Rule: Q[6, 9, 16, 66]

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)
-14.40673561483361294 - 3.457883847538860705e-1296j 8.81e-492, 8.81e-492j
-12.391486133499333228 + 1.864048312274718057e-1293j 8.43e-490, 8.43e-490j
-14.982055604227382293 + 8.8266976045872290015e-1302j 1.45e-492, 1.45e-492j
-16.308732178050861921 + 1.0580917481465840104e-1299j 1.07e-494, 1.07e-494j
-12.864132707639172954 - 1.1355354485117040525e-1295j 3.42e-490, 3.42e-490j
-15.607007568072038214 + 9.5651949811080254362e-1299j 1.67e-493, 1.67e-493j
-17.15834929874866103 - 9.3677564696013463164e-1302j 2.68e-496, 2.68e-496j
-13.354710699766731748 + 1.1760929868444133688e-1296j 1.27e-490, 1.27e-490j
-13.867076967364494113 - 5.6320057271417696012e-1297j 3.8e-491, 3.8e-491j
14.982055604227382293 - 6.6787597296337013802e-1293j 1.53e-492, 1.53e-492j
-11.933942857794045918 - 7.9755429720949123142e-1301j 1.56e-489, 1.56e-489j
13.354710699766731748 - 4.9119987495348239095e-1326j 1.36e-490, 1.36e-490j
3.8796403220018030492 + 4.7984403709597984897e-1352j 1.16e-494, 1.16e-494j
16.308732178050861921 + 9.5602894348182157896e-1391j 1.04e-494, 1.04e-494j
12.391486133499333228 - 8.8532515222062967086e-1455j 7.86e-490, 7.86e-490j
10.632769803552642203 + 2.4038323630492579353e-1531j 5.37e-489, 5.37e-489j
11.056071377216411899 + 1.356755311857114482e-1586j 4.09e-489, 4.09e-489j
17.15834929874866103 + 3.0805813533862734826e-1613j 2.79e-496, 2.79e-496j
-6.9079808502790681636 - 1.4063254673046199148e-1606j 1.8e-489, 1.8e-489j
-11.056071377216411899 - 1.1432771219337394706e-1609j 4.05e-489, 4.05e-489j
-2.4440098084964929991 - 2.6762555354086429698e-1633j 1.54e-498, 1.54e-498j
-2.5464506048208119765 - 2.842732033404085991e-1633j 3.75e-498, 3.75e-498j
10.21842085994737678 - 3.4430812771780513421e-1622j 6.24e-489, 6.24e-489j
13.867076967364494113 + 1.4725441488202921655e-1636j 4.01e-491, 4.01e-491j
11.933942857794045918 - 1.0197871452570913304e-1635j 1.61e-489, 1.61e-489j
6.6202082644312599908 - 5.6482768875991622569e-1641j 1.21e-489, 1.21e-489j
-7.2094666581229529422 - 4.6263875623789616179e-1643j 2.35e-489, 2.35e-489j
-11.489359163864077573 - 7.6335209743374797775e-1643j 2.94e-489, 2.94e-489j
-6.3323533291181535762 - 6.9333092660665506521e-1644j 7.61e-490, 7.61e-490j
6.9079808502790681636 - 1.6208013127356289855e-1641j 1.79e-489, 1.79e-489j
-1.3498952853218839149 + 9.792586294559996323e-1657j 2.12e-503, 2.12e-503j
15.607007568072038214 - 1.2831191747149538092e-1646j 1.66e-493, 1.66e-493j
1.8891758777537106755 + 5.8814694866899458765e-1654j 2.79e-501, 2.79e-501j
2.8523776395080470908 - 1.1289666575909104986e-1650j 1.55e-497, 1.55e-497j
8.2615823390255012477 + 2.2037494989452692835e-1641j 4.65e-489, 4.65e-489j
-7.8945557704634280356 + 8.1606725171567854782e-1647j 3.51e-489, 3.51e-489j
-8.2615823390255012477 + 2.0033738306096408203e-1647j 4.23e-489, 4.23e-489j
-5.5533529315903281928 - 6.757083787931593476e-1650j 2.72e-491, 2.72e-491j
3.5412019097483905709 - 3.7583713266129726977e-1653j 2.27e-495, 2.27e-495j
