Rule: Q[2, 3, 12, 74]

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)
1.3274606821993699152 + 1.0511292559194042514e-1216j 2.64e-504, 2.64e-504j
1 - 2.1246902096969462196e-1217j 2.08e-505, 2.08e-505j
0.72406087233068678388 - 1.2624380533653145745e-1219j 7.23e-506, 7.23e-506j
0.68116796980456124178 - 3.478412520712821232e-1219j 4.87e-506, 4.87e-506j
-0.68116796980456124178 + 9.3074807482833587735e-1219j 4.95e-506, 4.95e-506j
-0.72406087233068678388 - 3.3537133566163404518e-1218j 7.48e-506, 7.48e-506j
-0.33662337323955133113 + 7.1005807967297567405e-1221j 1.49e-508, 1.49e-508j
0.33662337323955133113 - 1.90490546333685145e-1250j 1.34e-508, 1.34e-508j
-2.4213946377471860095e-1253 + 1.8005944627635420218e-1252j 1.65e-1250, 1.65e-1250j
11.924609267059272587 - 3.9618402781573804514e-1201j 4.09e-490, 4.09e-490j
-11.924609267059272587 - 2.1607411143373187142e-1208j 4.51e-490, 4.51e-490j
-12.857853542582256491 + 1.0611468650076216225e-1216j 1.13e-490, 1.13e-490j
-16.310186268543438963 + 4.2486814935885681945e-1228j 4.67e-495, 4.67e-495j
2.7519945104752203117 - 1.0652213292364381272e-1236j 3.52e-499, 3.52e-499j
-15.607180856147852839 + 4.6467371066141662816e-1226j 7.05e-494, 7.05e-494j
-14.980987275767587465 - 2.0546229363166416984e-1233j 5.53e-493, 5.53e-493j
-12.383736399807930002 - 3.8276817923922378521e-1279j 2.32e-490, 2.32e-490j
-2.4494897427831780982 - 1.3909443793402735698e-1323j 4.01e-500, 4.01e-500j
12.857853542582256491 + 1.0874782220579038834e-1359j 1.07e-490, 1.07e-490j
10.200795355541717484 - 1.6310224517309475049e-1437j 1.03e-489, 1.03e-489j
14.980987275767587465 + 1.3331873355204106806e-1475j 5.98e-493, 5.98e-493j
-2.7519945104752203117 - 8.567508762454816663e-1509j 3.37e-499, 3.37e-499j
11.043092702719981466 + 1.1141740302002295132e-1501j 9.22e-490, 9.22e-490j
-9.7916649281929700354 - 2.8333940526268434297e-1514j 9.45e-490, 9.45e-490j
14.404424483838470206 - 7.1638386973544201e-1517j 3.13e-492, 3.13e-492j
-7.0933780743673768171 + 1.0001269270605746451e-1521j 3.11e-491, 3.11e-491j
3.8762133430273856642 + 4.2584452531094336112e-1528j 2.41e-496, 2.41e-496j
-1 + 1.4650114139299059184e-1537j 2.08e-505, 2.08e-505j
6.3696949628033208131 + 1.773579911970406582e-1521j 6.72e-492, 6.72e-492j
5.4590284882793065198 - 1.5804442738867327328e-1529j 7.9e-493, 7.9e-493j
8.9935670190583295153 + 2.0182146084974451761e-1546j 5.34e-490, 5.34e-490j
10.617630657083192649 + 7.9520197701955966519e-1570j 1.07e-489, 1.07e-489j
16.310186268543438963 - 5.4929315710758425139e-1590j 4.55e-495, 4.55e-495j
-6.7282482942979473817 + 1.0678460117367263136e-1584j 1.51e-491, 1.51e-491j
-7.4636728773117972164 - 5.6768867702325159445e-1584j 6.84e-491, 6.84e-491j
13.86349656921300961 - 1.9753804236778197257e-1583j 1.27e-491, 1.27e-491j
-7.8387410182551905286 + 4.5008419047131443314e-1584j 1.39e-490, 1.39e-490j
