Rule: Q[5, 8, 90]

Type

This rule is of Hermite (physicists') 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)
-12.124637590209038289 + 9.0865900156180723486e-1012j 6.02e-492, 6.02e-492j
-11.298703664325001961 + 7.8215194560551053167e-1008j 4.1e-490, 4.1e-490j
-11.694474293109052187 + 9.8362192758018762156e-1011j 6.28e-491, 6.28e-491j
10.238573404635680166 + 4.0244854068003875897e-1010j 2.41e-488, 2.41e-488j
5.8064025591316233008 + 1.07117780486538539e-1008j 2.38e-488, 2.38e-488j
-8.9975132288858578076 + 3.881069238493608266e-1005j 4.24e-487, 4.24e-487j
-9.5999549533950681207 - 3.2595297779230326647e-1015j 1.32e-487, 1.32e-487j
-1.3117288375389628728 - 3.1202797518949911128e-1049j 1.43e-501, 1.43e-501j
6.5572654241420281747 + 1.5340099367775122626e-1033j 1.93e-487, 1.93e-487j
6.054187287383318068 + 1.7517074105895725667e-1034j 5.14e-488, 5.14e-488j
6.8129580230115353535 + 5.0599461028421566829e-1035j 3.04e-487, 3.04e-487j
-9.2948431719078230196 - 1.7486388873490796745e-1037j 2.61e-487, 2.61e-487j
12.124637590209038289 + 7.3436337048178255614e-1056j 6.14e-492, 6.14e-492j
9.5999549533950681207 - 1.5605624273945509311e-1050j 1.43e-487, 1.43e-487j
9.2948431719078230196 - 1.6429873643343372849e-1050j 2.58e-487, 2.58e-487j
8.9975132288858578076 + 8.561109616438646519e-1056j 4.31e-487, 4.31e-487j
3.4375671008498786428 - 1.2217336466809922624e-1080j 1.76e-493, 1.76e-493j
-6.8129580230115353535 + 2.4664768052024047057e-1088j 2.94e-487, 2.94e-487j
-13.193570058343994324 - 3.340455278832747983e-1121j 7.31e-495, 7.31e-495j
-7.5996111025637796792 - 2.713921269193374478e-1145j 7.23e-487, 7.23e-487j
2.4028713171883968066 - 3.7359031958619626232e-1180j 1.5e-496, 1.5e-496j
7.5996111025637796792 - 5.0752730761109784913e-1169j 7.65e-487, 7.65e-487j
10.927662630233062916 - 3.148001003970978354e-1181j 2.07e-489, 2.07e-489j
-8.4225751349710499795 - 3.6137664095248955439e-1197j 7.36e-487, 7.36e-487j
-4.3629954339102545171 - 1.2324969579877931248e-1216j 4.2e-491, 4.2e-491j
-6.054187287383318068 - 9.1720493632480417311e-1212j 5.2e-488, 5.2e-488j
0.90898686827108110578 + 9.4400210426137899396e-1237j 2.65e-503, 2.65e-503j
8.1435438925872863314 + 3.0516771483422207077e-1219j 8.51e-487, 8.51e-487j
-5.8064025591316233008 + 2.3843210133321880377e-1219j 2.53e-488, 2.53e-488j
-2.0201828704560856329 - 8.6036822368657799147e-1238j 3.72e-498, 3.72e-498j
-2.4028713171883968066 + 3.1406418173371692076e-1235j 1.41e-496, 1.41e-496j
4.8366251451094131591 - 1.6057208858510083482e-1229j 5.07e-490, 5.07e-490j
4.5989357788845026117 + 5.609872875488620623e-1231j 1.51e-490, 1.51e-490j
11.298703664325001961 + 7.028066800318849202e-1230j 4.1e-490, 4.1e-490j
-5.0761203770616080491 - 3.2268736713160211887e-1225j 1.49e-489, 1.49e-489j
-9.9140300047181455717 - 1.4021643404824808843e-1240j 6.31e-488, 6.31e-488j
5.5608913262640844827 + 7.0280646905737692761e-1259j 9.75e-489, 9.75e-489j
-8.7070161705033709018 + 1.0759402439939949639e-1265j 5.79e-487, 5.79e-487j
-1.5091147786039928404 + 6.0597637037267105203e-1291j 1.92e-500, 1.92e-500j
5.0761203770616080491 - 8.3359055975643397582e-1281j 1.55e-489, 1.55e-489j
-10.927662630233062916 - 7.7957394477531941781e-1277j 2.16e-489, 2.16e-489j