6.0948601215851390862 + 3.2597618304779995911e-1646j 3.98e-490, 3.98e-490j
11.489359163864077573 - 4.6052094507232260534e-1646j 2.66e-489, 2.66e-489j
-4.4722404253994967677 + 3.487064864050273199e-1656j 1.35e-492, 1.35e-492j
-7.540987469012572874 + 7.4881481673613374098e-1653j 2.88e-489, 2.88e-489j
5.2158274060059501299 - 1.0024583425185256555e-1654j 5.29e-492, 5.29e-492j
-0.90757615332791102264 - 1.5533135747564064268e-1668j 1.57e-505, 1.57e-505j
-5.8625190572520782833 + 6.7170438701153367295e-1654j 1.45e-490, 1.45e-490j
12.864132707639172954 + 2.1466307429823971217e-1653j 3.7e-490, 3.7e-490j
-9.0221929532516584298 + 3.3221713081234285596e-1652j 6.43e-489, 6.43e-489j
1.5986770873680431564 - 5.8056054431878772715e-1665j 2.3e-502, 2.3e-502j
1.3498952853218839149 + 1.8964029529843518019e-1667j 2.37e-503, 2.37e-503j
-3.3242574335521189524 - 3.1491207117590118252e-1659j 9.43e-496, 9.43e-496j
-3.8796403220018030492 - 1.1426167343015513499e-1657j 1.27e-494, 1.27e-494j
-6.0948601215851390862 - 2.8026490440042001111e-1654j 4.19e-490, 4.19e-490j
14.40673561483361294 - 3.6070036820811620599e-1655j 7.86e-492, 7.86e-492j
2.5464506048208119765 - 6.5977155973890784157e-1662j 3.85e-498, 3.85e-498j
8.6380086682866644355 - 1.7153234212053974472e-1650j 5.34e-489, 5.34e-489j
7.8945557704634280356 + 9.6819718706267191729e-1655j 3.76e-489, 3.76e-489j
-1.1542189542438079954 - 9.4117837561100820681e-1679j 2.82e-504, 2.82e-504j
-5.2158274060059501299 + 2.6896440083888472863e-1668j 5.24e-492, 5.24e-492j
6.3323533291181535762 + 1.6465860373803717511e-1663j 7.67e-490, 7.67e-490j
3.1705262358161719064 + 4.4023789429150991999e-1687j 2.74e-496, 2.74e-496j
1.1542189542438079954 + 3.8115808612270753519e-1696j 2.91e-504, 2.91e-504j
-1.5986770873680431564 + 4.3402765093740094335e-1694j 2e-502, 2e-502j
-1.8891758777537106755 + 9.7123349791096294086e-1692j 3.04e-501, 3.04e-501j
-4.292421794229282496 + 2.5113617628422750759e-1683j 1.39e-492, 1.39e-492j
-3.1705262358161719064 - 1.4534592903707315287e-1687j 2.83e-496, 2.83e-496j
9.0221929532516584298 - 2.4379704277758318812e-1678j 6.3e-489, 6.3e-489j
5.8625190572520782833 - 1.3025484671375763022e-1697j 1.39e-490, 1.39e-490j
4.8675088466804980907 - 1.2637028786136562399e-1711j 1.45e-492, 1.45e-492j
9.8122232101871723388 - 3.4695965385744585647e-1720j 6.69e-489, 6.69e-489j
7.2094666581229529422 - 1.3137094934994421643e-1738j 2.36e-489, 2.36e-489j
-2.8523776395080470908 - 7.9858226932531749326e-1753j 1.58e-497, 1.58e-497j
-4.8675088466804980907 + 6.173187493462216549e-1748j 1.43e-492, 1.43e-492j
-10.21842085994737678 - 7.8815627211871369617e-1744j 6.42e-489, 6.42e-489j
3.3242574335521189524 - 1.8123682138723588893e-1751j 9.49e-496, 9.49e-496j
0.90757615332791102264 - 6.6012063523661556367e-1762j 1.43e-505, 1.43e-505j
-10.632769803552642203 - 1.4856964691063319254e-1743j 5.45e-489, 5.45e-489j
-2.1942479720396902511 - 4.1018813384042417303e-1761j 7.19e-500, 7.19e-500j
-9.8122232101871723388 - 5.2517282859396502481e-1749j 7e-489, 7e-489j
