j 2.91e-2006, 2.91e-2006j
166.98284276060599125 - 2.8682073916070613575e-5934j 1.02e-1999, 1.02e-1999j
13.683479078224191284 + 3.7072329868687077794e-5986j 4.36e-2016, 4.36e-2016j
3.2602953004851743777 - 1.5523107860602317861e-5997j 1.2e-2027, 1.2e-2027j
62.214750193102629251 + 4.6186820528942822751e-5968j 1.64e-2001, 1.64e-2001j
116.03831149193600772 - 1.1298433847766882966e-5988j 1.96e-1998, 1.96e-1998j
71.276499756930710329 - 4.1290820104869286988e-6008j 1.67e-2000, 1.67e-2000j
6.0063451732499199476 + 4.1350652776455801056e-6042j 3.52e-2023, 3.52e-2023j
1.014467512264252937 + 2.2881302157710157316e-6052j 1.56e-2033, 1.56e-2033j
29.113244606431013903 + 2.887112827697483699e-6027j 1e-2008, 1e-2008j
59.360508941175456275 + 7.2649184019542221728e-6018j 6.76e-2002, 6.76e-2002j
150.03616704135380641 + 1.11035003023868029e-6015j 5.19e-1999, 5.19e-1999j
161.14982252049638254 + 2.6705473850492204611e-6037j 1.8e-1999, 1.8e-1999j
230.64825979438835108 + 8.3841295676976581185e-6051j 3.38e-2004, 3.38e-2004j
77.751531125455833803 - 2.1552594158388018401e-6053j 6.29e-2000, 6.29e-2000j
179.26097304948663968 - 9.1669885196347196646e-6052j 2.11e-2000, 2.11e-2000j
11.393723925182386094 + 2.6792783017552101412e-6081j 1.02e-2017, 1.02e-2017j
99.451209974547710351 + 3.130206834365638705e-6061j 9.6e-1999, 9.6e-1999j
5.24917212132470625 + 7.8324485529839305732e-6089j 3.02e-2024, 3.02e-2024j
0.50562192283083319978 - 5.9220585139935681922e-6101j 5.86e-2036, 5.86e-2036j
2.1957364691994057984 - 6.7356483720736407848e-6095j 5.7e-2030, 5.7e-2030j
41.55649682386609951 - 1.9599893048659267989e-6069j 4.38e-2005, 4.38e-2005j
23.802009578543435641 + 1.5832835428184233023e-6075j 9.3e-2011, 9.3e-2011j
81.125220320573785812 - 1.2369129916650330232e-6064j 1.21e-1999, 1.21e-1999j
48.73382459223792585 - 6.5113717748457027557e-6068j 1.32e-2003, 1.32e-2003j
56.58673755523699827 + 1.2996373849225506873e-6066j 2.67e-2002, 2.67e-2002j
20.587182727930308557 - 4.8181695888830110204e-6077j 3.24e-2012, 3.24e-2012j
74.469334376554250111 - 2.9528985558867721013e-6064j 3.23e-2000, 3.23e-2000j
95.582706530508906684 - 7.2135889372619348449e-6063j 6.92e-1999, 6.92e-1999j
88.156313350845912549 + 2.6342090108223011531e-6062j 3.06e-1999, 3.06e-1999j
0.7335800468594778107 + 5.9349488905589116823e-6100j 1.04e-2034, 1.04e-2034j
14.932868611156513629 - 1.5127524496546875709e-6080j 2.73e-2015, 2.73e-2015j
84.59267298315571037 - 2.1167882516006325172e-6063j 2.02e-1999, 2.02e-1999j
248.82376974200564179 - 1.0333123205976782341e-6070j 7.75e-2006, 7.75e-2006j
10.358505528625318161 - 1.9687989700089024918e-6083j 1.72e-2018, 1.72e-2018j
4.5366202969211279833 + 2.2207039461951072612e-6090j 2.16e-2025, 2.16e-2025j
3.8727785520965323331 - 2.4119736049075638813e-6091j 1.58e-2026, 1.58e-2026j
192.45573279059095142 - 4.4495112091242243849e-6063j 2.93e-2001, 2.93e-2001j
111.7171262987350428 + 9.0975811381614156351e-6065j 1.73e-1998, 1.73e-1998j
65.151033470920244776 + 3.0930085119593149513e-6076j 3.94e-2001, 3.94e-2001j
