Rule: Q[1, 6, 12, 78]

Type

This rule is of Hermite (probabilists') 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)
14.984531579537664168 - 6.0075687230805455233e-2844j 8.45e-1002, 8.45e-1002j
17.711708919574111926 + 1.2343858214307634329e-2848j 2.11e-1006, 2.11e-1006j
8.5050960359709134305 + 3.2489888108714954216e-2841j 1.45e-999, 1.45e-999j
15.554594710866501729 - 3.8398329674500343794e-2845j 1.32e-1002, 1.32e-1002j
6.3515005081010183957 + 5.9186368162086422077e-2844j 2.71e-1001, 2.71e-1001j
14.449858647745422888 - 6.1358348660172163252e-2844j 3.64e-1001, 3.64e-1001j
1.4767692380455223972 + 2.7996555211597047632e-2856j 4.09e-1014, 4.09e-1014j
5.2597437027142124159 - 8.8110168955709358699e-2845j 9.25e-1003, 9.25e-1003j
2.3667594107345412886 + 1.1887000996850227615e-2851j 1.46e-1009, 1.46e-1009j
13.456264760990632416 + 7.0271701393887546129e-2841j 3.59e-1000, 3.59e-1000j
3.2792325148601542428 + 8.5745853848773052677e-2851j 4.51e-1007, 4.51e-1007j
3.5585535513977715865 + 2.1377616734600473006e-2849j 4.19e-1006, 4.19e-1006j
5.7587833040149909341 + 3.8685259823705248669e-2844j 1.22e-1001, 1.22e-1001j
6.6975325765071699672 + 1.1061470991528334929e-2843j 5.59e-1001, 5.59e-1001j
16.173918051193330197 + 2.1980566512450778114e-2846j 1.49e-1003, 1.49e-1003j
2.9944098549047408593 + 9.4053258602797619752e-2851j 6.75e-1008, 6.75e-1008j
4.9318879021050250625 + 6.7606959151758150661e-2847j 1.59e-1003, 1.59e-1003j
9.6435380617639221 - 7.0672899803121453276e-2841j 4.69e-999, 4.69e-999j
13.942261367617830268 + 8.356714697245798591e-2845j 1.27e-1000, 1.27e-1000j
0.51169598081006329412 - 6.0801053581722587049e-2865j 2.53e-1018, 2.53e-1018j
2.7166491645327282608 + 8.5053591534355859136e-2856j 1.09e-1008, 1.09e-1008j
1.7959546094993769997 - 1.5712110784881483035e-2860j 1.09e-1012, 1.09e-1012j
7.0502571090166917693 - 5.5684928500173812227e-2848j 1.19e-1000, 1.19e-1000j
16.869435742022726545 - 1.489491118276979604e-2852j 9.59e-1005, 9.59e-1005j
4.2756785879284246907 + 2.5742063656066902062e-2851j 9.12e-1005, 9.12e-1005j
7.4076946023773519875 + 6.709180713976874772e-2849j 2.35e-1000, 2.35e-1000j
10.430355608785004103 - 8.2369830751644666014e-2845j 6.46e-999, 6.46e-999j
1.1544053947399681272 + 5.3389208867870298512e-2865j 1.35e-1015, 1.35e-1015j
5.5561100339252269112 - 4.3379656884402291344e-2851j 5.02e-1002, 5.02e-1002j
8.135080633780030714 - 1.9902026851856675489e-2848j 8.85e-1000, 8.85e-1000j
3.7504397177257422563 + 1.6006375295259561115e-2854j 1.53e-1005, 1.53e-1005j
6.0213238265065084814 - 3.1107674171208340975e-2850j 1.79e-1001, 1.79e-1001j
3.969025615860502215 - 3.7380058242077335449e-2854j 3.24e-1005, 3.24e-1005j
4.6012893561154538599 - 4.4968588956564933802e-2853j 3.5e-1004, 3.5e-1004j
0.83152531436927173774 - 1.1358209823278880814e-2866j 5.01e-1017, 5.01e-1017j
8.879614408850154648 - 3.6416690173572892768e-2847j 2.26e-999, 2.26e-999j
10.033816869113836289 - 2.2480054902541275316e-2852j 5.43e-999, 5.43e-999j
-0.51169598081006329412 + 3.4677446469668950219e-2888j 2.78e-1018, 2.78e-1018j
