Rule: Q[2, 3, 12, 82]

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)
-15.314904483538265668 + 4.1468199363110572986e-1169j 7.8e-491, 7.8e-491j
15.314904483538265668 + 2.2610579318150325661e-1178j 8.02e-491, 8.02e-491j
-14.783399569231404891 + 1.0253080510248422596e-1178j 3.91e-490, 3.91e-490j
13.795833436519735739 - 2.7401163192897624481e-1190j 3.76e-489, 3.76e-489j
-17.189081891103674398 - 6.0569962113244038862e-1197j 7.96e-494, 7.96e-494j
16.497441656394839913 - 7.4748330391569192618e-1200j 1.29e-492, 1.29e-492j
-14.278859478878513082 + 1.1666159189632189398e-1208j 1.3e-489, 1.3e-489j
18.026784834214697398 + 3.7565106138924888871e-1224j 1.8e-495, 1.8e-495j
12.015529443887114537 - 8.6943638673251114937e-1230j 4.99e-488, 4.99e-488j
6.0218513061451774811 + 7.0952539945158865685e-1246j 5.49e-491, 5.49e-491j
-11.189962110712511445 - 1.5255247016211856486e-1260j 7.6e-488, 7.6e-488j
-12.88001911945555673 - 6.5674376523235502857e-1290j 1.83e-488, 1.83e-488j
17.189081891103674398 - 1.2955262410829844494e-1305j 8.34e-494, 8.34e-494j
9.2478681753193681524 - 4.8723523349750320803e-1298j 3.44e-488, 3.44e-488j
14.783399569231404891 - 1.2797667432228014228e-1297j 3.59e-490, 3.59e-490j
4.4445314428073408291 + 4.802303858739176186e-1307j 1.73e-493, 1.73e-493j
14.278859478878513082 - 1.0711239926635414272e-1300j 1.29e-489, 1.29e-489j
-1.9570793272785218266 + 9.7859633965222919431e-1332j 1.09e-501, 1.09e-501j
-2.9802876226142240763 + 7.7923712216801071421e-1328j 4.43e-498, 4.43e-498j
12.88001911945555673 - 1.8496694106277558581e-1316j 1.8e-488, 1.8e-488j
-13.795833436519735739 - 7.9934550148033821281e-1345j 3.66e-489, 3.66e-489j
2.9802876226142240763 + 1.8928767278119808294e-1374j 4.54e-498, 4.54e-498j
6.7189925345537940073 + 1.1711779002428412554e-1363j 3.01e-490, 3.01e-490j
3.3008613454059083377 + 3.9852191652344842603e-1380j 4.45e-497, 4.45e-497j
-5.1972139741334507794 + 5.2933014943009107473e-1369j 2.88e-490, 2.88e-490j
-11.598467076114231611 + 9.0165307619792668912e-1390j 6.33e-488, 6.33e-488j
10.395066553463989953 - 2.8527620912864421361e-1403j 7.07e-488, 7.07e-488j
9.6249952922651289788 + 1.5861310431614021657e-1404j 4.69e-488, 4.69e-488j
8.1429435497375842057 - 2.4448528066869919689e-1408j 6.3e-489, 6.3e-489j
-7.0705978488736175516 - 1.465228410810039555e-1418j 7.05e-490, 7.05e-490j
-13.330492871772509116 + 3.9866504442125222376e-1449j 8.95e-489, 8.95e-489j
15.881650436438002231 + 1.0870481513592636614e-1471j 1.24e-491, 1.24e-491j
1.3174441424841858247 + 2.6733389399354914657e-1507j 2.55e-504, 2.55e-504j
-12.015529443887114537 - 7.6740566943557089655e-1489j 4.52e-488, 4.52e-488j
-5.6722419346222860645 - 6.2184938354731149507e-1509j 2.5e-491, 2.5e-491j
-4.2051618777898870634 - 9.8685178738275581083e-1518j 3.75e-494, 3.75e-494j
-18.026784834214697398 - 1.8559593338604101323e-1511j 1.81e-495, 1.81e-495j
12.442261017176592161 + 2.1498204394456527883e-1510j 3.14e-488, 3.14e-488j
-12.442261017176592161 + 7.0700089152289456064e-1528j 3.09e-488, 3.09e-488j
-15.881650436438002231 - 7.4040304554365061901e-1539j 1.3e-491, 1.3e-491j