13.349814175479031554 + 5.0082979491978019094e-1582j 4.41e-491, 4.41e-491j
-3.0431441685578692994 + 1.3792180972749843598e-1596j 2.59e-498, 2.59e-498j
17.16121946408110236 + 2.2747778778327629271e-1594j 1.29e-496, 1.29e-496j
3.3008613454059083377 + 5.1346588013455795625e-1597j 1.41e-497, 1.41e-497j
-3.5682507595846786238 - 9.4087366644470881866e-1596j 6.01e-497, 6.01e-497j
-4.2051618777898870634 - 1.091851986196735376e-1593j 1.26e-495, 1.26e-495j
11.47829656241685731 - 1.0643718458477936414e-1586j 6.6e-490, 6.6e-490j
12.383736399807930002 + 1.9316701537939963558e-1596j 2.51e-490, 2.51e-490j
4.5416322047592374899 + 1.5918682726879728823e-1603j 6.79e-495, 6.79e-495j
-4.8775632543998467282 - 1.5540714824348160121e-1602j 3.81e-494, 3.81e-494j
-9.3894701994864985942 - 1.1856104605122474795e-1596j 7.6e-490, 7.6e-490j
-17.16121946408110236 - 1.6170861501761355698e-1607j 1.2e-496, 1.2e-496j
-14.404424483838470206 - 8.8235246368233769277e-1602j 3.13e-492, 3.13e-492j
-13.349814175479031554 - 1.7317492239935176812e-1600j 4.38e-491, 4.38e-491j
4.2051618777898870634 - 1.2364373418990459033e-1618j 1.27e-495, 1.27e-495j
-2.1678805849940762902 - 2.3025233666361879864e-1624j 4.97e-501, 4.97e-501j
-5.4590284882793065198 + 1.3803326480802821706e-1615j 8.76e-493, 8.76e-493j
-11.043092702719981466 + 5.4086520560453237052e-1612j 8.74e-490, 8.74e-490j
9.7916649281929700354 + 1.0103295246218260506e-1631j 9.8e-490, 9.8e-490j
15.607180856147852839 + 1.427057775102293473e-1649j 6.65e-494, 6.65e-494j
-13.86349656921300961 + 8.7306349174171703588e-1649j 1.34e-491, 1.34e-491j
5.1972139741334507794 - 8.3479459760087733648e-1649j 2.21e-493, 2.21e-493j
-10.617630657083192649 - 7.8679704033668152431e-1646j 9.64e-490, 9.64e-490j
1.9151245624950066718 - 1.591993018338725928e-1657j 6.29e-502, 6.29e-502j
3.0431441685578692994 - 6.2611067656771592302e-1654j 2.87e-498, 2.87e-498j
-3.3008613454059083377 + 2.474841706956678358e-1654j 1.38e-497, 1.38e-497j
-1.638713499982732431 + 4.6294607331229659905e-1659j 4.97e-503, 4.97e-503j
7.0933780743673768171 - 8.1906286055009724077e-1645j 3.32e-491, 3.32e-491j
9.3894701994864985942 + 1.9891595307829745569e-1646j 7.57e-490, 7.57e-490j
7.8387410182551905286 - 4.39524716552874332e-1650j 1.23e-490, 1.23e-490j
-10.200795355541717484 + 8.0567480073546597883e-1656j 1.02e-489, 1.02e-489j
-3.8762133430273856642 - 5.7299736331039420872e-1661j 2.61e-496, 2.61e-496j
2.4494897427831780982 + 1.1934629384762708868e-1664j 3.58e-500, 3.58e-500j
8.218587904759846717 + 6.2038318313361296641e-1653j 2.21e-490, 2.21e-490j
-5.1972139741334507794 - 5.3732171533454605667e-1664j 2.19e-493, 2.19e-493j
-4.5416322047592374899 - 6.912223085225643118e-1666j 6.66e-495, 6.66e-495j
-1.9151245624950066718 - 1.4358091396811880184e-1672j 6.56e-502, 6.56e-502j