0.20974885121610866756 + 9.7341771468342417226e-2889j 1.34e-1019, 1.34e-1019j
9.2589619533377983269 - 2.2098830357254033749e-2867j 3.29e-999, 3.29e-999j
-0.20974885121610866756 + 2.3456787241324086403e-2919j 1.34e-1019, 1.34e-1019j
2.1069178697540078011 + 1.5619659963064825004e-2902j 4.38e-1011, 4.38e-1011j
2.4600217170137661289 + 2.7065504807326414004e-2899j 3.61e-1009, 3.61e-1009j
7.7693241029372297341 + 9.5290700728030188253e-2902j 4.7e-1000, 4.7e-1000j
-1.146955009950512223e-2945 - 2.5676149323690784765e-2945j 2.73e-2943, 2.73e-2943j
-12.988024345400805771 - 4.1957400564828008457e-2952j 8.46e-1000, 8.46e-1000j
12.988024345400805771 + 1.0224552639617748965e-2991j 8.33e-1000, 8.33e-1000j
-12.09416675977880667 - 4.6416428944989509048e-3013j 2.65e-999, 2.65e-999j
-17.711708919574111926 - 8.63780869040613475e-3034j 2.39e-1006, 2.39e-1006j
-14.984531579537664168 - 9.8403683325950107811e-3044j 8.29e-1002, 8.29e-1002j
-12.534710944079195814 - 4.7132532024979858833e-3069j 1.64e-999, 1.64e-999j
-8.135080633780030714 - 7.0591032300148720853e-3090j 8.52e-1000, 8.52e-1000j
-13.456264760990632416 + 4.1029238600318917813e-3110j 3.69e-1000, 3.69e-1000j
-7.4076946023773519875 + 1.0430862302624820209e-3142j 2.38e-1000, 2.38e-1000j
-8.879614408850154648 - 3.8774752397002438132e-3171j 2.38e-999, 2.38e-999j
12.09416675977880667 + 1.1020648822621883943e-3188j 2.7e-999, 2.7e-999j
-3.5585535513977715865 - 3.3118678504729402841e-3218j 3.99e-1006, 3.99e-1006j
-2.4600217170137661289 - 2.1053760505912108661e-3224j 3.7e-1009, 3.7e-1009j
12.534710944079195814 - 4.0823776756709745669e-3214j 1.65e-999, 1.65e-999j
-11.244949828616738723 + 1.7702452996434830027e-3236j 5.04e-999, 5.04e-999j
-7.0502571090166917693 + 6.3560475442803061554e-3249j 1.1e-1000, 1.1e-1000j
-3.7504397177257422563 + 2.0259802490404806898e-3255j 1.47e-1005, 1.47e-1005j
-8.5050960359709134305 - 8.1780209399444694342e-3249j 1.42e-999, 1.42e-999j
11.664699111597654265 + 2.5123752394087592749e-3258j 4.26e-999, 4.26e-999j
-7.7693241029372297341 + 6.6913135005745937169e-3258j 4.39e-1000, 4.39e-1000j
-13.942261367617830268 - 2.0065433211488671321e-3270j 1.26e-1000, 1.26e-1000j
10.833808928849675222 - 3.7648853733292526182e-3284j 6.62e-999, 6.62e-999j
-15.554594710866501729 + 5.821126406863705124e-3292j 1.32e-1002, 1.32e-1002j
-14.449858647745422888 + 5.2573144222744474675e-3313j 3.71e-1001, 3.71e-1001j
-16.869435742022726545 + 1.0221099411345383462e-3324j 9.33e-1005, 9.33e-1005j
-4.9318879021050250625 - 1.8502664856178781585e-3336j 1.53e-1003, 1.53e-1003j
-5.7587833040149909341 - 1.0705231573835990115e-3341j 1.09e-1001, 1.09e-1001j
-16.173918051193330197 + 1.0212697669571755428e-3381j 1.42e-1003, 1.42e-1003j
-10.033816869113836289 + 9.4441966997908756928e-3421j 5.31e-999, 5.31e-999j
-6.3515005081010183957 - 2.4544295843434363798e-3452j 2.74e-1001, 2.74e-1001j
-6.6975325765071699672 + 3.4621728636233575851e-3483j 5.49e-1001, 5.49e-1001j
-4.6012893561154538599 - 6.0854307804596548563e-3522j 3.81e-1004, 3.81e-1004j
11.244949828616738723 - 3.2171026116626190328e-3578j 5.08e-999, 5.08e-999j