-4.4445314428073408291 + 1.5264580575674040431e-1544j 1.58e-493, 1.58e-493j
-7.42492479354365843 - 1.4421112917593258216e-1538j 1.54e-489, 1.54e-489j
7.7822786455193568437 + 6.4889490476272849669e-1539j 3.34e-489, 3.34e-489j
7.42492479354365843 + 1.053005909917858041e-1539j 1.56e-489, 1.56e-489j
-2.6735143535111018002 - 7.9104419583535531647e-1550j 7.05e-499, 7.05e-499j
13.330492871772509116 + 1.9762074835502901149e-1538j 8.81e-489, 8.81e-489j
-2.4494897427831780982 + 8.7766963589572283686e-1556j 1.53e-499, 1.53e-499j
8.5072131011482296202 + 3.6973765812476955102e-1543j 1.22e-488, 1.22e-488j
0.72406087233068678388 + 1.0050836168353526902e-1565j 2.15e-506, 2.15e-506j
-16.497441656394839913 - 3.5549738141398719342e-1547j 1.4e-492, 1.4e-492j
4.2051618777898870634 - 5.7427128436953649513e-1560j 3.51e-494, 3.51e-494j
5.6722419346222860645 - 5.1294379038940208717e-1557j 2.68e-491, 2.68e-491j
-5.1387830879005830418 - 3.9521917009828403188e-1555j 8.76e-491, 8.76e-491j
-10.789089009653512602 - 2.473769480343084448e-1552j 7.87e-488, 7.87e-488j
2.6735143535111018002 + 1.7464248184587439714e-1572j 7.22e-499, 7.22e-499j
5.2390248951021214092 - 3.8482534404753028113e-1564j 1.92e-490, 1.92e-490j
3.9316092693297583224 - 1.2187437875762104296e-1571j 4.69e-495, 4.69e-495j
-6.7189925345537940073 + 3.2055042527658765997e-1563j 3e-490, 3e-490j
-4.7316953251106660193 + 4.590289376269783006e-1566j 8.71e-493, 8.71e-493j
7.0705978488736175516 + 6.6265611663119010851e-1564j 6.81e-490, 6.81e-490j
11.189962110712511445 - 8.7807010177494286955e-1561j 7.48e-488, 7.48e-488j
4.7316953251106660193 - 1.6766204800709965315e-1566j 8.64e-493, 8.64e-493j
-0.58242593157741849169 - 3.5866241835969912423e-1580j 6.17e-507, 6.17e-507j
5.1972139741334507794 + 1.730232059742311091e-1563j 2.71e-490, 2.71e-490j
1 - 5.7547625926995248664e-1579j 1.54e-505, 1.54e-505j
8.8754045718181084703 - 9.2083746318328367607e-1562j 2.05e-488, 2.05e-488j
5.1387830879005830418 + 1.7925547380470677873e-1564j 9.04e-491, 9.04e-491j
-1.3174441424841858247 + 1.9109700978194271102e-1577j 2.64e-504, 2.64e-504j
-0.72406087233068678388 + 4.2572142207285908318e-1580j 2.39e-506, 2.39e-506j
-3.3008613454059083377 + 3.7639347410002728946e-1570j 4.47e-497, 4.47e-497j
-8.5072131011482296202 - 1.2382104651232170044e-1561j 1.17e-488, 1.17e-488j
-8.1429435497375842057 + 2.6375605256677317779e-1562j 6.97e-489, 6.97e-489j
0.31503134522768178133 - 1.0973682922016026874e-1585j 1.56e-508, 1.56e-508j
-2.2535044010417626129 + 4.5652357087235191361e-1578j 2.67e-500, 2.67e-500j
1.9570793272785218266 + 6.3415887055771031771e-1579j 1.11e-501, 1.11e-501j
3.6214835307306586622 - 6.4921728158717899798e-1574j 4.3e-496, 4.3e-496j
0.58242593157741849169 - 4.5559769557501444635e-1584j 6.19e-507, 6.19e-507j
-7.7822786455193568437 - 4.954621671582338474e-1565j 3.29e-489, 3.29e-489j
-0.31503134522768178133 - 3.122614245946387539e-1590j 1.56e-508, 1.56e-508j
-5.2390248951021214092 - 9.1731386241003531258e-1572j 1.78e-490, 1.78e-490j
10.007227370103011681 - 3.708352316606462392e-1570j 5.95e-488, 5.95e-488j