1.7457611011583465757 + 3.8160501587291437536e-6106j 3.76e-2031, 3.76e-2031j
0.3225476896193923118 + 7.6064719063257028066e-6112j 4.04e-2037, 4.04e-2037j
12.502826912416793476 + 3.9094394590504632883e-6092j 6.48e-2017, 6.48e-2017j
16.248927939045506543 - 4.6621612574605650127e-6089j 1.72e-2014, 1.72e-2014j
206.72400822877565013 + 6.3895182867981621297e-6073j 2.92e-2002, 2.92e-2002j
51.274755809759245642 + 1.3050415943774654232e-6089j 3.94e-2003, 3.94e-2003j
1.351780020056830361 - 2.5581557538642455585e-6119j 2.51e-2032, 2.51e-2032j
6.802351780341720644 + 1.7869808003859576629e-6108j 3.93e-2022, 3.93e-2022j
134.60512394262183236 - 1.0207160455571698796e-6083j 1.3e-1998, 1.3e-1998j
222.28591071446732159 + 1.792247599691827231e-6095j 1.78e-2003, 1.78e-2003j
7.6311390428973281875 - 4.168877930591191514e-6123j 3.63e-2021, 3.63e-2021j
0.073860058705502177581 - 7.1197701934408856646e-6142j 5.52e-2040, 5.52e-2040j
2.7008872471622632096 - 2.9429710990092898836e-6131j 7.91e-2029, 7.91e-2029j
199.44378240745629966 + 2.3317632349541885739e-6100j 1.06e-2001, 1.06e-2001j
9.3950709123011331292 + 4.653074503417247885e-6127j 2.41e-2019, 2.41e-2019j
0.014159778078285904001 - 5.5728326249818275108e-6230j 9.02e-2042, 9.02e-2042j
46.267911846490828168 + 2.7854335338979382044e-6111j 4.18e-2004, 4.18e-2004j
8.4918235408483385477 - 2.5787144498547471205e-6127j 3.56e-2020, 3.56e-2020j
0.17815438432706083391 - 2.4635248367967151978e-6145j 1.6e-2038, 1.6e-2038j
35.024663779887878055 + 1.7713752709812982729e-6113j 7.7e-2007, 7.7e-2007j
107.51484156180690548 + 3.1882921154369223726e-6104j 1.5e-1998, 1.5e-1998j

Weights

Weight (ball center) Error (ball radius)
6.4984519592749931374e-134 - 1.1933843950111321016e-4190j 1.17e-378, 2.77e-1881j
1.9556626844098186856e-127 + 2.3745728347564354777e-4187j 7.6e-377, 1.8e-1879j
4.1203075284778552854e-112 - 1.5577554363147820848e-4179j 7.56e-373, 1.79e-1875j
9.4867448425376665695e-117 + 6.672549101517552544e-4182j 3.63e-374, 8.58e-1877j
6.5594429603503603108e-14 + 7.7402107773501431016e-4128j 2.54e-297, 6e-1800j
1.0452296116688247771e-23 + 3.1717617756927434085e-4132j 9.51e-318, 2.25e-1820j
4.8693356382524137208e-45 + 9.1892454348592070042e-4145j 7.52e-342, 1.78e-1844j
1.621731026757462786e-67 - 1.3663460743427390151e-4156j 7.34e-357, 1.74e-1859j
2.1110632897772187788e-56 + 8.7515697932255760442e-4151j 2.1e-350, 4.97e-1853j
9.0664399626530493876e-104 - 2.8741141226561110706e-4175j 8.25e-372, 1.95e-1874j
3.1162825218511328261e-08 + 8.2101824176517352592e-4125j 3.45e-284, 8.15e-1787j
8.2896578381606880347e-122 - 1.7524686752464650879e-4184j 1.1e-376, 2.6e-1879j
1.875882513834137968e-17 + 8.9490502951483668294e-4130j 3.72e-311, 8.8e-1814j
2.0751672268227694365e-19 + 1.1674667391840924975e-4130j 6.75e-315, 1.6e-1817j
2.5690972976516467086e-12 + 4.6263039009061094949e-4127j 8.37e-300, 1.98e-1802j
2.1328108729203136343e-52 + 1.1255541705246159071e-4148j 2.72e-349, 6.43e-1852j