9.413594095107672465 + 3.1397844183122983459e-1758j 7.27e-489, 7.27e-489j
-6.6202082644312599908 - 9.4475706956340497763e-1771j 1.32e-489, 1.32e-489j
2.1942479720396902511 + 3.4339930251140580935e-1780j 7.89e-500, 7.89e-500j
4.3418637687749014379 - 5.3232533582900864815e-1773j 2.33e-492, 2.33e-492j
-0.61670659019259415219 + 5.2819749011097870279e-1790j 3.69e-507, 3.69e-507j
4.4722404253994967677 + 1.0675552158818279e-1772j 1.32e-492, 1.32e-492j
2.4440098084964929991 + 1.0440109143227613313e-1778j 1.55e-498, 1.55e-498j
0.61670659019259415219 + 3.4024605978241865796e-1788j 3.91e-507, 3.91e-507j
5.5533529315903281928 - 2.8168666765453911387e-1770j 2.64e-491, 2.64e-491j
-8.6380086682866644355 - 3.846718311629995608e-1774j 5.13e-489, 5.13e-489j
-0.31101948466043283062 + 1.1712138346112637288e-1793j 1.09e-508, 1.09e-508j
7.540987469012572874 + 1.9444953394044476864e-1773j 2.94e-489, 2.94e-489j
2.1577601285831736289e-1839 - 1.9808939107332864845e-1840j 2.1e-1837, 2.1e-1837j
0.31101948466043283062 - 3.4175877602372202676e-1802j 1.09e-508, 1.09e-508j
-9.413594095107672465 - 4.946847461726213639e-1782j 6.82e-489, 6.82e-489j
-3.5412019097483905709 - 5.961595110982005188e-1792j 2.25e-495, 2.25e-495j
-4.3418637687749014379 - 5.8581788781720135693e-1789j 2.24e-492, 2.24e-492j
4.292421794229282496 - 9.2788868540391012683e-1789j 1.31e-492, 1.31e-492j

Weights

Weight (ball center) Error (ball radius)
1.888044206889719555e-46 - 1.0678100509032330122e-1334j 1.14e-120, 1.17e-367j
8.4206651808166467832e-35 + 1.1778312458455706488e-1326j 5.48e-115, 5.59e-362j
4.3217509472021609905e-50 + 6.8381924262801221656e-1337j 1.4e-122, 1.43e-369j
5.2963694954304958055e-59 + 4.2923479170484044211e-1342j 4.22e-127, 4.31e-374j
2.2300233287349434667e-37 + 9.3337028954156839263e-1329j 2.21e-117, 2.26e-364j
3.3575447591689951864e-54 - 2.5842587949474331406e-1339j 4.83e-125, 4.93e-372j
4.6056117388422197962e-65 - 1.5171512915003081718e-1345j 2.38e-130, 2.43e-377j
3.7382086438026944533e-40 - 1.0360463256774048006e-1330j 2.08e-119, 2.12e-366j
3.6685048615378998495e-43 + 1.1688938739397769422e-1332j 1.1e-120, 1.12e-367j
4.3217509472021609905e-50 - 6.4681402897508304252e-1338j 5.19e-135, 5.3e-382j
2.1328286526874670955e-32 - 1.0081062093462890615e-1325j 2.51e-117, 2.57e-364j
3.7382086438026944533e-40 + 3.8739586409688035456e-1332j 9.67e-131, 9.88e-378j
7.5167411030317414291e-05 - 3.9521477104797828544e-1306j 2.14e-85, 2.19e-332j
5.2963694954304958055e-59 - 5.8582373104661358935e-1343j 5.91e-139, 6.04e-386j
8.4206651808166467832e-35 + 6.3370843958615560608e-1329j 2.45e-128, 2.5e-375j
4.7101167410705843876e-26 + 2.0662457095639291711e-1323j 1.08e-123, 1.11e-370j
4.8875525066045173561e-28 - 1.0619089490122134452e-1324j 6.11e-125, 6.24e-372j
4.6056117388422197962e-65 + 2.447282991958009303e-1346j 2.05e-142, 2.09e-389j
5.0115511906436774332e-12 + 7.9525667396735210504e-1312j 9.78e-110, 9.99e-357j
4.8875525066045173561e-28 - 1.8642843786678453185e-1323j 2.65e-122, 2.7e-369j