6.0226029378228081637 + 5.2960951637926279177e-1662j 3.3e-492, 3.3e-492j
-6.3696949628033208131 + 2.9800203885626689286e-1663j 7.08e-492, 7.08e-492j
7.4636728773117972164 + 8.7443090389191467994e-1660j 6.42e-491, 6.42e-491j
-8.9935670190583295153 - 4.369886725286645866e-1661j 5.4e-490, 5.4e-490j
3.5682507595846786238 - 5.6570363959506238993e-1671j 6.03e-497, 6.03e-497j
2.1678805849940762902 - 1.9725410367054088468e-1675j 4.99e-501, 4.99e-501j
-11.47829656241685731 + 2.4412828406004114127e-1662j 7e-490, 7e-490j
-6.0226029378228081637 - 2.7625191721014786611e-1668j 3.08e-492, 3.08e-492j
-1.3274606821993699152 + 1.5253312841559451886e-1680j 2.87e-504, 2.87e-504j
4.8775632543998467282 + 1.6799136879973746482e-1669j 3.91e-494, 3.91e-494j
6.7282482942979473817 + 2.1779636569409647155e-1665j 1.37e-491, 1.37e-491j
8.6034186757773965373 + 2.187680475812747126e-1675j 3.59e-490, 3.59e-490j
-5.7066512574330546857 + 6.4578352358845469891e-1682j 1.74e-492, 1.74e-492j
1.638713499982732431 + 1.6462967944230568502e-1694j 4.83e-503, 4.83e-503j
5.7066512574330546857 + 8.780294245683473641e-1683j 1.7e-492, 1.7e-492j
-8.6034186757773965373 + 2.3252693304685685426e-1678j 3.65e-490, 3.65e-490j
-8.218587904759846717 - 9.8472098448329110276e-1685j 2.23e-490, 2.23e-490j

Weights

Weight (ball center) Error (ball radius)
0.053142621190335311427 - 4.712270062512248446e-1213j 4.39e-42, 4.35e-286j
0.080552715302929798102 + 1.0735517219901959706e-1212j 6.39e-33, 6.33e-277j
-0.029209844476523484206 - 7.6555188359650976287e-1212j 3.19e-33, 3.17e-277j
0.13370216235239159245 + 7.4738492908119796378e-1212j 2.92e-33, 2.9e-277j
0.13370216235239159245 + 6.6660186409785031598e-1212j 1.29e-34, 1.28e-278j
-0.029209844476523484206 - 6.7789221334580822225e-1212j 1.1e-34, 1.09e-278j
0.12697250894070375603 - 9.2615086896729737209e-1213j 1.79e-35, 1.78e-279j
0.12697250894070375603 - 9.7996495008672170586e-1213j 7.99e-36, 7.92e-280j
0.13427248972539462283 + 7.4242332930476597602e-1213j 5.22e-36, 5.17e-280j
2.3927585370099371735e-32 + 6.3483730127162659921e-1232j 3.74e-128, 3.7e-372j
2.3927585370099371735e-32 + 3.9748330549766160415e-1234j 1.45e-128, 1.44e-372j
2.424733247491234903e-37 + 6.3244347975584895568e-1237j 1.28e-131, 1.27e-375j
5.1812869890208673845e-59 + 1.0774079894130628221e-1248j 8.49e-142, 8.42e-386j
0.0027344386034626175874 + 4.3053007636269655696e-1214j 5.46e-77, 5.41e-321j
3.3548589026196936456e-54 - 4.0688980841515015216e-1246j 4.08e-140, 4.05e-384j
4.400577462168486788e-50 + 6.7339135995409220361e-1244j 1e-138, 9.96e-383j
9.2994999445772085179e-35 - 1.7484964648459267192e-1235j 2.06e-131, 2.04e-375j
0.0059057841626707118034 - 5.5897381512282765151e-1214j 1.96e-77, 1.94e-321j
2.424733247491234903e-37 - 1.6794585972072491777e-1235j 7.24e-138, 7.18e-382j
4.1799654430966487007e-24 + 2.9287429954415248582e-1228j 5.89e-130, 5.84e-374j