-3.8963503032273427285 - 4.1359693053375384153e-1300j 3.1e-492, 3.1e-492j
1.3117288375389628728 + 9.2987284002857786199e-1314j 1.51e-501, 1.51e-501j
2.2177775884631715445 + 2.7219580718682265052e-1309j 2.58e-497, 2.58e-497j
-3.6658316928775966507 - 1.1857080614009379487e-1304j 8.08e-493, 8.08e-493j
-10.575557707186752577 - 4.9260740791015121269e-1298j 8.18e-489, 8.18e-489j
3.2122961187372970202 + 2.972214684558139461e-1332j 3.85e-494, 3.85e-494j
-7.071727979716352038 - 4.1426300113788309698e-1329j 4.68e-487, 4.68e-487j
2.0201828704560856329 + 9.3349161132604425212e-1361j 3.91e-498, 3.91e-498j
5.3175065293158976059 + 1.0237964130062759397e-1348j 3.75e-489, 3.75e-489j
-12.60795535589976807 - 1.9416523411372550411e-1353j 3.58e-493, 3.58e-493j
-10.238573404635680166 + 4.7723715352375932851e-1379j 2.44e-488, 2.44e-488j
-3.2122961187372970202 - 1.7773725023434016418e-1417j 3.96e-494, 3.96e-494j
3.6658316928775966507 - 1.5259873893594921716e-1411j 7.84e-493, 7.84e-493j
2.7796084806380178827 - 7.1100213167044920383e-1420j 2.27e-495, 2.27e-495j
13.193570058343994324 - 1.0641524805348903318e-1410j 6.94e-495, 6.94e-495j
-0.90898686827108110578 - 4.1549673283904470734e-1435j 2.67e-503, 2.67e-503j
-2.5832564688314974155 - 5.8906108157905854285e-1423j 5.9e-496, 5.9e-496j
-8.1435438925872863314 - 2.8156604133448819231e-1417j 8.61e-487, 8.61e-487j
-1.6649033675753824191 - 4.9186350154124692044e-1451j 1.29e-499, 1.29e-499j
7.3338440124764870388 - 1.1405115306263404998e-1437j 5.97e-487, 5.97e-487j
-1.0902150130568489009 + 8.1023041952115209332e-1454j 1.54e-502, 1.54e-502j
-4.8366251451094131591 + 1.4892287414390054253e-1438j 5.13e-490, 5.13e-490j
10.575557707186752577 - 5.8837329250608267022e-1440j 8.23e-489, 8.23e-489j
1.8259424063458765091 - 7.5866615495450932564e-1451j 6.43e-499, 6.43e-499j
6.3044122792519100788 - 6.4062172155875864629e-1439j 1.06e-487, 1.06e-487j
3.8963503032273427285 - 1.396153852119550103e-1443j 3.12e-492, 3.12e-492j
-1.8259424063458765091 + 5.2663629490316522619e-1451j 6.43e-499, 6.43e-499j
8.7070161705033709018 - 7.3475666106825746675e-1437j 6.02e-487, 6.02e-487j
8.4225751349710499795 - 8.6911754636504094967e-1438j 7.39e-487, 7.39e-487j
11.694474293109052187 + 3.8013326895678525809e-1451j 6.25e-491, 6.25e-491j
12.60795535589976807 + 2.37528947041987633e-1454j 3.65e-493, 3.65e-493j
-4.1287862565931199963 - 9.9393100220229817866e-1448j 1.26e-491, 1.26e-491j
-6.3044122792519100788 + 2.4935344398322155139e-1450j 9.96e-488, 9.96e-488j
-7.8693775157807601356 - 1.286321205666028379e-1466j 7.8e-487, 7.8e-487j
9.9140300047181455717 - 2.1411806525159679813e-1485j 6.54e-488, 6.54e-488j
7.8693775157807601356 + 1.9688012810979211156e-1484j 7.88e-487, 7.88e-487j
-5.3175065293158976059 - 1.9723622480700182565e-1485j 4.01e-489, 4.01e-489j
-2.7796084806380178827 + 6.9589379396020325227e-1492j 2.31e-495, 2.31e-495j
-5.5608913262640844827 - 5.5519224830949175178e-1485j 9.75e-489, 9.75e-489j
2.5832564688314974155 + 2.3030253684757503179e-1493j 5.74e-496, 5.74e-496j
0.44571214628839900629 - 8.3285255692362725893e-1505j 2.49e-507, 2.49e-507j
-7.3338440124764870388 + 1.543336311115566198e-1483j 5.97e-487, 5.97e-487j
-3.4375671008498786428 - 3.366079596537247718e-1498j 1.72e-493, 1.72e-493j
