Rule: Q[1, 4, 8, 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)
-15.454048696713245619 + 2.4024030352541502743e-1124j 3.2e-489, 3.2e-489j
15.454048696713245619 - 3.5023289546246891297e-1121j 3.09e-489, 3.09e-489j
-17.830341458107954436 + 3.7761777837641853341e-1135j 5.25e-493, 5.25e-493j
17.830341458107954436 - 3.3340586051457907943e-1133j 5.39e-493, 5.39e-493j
-11.516710218677527505 + 2.5252822235903028024e-1128j 1.19e-486, 1.19e-486j
3.653276333309534136 + 6.1709624663020323706e-1145j 8.02e-496, 8.02e-496j
13.576386493262078942 + 1.8020412641970591894e-1135j 2.01e-487, 2.01e-487j
10.371621666772966066 - 3.0238599292475356868e-1154j 8.99e-487, 8.99e-487j
-9.2733736949018217222 + 3.8883699035399984164e-1170j 2.79e-487, 2.79e-487j
11.516710218677527505 + 2.0287578126595198775e-1169j 1.15e-486, 1.15e-486j
-11.911319985982460439 - 9.1144783996567585719e-1177j 1.13e-486, 1.13e-486j
-15.978779959323003744 + 4.5182607550885685518e-1180j 6.83e-490, 6.83e-490j
16.538484150138335205 + 1.4319187829729456083e-1180j 9.23e-491, 9.23e-491j
14.479529368188253605 - 1.4063529252512454626e-1175j 3.7e-488, 3.7e-488j
9.6349776360215022855 - 2.3533748831517243594e-1186j 4.25e-487, 4.25e-487j
10.747473090006847468 - 9.6487222818178955348e-1212j 1.05e-486, 1.05e-486j
-8.9157853481095274549 - 7.1895179335545774114e-1226j 1.43e-487, 1.43e-487j
-18.6585257126296637 - 2.4234715490482737679e-1240j 1.08e-494, 1.08e-494j
13.144893273843000695 + 1.2220802690128560386e-1230j 3.84e-487, 3.84e-487j
-5.1842688974446196437 - 2.505672935786435476e-1247j 9.67e-493, 9.67e-493j
7.8642879324855760898 + 1.1613210769338413834e-1240j 1.46e-488, 1.46e-488j
-10.000933618328152661 + 2.9105989838445369859e-1242j 6.16e-487, 6.16e-487j
-12.313580356127716119 - 1.0893884294855837183e-1249j 9.06e-487, 9.06e-487j
17.146826918932262617 - 5.2851454418023302971e-1262j 9.1e-492, 9.1e-492j
-7.5200898517660283346 + 2.427668678401216937e-1269j 5.65e-489, 5.65e-489j
10.000933618328152661 - 4.6607171833535611002e-1285j 6.05e-487, 6.05e-487j
-12.724405235921717899 - 3.1836350396925774287e-1318j 6.61e-487, 6.61e-487j
-13.576386493262078942 - 7.0090623618238135229e-1370j 2.04e-487, 2.04e-487j
12.313580356127716119 - 2.0943583807915581988e-1414j 8.51e-487, 8.51e-487j
-16.538484150138335205 - 1.1522404472519598666e-1435j 9.45e-491, 9.45e-491j
-13.144893273843000695 + 3.4670530747383454787e-1438j 3.96e-487, 3.96e-487j
-14.956097139162818581 + 8.4589154716451309701e-1459j 1.38e-488, 1.38e-488j
1.8550647122034955328 - 2.6268800704111437638e-1485j 2.14e-501, 2.14e-501j
11.911319985982460439 + 4.8710350370041205745e-1469j 1.06e-486, 1.06e-486j
14.956097139162818581 + 2.1428485330517348465e-1490j 1.2e-488, 1.2e-488j
5.1842688974446196437 - 1.8812853356835728127e-1511j 1.03e-492, 1.03e-492j
9.2733736949018217222 - 8.334604374296130919e-1505j 2.72e-487, 2.72e-487j
0.9467370109606977334 - 3.7733493193671741361e-1537j 1.66e-505, 1.66e-505j
4.8614540155894593679 + 1.2489058093472956693e-1524j 2.37e-493, 2.37e-493j
-3.653276333309534136 - 3.8428156395292803068e-1527j 8.16e-496, 8.16e-496j
18.6585257126296637 - 1.6072565762223866508e-1519j 1.02e-494, 1.02e-494j
-6.1702073152077686232 - 6.5460421208711390912e-1526j 5.72e-491, 5.72e-491j