4.400577462168486788e-50 - 5.927916908986662528e-1243j 9.76e-145, 9.68e-389j
0.0027344386034626175874 + 2.6907696100577432054e-1214j 6.55e-86, 6.49e-330j
5.6679326889322524057e-28 + 3.2619153295872435863e-1230j 4.91e-133, 4.86e-377j
2.4523834346853044039e-22 - 2.5933833994266616977e-1228j 2.51e-129, 2.48e-373j
1.956360087863889415e-46 + 6.7918977795261636876e-1241j 3.97e-143, 3.93e-387j
1.7400941260505646461e-12 + 6.5348184239503558001e-1222j 1.2e-119, 1.19e-363j
7.0099686448379031184e-05 + 1.4917096080276636252e-1215j 8.59e-101, 8.51e-345j
0.080552715302929798102 + 9.0739328376961201444e-1213j 4.16e-63, 4.13e-307j
2.1869817556968751685e-10 + 9.8119599053912935496e-1220j 2.62e-117, 2.6e-361j
3.097014744333189394e-08 - 1.181758190185596532e-1217j 1.98e-112, 1.96e-356j
4.2800640105614913317e-19 - 1.7864301692730286272e-1225j 2.5e-128, 2.48e-372j
5.5625954837340328076e-26 - 3.0994494002843316281e-1229j 1.84e-132, 1.83e-376j
5.1812869890208673845e-59 - 6.9364494932027489971e-1248j 1.28e-150, 1.26e-394j
2.1368757533890987904e-11 - 4.412260644604535399e-1221j 1.7e-122, 1.68e-366j
1.1906380701612549439e-13 - 9.4654509837895971648e-1223j 1.35e-125, 1.34e-369j
3.8646879323941681269e-43 - 5.347672820487819158e-1239j 2.45e-143, 2.43e-387j
6.8384034011238680811e-15 + 1.3185973381701116242e-1223j 5.34e-128, 5.29e-372j
4.0015060183281790192e-40 + 3.2337561485873644416e-1237j 5.8e-142, 5.75e-386j
0.0010679437076922325569 - 1.4585639884906912926e-1214j 1.67e-103, 1.66e-347j
4.3912428656027334022e-65 + 3.4971999514636691144e-1251j 1.91e-154, 1.89e-398j
0.00043009529571036115056 + 1.3015880708952337067e-1214j 3.3e-106, 3.27e-350j
0.00019868997078715998976 - 2.6992878792850120168e-1215j 6.7e-109, 6.65e-353j
1.9272985536675071725e-05 - 2.1539727414322965875e-1216j 1.02e-113, 1.01e-357j
4.3194859036450111256e-30 - 3.9913520308625423642e-1231j 1.49e-137, 1.48e-381j
9.2994999445772085179e-35 + 9.2573140186108462908e-1234j 8.25e-140, 8.18e-384j
4.4706071147219223166e-06 + 1.4916800557003229554e-1216j 5.38e-117, 5.33e-361j
9.0388518767812722409e-07 - 2.3636994094272876236e-1217j 7.75e-120, 7.68e-364j
1.1419887820458072101e-20 + 2.666063151380748474e-1227j 1.43e-137, 1.42e-381j
4.3912428656027334022e-65 - 6.2963676704375454724e-1252j 1.07e-159, 1.06e-403j
1.956360087863889415e-46 - 6.3970037806236805629e-1242j 5.42e-152, 5.37e-396j
4.0015060183281790192e-40 - 1.8234951889233699881e-1238j 2.2e-149, 2.18e-393j
1.9272985536675071725e-05 - 4.5006906751151517934e-1216j 1.63e-120, 1.61e-364j
0.0099884144884848634929 + 1.1658352093819428188e-1213j 3.8e-107, 3.77e-351j
3.097014744333189394e-08 - 4.395393000544705934e-1218j 8.03e-127, 7.96e-371j
5.6679326889322524057e-28 + 1.2519473742204351885e-1231j 1.45e-144, 1.44e-388j
2.4523834346853044039e-22 - 2.6404110716578200691e-1227j 7.28e-141, 7.22e-385j
