Rule: Q[8, 11, 64]

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.

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)
13.91825944594843877 - 5.6133124370274540792e-1376j 3.02e-494, 3.02e-494j
-13.329805885861729807 - 2.3298743464867507422e-1381j 1.41e-493, 1.41e-493j
12.777542864560785662 + 3.6542016493487386569e-1380j 5.11e-493, 5.11e-493j
13.329805885861729807 + 3.4555824562704256281e-1402j 1.41e-493, 1.41e-493j
-16.141410269036616933 + 6.3120088104798417044e-1418j 1.15e-497, 1.15e-497j
11.266160679587900685 + 1.7175231426702398058e-1429j 5.98e-492, 5.98e-492j
-13.91825944594843877 - 9.969752194695422363e-1445j 2.98e-494, 2.98e-494j
-14.557120870123317529 - 3.0307488744391799999e-1449j 4.12e-495, 4.12e-495j
-6.6792660972764417418 - 3.6510609541163238701e-1465j 1.94e-492, 1.94e-492j
2.3357675405796096575 - 3.8689510517866505971e-1494j 3.76e-501, 3.76e-501j
10.341381882689491384 - 1.679829537947421094e-1483j 1.26e-491, 1.26e-491j
11.750468937884245672 - 1.5932299135806703766e-1506j 3.42e-492, 3.42e-492j
10.797181273763669295 - 2.5327928140416837649e-1521j 9.2e-492, 9.2e-492j
-10.797181273763669295 + 2.6593124347667106698e-1526j 9.15e-492, 9.15e-492j
-11.266160679587900685 - 1.1840008015272574425e-1534j 5.91e-492, 5.91e-492j
4.1445471861258943321 - 2.9829704431999396616e-1542j 1.13e-496, 1.13e-496j
-10.341381882689491384 - 3.1306762223269783523e-1536j 1.36e-491, 1.36e-491j
-3.7940450031870709171 - 2.0678035509839952568e-1546j 1.97e-497, 1.97e-497j
16.141410269036616933 + 8.0802013320999907977e-1548j 1.08e-497, 1.08e-497j
-4.8284526110215766315 - 4.2785290290874709953e-1544j 4.15e-495, 4.15e-495j
-8.6223459487395071092 + 1.591506476590014237e-1540j 1.3e-491, 1.3e-491j
-9.8971149995751123383 - 7.7907579199728178511e-1545j 1.54e-491, 1.54e-491j
2.6377551349274913907 + 6.658521069867337129e-1557j 6.18e-500, 6.18e-500j
6.0484198139075743475 + 9.3318815295343284473e-1549j 7.37e-493, 7.37e-493j
14.557120870123317529 + 1.0008959348065756223e-1552j 4.09e-495, 4.09e-495j
-6.3410934275277093073 - 2.5278346809162012642e-1552j 1.25e-492, 1.25e-492j
5.1580652270647994111 - 6.363660754332985256e-1566j 1.98e-494, 1.98e-494j
7.4242573168896287954 + 2.1319583798152219966e-1562j 4.68e-492, 4.68e-492j
-3.0950955154405111371 + 1.4694823239644087592e-1578j 5.8e-499, 5.8e-499j
-1.6365190424351079992 - 2.5309098467532245392e-1583j 9.86e-504, 9.86e-504j
-1.9881386728790509247 - 2.6533522365712106706e-1582j 1.84e-502, 1.84e-502j
-2.8024858612875416991 - 1.5667645004596514692e-1578j 1.98e-499, 1.98e-499j
3.7940450031870709171 + 1.7597567892014499805e-1577j 1.83e-497, 1.83e-497j
-15.274024218504320714 + 1.2485121380503227299e-1574j 3.2e-496, 3.2e-496j
4.490222781093641142 + 3.3658967277509104197e-1576j 7.19e-496, 7.19e-496j
-11.750468937884245672 + 1.6407095015012409094e-1569j 3.71e-492, 3.71e-492j
-12.777542864560785662 + 1.6835078629461299937e-1578j 5.36e-493, 5.36e-493j
2.8024858612875416991 - 1.942166452587168353e-1597j 1.77e-499, 1.77e-499j
3.4420239350883741819 + 2.708432556439343435e-1596j 2.98e-498, 2.98e-498j
5.4778702791043117106 + 9.9214712695454116376e-1590j 9.93e-494, 9.93e-494j
-0.97448664368721513187 - 8.3832560154447128448e-1617j 1.09e-505, 1.09e-505j