-2.3667594107345412886 - 4.5365156605286481336e-3630j 1.41e-1009, 1.41e-1009j
-2.7166491645327282608 + 2.4592685776216253346e-3624j 1.14e-1008, 1.14e-1008j
-3.2792325148601542428 - 1.5083723867741179855e-3624j 4.64e-1007, 4.64e-1007j
-1.4767692380455223972 + 1.5469332590868460179e-3665j 4.07e-1014, 4.07e-1014j
-10.833808928849675222 + 3.2889097093834974983e-3658j 6.28e-999, 6.28e-999j
-5.5561100339252269112 + 6.0054324768256899779e-3697j 4.75e-1002, 4.75e-1002j
-1.1544053947399681272 + 3.4372277905837782338e-3736j 1.17e-1015, 1.17e-1015j
-0.83152531436927173774 - 1.4292831005068185988e-3737j 5.12e-1017, 5.12e-1017j
-11.664699111597654265 - 2.3079284684037302335e-3718j 3.95e-999, 3.95e-999j
-1.7959546094993769997 - 6.7355599329376740308e-3761j 1.14e-1012, 1.14e-1012j
-10.430355608785004103 - 2.1304008398611038087e-3745j 6.05e-999, 6.05e-999j
-2.9944098549047408593 + 7.3634024089994489682e-3784j 6.61e-1008, 6.61e-1008j
-4.2756785879284246907 + 1.3800786826832295324e-3780j 9.3e-1005, 9.3e-1005j
-9.2589619533377983269 + 1.8179487666929119798e-3774j 3.36e-999, 3.36e-999j
-6.0213238265065084814 + 5.0504422326954662018e-3803j 1.74e-1001, 1.74e-1001j
-9.6435380617639221 + 8.3462744365308336931e-3829j 4.62e-999, 4.62e-999j
-3.969025615860502215 - 7.5979349512655401148e-3850j 3.18e-1005, 3.18e-1005j
-5.2597437027142124159 + 1.1272787476753166236e-3847j 8.27e-1003, 8.27e-1003j
-2.1069178697540078011 + 7.0696633102470910171e-3856j 4.21e-1011, 4.21e-1011j

Weights

Weight (ball center) Error (ball radius)
3.8403672989995792476e-50 - 1.6047232219977790201e-2877j 1.12e-351, 6.54e-851j
2.9482376670899609285e-69 - 6.3554271896897933309e-2888j 2.72e-359, 1.59e-858j
2.9107343684241364442e-17 - 5.3757828594430151366e-2857j 3.12e-330, 1.82e-829j
6.8426064004231168227e-54 + 1.3749639030568373269e-2879j 1.6e-353, 9.34e-853j
2.3637792597722458433e-10 - 9.3463666554916409249e-2853j 8.53e-320, 4.99e-819j
9.4830529563616453189e-47 + 1.2350640091922354576e-2875j 8.33e-351, 4.88e-850j
0.043076319905983359913 - 3.3948489412903183177e-2847j 5.31e-265, 3.11e-764j
1.2608537732623687696e-07 - 8.7302000549339826561e-2851j 2.76e-313, 1.61e-812j
0.0034936719076039939606 + 1.9512231559836237865e-2846j 1.04e-283, 6.11e-783j
9.1193057657073426616e-41 + 2.9156924402685636338e-2872j 6.49e-349, 3.8e-848j
0.00053166431562532378121 + 1.1329879120025178508e-2847j 2.19e-296, 1.28e-795j
0.0001801905302594343597 - 7.9750511717418173077e-2848j 1.78e-299, 1.04e-798j
5.0068677647842951028e-09 - 4.2469330184390836684e-2851j 2.88e-317, 1.69e-816j
2.5371542862823302319e-11 - 1.0126265043063139919e-2853j 3.44e-323, 2.02e-822j
4.0718908682549253542e-58 - 6.7119844034391624801e-2882j 1.49e-357, 8.69e-857j
0.0012630749872770524089 - 2.1405234673291541644e-2847j 9.94e-296, 5.81e-795j
6.9000416589348631216e-07 + 1.8653629200475101614e-2850j 8.79e-313, 5.14e-812j
9.8805956081350559859e-22 - 7.6083557847540601829e-2861j 4.52e-339, 2.64e-838j
1.2187314712167533036e-43 - 6.855894390836467224e-2874j 2.07e-351, 1.21e-850j
0.11051107290059515782 + 5.659722716144895068e-2847j 2.55e-277, 1.49e-776j