7.6727786322404260591e-09 - 3.5810688622283584138e-4125j 2.84e-290, 6.72e-1793j
7.5816792710982845742e-30 + 2.4436531091409235365e-4136j 9.94e-332, 2.35e-1834j
2.2675984631180807436e-54 - 1.0244824808687854532e-4149j 2.28e-350, 5.39e-1853j
5.7552415684015129006e-142 + 9.5582325271387091168e-4195j 6.67e-387, 1.58e-1889j
1.4302947707728477283e-80 + 2.3755316508246433303e-4163j 2.58e-366, 6.1e-1869j
9.4652903419203629343e-15 - 1.348158963967836508e-4128j 1.32e-310, 3.12e-1813j
3.8121118913530803175e-10 - 6.4928817756761632291e-4126j 6.37e-300, 1.51e-1802j
4.4494362443696705417e-75 + 1.6363246934239314954e-4160j 7.45e-364, 1.76e-1866j
6.45722100953366171e-93 + 1.0520308264236165128e-4169j 9.81e-372, 2.32e-1874j
1.4536295941644237161e-11 - 1.1157987954228647978e-4126j 7.55e-306, 1.78e-1808j
1.2005700849450252823e-60 + 5.2132266855837243492e-4153j 1e-356, 2.37e-1859j
4.9354062341820109243e-40 - 4.6129566883569648736e-4142j 2.91e-346, 6.88e-1849j
7.24001155263366249e-63 - 3.6101609192517264402e-4154j 4.13e-358, 9.76e-1861j
1.6019233488903629244e-16 - 2.4349262836279232056e-4129j 6.9e-320, 1.63e-1822j
1.7921603017352444258e-72 - 3.6545144226223126641e-4159j 2.19e-363, 5.18e-1866j
1.3869907620474380507e-06 - 8.97372481964173171e-4124j 4.91e-299, 1.16e-1801j
0.02249572736675078549 - 1.9300989256262196611e-4121j 4.77e-267, 1.13e-1769j
2.7677364992159238755e-27 + 9.2813618781690106259e-4135j 2.34e-336, 5.53e-1839j
1.7655483939467810008e-50 - 1.1639257696874032053e-4147j 1.71e-352, 4.04e-1855j
3.4926798992668966005e-31 - 4.0103968070187088222e-4137j 1.62e-340, 3.84e-1843j
0.0019154741938492146173 - 6.6523677630888905618e-4122j 1.65e-286, 3.91e-1789j
0.11200392217592388393 + 5.8873884602912466443e-4121j 1.01e-249, 2.39e-1752j
4.2461425509122597426e-13 - 2.0574284099921393778e-4127j 2.8e-316, 6.62e-1819j
4.6702951635427759463e-26 - 6.25941826677932933e-4134j 2.36e-335, 5.59e-1838j
3.7251064186080162723e-65 + 2.3137112621914080285e-4155j 2.9e-360, 6.86e-1863j
5.9187795450315745972e-70 + 7.3991520593526557207e-4158j 4.87e-363, 1.15e-1865j
5.8114310722143829098e-100 + 2.5591261868872454157e-4173j 7.46e-379, 1.76e-1881j
5.6894087006000562116e-34 - 1.0259903561329268296e-4138j 8.07e-346, 1.91e-1848j
8.9373178346097243476e-78 - 6.5965460836764427888e-4162j 4.96e-368, 1.17e-1870j
1.208128629006454487e-05 - 3.9746430119981501977e-4123j 6.47e-310, 1.53e-1812j
2.5256627157084192149e-43 - 7.6554085700719999851e-4144j 8.36e-352, 1.98e-1854j
0.0038637388624816513237 + 8.785897013883569397e-4122j 5.83e-298, 1.38e-1800j
0.12302467093902870368 + 6.6349715247204567745e-4121j 1.16e-271, 2.75e-1774j
0.053157296393166548446 - 3.1454252237070665404e-4121j 5.94e-284, 1.4e-1786j
2.0455588714956623376e-18 - 3.2303938058780875446e-4130j 1.54e-331, 3.65e-1834j
7.6948911084155831992e-11 + 2.7044072850649532935e-4126j 3.97e-321, 9.39e-1824j
2.0036111839342951293e-35 + 1.5762172366948693919e-4139j 1.61e-347, 3.81e-1850j
1.7126472307130607327e-21 + 1.7458379341655782213e-4131j 2.16e-336, 5.11e-1839j
