Rule: Q[4, 10, 90]

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.

Warning: This rule has negative weights!

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)
-17.841350172445434241 + 4.7377502135777803924e-1035j 5.64e-493, 5.64e-493j
17.841350172445434241 - 4.7229618065235463473e-1036j 5.41e-493, 5.41e-493j
15.466606941299169974 + 2.6250894035097378515e-1032j 3.45e-489, 3.45e-489j
18.6690767166952104 + 1.9004672503285106731e-1037j 1.16e-494, 1.16e-494j
-15.466606941299169974 + 1.8677035966654509682e-1027j 3.57e-489, 3.57e-489j
16.5502873244877788 + 7.068983990048859056e-1033j 9.86e-491, 9.86e-491j
-13.159395957509461707 - 4.6345941128095462236e-1027j 4.71e-487, 4.71e-487j
-16.5502873244877788 + 3.3767590012013826867e-1038j 1.06e-490, 1.06e-490j
-14.492846740811774787 + 9.3382056237952504911e-1033j 4.55e-488, 4.55e-488j
11.145660295337363597 + 3.756392471447082826e-1038j 1.45e-486, 1.45e-486j
4.8777383426293956602 + 6.5481189213190242233e-1044j 6.26e-493, 6.26e-493j
-13.59048587795397733 + 7.945578257243971339e-1036j 2.59e-487, 2.59e-487j
5.861282930748284794 + 1.4990285862844123456e-1046j 2.68e-491, 2.68e-491j
-18.6690767166952104 + 1.257757596568206501e-1048j 1.22e-494, 1.22e-494j
-15.990961885302351759 - 4.7263166308664362638e-1044j 7.07e-490, 7.07e-490j
17.158242875442731792 - 5.3425801148125227364e-1048j 1.02e-491, 1.02e-491j
1.4656205133845524345 + 2.8848104077688192612e-1057j 3.53e-503, 3.53e-503j
9.6536143000939655259 + 1.2049475733616494046e-1041j 5.36e-487, 5.36e-487j
-4.3114957943342286437 + 2.2650594195590773334e-1047j 2.42e-493, 2.42e-493j
-7.5417685234794781136 + 1.1959355632815213785e-1043j 8.01e-489, 8.01e-489j
-9.6536143000939655259 - 2.8986811386972650196e-1039j 5.23e-487, 5.23e-487j
1.7268020678184978106 - 1.7946735437142622023e-1061j 1.98e-502, 1.98e-502j
11.927094061724505373 - 2.2018329425943027105e-1046j 1.26e-486, 1.26e-486j
13.159395957509461707 - 4.9217850772180496774e-1047j 4.69e-487, 4.69e-487j
-9.292525046090957295 - 2.516514955678663237e-1044j 3.34e-487, 3.34e-487j
14.03426039200148639 - 5.8755278198043046423e-1055j 1.12e-487, 1.12e-487j
3.2762772292662860982 - 6.2372499428419797663e-1065j 2.06e-497, 2.06e-497j
-6.8624525804885148647 + 1.5213890998633463691e-1056j 9.98e-490, 9.98e-490j
6.1928371626031745593 - 6.1949995217938096261e-1058j 9.7e-491, 9.7e-491j
-1.7268020678184978106 - 3.2781499067062640123e-1069j 2.07e-502, 2.07e-502j
-5.2039464137006749883 - 1.6312625297939129363e-1059j 2.07e-492, 2.07e-492j
7.2008152553598648465 - 1.8487856787257209353e-1056j 3.01e-489, 3.01e-489j
-17.158242875442731792 - 1.3478873651636827553e-1055j 9.76e-492, 9.76e-492j
8.9354570078165330708 + 4.788504788610696038e-1055j 1.83e-487, 1.83e-487j
-14.969033004429639139 + 2.1727672996213060641e-1052j 1.4e-488, 1.4e-488j
-6.5265117534425207543 - 1.2053534550574502092e-1067j 3.12e-490, 3.12e-490j
14.969033004429639139 + 2.0099272693119682104e-1071j 1.46e-488, 1.46e-488j
10.019064236914553576 + 5.6561952877705244566e-1071j 8.29e-487, 8.29e-487j
14.492846740811774787 + 1.4485687736208643424e-1070j 4.5e-488, 4.5e-488j
9.292525046090957295 - 2.674280010929419301e-1073j 3.39e-487, 3.39e-487j
-1.2846313116765794358 - 1.9984857210250926281e-1088j 7.38e-504, 7.38e-504j