0.0019241059380669017567 - 1.1442086660810670675e-1303j 9.22e-77, 9.41e-324j
0.0038835231007577279539 + 9.1171990398986797424e-1304j 4.19e-78, 4.28e-325j
3.4688674613619952326e-24 - 3.6072895642285563117e-1322j 3.26e-124, 3.33e-371j
3.6685048615378998495e-43 - 6.5664224218120764828e-1334j 9.81e-135, 1e-381j
2.1328286526874670955e-32 - 1.8966832627746423065e-1327j 4.79e-129, 4.89e-376j
3.5211742925965751191e-11 - 1.4720491180554914117e-1311j 1.27e-113, 1.29e-360j
6.5316911948514962896e-13 - 1.0589440089536637398e-1312j 1.08e-114, 1.1e-361j
3.7884909024325835308e-30 + 1.2755069509840279944e-1324j 2.61e-126, 2.66e-373j
2.1530663568237740161e-10 + 2.8935209235887247054e-1310j 2.01e-111, 2.05e-358j
5.0115511906436774332e-12 + 2.2596060852177377966e-1312j 8.39e-116, 8.57e-363j
0.03400093507021686504 - 1.3975680746481457521e-1303j 7.84e-70, 8.01e-317j
3.3575447591689951864e-54 + 2.9678129418628117457e-1340j 5.47e-141, 5.58e-388j
0.020112827091239477243 - 4.3811913695164893674e-1304j 5.21e-81, 5.32e-328j
0.0022018112571773995628 - 1.2252968574620106028e-1304j 2e-92, 2.04e-339j
2.2413153884882609898e-16 + 1.8838092646860911885e-1316j 1.74e-121, 1.78e-368j
4.2195804142042688759e-15 - 1.0329760015851244643e-1314j 2.41e-119, 2.46e-366j
2.2413153884882609898e-16 + 9.4202708528429974288e-1316j 6.51e-121, 6.65e-368j
2.6400098000534541424e-08 - 6.0660317300704874628e-1309j 1.18e-111, 1.2e-358j
0.00024416453977816729142 + 1.9472989334540716211e-1305j 3.88e-101, 3.97e-348j
7.0348119645354820972e-10 - 3.6542534465182984692e-1310j 2.04e-115, 2.09e-362j
3.7884909024325835308e-30 + 4.8192274167765373309e-1326j 5.3e-131, 5.41e-378j
1.2025833918442875206e-05 + 3.6378776111033366891e-1306j 3.7e-108, 3.78e-355j
6.1601881221849597828e-14 + 1.1001690885264598137e-1313j 1.43e-118, 1.46e-365j
1.6958763588048168144e-07 + 8.8715235094820965979e-1309j 1.58e-113, 1.61e-360j
0.073144889359445382212 - 9.4529562963722315339e-1304j 9.08e-83, 9.28e-330j
3.8140040405264255726e-09 + 2.4454042909299891795e-1309j 1.57e-113, 1.61e-360j
2.2300233287349434667e-37 - 1.7513607124453036912e-1330j 5.03e-136, 5.14e-383j
3.2641421962382536773e-19 + 7.3393244744614181431e-1318j 2.99e-125, 3.06e-372j
0.030788766064810242379 + 6.7991613868543117267e-1304j 4.96e-89, 5.07e-336j
0.03400093507021686504 - 1.1229901785986031434e-1303j 7.09e-86, 7.24e-333j
0.00011617247027471546061 - 1.1809139656495381825e-1304j 1.87e-104, 1.91e-351j
7.5167411030317414291e-05 - 7.554776335426872259e-1306j 1.59e-107, 1.62e-354j
7.0348119645354820972e-10 - 1.0728321839586442326e-1309j 1.29e-114, 1.32e-361j
1.888044206889719555e-46 + 8.0278858109852619773e-1336j 4.66e-142, 4.76e-389j
0.0038835231007577279539 + 6.0088515213284127142e-1304j 6.58e-102, 6.72e-349j
9.5221748625762369565e-18 - 1.5042153228968535961e-1317j 2.16e-128, 2.2e-375j
4.2195804142042688759e-15 - 2.2899469352888446484e-1315j 2.84e-126, 2.9e-373j
0.042740201896048603716 + 1.4354763170086305141e-1303j 1.05e-91, 1.07e-338j
1.6958763588048168144e-07 + 2.1768152210604574099e-1308j 1.61e-115, 1.65e-362j