4.5428727374767395384 - 5.0435501570913724135e-1530j 5.93e-494, 5.93e-494j
-11.128980410251045791 + 3.8716298292935288297e-1521j 1.21e-486, 1.21e-486j
3.1364111439317137733 - 5.3221400017942258069e-1546j 3.45e-497, 3.45e-497j
-3.1364111439317137733 - 8.890237707323903729e-1546j 3.46e-497, 3.46e-497j
-7.1787182814789551924 + 2.5469471572275514179e-1535j 2.14e-489, 2.14e-489j
14.020555690446200963 - 1.2586768672572715959e-1555j 9.83e-488, 9.83e-488j
11.128980410251045791 + 5.4184923025842000116e-1583j 1.2e-486, 1.2e-486j
-4.2308850559071941351 + 6.8559034848204158455e-1609j 1.27e-494, 1.27e-494j
6.5038773509813369939 + 9.4377917381643401441e-1601j 2.19e-490, 2.19e-490j
-3.9309600114055574643 + 1.1787840819217083234e-1611j 3.17e-495, 3.17e-495j
3.3981732054051338643 + 1.6277818713349170207e-1612j 1.99e-496, 1.99e-496j
-2.1342105870794373542 - 2.6980259949956277534e-1616j 2.85e-500, 2.85e-500j
12.724405235921717899 + 2.1560866380097990933e-1603j 6.37e-487, 6.37e-487j
8.2114932477217886613 + 1.0719862989937108199e-1608j 3.53e-488, 3.53e-488j
3.9309600114055574643 - 1.7096185364394085127e-1617j 3.02e-495, 3.02e-495j
-6.5038773509813369939 + 2.1124945647080748604e-1612j 2.15e-490, 2.15e-490j
-2.5822725151665034932 - 4.5110688709241805679e-1620j 9.23e-499, 9.23e-499j
15.978779959323003744 - 1.1573030312607492198e-1607j 6.13e-490, 6.13e-490j
-14.020555690446200963 + 6.8724917370648656545e-1610j 9.18e-488, 9.18e-488j
-1.3556261799742658658 - 6.24518960551149568e-1626j 1.11e-502, 1.11e-502j
-3.3981732054051338643 + 2.3536703480066712305e-1618j 1.85e-496, 1.85e-496j
-4.5428727374767395384 + 5.3300235171185727201e-1616j 5.52e-494, 5.52e-494j
-17.146826918932262617 - 4.5496522595076591743e-1616j 8.8e-492, 8.8e-492j
6.840020876328470958 - 3.7169005719023050174e-1610j 6.97e-490, 6.97e-490j
-14.479529368188253605 + 3.128731590290438039e-1610j 3.92e-488, 3.92e-488j
-0.63033216219547483293 + 3.5234586463201390133e-1629j 3.65e-507, 3.65e-507j
-10.371621666772966066 - 1.0586700835359238822e-1608j 9.62e-487, 9.62e-487j
2.8569700138728056542 + 6.4646304691685117333e-1619j 5.44e-498, 5.44e-498j
1.2855015571280088922 + 2.9409383310074799037e-1624j 3.91e-503, 3.91e-503j
8.9157853481095274549 - 4.3196282763304770897e-1607j 1.51e-487, 1.51e-487j
-1.8550647122034955328 - 3.8719480603313504928e-1625j 2.24e-501, 2.24e-501j
-6.840020876328470958 + 1.0236704946327094339e-1613j 6.68e-490, 6.68e-490j
-2.8569700138728056542 + 6.7083897773564522946e-1622j 5.52e-498, 5.52e-498j
5.8389855202136319245 + 4.5046814321906507792e-1614j 1.69e-491, 1.69e-491j
4.2308850559071941351 + 2.2569846417277151181e-1617j 1.28e-494, 1.28e-494j
-5.8389855202136319245 + 5.6474673849178564599e-1616j 1.71e-491, 1.71e-491j
-0.9467370109606977334 - 1.0848209675024403388e-1628j 1.72e-505, 1.72e-505j
-8.5619137709642672058 + 2.244540305767159861e-1611j 7.67e-488, 7.67e-488j
-7.8642879324855760898 + 1.6402202677808804136e-1612j 1.48e-488, 1.48e-488j
0.63033216219547483293 + 8.9637698520399097425e-1631j 4.04e-507, 4.04e-507j
7.5200898517660283346 + 8.815450200255679714e-1611j 5.96e-489, 5.96e-489j
-9.6349776360215022855 - 8.1272384378481740598e-1610j 4.17e-487, 4.17e-487j