2.4494897427831780982 + 9.0589432238299627817e-1581j 1.64e-499, 1.64e-499j
-9.6249952922651289788 + 1.1204838000186901159e-1567j 4.52e-488, 4.52e-488j
-8.8754045718181084703 - 2.6041261837933557412e-1591j 2.13e-488, 2.13e-488j
-6.3696949628033208131 - 2.6079988246449326624e-1605j 1.27e-490, 1.27e-490j
-10.007227370103011681 + 1.8486904748332769765e-1616j 6.18e-488, 6.18e-488j
11.598467076114231611 + 3.3562613262718949586e-1634j 6.18e-488, 6.18e-488j
-10.395066553463989953 + 1.2302933100243847965e-1644j 7.45e-488, 7.45e-488j
-6.0218513061451774811 + 9.0605683193524053451e-1660j 5.48e-491, 5.48e-491j
-1.638713499982732431 + 1.6645386954808953199e-1672j 4.76e-503, 4.76e-503j
-3.9316092693297583224 - 4.5578286897854371581e-1664j 4.92e-495, 4.92e-495j
6.3696949628033208131 + 1.9340002338717487754e-1660j 1.16e-490, 1.16e-490j
2.2535044010417626129 - 2.3129757673807492372e-1670j 2.76e-500, 2.76e-500j
10.789089009653512602 - 4.9820906684079303034e-1657j 7.97e-488, 7.97e-488j
-3.6214835307306586622 - 2.166464468608292181e-1665j 4e-496, 4e-496j
-1 - 1.339684491722324721e-1674j 1.71e-505, 1.71e-505j
-9.2478681753193681524 + 7.0545244926543597767e-1657j 3.4e-488, 3.4e-488j
1.638713499982732431 - 6.2012103864880021731e-1676j 4.75e-503, 4.75e-503j
1.2497895675333150241e-1724 - 1.7048142325365819967e-1723j 1.69e-1721, 1.69e-1721j

Weights

Weight (ball center) Error (ball radius)
2.5594450394246330659e-52 + 8.7153546285140562322e-1220j 4.97e-136, 1.7e-379j
2.5594450394246330659e-52 + 3.4651068723237117892e-1222j 6.03e-137, 2.07e-380j
7.1953776462791016264e-49 - 1.4298006935933133081e-1218j 9.99e-135, 3.42e-378j
8.8676599600525200927e-43 - 5.2404315166738374901e-1217j 6.07e-133, 2.08e-376j
2.0612648742488581491e-65 + 6.0732148358705651802e-1228j 4.7e-142, 1.61e-385j
2.0516429766997323341e-60 + 1.59349127866746334e-1226j 1.32e-140, 4.51e-384j
1.0484416721113395817e-45 + 3.767289260065857803e-1217j 3.2e-134, 1.1e-377j
1.0521637729265030645e-71 + 1.6386047737997010586e-1232j 4.24e-145, 1.45e-388j
7.5113800283787913903e-33 - 1.7140906637949582892e-1211j 3.03e-129, 1.04e-372j
1.8526994105183936343e-09 - 1.7684749961048377716e-1196j 7.78e-104, 2.66e-347j
1.0416782486547731039e-28 - 2.5233542091100419694e-1208j 1.35e-127, 4.63e-371j
1.6767964394761047928e-37 - 4.9399197929206880586e-1213j 3.46e-132, 1.19e-375j
2.0612648742488581491e-65 - 3.5018111275687344205e-1229j 5.1e-143, 1.75e-386j
4.0135293218232484271e-20 + 4.8800158798696452244e-1204j 1.03e-121, 3.52e-365j
7.1953776462791016264e-49 - 2.5248801197226728915e-1220j 4.82e-137, 1.65e-380j
5.2022534168331593179e-06 - 3.4390690466108826223e-1193j 8.77e-98, 3.01e-341j
1.0484416721113395817e-45 + 1.3188863716236198004e-1218j 6.75e-136, 2.31e-379j
0.018437380681619400988 + 6.34906854155044598e-1191j 1.35e-71, 4.62e-315j
0.001493058526709152699 + 1.0690988804426930981e-1191j 1.22e-86, 4.18e-330j
1.6767964394761047928e-37 - 4.2660652420545699703e-1214j 2.84e-133, 9.74e-377j
8.8676599600525200927e-43 - 1.0042507859621338225e-1215j 1.42e-138, 4.86e-382j