2.9916875134528022128 + 8.8309830758671338771e-1500j 9.38e-495, 9.38e-495j
7.071727979716352038 - 1.4660300788211580298e-1492j 4.56e-487, 4.56e-487j
0.22279197965133039658 + 3.0339949915864445243e-1583j 7.75e-509, 7.75e-509j
-0.44571214628839900629 - 4.8111356262394195326e-1512j 2.8e-507, 2.8e-507j
-4.5989357788845026117 + 3.9885764240679656546e-1495j 1.6e-490, 1.6e-490j
-2.9916875134528022128 + 5.1084716938635099324e-1501j 9.31e-495, 9.31e-495j
-6.5572654241420281747 + 5.9998317775537449292e-1492j 1.93e-487, 1.93e-487j
0.95857246461381850711 - 9.6246739501220886869e-1512j 7.08e-503, 7.08e-503j
1.5091147786039928404 - 2.5127676913727440419e-1509j 2.03e-500, 2.03e-500j
-0.66944416045059213743 - 8.7479394533733316679e-1514j 1.33e-505, 1.33e-505j
-0.22279197965133039658 - 5.7116468288676292075e-1518j 7.75e-509, 7.75e-509j
-1.0298349984446422476e-1570 - 4.305217455369872138e-1570j 4.56e-1568, 4.56e-1568j
4.1287862565931199963 + 2.8745349008662270961e-1500j 1.23e-491, 1.23e-491j
0.66944416045059213743 + 1.1660859986144538218e-1514j 1.2e-505, 1.2e-505j
4.3629954339102545171 - 1.6012447971666577612e-1500j 4.38e-491, 4.38e-491j
-2.2177775884631715445 + 9.3618176595625831975e-1507j 2.46e-497, 2.46e-497j
-0.95857246461381850711 + 1.556953156016791763e-1512j 7.4e-503, 7.4e-503j
1.6649033675753824191 + 2.0817385168330282509e-1508j 1.39e-499, 1.39e-499j
1.0902150130568489009 - 2.7673895329104995493e-1511j 1.69e-502, 1.69e-502j

Weights

Weight (ball center) Error (ball radius)
3.6528920647331769543e-65 + 6.1207228208610126029e-1052j 2.16e-133, 1.69e-376j
7.7872964851026269897e-57 + 1.4451618695102746571e-1047j 1.02e-130, 7.94e-374j
9.3476302914937275866e-61 - 1.2522491989253708193e-1049j 5.51e-132, 4.3e-375j
5.546996218647832434e-47 + 8.6464421092563201056e-1042j 3.45e-130, 2.69e-373j
3.1733468541598382796e-16 - 2.3629891361681453842e-1023j 1.11e-105, 8.65e-349j
1.1508496073439196902e-36 - 8.3229784430245420651e-1037j 6.96e-123, 5.43e-366j
1.6508362235331238603e-41 - 2.0236144174214086902e-1039j 2.52e-125, 1.97e-368j
0.021271989003343203834 + 2.8764274239070789975e-1014j 1.59e-45, 1.24e-288j
3.0407768802821608824e-20 - 1.9938316055890022486e-1026j 4.45e-112, 3.47e-355j
1.6955320965161126909e-17 - 2.4047378511538116332e-1024j 9.42e-109, 7.35e-352j
1.0076105943082754904e-21 + 2.113618910353006819e-1027j 5.55e-114, 4.33e-357j
5.1235678878184675112e-39 + 4.4474112220064325765e-1038j 4.5e-126, 3.51e-369j
3.6528920647331769543e-65 - 1.7370161916804016963e-1051j 4.77e-141, 3.72e-384j
1.6508362235331238603e-41 + 8.2175382217166001923e-1039j 2.57e-130, 2e-373j
5.1235678878184675112e-39 - 1.9249041868854471081e-1037j 3.11e-129, 2.42e-372j
1.1508496073439196902e-36 + 3.8631326163639902541e-1036j 3.2e-128, 2.49e-371j
9.4480137332659353181e-07 - 3.6629884409025614454e-1018j 1.65e-94, 1.29e-337j
1.0076105943082754904e-21 - 1.6858458325120524011e-1028j 1.52e-121, 1.19e-364j
9.6536126222175170269e-77 + 5.5976930075148147477e-1058j 1.49e-147, 1.17e-390j
1.2496802786226781883e-26 + 2.7760583935328899089e-1031j 2.6e-125, 2.03e-368j
0.00031351045267912587243 + 1.2465514298829336681e-1015j 8.64e-85, 6.74e-328j
1.2496802786226781883e-26 - 2.0754521063840231021e-1030j 8.72e-124, 6.8e-367j