6.5265117534425207543 + 7.7603484241362165079e-1076j 3.12e-490, 3.12e-490j
-10.76462446183960101 - 2.1922980859077718074e-1070j 1.28e-486, 1.28e-486j
5.2039464137006749883 - 1.163163100701214947e-1099j 1.99e-492, 1.99e-492j
13.59048587795397733 + 2.3461197613504957673e-1093j 2.33e-487, 2.33e-487j
-8.9354570078165330708 + 6.2077761609838363177e-1100j 1.89e-487, 1.89e-487j
-4.2426434457926842072 + 2.5587360227419220451e-1138j 1.43e-493, 1.43e-493j
-2.9602846865921801883 - 2.1761607069440960654e-1143j 1.82e-498, 1.82e-498j
-3.9139904759243095378 + 1.3221362905769301671e-1140j 3.39e-495, 3.39e-495j
7.5417685234794781136 - 6.4564905454770505292e-1131j 8.66e-489, 8.66e-489j
2.6461917682568042197 - 8.0161947211975838404e-1148j 1.87e-499, 1.87e-499j
-12.73932089583590749 - 1.4186267762132613331e-1134j 7.27e-487, 7.27e-487j
8.5821067994481451968 - 3.996574882360894234e-1159j 1e-487, 1e-487j
6.8624525804885148647 - 8.8180583361776774234e-1162j 1.05e-489, 1.05e-489j
7.8854966983313378109 + 4.7108747055121906835e-1161j 2.16e-488, 2.16e-488j
-2.3344142183389772393 + 1.343092803843817789e-1172j 1.77e-500, 1.77e-500j
8.2322019505802571193 - 1.7462146503664116043e-1159j 4.78e-488, 4.78e-488j
-1.0391137051328423322 + 1.5131394479202384983e-1178j 3.95e-505, 3.95e-505j
12.328919565960647878 - 2.4644053086106465328e-1157j 1.05e-486, 1.05e-486j
4.2426434457926842072 - 9.1183599448091065249e-1185j 1.43e-493, 1.43e-493j
2.0262440341796949676 + 9.3685380521370383144e-1193j 1.81e-501, 1.81e-501j
-8.2322019505802571193 - 8.60613296955962184e-1177j 4.53e-488, 4.53e-488j
-2.0262440341796949676 + 5.4276437700079124231e-1227j 1.82e-501, 1.82e-501j
-11.927094061724505373 + 3.9754807339787420081e-1227j 1.36e-486, 1.36e-486j
1.0391137051328423322 - 1.04512283543861898e-1276j 4.24e-505, 4.24e-505j
2.3344142183389772393 + 1.4723378224486527302e-1272j 1.73e-500, 1.73e-500j
-4.5518757004809729224 + 2.1533503625326208911e-1264j 2.37e-493, 2.37e-493j
-5.5317060638607199116 + 2.6125729195336900354e-1259j 7.93e-492, 7.93e-492j
12.73932089583590749 + 7.7586492271395156594e-1269j 7.33e-487, 7.33e-487j
4.3114957943342286437 + 4.6617341996936828581e-1307j 2.31e-493, 2.31e-493j
-10.389256807496844246 + 1.5213629502470619341e-1313j 1.12e-486, 1.12e-486j
3.9139904759243095378 + 2.9569529397689230602e-1362j 3.5e-495, 3.5e-495j
1.2846313116765794358 - 2.2080003113716689587e-1372j 6.9e-504, 6.9e-504j
15.990961885302351759 - 1.881510746625995607e-1353j 6.98e-490, 6.98e-490j
-11.145660295337363597 - 7.6015228955247624587e-1384j 1.43e-486, 1.43e-486j
2.9602846865921801883 - 4.3174522880507050548e-1433j 2.03e-498, 2.03e-498j
5.5317060638607199116 + 3.3715264155081701846e-1427j 7.35e-492, 7.35e-492j
4.5518757004809729224 + 4.2249750400701139159e-1429j 2.4e-493, 2.4e-493j
-10.019064236914553576 + 1.7787233016936016821e-1420j 8.22e-487, 8.22e-487j
-5.861282930748284794 - 3.8570085303429491348e-1443j 2.62e-491, 2.62e-491j
-0.74196378430272585765 - 9.5426116614726294494e-1460j 1.77e-506, 1.77e-506j
10.389256807496844246 + 2.5036726935929006013e-1440j 1.13e-486, 1.13e-486j
-7.8854966983313378109 - 2.0686517680004053016e-1439j 2.04e-488, 2.04e-488j
-1.4656205133845524345 + 1.3940019442232060148e-1474j 3.74e-503, 3.74e-503j