0.0027561255034160460408 + 4.971956107948351793e-2847j 9.06e-295, 5.3e-794j
0.025159670016353841084 + 4.0895166736742555484e-2847j 1.24e-283, 7.25e-783j
2.2796794035808725979e-12 - 2.7056602446416404981e-2854j 1.79e-326, 1.05e-825j
4.7922936278680514267e-63 + 1.4143684618944025754e-2884j 1.14e-360, 6.68e-860j
1.369942325433765542e-05 + 3.1365012550028034567e-2849j 8.32e-311, 4.87e-810j
1.7416752753847846357e-13 + 4.1126110632610650756e-2855j 6.12e-329, 3.58e-828j
3.7921519419377724184e-25 - 6.664028574623193477e-2863j 2.38e-343, 1.39e-842j
0.066187468187488072369 + 3.4060524227537799096e-2847j 7.59e-279, 4.44e-778j
1.9858977113191403874e-08 + 9.2008488007500447565e-2851j 7.25e-319, 4.24e-818j
6.2500088993251097073e-16 + 2.0505329715037027749e-2856j 4.01e-333, 2.34e-832j
5.5157274152903263036e-05 + 5.1021300646909214107e-2848j 7.34e-309, 4.29e-808j
1.6698609581812671811e-09 + 5.2141434038190832433e-2852j 7.87e-322, 4.61e-821j
4.2795467650055482692e-05 - 1.6422731692906834321e-2848j 7.43e-311, 4.35e-810j
3.3189557892088967168e-06 - 7.3585694445639073212e-2850j 5.66e-315, 3.31e-814j
0.090975535157811571018 - 3.999530448460427558e-2847j 1.94e-286, 1.14e-785j
1.1365172403660697743e-18 - 2.9519217532189859057e-2858j 4.75e-338, 2.78e-837j
2.1567808983795420237e-23 + 1.0177138201142289988e-2861j 9.82e-344, 5.74e-843j
0.11051107290059515782 + 5.0334499585172746979e-2847j 1.09e-291, 6.38e-791j
0.10746198946079547541 - 1.0387861554448660007e-2846j 2.18e-291, 1.28e-790j
3.6911954006221127703e-20 + 1.5221629309597265176e-2859j 4.6e-340, 2.69e-839j
0.10746198946079547541 - 9.9007398899319834184e-2847j 3.24e-292, 1.9e-791j
0.013128070614597435143 - 6.8751620664621606034e-2847j 5.12e-303, 3e-802j
0.0034420784441346515422 - 1.8505541851272609008e-2846j 2.53e-305, 1.48e-804j
1.1326748644578494764e-14 - 7.9590406420013670955e-2856j 1.33e-333, 7.76e-833j
0.063434508115495452349 + 1.3723986732554171815e-2846j 4.6e-294, 2.69e-793j
4.3024378677206385989e-38 + 1.9927827327869580447e-2871j 6e-377, 3.51e-876j
4.3024378677206385989e-38 - 9.9231187492598212965e-2871j 1.11e-356, 6.5e-856j
3.001050575339669083e-33 + 1.1116329402450934232e-2868j 9.81e-375, 5.74e-874j
2.9482376670899609285e-69 + 2.1860262579392773312e-2888j 8.15e-390, 4.77e-889j
3.8403672989995792476e-50 + 4.3059191761598580359e-2878j 2.56e-383, 1.5e-882j
1.3579662772949004542e-35 - 5.1323699198023335261e-2870j 3.66e-376, 2.14e-875j
6.2500088993251097073e-16 + 4.3786568191412326428e-2858j 6.6e-365, 3.86e-864j
9.1193057657073426616e-41 - 6.3492386726875117022e-2873j 3.54e-379, 2.07e-878j
1.7416752753847846357e-13 + 2.4495955989524605148e-2856j 1.04e-362, 6.07e-862j
1.1365172403660697743e-18 + 6.5198168563630053366e-2860j 7.26e-368, 4.25e-867j
3.001050575339669083e-33 - 6.6845383193241623728e-2868j 4.4e-362, 2.57e-861j
0.0001801905302594343597 - 3.3956070135550117995e-2848j 2.16e-345, 1.27e-844j
0.0034420784441346515422 - 1.0395157342606699795e-2846j 2.56e-336, 1.5e-835j
1.3579662772949004542e-35 + 2.7939963946035232545e-2869j 4.44e-363, 2.6e-862j
5.7705793689224668979e-29 + 3.3440029943835504528e-2866j 2.27e-375, 1.33e-874j