-1.5417968573677566162 + 2.491401829895312291e-1626j 2.34e-502, 2.34e-502j
8.5619137709642672058 + 5.3437462458002205115e-1610j 7.37e-488, 7.37e-488j
2.1342105870794373542 + 2.164725628565761273e-1625j 2.77e-500, 2.77e-500j
2.5822725151665034932 + 2.8155646698774224904e-1624j 9.51e-499, 9.51e-499j
-4.8614540155894593679 + 1.8776836475146130398e-1618j 2.58e-493, 2.58e-493j
-5.5102714425806294624 - 3.4353880943992223682e-1617j 4.04e-492, 4.04e-492j
1.5417968573677566162 - 1.7831390319731602145e-1627j 2.27e-502, 2.27e-502j
-8.2114932477217886613 + 1.7833349557122582128e-1612j 3.63e-488, 3.63e-488j
5.5102714425806294624 + 1.7851151276290370157e-1616j 4.23e-492, 4.23e-492j
6.1702073152077686232 - 5.5606402832067753259e-1615j 6.02e-491, 6.02e-491j
-1.2855015571280088922 + 2.2244045551982551728e-1627j 3.9e-503, 3.9e-503j
-2.3545289224657440776 - 8.4585021237050142804e-1626j 1.89e-499, 1.89e-499j
7.1787182814789551924 + 6.9043018187976653657e-1614j 2.21e-489, 2.21e-489j
0.31507543921086206271 + 1.6573012906120444106e-1632j 1.28e-508, 1.28e-508j
1.3556261799742658658 + 1.0714904843424162535e-1627j 9.94e-503, 9.94e-503j
-10.747473090006847468 + 3.5000767044789267803e-1611j 1.01e-486, 1.01e-486j
-0.31507543921086206271 + 4.2978323019671084094e-1633j 1.32e-508, 1.32e-508j
1.1889534340544512548e-1679 + 9.6343508773899415937e-1680j 1.58e-1677, 1.58e-1677j
2.3545289224657440776 - 6.7442749563780646656e-1624j 1.89e-499, 1.89e-499j

Weights

Weight (ball center) Error (ball radius)
2.8050819530312969665e-53 + 8.5403770925871692463e-1171j 1.52e-137, 2.05e-380j
2.8050819530312969665e-53 - 1.166883608188068073e-1171j 4.38e-138, 5.89e-381j
2.7083433832334531463e-70 + 6.8731484586917188697e-1180j 2.84e-144, 3.82e-387j
2.7083433832334531463e-70 - 1.4839210049257872631e-1180j 1.19e-144, 1.6e-387j
2.4654921622943330932e-30 + 3.9261987148849984808e-1157j 5.64e-126, 7.59e-369j
0.00013302051816820357059 - 1.9670273743542212919e-1142j 3.55e-78, 4.78e-321j
1.6508362235331238603e-41 - 2.2832402691622740863e-1165j 4.89e-133, 6.57e-376j
6.5192644614489559115e-25 - 2.8392372597884631723e-1156j 1.54e-122, 2.08e-365j
3.0407768802821608824e-20 + 1.2312907411956028286e-1152j 5.3e-119, 7.13e-362j
2.4654921622943330932e-30 + 2.6984356418733832665e-1159j 7.66e-127, 1.03e-369j
2.4681091521996607884e-32 + 1.2768536251640783596e-1158j 1.21e-128, 1.62e-371j
7.7872964851026269897e-57 - 1.022183921369391222e-1172j 5.46e-141, 7.35e-384j
9.3476302914937275866e-61 - 1.452161510220502893e-1175j 1.76e-142, 2.37e-385j
5.546996218647832434e-47 - 2.6987479594420261257e-1168j 1.42e-136, 1.91e-379j
1.0076105943082754904e-21 - 1.8538527396619897526e-1154j 7.06e-122, 9.49e-365j
1.2496802786226781883e-26 + 3.0800187599485968172e-1157j 3.91e-125, 5.26e-368j
7.7744971466165397429e-19 - 6.9382020550424159107e-1152j 1.68e-121, 2.26e-364j
9.6536126222175170269e-77 - 2.747832298205678882e-1183j 1.28e-151, 1.72e-394j
5.1235678878184675112e-39 + 5.0081692812081501639e-1164j 1.5e-133, 2.02e-376j
1.8878348777655657322e-07 + 6.6310614880711655289e-1145j 2.81e-104, 3.78e-347j
5.1246015893505141857e-15 + 1.7344136159140325052e-1150j 2.8e-117, 3.77e-360j
2.8072392015990304499e-23 + 3.4005135392197385042e-1154j 3.45e-126, 4.65e-369j