-8.5821067994481451968 - 7.0150686711835768892e-1458j 9.46e-488, 9.46e-488j
10.76462446183960101 + 5.4372815002212515315e-1473j 1.35e-486, 1.35e-486j
-3.5940812584752307271 + 9.3467959366905591818e-1487j 2.44e-496, 2.44e-496j
-11.53293105609684988 - 6.8178389452694786997e-1475j 1.42e-486, 1.42e-486j
3.5940812584752307271 + 6.6621348720784287352e-1503j 2.5e-496, 2.5e-496j
-0.41481227420381804714 + 5.341626759068172389e-1514j 1.1e-507, 1.1e-507j
0.18790942540245570008 + 2.5743972000102494382e-1515j 1.2e-508, 1.2e-508j
11.53293105609684988 - 9.0482826058056381911e-1493j 1.4e-486, 1.4e-486j
0.74196378430272585765 + 1.80749351291603635e-1512j 1.67e-506, 1.67e-506j
-6.1928371626031745593 + 1.7069257523685618554e-1496j 9.76e-491, 9.76e-491j
-4.8777383426293956602 - 7.8557217944084339559e-1500j 6.01e-493, 6.01e-493j
-2.6461917682568042197 - 2.6458868214225608588e-1505j 1.63e-499, 1.63e-499j
0.41481227420381804714 - 1.04460365204035532e-1514j 1.21e-507, 1.21e-507j
-7.2008152553598648465 - 6.8287749299086731097e-1495j 3.21e-489, 3.21e-489j
-14.03426039200148639 + 3.4292095086514422242e-1492j 1.19e-487, 1.19e-487j
-0.18790942540245570008 - 2.8223719583602106604e-1572j 1.39e-508, 1.39e-508j
-0.26842728825432580984 - 7.0337877031919768365e-1528j 4.23e-508, 4.23e-508j
-3.2762772292662860982 + 3.8695135406215495065e-1517j 2.14e-497, 2.14e-497j
-12.328919565960647878 - 4.5173280391996326069e-1505j 1.05e-486, 1.05e-486j
0.26842728825432580984 - 3.1835256033405701812e-1539j 4.23e-508, 4.23e-508j

Weights

Weight (ball center) Error (ball radius)
2.2242320373546175114e-70 - 1.0878263584929997592e-1082j 8.19e-140, 8.35e-383j
2.2242320373546175114e-70 - 1.6427023931635025147e-1083j 4.11e-140, 4.19e-383j
2.3083664669026099558e-53 - 1.586156965550927171e-1074j 2.51e-134, 2.56e-377j
7.9238689112195113766e-77 + 6.9198141867012356068e-1087j 2e-142, 2.04e-385j
2.3083664669026099558e-53 - 1.9682761246652402035e-1073j 4.32e-136, 4.4e-379j
7.6843784760767863974e-61 - 1.7196582082305356388e-1078j 2.01e-137, 2.05e-380j
4.2297840119909140988e-39 - 1.397297269862526379e-1064j 5.94e-130, 6.05e-373j
7.6843784760767863974e-61 - 1.5069113849741503735e-1077j 1.97e-139, 2.01e-382j
4.5700133005207940038e-47 - 7.633055259472430593e-1070j 3.89e-134, 3.96e-377j
1.6218944314862408201e-28 - 5.2738602476177109785e-1061j 4.03e-126, 4.11e-369j
8.85476631758619514e-07 - 6.3675926236021377472e-1047j 2.36e-93, 2.4e-336j
1.3618607379419571076e-41 - 2.0454840395823580645e-1066j 9.27e-132, 9.45e-375j
4.572156800451468725e-09 + 5.734876730702898803e-1049j 2.29e-101, 2.34e-344j
7.9238689112195113766e-77 + 3.9979414298205152167e-1086j 1.71e-146, 1.74e-389j
6.4049423601347529267e-57 + 2.1053592424171552302e-1075j 4.02e-138, 4.1e-381j
3.0014325571500269395e-65 + 8.1453843381395415447e-1081j 2.54e-143, 2.59e-386j
0.029228653625810267497 - 3.5136651393257848814e-1042j 4.48e-52, 4.57e-295j
8.4066587431759161919e-22 - 3.6938568263340747202e-1057j 1.35e-122, 1.38e-365j
-2.0628224419483233797e-06 + 8.4626448957262266623e-1045j 6.98e-94, 7.12e-337j
6.0865042745350803332e-14 + 8.4317914240281161508e-1052j 1.75e-114, 1.79e-357j
8.4066587431759161919e-22 + 2.4029499388244188663e-1056j 3.76e-123, 3.83e-366j
0.026052658872880845959 + 1.0275894351235035652e-1042j 8.82e-62, 8.99e-305j