3.3548589026196936456e-54 + 3.0419994851788655384e-1245j 1.05e-155, 1.04e-399j
3.8646879323941681269e-43 + 4.0206763422099497417e-1240j 2.95e-151, 2.93e-395j
1.6271178300322721206e-07 + 2.5491583543557649322e-1217j 1.19e-128, 1.18e-372j
5.5625954837340328076e-26 - 1.7970355669833134932e-1230j 7.5e-144, 7.43e-388j
0.016299830704538835758 - 2.6007784180685074775e-1213j 1.12e-112, 1.11e-356j
0.0010679437076922325569 - 2.4580619815911360486e-1214j 5.34e-119, 5.29e-363j
0.00043009529571036115056 + 7.3723007401327058398e-1215j 4.28e-120, 4.24e-364j
0.030983442384041894137 + 2.5189882375079081184e-1213j 3.96e-111, 3.93e-355j
1.7400941260505646461e-12 + 2.572400832998346508e-1221j 5.96e-137, 5.91e-381j
1.1419887820458072101e-20 + 2.2414763085530161975e-1226j 1.89e-141, 1.87e-385j
6.8384034011238680811e-15 + 6.3780362283731369551e-1223j 1.89e-139, 1.88e-383j
4.1799654430966487007e-24 + 2.281819485426404812e-1229j 1.66e-145, 1.64e-389j
7.0099686448379031184e-05 + 7.5983147624527832541e-1216j 7.66e-127, 7.59e-371j
0.0059057841626707118034 - 8.4797058621868422639e-1214j 6.82e-122, 6.76e-366j
3.281367938941639968e-16 - 9.4786252630365573988e-1224j 2.09e-140, 2.07e-384j
1.6271178300322721206e-07 + 1.0016041107160264185e-1217j 9.52e-133, 9.44e-377j
4.4706071147219223166e-06 + 6.6882959229490169595e-1217j 2.28e-130, 2.26e-374j
0.016299830704538835758 - 1.8810427419343534675e-1213j 4.97e-121, 4.93e-365j
1.7890363235084701454e-09 - 6.2426785679732484847e-1219j 1.83e-136, 1.82e-380j
2.1869817556968751685e-10 + 2.9793331834589835062e-1220j 2.1e-137, 2.08e-381j
1.1906380701612549439e-13 - 4.1138939778646201221e-1222j 5e-140, 4.95e-384j
4.2800640105614913317e-19 - 2.5031418723987084541e-1226j 5.7e-145, 5.65e-389j
0.00019868997078715998976 - 5.0044919818821598282e-1215j 4.31e-130, 4.28e-374j
0.0099884144884848634929 + 1.6838844093553206221e-1213j 1.6e-126, 1.59e-370j
4.3194859036450111256e-30 - 7.6118336311158227239e-1233j 1.93e-150, 1.91e-394j
1.7890363235084701454e-09 - 2.0529369742801232168e-1219j 3.77e-137, 3.74e-381j
0.053142621190335311427 - 3.7681835783816273906e-1213j 2.12e-125, 2.1e-369j
9.0388518767812722409e-07 - 5.6357023642450709339e-1217j 1.31e-134, 1.3e-378j
2.1368757533890987904e-11 - 1.5838188089147712969e-1220j 2.36e-139, 2.34e-383j
1.3065216704986976549e-17 + 1.3389441055544679487e-1224j 1.13e-143, 1.12e-387j
9.6328895430104721972e-09 + 1.2964254804452476053e-1218j 1.71e-137, 1.7e-381j
0.030983442384041894137 + 3.3214713089786871622e-1213j 1.58e-130, 1.57e-374j
9.6328895430104721972e-09 + 3.6760468398688509717e-1218j 4.79e-137, 4.75e-381j
1.3065216704986976549e-17 + 2.1662584178541960322e-1225j 5.78e-146, 5.74e-390j
3.281367938941639968e-16 - 1.7439185678046961279e-1224j 4.77e-145, 4.73e-389j

Download

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