1.9277410900616273973e-34 - 4.5365603576531574918e-1160j 6.99e-134, 9.41e-377j
3.6528920647331769543e-65 + 7.0995389915263049509e-1178j 3.9e-147, 5.25e-390j
7.1752925459390373152e-14 - 4.2613096711759454335e-1149j 9.15e-119, 1.23e-361j
2.8072392015990304499e-23 + 2.3927177407762594581e-1155j 1.72e-125, 2.31e-368j
1.1508496073439196902e-36 + 1.8834190626572183115e-1161j 7.98e-136, 1.07e-378j
1.6508362235331238603e-41 + 2.7926414820724971237e-1164j 1.83e-139, 2.46e-382j
1.9277410900616273973e-34 + 1.5136212383499758604e-1161j 1.83e-135, 2.46e-378j
9.3476302914937275866e-61 + 8.052305991538740462e-1175j 2.08e-148, 2.79e-391j
5.1235678878184675112e-39 - 7.6095988090712195209e-1163j 4.7e-138, 6.33e-381j
5.1935616568634855825e-50 - 5.1492187562369339876e-1169j 6.35e-144, 8.55e-387j
0.021271989003343203834 + 2.824389645517324392e-1140j 1.67e-84, 2.25e-327j
2.4681091521996607884e-32 - 2.1465019642163620045e-1160j 8.32e-136, 1.12e-378j
5.1935616568634855825e-50 + 6.5857994089343122078e-1170j 4.45e-145, 5.99e-388j
1.8878348777655657322e-07 + 2.513301992822044609e-1145j 1.91e-116, 2.58e-359j
3.0407768802821608824e-20 + 1.3273430838237273131e-1153j 5.51e-130, 7.42e-373j
0.081092691739480570341 + 3.5247299744799019452e-1140j 2.73e-89, 3.68e-332j
9.4480137332659353181e-07 - 9.770720849140008563e-1145j 8.62e-115, 1.16e-357j
0.00013302051816820357059 - 3.7957143169968057111e-1142j 7.27e-110, 9.78e-353j
9.6536126222175170269e-77 + 6.5210110176919522321e-1184j 4.09e-158, 5.51e-401j
7.1700292671818446511e-10 - 1.2775542316735826504e-1146j 1.3e-123, 1.75e-366j
4.1588755082024196059e-06 + 3.7947108396657899342e-1144j 6.33e-114, 8.51e-357j
1.9553609809056901428e-28 - 1.7755602950871145603e-1156j 2.89e-137, 3.88e-380j
0.0007944412591962034656 - 1.1839004939247898658e-1141j 4.44e-106, 5.98e-349j
0.0007944412591962034656 - 2.0705193183007924722e-1141j 6.03e-108, 8.12e-351j
8.7486029563406878422e-13 + 1.8838313298191332222e-1148j 3.3e-127, 4.43e-370j
3.7095242845118275612e-44 + 8.6965196123451480953e-1167j 2.71e-143, 3.64e-386j
1.9553609809056901428e-28 - 3.0353998017562406175e-1158j 2.73e-135, 3.67e-378j
1.5900320093392695451e-05 - 3.2051405426349156189e-1143j 4.8e-117, 6.45e-360j
8.7177033606316996012e-11 + 9.0350258652510003705e-1148j 2.24e-126, 3.01e-369j
5.1163771451049529598e-05 + 1.1653810535395021039e-1142j 2.64e-115, 3.55e-358j
0.00031351045267912587243 + 5.24664555207815792e-1142j 2.51e-111, 3.37e-354j
0.010336661715702960965 - 3.0300693555206544702e-1140j 7.79e-105, 1.05e-347j
1.1508496073439196902e-36 - 9.3587473211231410067e-1163j 6.77e-140, 9.1e-383j
3.1733468541598382796e-16 - 3.1766050873644902351e-1151j 6.39e-131, 8.59e-374j
5.1163771451049529598e-05 + 5.7211367708001647208e-1143j 3.09e-115, 4.16e-358j
8.7177033606316996012e-11 + 3.249689122052019885e-1147j 1.19e-128, 1.6e-371j
0.0036764440491090539153 - 1.1575271641431609845e-1140j 1.57e-110, 2.11e-353j
7.7872964851026269897e-57 + 1.6841667522717190091e-1173j 1.13e-150, 1.52e-393j
3.7095242845118275612e-44 - 8.9164052027476702356e-1166j 4.6e-149, 6.19e-392j
-0.036560101135249789308 + 3.3352721568742732936e-1139j 6.63e-102, 8.92e-345j
0.00031351045267912587243 + 9.6410758885182431559e-1142j 6.96e-116, 9.37e-359j