2.0428189686462970925e-32 - 3.4985388933139681342e-1063j 2.58e-131, 2.63e-374j
4.2297840119909140988e-39 + 7.4469310724069280989e-1067j 2.03e-135, 2.06e-378j
2.5418443644989321771e-20 - 1.5933612062592870907e-1055j 1.89e-123, 1.92e-366j
3.0581057469775841934e-44 + 1.219352033684253185e-1069j 6.3e-138, 6.42e-381j
0.00059015327211331673684 - 8.2085405619106430164e-1045j 2.51e-89, 2.56e-332j
7.9896967960131826001e-12 + 1.8439631973871280583e-1050j 3.37e-114, 3.44e-357j
6.2367147880162702488e-10 - 1.2669207536038850659e-1049j 3.47e-110, 3.54e-353j
0.026052658872880845959 - 1.3377998066776908837e-1042j 6.52e-71, 6.65e-314j
1.7170159245967418876e-07 - 2.807980602101283869e-1047j 6.74e-106, 6.87e-349j
7.4550746970292416281e-13 + 1.1769812783771857942e-1051j 3.46e-116, 3.52e-359j
3.0014325571500269395e-65 + 6.1763336003787901436e-1080j 6.47e-146, 6.59e-389j
6.5129923776533568867e-19 - 1.8957687003196665813e-1055j 8.85e-124, 9.02e-367j
4.2762919934626482057e-50 + 1.364762079753123312e-1071j 8.91e-140, 9.08e-383j
7.5198494941384824527e-11 - 8.1658510967568001736e-1050j 3.41e-114, 3.48e-357j
4.2762919934626482057e-50 + 8.7787491003226569084e-1073j 1.56e-141, 1.59e-384j
2.3381326913891779843e-23 + 4.6001848368744241531e-1058j 3.76e-128, 3.84e-371j
4.5700133005207940038e-47 - 3.6801978725839997311e-1071j 1.96e-140, 1.99e-383j
2.5418443644989321771e-20 + 2.7442911241018007723e-1056j 7.35e-126, 7.5e-369j
0.033587282258781753836 - 7.1869911566655393397e-1042j 3.31e-75, 3.37e-318j
7.5198494941384824527e-11 + 2.749681191306969527e-1050j 2.03e-116, 2.07e-359j
1.0378426794384682475e-26 - 5.5263558652909141019e-1059j 3.4e-135, 3.47e-378j
1.7170159245967418876e-07 + 1.2129790137078111427e-1047j 3.31e-111, 3.37e-354j
1.3618607379419571076e-41 - 3.2957952069195162783e-1068j 1.98e-138, 2.02e-381j
6.5129923776533568867e-19 + 9.9179035899607477426e-1055j 7.63e-129, 7.78e-372j
1.8146394146813614011e-05 - 9.5920921369061638715e-1045j 4.2e-109, 4.28e-352j
0.0015718291405253260663 - 2.8645575890507574739e-1044j 3.37e-102, 3.44e-345j
6.0443913019392748447e-05 + 4.6254391946700575138e-1045j 4.56e-108, 4.64e-351j
6.0865042745350803332e-14 - 2.2871777302498149587e-1052j 3.16e-123, 3.22e-366j
0.0037668650367335293651 - 4.2926253190427126356e-1044j 6.57e-101, 6.69e-344j
9.5083891347958357506e-37 + 8.8439789734073778225e-1064j 3.57e-142, 3.64e-385j
1.4239350524166428046e-17 + 1.2234293672894552688e-1054j 6.68e-127, 6.81e-370j
7.9896967960131826001e-12 - 5.7963338358274592302e-1051j 2.97e-121, 3.02e-364j
4.3302427451911070056e-15 + 4.2301062657823077587e-1053j 3.32e-125, 3.38e-368j
0.0081168249556428289939 - 1.5652964758445934321e-1043j 7.6e-105, 7.75e-348j
2.6726223190772580483e-16 - 7.4093657022467582689e-1054j 2.21e-126, 2.25e-369j
0.065910651523772700483 + 8.2875246503986686099e-1042j 2.4e-90, 2.45e-333j
1.5940758932912421886e-34 + 2.3915765576867585871e-1064j 1.38e-140, 1.4e-383j
1.8146394146813614011e-05 + 4.9215975001965059073e-1045j 6.98e-116, 7.12e-359j
0.015638133193231573865 - 3.0910337329827611225e-1043j 8.72e-103, 8.89e-346j
2.6726223190772580483e-16 + 3.2176346038313886799e-1053j 9.85e-135, 1e-377j
0.015638133193231573865 + 4.2153636805592958535e-1043j 6.81e-105, 6.95e-348j