-7.8149621932413285045 + 1.2564345186190644736e-1613j 7.2e-492, 7.2e-492j
3.0950955154405111371 + 5.4645171556799724015e-1644j 6.26e-499, 6.26e-499j
9.4631122084662877704 - 5.9670953598918393349e-1637j 1.67e-491, 1.67e-491j
-7.4242573168896287954 - 2.1582391610827097732e-1650j 4.95e-492, 4.95e-492j
-5.7772641382621960874 - 1.2941048946214232176e-1651j 3.12e-493, 3.12e-493j
-2.6377551349274913907 - 2.3561979928463831031e-1658j 6.41e-500, 6.41e-500j
12.252960379752258277 + 5.4225526400136681773e-1649j 1.4e-492, 1.4e-492j
0.79867251029900488539 + 2.2529211257386302662e-1667j 2.42e-506, 2.42e-506j
-1.2891379487562259014 - 1.2922521593244387533e-1666j 7.35e-505, 7.35e-505j
6.6792660972764417418 + 1.4102825974411732089e-1653j 1.89e-492, 1.89e-492j
15.274024218504320714 - 1.1017344380552605273e-1660j 3.29e-496, 3.29e-496j
-12.252960379752258277 - 2.1459417034769024556e-1656j 1.68e-492, 1.68e-492j
1.9881386728790509247 + 6.1781713414595087402e-1668j 1.81e-502, 1.81e-502j
5.7772641382621960874 + 1.9880722295535547352e-1656j 3.07e-493, 3.07e-493j
-8.2145271776605993101 + 9.0417858584383831034e-1655j 9.63e-492, 9.63e-492j
-2.3357675405796096575 - 8.4069689913795174208e-1670j 4.08e-501, 4.08e-501j
-6.0484198139075743475 - 4.3825557293521183191e-1660j 6.93e-493, 6.93e-493j
7.0441857491954455192 - 6.0592599188299979485e-1689j 3.02e-492, 3.02e-492j
7.8149621932413285045 - 1.389741276495946479e-1748j 6.98e-492, 6.98e-492j
9.0384148030261830466 + 2.2494168370788806048e-1806j 1.46e-491, 1.46e-491j
-5.1580652270647994111 + 1.1263551194251589919e-1828j 2.11e-494, 2.11e-494j
-4.1445471861258943321 - 1.3550574686166779899e-1836j 1.13e-496, 1.13e-496j
-3.4420239350883741819 - 8.3109816456591561902e-1843j 3.03e-498, 3.03e-498j
1.2891379487562259014 - 1.7888177047963212836e-1851j 8.33e-505, 8.33e-505j
9.8971149995751123383 - 8.2144319791362075546e-1837j 1.65e-491, 1.65e-491j
-4.490222781093641142 + 1.3483832872654754895e-1856j 7.21e-496, 7.21e-496j
-7.0441857491954455192 + 5.0926435779675777558e-1851j 2.97e-492, 2.97e-492j
-0.53907981135137510807 - 1.5319033753210840151e-1869j 1.07e-507, 1.07e-507j
8.6223459487395071092 + 1.8157229161214626768e-1853j 1.3e-491, 1.3e-491j
0.97448664368721513187 + 8.0019755157463278026e-1876j 1.16e-505, 1.16e-505j
1.6365190424351079992 + 3.779893134116766844e-1873j 1.01e-503, 1.01e-503j
-9.0384148030261830466 + 3.5309009209246250959e-1860j 1.63e-491, 1.63e-491j
-0.79867251029900488539 - 5.0784658838603071428e-1878j 2.65e-506, 2.65e-506j
0.53907981135137510807 + 1.7012859377099874462e-1878j 1.2e-507, 1.2e-507j
-9.4631122084662877704 - 2.667539118219224742e-1861j 1.62e-491, 1.62e-491j
8.2145271776605993101 + 1.830754227628030079e-1863j 1.01e-491, 1.01e-491j
-4.9893200329122219338e-1919 - 8.4294107642652013001e-1919j 8.13e-1917, 8.13e-1917j
-0.22411440845419671995 - 4.1437844873601476725e-1887j 4.8e-509, 4.8e-509j
-5.4778702791043117106 + 2.3068195931558909041e-1872j 8.76e-494, 8.76e-494j
6.3410934275277093073 - 3.1256598429733007921e-1869j 1.18e-492, 1.18e-492j
0.22411440845419671995 + 6.313177852388329949e-1888j 4.8e-509, 4.8e-509j
4.8284526110215766315 + 8.5682481414592593794e-1875j 3.85e-495, 3.85e-495j

Weights

Weight (ball center) Error (ball radius)
2.0957063654368029323e-43 + 9.0724574233876634884e-1418j 2.41e-158, 1.15e-405j