2.8050819530312969665e-53 + 3.5584991723949796916e-1045j 5.37e-139, 4.19e-382j
2.4681091521996607884e-32 - 1.9190085348499633173e-1034j 5.44e-129, 4.25e-372j
7.1700292671818446511e-10 - 3.6405275025049945989e-1021j 2.99e-109, 2.33e-352j
1.6955320965161126909e-17 + 5.0237685270475983505e-1026j 1.36e-119, 1.06e-362j
0.084543024415864831337 - 3.1105888011208174814e-1013j 4.93e-71, 3.85e-314j
2.4654921622943330932e-30 - 1.4438139384195226818e-1032j 1.43e-128, 1.11e-371j
3.1733468541598382796e-16 - 2.9270911408549595615e-1025j 2.04e-118, 1.59e-361j
0.0019040404634880607638 + 2.7626207278944419297e-1015j 3.28e-86, 2.56e-329j
0.00031351045267912587243 + 5.168133325570073734e-1016j 6.62e-92, 5.16e-335j
9.3246713882115973256e-12 - 2.0857282166321182619e-1021j 1.79e-113, 1.39e-356j
8.7177033606316996012e-11 + 7.3124206498991092446e-1021j 4.01e-112, 3.13e-355j
7.7872964851026269897e-57 - 4.5006647175823292531e-1047j 5.12e-144, 3.99e-387j
8.7486029563406878422e-13 + 4.0700861703421679363e-1023j 4.08e-117, 3.18e-360j
3.7095242845118275612e-44 + 7.6971744630063165397e-1041j 1.8e-138, 1.4e-381j
5.1246015893505141857e-15 + 7.4261571360486065908e-1023j 1.16e-118, 9.07e-362j
1.9277410900616273973e-34 + 1.3497230969497285372e-1035j 2.29e-133, 1.78e-376j
0.010336661715702960965 - 2.106564323884574371e-1014j 4.15e-86, 3.24e-329j
8.7486029563406878422e-13 + 6.0652230364593645324e-1022j 8.82e-117, 6.88e-360j
2.8050819530312969665e-53 - 1.0890692132518753564e-1045j 2.09e-143, 1.63e-386j
3.3322618020446188699e-08 - 6.0830576843137811195e-1020j 2.72e-113, 2.12e-356j
0.021271989003343203834 + 4.5553528434767097849e-1014j 2.86e-86, 2.23e-329j
0.0007944412591962034656 - 2.6208967885348470202e-1015j 4.74e-99, 3.7e-342j
1.8878348777655657322e-07 + 2.4028155181596067794e-1019j 5.52e-112, 4.3e-355j
5.1935616568634855825e-50 + 5.859227194932352018e-1044j 2.52e-142, 1.97e-385j
4.1588755082024196059e-06 + 1.2737797972815344664e-1017j 4.11e-110, 3.21e-353j
2.8072392015990304499e-23 + 2.1692339056065712568e-1029j 6.17e-130, 4.82e-373j
0.0019040404634880607638 + 5.7106955930113804402e-1015j 1.09e-99, 8.48e-343j
7.1752925459390373152e-14 - 1.8880956283419296547e-1022j 3.42e-121, 2.67e-364j
2.7083433832334531463e-70 - 1.2771590710583661918e-1054j 1.96e-152, 1.53e-395j
5.546996218647832434e-47 - 2.3885527674280149124e-1042j 5.58e-141, 4.35e-384j
4.1588755082024196059e-06 + 3.6638370720675911986e-1018j 4.54e-113, 3.54e-356j
1.8878348777655657322e-07 + 1.063275426980766622e-1018j 2.73e-115, 2.13e-358j
5.1163771451049529598e-05 + 1.5831255475218600721e-1016j 2.02e-109, 1.58e-352j
9.6536126222175170269e-77 - 1.4397182318083304202e-1057j 3.96e-161, 3.09e-404j
0.084543024415864831337 - 2.2684950781559563264e-1013j 1.56e-96, 1.21e-339j
0.00013302051816820357059 - 1.9283865834756478148e-1016j 7.31e-110, 5.7e-353j
2.4654921622943330932e-30 + 2.418787671335780344e-1033j 5.81e-135, 4.54e-378j
0.0050594192066525087231 + 1.5165959411859715209e-1014j 1.25e-101, 9.75e-345j
6.5192644614489559115e-25 + 2.2078994816106918121e-1029j 1.51e-134, 1.17e-377j
0.041701319994551406 - 8.5241640798518505354e-1014j 4.93e-97, 3.85e-340j
9.3246713882115973256e-12 - 1.9004684951381569937e-1022j 5.88e-124, 4.59e-367j