2.0428189686462970925e-32 + 7.1238858303280154507e-1062j 1.16e-145, 1.19e-388j
0.065910651523772700483 - 7.0750920714055816816e-1042j 1.74e-95, 1.78e-338j
0.0081168249556428289939 + 1.0938779299503326573e-1043j 3.53e-107, 3.6e-350j
4.1416161202311286863e-06 - 1.0075763921330541531e-1045j 6.05e-121, 6.17e-364j
2.9714738043572782085e-08 + 6.370781356781067059e-1048j 5.79e-126, 5.9e-369j
9.5083891347958357506e-37 - 1.4341819573284786776e-1065j 5.09e-143, 5.19e-386j
-2.0628224419483233797e-06 - 4.2923372512181587153e-1045j 2.28e-118, 2.32e-361j
5.4215829909174734519e-25 + 4.4839083924263389698e-1058j 5.75e-142, 5.86e-385j
6.0443913019392748447e-05 - 2.5068608643605102528e-1045j 3.49e-118, 3.56e-361j
0.033587282258781753836 + 5.9092326212841535292e-1042j 1.3e-104, 1.32e-347j
6.4049423601347529267e-57 + 2.0447704951335772087e-1076j 6.7e-154, 6.83e-397j
1.6218944314862408201e-28 + 6.3666481027088711193e-1060j 2.28e-144, 2.33e-387j
0.0015718291405253260663 + 1.8131044160694504826e-1044j 2.47e-115, 2.52e-358j
2.9714738043572782085e-08 - 2.5960317519738638104e-1048j 2.39e-123, 2.44e-366j
4.1416161202311286863e-06 + 4.9100604567296390679e-1046j 3.14e-120, 3.2e-363j
2.3381326913891779843e-23 - 3.3970989055323422551e-1057j 1.41e-141, 1.44e-384j
4.572156800451468725e-09 - 1.4961508525450551673e-1048j 1.33e-131, 1.36e-374j
0.092995740813785467044 - 1.300675187647268272e-1041j 2.28e-111, 2.32e-354j
5.4215829909174734519e-25 - 5.2727659074688966084e-1059j 2.2e-139, 2.24e-382j
4.3302427451911070056e-15 - 1.6884714812983053918e-1052j 3.42e-137, 3.49e-380j
0.029228653625810267497 + 4.3938663679023301715e-1042j 1.51e-116, 1.54e-359j
1.4239350524166428046e-17 - 5.8123458034152909506e-1054j 1.06e-138, 1.08e-381j
1.0378426794384682475e-26 + 5.530283578770280177e-1060j 2.75e-141, 2.8e-384j
0.00019924034777574045714 - 7.162416983591613089e-1045j 1.46e-127, 1.49e-370j
2.0427368147150793241e-30 - 6.8932820073699268845e-1061j 2.84e-147, 2.89e-390j
0.00019924034777574045714 + 4.0918534722952583514e-1045j 1.21e-124, 1.23e-367j
0.14677663487395646589 + 6.8230149523727360728e-1041j 2.33e-119, 2.37e-362j
0.2117097843206967301 - 1.6544448517490249677e-1040j 1.27e-120, 1.3e-363j
2.0427368147150793241e-30 + 4.5392771375336362569e-1062j 1.23e-143, 1.26e-386j
0.092995740813785467044 + 1.1619017404580791794e-1041j 4.93e-121, 5.02e-364j
6.2367147880162702488e-10 + 3.5220083896997761648e-1049j 2.2e-134, 2.24e-377j
8.85476631758619514e-07 + 1.3801304530410301344e-1046j 2.3e-131, 2.34e-374j
0.0037668650367335293651 + 6.4516498610508376625e-1044j 1.03e-127, 1.05e-370j
0.14677663487395646589 - 6.4062192023256278282e-1041j 1.35e-123, 1.38e-366j
7.4550746970292416281e-13 - 4.0235238207471122421e-1051j 8.31e-137, 8.47e-380j
3.0581057469775841934e-44 + 3.7908240593856114751e-1068j 7.15e-156, 7.26e-399j
0.2117097843206967301 + 1.7023532742195312454e-1040j 1.87e-125, 1.6e-368j
-0.13622621350934075325 - 2.040929368952143021e-1040j 1.92e-125, 1.63e-368j
0.00059015327211331673684 + 1.3644476311918544407e-1044j 1.72e-130, 1.56e-373j
1.5940758932912421886e-34 - 7.3417039439003919322e-1063j 1.28e-150, 1.32e-393j
-0.13622621350934075325 + 1.9593731125101935768e-1040j 1.25e-125, 9.54e-369j

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