7.2693681583882170397e-25 + 5.4139544303076334684e-4133j 4.58e-339, 1.08e-1841j
1.767742095652514699e-09 + 1.5369215318672084853e-4125j 4.67e-322, 1.1e-1824j
1.4741604480236652287e-32 + 6.488342367342954709e-4138j 1.94e-345, 4.58e-1848j
1.1763458668643561263e-41 + 6.079157268445688242e-4143j 2.08e-352, 4.91e-1855j
1.8707572811327016306e-38 + 3.3534778612452669651e-4141j 9.57e-351, 2.26e-1853j
0.12168468213217270391 - 6.6419704345269491017e-4121j 2.57e-295, 6.08e-1798j
4.1972340774688092064e-07 + 4.1121386992850786338e-4124j 1.06e-324, 2.5e-1827j
6.4240688300643526205e-37 - 2.3420320367647696945e-4140j 9.86e-350, 2.33e-1852j
8.3517390250972667967e-108 + 2.4762894599786359588e-4177j 2.22e-388, 5.24e-1891j
3.1667993221336848302e-05 + 7.8360793726251304154e-4123j 5.71e-324, 1.35e-1826j
0.0073760946333451261027 - 1.1496474329603368547e-4121j 6.91e-316, 1.63e-1818j
0.013281335502765712398 + 1.4944773863930216302e-4121j 1.75e-314, 4.13e-1817j
1.7916912767207263844e-83 - 7.5730285952136019706e-4165j 7.83e-378, 1.85e-1880j
1.2923614366955093879e-48 + 1.1359610546699568646e-4146j 3.42e-359, 8.09e-1862j
1.5106670860700785144e-28 - 1.4880362752823704538e-4135j 1.04e-346, 2.45e-1849j
0.073663356041031515087 + 3.9562292048681359889e-4121j 1.6e-314, 3.79e-1817j
0.11831230903758622449 - 5.5300200477969884556e-4121j 2.37e-311, 5.61e-1814j
4.2563904401794454197e-06 + 1.9177029027202737842e-4123j 7.06e-329, 1.67e-1831j
1.1834892643053149119e-07 - 1.8522922521004910726e-4124j 1.49e-330, 3.51e-1833j
1.2365930320775493342e-89 - 5.1089570980392554548e-4168j 1.25e-381, 2.94e-1884j
1.3902170892718873762e-22 - 9.0364996219804335504e-4132j 5.26e-345, 1.24e-1847j
0.094643247748468309119 - 4.8983843811696705178e-4121j 1.86e-320, 4.41e-1823j
0.00090363445804891383233 + 4.9450506676350865841e-4122j 2.26e-329, 5.33e-1832j
1.7113145507917354358e-58 - 6.9946988778534018491e-4152j 3.7e-365, 8.75e-1868j
2.3646286334912097524e-96 - 1.81347927097649977e-4171j 4.64e-385, 1.1e-1887j
0.0004094516858691375628 - 3.5361966084478612235e-4122j 2.48e-333, 5.85e-1836j
0.076794549548254085259 - 1.8963733699254499197e-4121j 4.18e-327, 9.89e-1830j
0.035751649503190653185 + 2.4744568833831393601e-4121j 7.66e-329, 1.81e-1831j
1.7208141032880184185e-86 + 2.1147585516886909228e-4166j 3.98e-380, 9.41e-1883j
7.735842139483023233e-05 - 1.4290175317921369186e-4122j 6.21e-335, 1.47e-1837j
0.035745198319695680038 + 5.3574808275081984055e-4122j 2.31e-330, 5.46e-1833j
1.9565604080110812949e-20 - 4.3352873425104346383e-4131j 3.51e-346, 8.3e-1849j
0.00018035409376689926807 + 2.3588412316117999097e-4122j 4.43e-335, 1.05e-1837j
0.1046659771310490463 + 3.7097792438490531316e-4121j 4.86e-332, 1.15e-1834j
1.2749710910614933672e-15 + 6.3134958479066083197e-4129j 1.74e-343, 4.11e-1846j
8.4011406081327295827e-47 - 1.0490831349668757709e-4145j 2.24e-360, 5.3e-1863j

Download

All rules of Laguerre type can be downloaded here packed into a single file.