0.001493058526709152699 + 7.2078636815111824612e-1192j 3.6e-87, 1.23e-330j
2.1996246369491829962e-11 - 3.9668483468573499039e-1198j 1.2e-113, 4.1e-357j
0.0005529014595031520792 - 2.9581322817425027639e-1192j 1.36e-90, 4.66e-334j
-2.2161585318034032876e-06 + 1.6382088528194219406e-1192j 4.14e-109, 1.42e-352j
1.0112184014124114968e-30 + 2.0014408355164228068e-1209j 3.84e-134, 1.32e-377j
5.3521392863127632101e-25 - 5.6722782417967782419e-1207j 3.25e-128, 1.11e-371j
1.1577142575324093617e-21 - 5.5744271164516585609e-1205j 1.27e-125, 4.35e-369j
5.7759192865390872026e-16 - 2.175899324268279536e-1201j 3.29e-121, 1.13e-364j
1.9618065064568072595e-12 + 1.7004273297126120022e-1198j 3.18e-122, 1.09e-365j
4.7184013763808136689e-40 + 2.3850328223779225359e-1214j 8.62e-140, 2.95e-383j
3.9822883766972640697e-56 - 3.1198305472794907687e-1224j 8.37e-144, 2.87e-387j
0.053724040181747627377 + 9.2701806371811164777e-1191j 1.17e-73, 4.02e-317j
7.5113800283787913903e-33 - 1.4198701632783086485e-1210j 4.16e-137, 1.43e-380j
1.456802580685757253e-08 + 3.2976007889496614166e-1195j 5.64e-117, 1.93e-360j
1.4163101883845628971e-05 + 1.0987611452762056271e-1192j 1.01e-108, 3.46e-352j
1.0521637729265030645e-71 - 2.0146114116233358656e-1231j 7.43e-156, 2.54e-399j
4.1672964960251239027e-35 + 9.2439145604143877909e-1213j 3.12e-137, 1.07e-380j
4.1672964960251239027e-35 + 8.9320122790363682931e-1212j 2.15e-138, 7.38e-382j
3.9822883766972640697e-56 + 1.7173119995778614582e-1222j 8.63e-149, 2.96e-392j
5.2022534168331593179e-06 - 6.2513093386448989634e-1193j 1.27e-112, 4.37e-356j
1.516654245622667485e-13 - 2.821181710119320123e-1199j 5.89e-127, 2.02e-370j
1.010841502193585849e-14 + 1.4887526689112695442e-1200j 6.18e-127, 2.12e-370j
1.516654245622667485e-13 - 9.7885811292027931076e-1200j 4.37e-126, 1.5e-369j
0.0032003916747687682481 - 3.212530732131864601e-1191j 8.7e-104, 2.98e-347j
4.7184013763808136689e-40 + 1.6522329040614233628e-1215j 7.93e-142, 2.72e-385j
0.003196786906909953437 + 6.819854611690426985e-1191j 2.75e-102, 9.44e-346j
2.8123690297000741468e-17 + 3.0261460245476197345e-1202j 1.52e-129, 5.22e-373j
0.060813884128018828909 + 5.5756293317590005085e-1190j 4.86e-92, 1.67e-335j
2.0516429766997323341e-60 - 4.2867739560268815621e-1225j 3.55e-153, 1.21e-396j
1.4163101883845628971e-05 + 6.2535409859724795822e-1193j 9.86e-116, 3.38e-359j
1.456802580685757253e-08 + 1.5151012422338171482e-1195j 5.25e-122, 1.8e-365j
1.449342730595595672e-06 - 9.1572196204614239593e-1193j 9e-121, 3.08e-364j
8.3782982660139833767e-27 + 2.8689318049667396248e-1207j 5.98e-140, 2.05e-383j
0.0032003916747687682481 - 2.2576111133953375434e-1191j 7.95e-109, 2.73e-352j
1.1819458498132040614e-06 - 3.9485443704330633604e-1193j 3.96e-121, 1.36e-364j
5.2309664489915798394e-05 - 9.2148671243975346574e-1193j 7.6e-116, 2.61e-359j
2.1996246369491829962e-11 - 1.016822049229909203e-1197j 4.06e-130, 1.39e-373j
1.7452665279993149552e-06 + 2.6091477131242553002e-1193j 3.92e-121, 1.34e-364j
1.9618065064568072595e-12 + 6.2624658120562753883e-1199j 1.81e-127, 6.2e-371j