2.1530663568237740161e-10 + 9.3638164683320460265e-1311j 5.25e-121, 5.36e-368j
0.00077518961148772643368 + 9.0122897042832532918e-1305j 2.09e-108, 2.13e-355j
0.042740201896048603716 + 1.1908487308657988061e-1303j 2.55e-93, 2.6e-340j
0.030788766064810242379 + 8.8133613806020482362e-1304j 2.13e-98, 2.17e-345j
0.020112827091239477243 - 5.9573521166004456145e-1304j 4.9e-101, 5e-348j
5.4114305718169150949e-05 + 1.9376525489181393933e-1305j 2.41e-113, 2.46e-360j
0.00077518961148772643368 + 1.5209694039650491291e-1304j 8.51e-109, 8.69e-356j
3.2641421962382536773e-19 + 1.1548529998571426447e-1318j 7.65e-131, 7.81e-378j
3.8140040405264255726e-09 + 8.7467444632425960261e-1310j 1.34e-121, 1.37e-368j
1.0130783980258476725e-06 - 5.1243658172944402686e-1308j 1.17e-119, 1.19e-366j
1.988938029868047504e-22 + 5.7333798096122393833e-1321j 2.48e-133, 2.53e-380j
6.5316911948514962896e-13 - 2.7996798292254892342e-1313j 9.31e-126, 9.51e-373j
0.0022018112571773995628 - 1.9580545214778184736e-1304j 7.64e-112, 7.81e-359j
1.0130783980258476725e-06 - 1.175441203330377496e-1307j 2.04e-119, 2.08e-366j
3.4688674613619952326e-24 - 3.7529302873523035176e-1321j 1.85e-138, 1.89e-385j
0.00011617247027471546061 - 6.813377026398038256e-1305j 3.02e-117, 3.09e-364j
0.073144889359445382212 - 8.1627701199882550422e-1304j 4.13e-109, 4.21e-356j
4.7101167410705843876e-26 + 2.7047507052906333508e-1322j 1.96e-139, 2e-386j
0.01100996655255640743 + 6.290557940717225349e-1304j 1.26e-113, 1.28e-360j
1.988938029868047504e-22 + 4.9352578970434582344e-1320j 1.07e-137, 1.09e-384j
9.0119815204286428291e-21 - 8.411698581142416799e-1320j 8.28e-136, 8.46e-383j
3.5211742925965751191e-11 - 4.8490827498409192243e-1311j 6.91e-127, 7.06e-374j
0.01100996655255640743 + 4.3979096744426631625e-1304j 2.65e-117, 2.7e-364j
-4.4733447743949759127e-05 - 9.9893692298844584512e-1306j 2.82e-124, 2.88e-371j
0.099164914290528031976 + 6.1473657245222056426e-1304j 3.4e-117, 3.48e-364j
1.2025833918442875206e-05 + 1.708381379698534787e-1306j 1.16e-124, 1.19e-371j
0.0019241059380669017567 - 7.672189063981411928e-1304j 1.89e-119, 1.93e-366j
0.099164914290528031976 + 5.5644937264546432475e-1304j 1.96e-118, 2e-365j
2.6400098000534541424e-08 - 2.3115930693018832295e-1309j 2.11e-127, 2.15e-374j
9.5221748625762369565e-18 - 8.427225797827725346e-1317j 5.11e-137, 5.22e-384j
0.11759483122143335988 - 4.7460553699393160536e-1304j 4.89e-120, 4.99e-367j
6.1601881221849597828e-14 + 2.6773147213531008696e-1314j 5.6e-133, 5.72e-380j
0.12439982718677942754 + 4.2925644869745525256e-1304j 1.69e-120, 1.73e-367j
0.11759483122143335988 - 4.5136461594101756705e-1304j 1.31e-120, 1.34e-367j
9.0119815204286428291e-21 - 6.1589400054208493254e-1319j 2.42e-139, 2.48e-386j
0.00024416453977816729142 + 3.5055776358838829823e-1305j 2.72e-125, 2.78e-372j
-4.4733447743949759127e-05 - 2.076534795691978135e-1305j 1e-125, 1.02e-372j
5.4114305718169150949e-05 + 9.4063063266408150317e-1306j 8.3e-126, 8.45e-373j

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