2.2796794035808725979e-12 - 1.7723719942871935339e-2855j 9.96e-363, 5.83e-862j
5.5157274152903263036e-05 + 2.0678223248726130213e-2848j 3.34e-348, 1.95e-847j
2.9107343684241364442e-17 - 5.4966756315481009845e-2859j 2.96e-367, 1.73e-866j
4.8140073216401111735e-31 + 1.3900151180474569186e-2866j 2.67e-363, 1.56e-862j
1.1326748644578494764e-14 - 3.3303677418900179425e-2857j 7.42e-365, 4.34e-864j
1.2187314712167533036e-43 + 1.6083168033395957985e-2874j 1.58e-383, 9.24e-883j
5.2941099596424257697e-27 + 4.2868721978968053496e-2864j 7.52e-362, 4.4e-861j
6.8426064004231168227e-54 - 3.9261769760762435615e-2880j 2.69e-389, 1.57e-888j
9.4830529563616453189e-47 - 3.1049808262912733233e-2876j 1.23e-385, 7.22e-885j
4.7922936278680514267e-63 - 4.5589123092703273617e-2885j 5.55e-394, 3.25e-893j
6.9000416589348631216e-07 + 6.7039441891393241556e-2851j 1.25e-362, 7.34e-862j
5.0068677647842951028e-09 - 4.570517298779083711e-2852j 1.49e-365, 8.71e-865j
4.0718908682549253542e-58 + 2.0361336042694323374e-2882j 7.16e-392, 4.19e-891j
2.1567808983795420237e-23 - 7.0772584231817336515e-2863j 1.73e-377, 1.01e-876j
2.3637792597722458433e-10 - 9.9182172103000333935e-2854j 1.24e-367, 7.23e-867j
2.5371542862823302319e-11 + 1.2936973018248928174e-2854j 3.58e-368, 2.1e-867j
3.3189557892088967168e-06 - 2.4768330162489201959e-2850j 1.12e-363, 6.53e-863j
5.7705793689224668979e-29 - 2.5645827069037811667e-2865j 4.94e-372, 2.89e-871j
0.0034936719076039939606 + 1.1213735599719053818e-2846j 1.61e-355, 9.42e-855j
0.0027561255034160460408 + 2.6213349853192793942e-2847j 1.56e-357, 9.13e-857j
0.00053166431562532378121 + 5.182651784440746109e-2848j 8.21e-360, 4.81e-859j
0.043076319905983359913 - 2.4139026459069239407e-2847j 1.62e-353, 9.47e-853j
5.2941099596424257697e-27 - 4.785044612421394104e-2865j 3.26e-381, 1.91e-880j
1.9858977113191403874e-08 + 1.0598685670827228174e-2851j 9.3e-368, 5.44e-867j
0.066187468187488072369 + 2.6113570304859743482e-2847j 5.17e-354, 3.02e-853j
0.090975535157811571018 - 3.3045052451814006253e-2847j 3.93e-353, 2.3e-852j
4.8140073216401111735e-31 - 2.0650677955550008223e-2867j 4.5e-383, 2.64e-882j
0.025159670016353841084 + 2.696612643507115909e-2847j 1.8e-358, 1.06e-857j
3.7921519419377724184e-25 + 6.1222339923621124654e-2864j 1.47e-380, 8.62e-880j
0.0012630749872770524089 - 1.0526541056202906282e-2847j 1.82e-362, 1.06e-861j
1.369942325433765542e-05 + 1.1170803889736950679e-2849j 1.57e-366, 9.21e-866j
3.6911954006221127703e-20 - 7.2400827103674475973e-2861j 1.88e-378, 1.1e-877j
1.6698609581812671811e-09 + 8.3008538995548368233e-2853j 9.49e-371, 5.55e-870j
9.8805956081350559859e-22 + 7.4621064888239816998e-2862j 1.71e-379, 1e-878j
4.2795467650055482692e-05 - 6.2958686925626740558e-2849j 4.53e-367, 2.65e-866j
1.2608537732623687696e-07 - 2.282849768385511661e-2851j 1.4e-369, 8.2e-869j
0.013128070614597435143 - 4.2082648879263021306e-2847j 4.27e-365, 2.47e-864j

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All rules of Hermite (probabilists') type can be downloaded here packed into a single file.