5.9181450150383572857e-40 - 3.4234142163401891769e-1418j 2.81e-158, 1.34e-405j
7.5588480618937130014e-37 + 2.5949487122238925093e-1415j 8.82e-156, 4.21e-403j
5.9181450150383572857e-40 - 1.7195637106037039441e-1416j 5.21e-157, 2.48e-404j
1.0605937370724814591e-57 - 6.2317749209836921014e-1428j 2.04e-166, 9.71e-414j
5.2121796348493248142e-29 - 5.3512417668791860829e-1411j 5.94e-153, 2.83e-400j
2.0957063654368029323e-43 + 3.9641057187520602607e-1420j 4.95e-161, 2.36e-408j
2.5838889176141577119e-47 - 2.7830137529402137515e-1422j 9.68e-163, 4.62e-410j
2.9035432829505253957e-11 + 3.5046027280779286118e-1400j 1.74e-139, 8.31e-387j
0.0088716967513372364347 + 1.5756428358839528322e-1393j 1.65e-106, 7.86e-354j
1.0747055660231125978e-24 - 1.4308595430054214886e-1408j 3.09e-152, 1.47e-399j
2.0480565188754178309e-31 + 2.6026792483653824048e-1412j 7.48e-156, 3.57e-403j
8.9287471072210494034e-27 + 9.3858610766258579151e-1410j 2.85e-153, 1.36e-400j
8.9287471072210494034e-27 + 1.2231435989141157188e-1410j 2.15e-157, 1.03e-404j
5.2121796348493248142e-29 - 5.8785403859221588051e-1412j 8.84e-159, 4.22e-406j
2.5895530038728769549e-05 + 9.0978789122584272946e-1396j 1.49e-131, 7.09e-379j
1.0747055660231125978e-24 - 2.1630032834972297655e-1409j 1.25e-156, 5.95e-404j
0.00010508179267642267021 - 1.4457189536967705278e-1395j 7.6e-131, 3.62e-378j
1.0605937370724814591e-57 + 8.6186492796438450013e-1427j 3.75e-173, 1.79e-420j
1.1535662400227943726e-06 + 6.880243269169628932e-1397j 1.07e-137, 5.12e-385j
1.1801347274055225983e-17 - 5.9816960630656782451e-1405j 2.49e-153, 1.19e-400j
9.3999848466084797344e-23 + 3.3170372068285418918e-1408j 4.17e-156, 1.99e-403j
0.002727867736927326899 - 1.766036364491487317e-1393j 1.17e-123, 5.56e-371j
1.2293709998517864682e-09 + 3.0828168006636249857e-1398j 1.6e-146, 7.61e-394j
2.5838889176141577119e-47 + 1.2924370674096037261e-1420j 9.42e-170, 4.49e-417j
2.3543133460670771175e-10 - 2.5065259164162084602e-1399j 7.67e-147, 3.66e-394j
2.1666291789813445918e-07 - 5.7356995690731245679e-1397j 1.01e-142, 4.84e-390j
1.6528713767728808938e-13 + 1.6579340767484165822e-1401j 3.47e-152, 1.66e-399j
0.0011216918664061762022 - 1.9388858136707184234e-1394j 1.98e-131, 9.45e-379j
0.036689760127904025961 + 2.9225501299698165004e-1393j 7.64e-118, 3.64e-365j
0.019420901044245486861 - 1.5432599250280925409e-1393j 2.8e-122, 1.33e-369j
0.0015954012829624161312 + 8.3124133077738723659e-1394j 1.22e-129, 5.8e-377j
0.00010508179267642267021 - 2.5205499683535670823e-1395j 1.48e-137, 7.07e-385j
6.74951125211134546e-52 + 8.7094951083126698976e-1425j 5.24e-174, 2.5e-421j
5.7171542038917779854e-06 - 3.5178525496499775952e-1396j 6.36e-142, 3.04e-389j
2.0480565188754178309e-31 + 2.343110110513140562e-1413j 7.21e-165, 3.44e-412j
7.5588480618937130014e-37 + 1.8609520035452611134e-1416j 1.38e-167, 6.57e-415j
0.0015954012829624161312 + 1.2473263750730233825e-1393j 8.74e-134, 4.17e-381j
0.00037490757469988068465 + 7.8431286666078935864e-1395j 3.9e-138, 1.86e-385j
3.8030505015657418585e-08 + 2.329456483879994693e-1397j 1.51e-148, 7.23e-396j
0.064063626725002173482 + 2.7299516151342145405e-1392j 7.2e-127, 3.43e-374j
8.627230064983422208e-15 - 5.7145905158164433275e-1403j 1.55e-157, 7.39e-405j