1.0416782486547731039e-28 - 3.9270866122217113444e-1209j 1.35e-140, 4.64e-384j
1.7452665279993149552e-06 + 1.3774483564847075419e-1193j 2.2e-120, 7.55e-364j
0.06091103778811356219 - 6.8588362540381033503e-1190j 7.1e-106, 2.43e-349j
-2.2161585318034032876e-06 + 8.080535507619239977e-1193j 1.82e-121, 6.24e-365j
0.075262154453923946963 - 1.960441728777773381e-1190j 4.78e-106, 1.64e-349j
1.1591145678454884507e-18 - 3.9712215416008586565e-1203j 6.7e-135, 2.3e-378j
1.449342730595595672e-06 - 4.5559011365424734921e-1193j 1.21e-121, 4.14e-365j
0.053724040181747627377 + 1.101520364741134029e-1190j 4.3e-110, 1.47e-353j
0.060813884128018828909 + 6.1290030979932607271e-1190j 2.82e-108, 9.67e-352j
0.0005529014595031520792 - 4.583627446058367268e-1192j 3.7e-122, 1.27e-365j
2.8123690297000741468e-17 + 1.0589376406618696083e-1201j 1.51e-142, 5.18e-386j
5.7759192865390872026e-16 - 7.1168702966832070425e-1201j 2.18e-141, 7.48e-385j
0.11708217460698662656 + 3.5943729330080731311e-1190j 9.98e-112, 3.42e-355j
0.0082709911224594815656 - 7.5348794276211512195e-1191j 3.74e-119, 1.28e-362j
0.018437380681619400988 + 4.9102493608159989635e-1191j 6.62e-119, 2.27e-362j
0.00017999514162850277089 + 1.4934220808760978575e-1192j 3.08e-123, 1.06e-366j
0.06091103778811356219 - 6.3562658232440337951e-1190j 4.42e-113, 1.52e-356j
1.010841502193585849e-14 + 4.5649410711545365474e-1200j 3.07e-140, 1.05e-383j
0.11708217460698662656 + 3.745352895403375273e-1190j 1.92e-114, 6.57e-358j
1.1819458498132040614e-06 - 8.0546915504100091753e-1193j 1.63e-130, 5.58e-374j
2.7554128191962999323e-23 + 5.8759610159135440231e-1206j 4.8e-143, 1.64e-386j
0.003196786906909953437 + 4.9391077968799272341e-1191j 1.38e-122, 4.73e-366j
1.1577142575324093617e-21 - 2.443372101796436182e-1204j 3.87e-147, 1.33e-390j
1.1591145678454884507e-18 - 1.4918095792445930005e-1202j 6.44e-145, 2.21e-388j
2.1508653235661400562e-10 + 6.1991177591547950913e-1197j 3.22e-138, 1.1e-381j
2.7554128191962999323e-23 + 2.803332920826298059e-1205j 2.31e-148, 7.91e-392j
1.0112184014124114968e-30 + 2.7637672802005243248e-1210j 4.86e-152, 1.66e-395j
5.3521392863127632101e-25 - 2.9642055620914203091e-1206j 2.89e-149, 9.9e-393j
1.8526994105183936343e-09 - 4.0604006473628090109e-1196j 2.36e-138, 8.07e-382j
0.033437083827080095737 - 7.3498449581858734918e-1191j 1.56e-130, 5.34e-374j
5.2309664489915798394e-05 - 1.5580204464000308837e-1192j 7.88e-135, 2.7e-378j
2.1508653235661400562e-10 + 2.5572253384814729677e-1197j 1.53e-139, 5.26e-383j
0.0082709911224594815656 - 5.601877517909507001e-1191j 1.33e-132, 4.54e-376j
8.3782982660139833767e-27 + 4.974049635114772739e-1208j 1.16e-150, 3.99e-394j
0.00017999514162850277089 + 2.4184556527113553795e-1192j 2.83e-135, 9.69e-379j
0.075262154453923946963 - 2.2343439029153423894e-1190j 9.25e-133, 3.17e-376j
4.0135293218232484271e-20 + 1.9757046884479444157e-1203j 3.39e-147, 1.16e-390j
0.033437083827080095737 - 5.9289930579727399799e-1191j 6.96e-134, 2.29e-377j
0.12672853484846303562 - 2.9044846512490149879e-1190j 7.2e-133, 2.62e-376j

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