5.1935616568634855825e-50 - 2.0125598497756663594e-1043j 1.02e-148, 7.94e-392j
0.0036764440491090539153 - 1.4083477515807050187e-1014j 9.92e-107, 7.74e-350j
7.7744971466165397429e-19 + 1.9639616424186447001e-1025j 1.66e-129, 1.3e-372j
3.3322618020446188699e-08 - 3.0901133858198050258e-1019j 3.22e-120, 2.51e-363j
0.0036764440491090539153 - 7.344869742376178493e-1015j 3.3e-108, 2.57e-351j
1.9277410900616273973e-34 - 6.7546586034703791283e-1035j 2.99e-141, 2.33e-384j
2.4681091521996607884e-32 + 1.0438146711827945405e-1033j 3.66e-140, 2.86e-383j
9.3476302914937275866e-61 + 3.7219258456129526054e-1049j 1.37e-155, 1.07e-398j
2.7083433832334531463e-70 + 3.4576893228700989363e-1054j 2.13e-160, 1.66e-403j
5.2004740815870994794e-09 + 1.5090917179424547235e-1020j 5.31e-128, 4.14e-371j
7.7744971466165397429e-19 - 8.0758887783602988103e-1027j 5.94e-137, 4.63e-380j
1.9553609809056901428e-28 - 2.7282096079115628777e-1032j 1.63e-141, 1.27e-384j
3.7095242845118275612e-44 - 2.9456295519884460494e-1040j 2.26e-148, 1.76e-391j
1.9553609809056901428e-28 + 1.8086525878504633349e-1031j 2.51e-140, 1.96e-383j
7.1752925459390373152e-14 - 8.2980942562149207192e-1024j 1.83e-134, 1.43e-377j
5.1163771451049529598e-05 + 5.5809100394676411063e-1017j 6.43e-126, 5.01e-369j
5.1246015893505141857e-15 + 1.60388503526978204e-1024j 2.56e-135, 2e-378j
0.00013302051816820357059 - 5.0194951801994527289e-1016j 1.02e-125, 7.93e-369j
0.10319069362451338453 - 2.8812646951284284802e-1014j 9.48e-121, 7.39e-364j
6.5192644614489559115e-25 - 2.5664202894322733687e-1030j 2.57e-141, 2.01e-384j
9.4480137332659353181e-07 - 9.3866281919365683826e-1019j 1.35e-129, 1.05e-372j
1.5900320093392695451e-05 - 4.4969750243821571782e-1017j 1.6e-128, 1.25e-371j
2.8072392015990304499e-23 - 2.2078401167421742494e-1028j 1.18e-140, 9.23e-384j
0.11962556296673448374 + 2.2899218166448528719e-1014j 3.07e-123, 2.4e-366j
0.10319069362451338453 - 2.4704535362726272199e-1014j 7.57e-124, 5.9e-367j
8.7177033606316996012e-11 + 8.4854830224848983877e-1022j 4.1e-133, 3.2e-376j
1.5900320093392695451e-05 - 1.4386812070044418185e-1017j 8.01e-129, 6.25e-372j
3.0407768802821608824e-20 + 1.2108594065502269309e-1027j 3.74e-139, 2.92e-382j
-0.036560101135249789308 + 3.7895038782289572916e-1013j 2.34e-125, 1.83e-368j
0.010336661715702960965 - 3.586131609235538023e-1014j 4.8e-127, 3.75e-370j
0.081092691739480570341 + 3.6775427361087055888e-1014j 2.54e-125, 1.98e-368j
0.11962556296673448374 + 2.1206859095857670253e-1014j 1.16e-125, 9.03e-369j
0.12568977167260647557 - 2.0836484585620423886e-1014j 3.63e-126, 2.83e-369j
5.2004740815870994794e-09 + 8.9372135003442765893e-1020j 2.41e-134, 1.88e-377j
0.081092691739480570341 + 4.6360555136312239732e-1014j 2.43e-127, 1.9e-370j
7.1700292671818446511e-10 - 2.5649210312470897367e-1020j 5.64e-135, 4.4e-378j
0.0007944412591962034656 - 1.1721304930038759185e-1015j 2.45e-129, 1.91e-372j
-0.036560101135249789308 + 2.7155824340660684828e-1013j 9.62e-127, 7.51e-370j
0.0050594192066525087231 + 2.73596048293549813e-1014j 3.97e-129, 3.12e-372j
0.041701319994551406 - 1.2465150166455359643e-1013j 2.32e-128, 1.77e-371j

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