0.0011216918664061762022 - 3.0394907590815392764e-1394j 8.83e-137, 4.21e-384j
6.1367948045329940205e-21 - 2.3143487219563218072e-1406j 1.56e-163, 7.42e-411j
1.6528713767728808938e-13 + 5.0919186740690863131e-1402j 1.44e-156, 6.85e-404j
6.3758557054351905819e-09 - 3.8458956460129170351e-1398j 3.75e-152, 1.79e-399j
0.002727867736927326899 - 1.2060549675370245336e-1393j 6.25e-138, 2.98e-385j
5.1218407125058198483e-34 - 1.0357172087498110103e-1413j 9.85e-170, 4.7e-417j
0.048470442216970764986 - 4.7352264881538551901e-1392j 3.82e-131, 1.82e-378j
0.05928219315562899617 - 7.5233879615737664275e-1393j 2.15e-130, 1.02e-377j
2.9035432829505253957e-11 + 9.8959403105041902863e-1400j 2.16e-156, 1.03e-403j
6.74951125211134546e-52 - 1.9319844578059671812e-1423j 5.48e-179, 2.61e-426j
5.1218407125058198483e-34 - 7.510269482165430905e-1415j 1.89e-169, 9.02e-417j
0.019420901044245486861 - 2.0541420330586152007e-1393j 4.77e-138, 2.28e-385j
6.3758557054351905819e-09 - 9.248376742690353919e-1398j 1.08e-152, 5.14e-400j
3.5829641608552262971e-16 + 6.0555702755193633399e-1404j 2.48e-160, 1.19e-407j
0.0088716967513372364347 + 1.1250485965636557974e-1393j 2.38e-141, 1.14e-388j
1.2293709998517864682e-09 + 1.2229884307400673538e-1398j 9.73e-155, 4.64e-402j
2.5025338390981203872e-12 - 1.3094225124937663623e-1400j 2.15e-158, 1.02e-405j
8.627230064983422208e-15 - 2.0138904608697609448e-1402j 1.58e-161, 7.56e-409j
3.0553856513135319037e-19 + 2.5195066324028107302e-1405j 4.64e-165, 2.21e-412j
2.1666291789813445918e-07 - 2.6474658930972502527e-1397j 2.4e-155, 1.14e-402j
2.5895530038728769549e-05 + 4.9418187893900307541e-1396j 4.21e-153, 2.01e-400j
0.00037490757469988068465 + 4.7476397636563286343e-1395j 7.48e-151, 3.57e-398j
0.05928219315562899617 - 9.0495084376619586655e-1393j 1.59e-145, 7.56e-393j
9.3999848466084797344e-23 + 1.9244926374881827544e-1407j 6.02e-168, 2.87e-415j
5.7171542038917779854e-06 - 1.8093831605808392526e-1396j 2.88e-154, 1.38e-401j
2.5025338390981203872e-12 - 4.3301834201217902017e-1401j 1.17e-160, 5.59e-408j
0.10932993981225860799 + 4.0437739423614296616e-1392j 2.74e-147, 1.31e-394j
1.1801347274055225983e-17 - 2.5126439044400860027e-1404j 4.66e-166, 2.22e-413j
0.064063626725002173482 + 3.1384481840969142322e-1392j 2.38e-148, 1.14e-395j
0.036689760127904025961 + 3.6962826710451596858e-1393j 4.48e-150, 2.14e-397j
3.0553856513135319037e-19 + 5.4368505174783494304e-1406j 8.11e-167, 3.87e-414j
0.048470442216970764986 - 4.2240896110938617178e-1392j 4.81e-150, 2.3e-397j
0.10932993981225860799 + 4.367679662528170568e-1392j 1.7e-150, 8.13e-398j
6.1367948045329940205e-21 - 4.487388296513074707e-1407j 6.39e-168, 3.05e-415j
3.5829641608552262971e-16 + 2.3226050899608324242e-1403j 6.91e-166, 3.3e-413j
0.068051349943459864373 + 7.2162470630878852464e-1392j 3.13e-152, 1.51e-399j
0.11388778612497470487 - 5.5704795157951727032e-1392j 4.13e-152, 1.98e-399j
3.8030505015657418585e-08 + 1.0193544626307328772e-1397j 8e-159, 3.81e-406j
2.3543133460670771175e-10 - 6.6550785262982265691e-1399j 1.19e-161, 5.7e-409j
0.11388778612497470487 - 5.7517371643159223383e-1392j 9.45e-154, 5.74e-401j
1.1535662400227943726e-06 + 1.4123439816159406593e-1396j 3.78e-159, 1.79e-406j

Download

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