4.1588755082024196059e-06 + 8.7414588316502651021e-1144j 4.77e-122, 6.42e-365j
3.6528920647331769543e-65 - 3.5968386201112867529e-1177j 1.6e-159, 2.15e-402j
9.3246713882115973256e-12 - 2.0318024146907185774e-1148j 1.02e-131, 1.37e-374j
5.546996218647832434e-47 + 2.3865954224192480453e-1167j 1.13e-150, 1.52e-393j
0.10319069362451338453 - 2.6172969380753606391e-1140j 7.8e-108, 1.05e-350j
6.5192644614489559115e-25 - 5.4339615557803800608e-1155j 2.55e-141, 3.44e-384j
0.0019040404634880607638 + 2.7745993242769504434e-1141j 2.03e-118, 2.73e-361j
0.084543024415864831337 - 2.1949681820417598829e-1139j 5.54e-109, 7.45e-352j
7.7744971466165397429e-19 - 8.824092938579292144e-1153j 3.07e-136, 4.13e-379j
0.021271989003343203834 + 3.9094571879316958532e-1140j 1.21e-113, 1.62e-356j
9.3246713882115973256e-12 - 7.9796210136110377246e-1148j 9.38e-133, 1.26e-375j
0.0019040404634880607638 + 4.6062552117837794613e-1141j 9.09e-119, 1.22e-361j
5.2004740815870994794e-09 + 1.5930933738917147862e-1146j 1.08e-130, 1.46e-373j
1.5900320093392695451e-05 - 1.4824502108661621899e-1143j 1.39e-124, 1.87e-367j
5.2004740815870994794e-09 + 4.871966764209946164e-1146j 1.33e-130, 1.79e-373j
0.081092691739480570341 + 4.1564002346059018395e-1140j 7.92e-114, 1.07e-356j
1.6955320965161126909e-17 + 3.7206309180332820472e-1151j 1.46e-138, 1.96e-381j
5.1246015893505141857e-15 + 9.2098421470832768645e-1150j 7.34e-137, 9.88e-380j
0.10319069362451338453 - 2.345566657649444024e-1140j 3.83e-118, 5.15e-361j
7.1752925459390373152e-14 - 8.9404516853126918608e-1150j 5.74e-137, 7.72e-380j
1.0076105943082754904e-21 - 2.0860004798772656521e-1153j 3.29e-141, 4.43e-384j
0.041701319994551406 - 1.0872344967721547827e-1139j 6.11e-120, 8.22e-363j
1.6955320965161126909e-17 + 5.4708639695137307213e-1152j 1.62e-138, 2.18e-381j
0.010336661715702960965 - 2.0823172248148988596e-1140j 9.83e-125, 1.32e-367j
0.0036764440491090539153 - 7.3338938738263603701e-1141j 8.84e-126, 1.19e-368j
9.4480137332659353181e-07 - 2.4053700804618006021e-1144j 9.55e-131, 1.28e-373j
3.3322618020446188699e-08 - 1.8129423650756754104e-1145j 2.23e-132, 3e-375j
0.041701319994551406 - 8.3041317850157314673e-1140j 2.67e-125, 3.59e-368j
3.1733468541598382796e-16 - 1.8974875406927017301e-1150j 1.44e-138, 1.94e-381j
3.3322618020446188699e-08 - 6.392669827828866767e-1146j 1.37e-134, 1.85e-377j
7.1700292671818446511e-10 - 3.8599924485068714888e-1147j 3.08e-136, 4.15e-379j
0.084543024415864831337 - 2.7467767613963820174e-1139j 8.06e-127, 1.07e-369j
0.0050594192066525087231 + 2.2815046460518836076e-1140j 1.1e-128, 1.46e-371j
8.7486029563406878422e-13 + 4.3685046122091655204e-1149j 2.29e-138, 3.1e-381j
0.11962556296673448374 + 1.9934474314369765785e-1140j 5.25e-129, 7.05e-372j
-0.036560101135249789308 + 2.632539025341114429e-1139j 2.55e-128, 3.41e-371j
1.2496802786226781883e-26 + 8.9268230127769538674e-1156j 9.26e-147, 1.18e-389j
0.11962556296673448374 + 2.1056331053225231824e-1140j 7.3e-130, 9.51e-373j
0.12568977167260647557 - 1.9378810907028892088e-1140j 5.81e-130, 6.99e-373j
0.0050594192066525087231 + 1.5067319015150740137e-1140j 1.19e-130, 2.13e-373j

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