Rule: Q[4, 10, 90]

Type

This rule is of Hermite (physicists') 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)
-10.584704745237436922 + 1.9087228965901779952e-1300j 9.18e-489, 9.18e-489j
10.247990209125360078 - 3.4415178161154437767e-1309j 2.83e-488, 2.83e-488j
-12.615739692459945611 - 4.8518257567018341464e-1313j 3.66e-493, 3.66e-493j
-10.936542650139569274 + 1.3509731348479146416e-1306j 2.29e-489, 2.29e-489j
11.307317586792911516 + 4.1436690280239921726e-1318j 4.85e-490, 4.85e-490j
11.70282039773107087 + 4.9902466695214711668e-1318j 6.79e-491, 6.79e-491j
-11.307317586792911516 - 9.5463995983086494865e-1313j 4.78e-490, 4.78e-490j
-1.22103345192141233 + 1.0109984563852414672e-1332j 1.44e-502, 1.44e-502j
7.8811719756347078181 - 1.8052423805091334318e-1319j 1.05e-486, 1.05e-486j
8.7178626297942803174 + 1.3032000509260774741e-1317j 7.84e-487, 7.84e-487j
2.3166778458612639611 - 1.2207036946956880241e-1328j 1.58e-497, 1.58e-497j
10.584704745237436922 - 1.6765496374892551331e-1320j 9.47e-489, 9.47e-489j
-1.8711401436544073366 - 9.7010340269295861953e-1330j 1.21e-499, 1.21e-499j
-5.3328356650915949362 + 1.6641719462081121082e-1316j 5.22e-489, 5.22e-489j
12.132709890471321759 + 1.6303950719955318791e-1331j 7.07e-492, 7.07e-492j
-9.9237206921220257213 + 1.7071898325687564756e-1327j 7.55e-488, 7.55e-488j
4.3789971524607627847 - 1.9291309814256107845e-1356j 6.39e-491, 6.39e-491j
-6.3183222432279279225 + 4.0710158192648332465e-1351j 1.25e-487, 1.25e-487j
-0.90837151181107858892 + 1.6414915434070670648e-1386j 5.3e-504, 5.3e-504j
-4.6149407183529111297 - 8.4018547741125265021e-1369j 2.2e-490, 2.2e-490j
-3.0000019506766675549 - 6.8589682580479211496e-1385j 9.53e-494, 9.53e-494j
-3.911506869286660161 + 2.0414062905628051172e-1377j 5.2e-492, 5.2e-492j
-9.6099247239212672721 - 1.2164533108509755674e-1394j 1.75e-487, 1.75e-487j
9.0080601931570536252 - 7.3130640026586467299e-1413j 5.13e-487, 5.13e-487j
6.0684659147569613041 + 5.1985361192953256076e-1430j 6.39e-488, 6.39e-488j
-6.8261361345558698097 + 6.2602287538474646768e-1457j 3.86e-487, 3.86e-487j
12.615739692459945611 - 2.5579224883619946267e-1475j 3.62e-493, 3.62e-493j
-13.201030744867069229 - 6.4608402261116823377e-1477j 7.29e-495, 7.29e-495j
8.4337290908952001737 + 3.9919506743612307652e-1467j 9.06e-487, 9.06e-487j
9.6099247239212672721 - 7.6711460606466296818e-1475j 1.58e-487, 1.58e-487j
-8.4337290908952001737 + 3.8107889408655844551e-1495j 8.73e-487, 8.73e-487j
1.22103345192141233 - 2.1243240347346236497e-1530j 1.52e-502, 1.52e-502j
2.7676092070256417508 + 7.4679559365178673007e-1524j 2.35e-495, 2.35e-495j
6.8261361345558698097 - 4.0689910126136188883e-1514j 3.86e-487, 3.86e-487j
-12.132709890471321759 + 1.1667752418946203351e-1514j 6.6e-492, 6.6e-492j
1.8711401436544073366 + 6.4743179059598210888e-1543j 1.35e-499, 1.35e-499j
-9.3050981178737809835 + 8.1285749036766508572e-1529j 3.2e-487, 3.2e-487j
9.9237206921220257213 + 1.3720222892650072126e-1549j 7.81e-488, 7.81e-488j
-8.1550137567230136803 + 4.2977437954281151238e-1550j 9.72e-487, 9.72e-487j
7.0845482630659029811 - 5.4994528226248547293e-1569j 5.74e-487, 5.74e-487j
-1.6506801238857845559 - 8.6819913614667319733e-1583j 1.16e-500, 1.16e-500j
-6.5708074744367509889 - 2.9429648343846855877e-1567j 2.39e-487, 2.39e-487j
5.5758881884142202387 + 3.2796807508199124768e-1583j 1.38e-488, 1.38e-488j
13.201030744867069229 - 6.7663188561853870802e-1591j 8.12e-495, 8.12e-495j
6.5708074744367509889 - 4.3788157732759373707e-1581j 2.33e-487, 2.33e-487j
-2.5413992300033163289 - 1.5099769983976561422e-1598j 1.66e-496, 1.66e-496j
2.5413992300033163289 + 4.4036100825352649296e-1598j 1.85e-496, 1.85e-496j
-9.0080601931570536252 + 6.8776788818874395516e-1587j 5.17e-487, 5.17e-487j
3.911506869286660161 + 3.0553225616133933051e-1607j 5.46e-492, 5.46e-492j
10.936542650139569274 - 3.0188331847475648849e-1605j 2.46e-489, 2.46e-489j
-4.8524867552345506937 - 2.3775691919444599997e-1603j 6.7e-490, 6.7e-490j
-2.7676092070256417508 - 1.9877576117692539142e-1610j 2.26e-495, 2.26e-495j
-11.70282039773107087 + 1.2168153543941973964e-1601j 7.21e-491, 7.21e-491j
2.0932373761320241725 - 2.8386112371847398261e-1617j 1.46e-498, 1.46e-498j
-3.6797457980591619091 + 2.6115966174914256355e-1611j 1.38e-492, 1.38e-492j
-5.5758881884142202387 + 2.4458014449495259906e-1605j 1.43e-488, 1.43e-488j
-0.52464762327529031788 - 1.516234191998140939e-1638j 1.27e-506, 1.27e-506j
-1.0363502036603261655 - 3.263540835160297578e-1635j 2.53e-503, 2.53e-503j
-3.4490818589268770537 - 2.1554681627024745184e-1625j 3.92e-493, 3.92e-493j
4.6149407183529111297 + 1.2953210132910417547e-1621j 2.22e-490, 2.22e-490j
5.3328356650915949362 + 3.8311580361680294323e-1618j 5.59e-489, 5.59e-489j
5.0917452971365012878 + 9.6859820857730835469e-1621j 2.01e-489, 2.01e-489j
1.6506801238857845559 + 1.2011277017779584583e-1639j 1.09e-500, 1.09e-500j
-7.6117389538933716533 - 1.9665560850007562963e-1623j 9.05e-487, 9.05e-487j
4.1445529067850734025 + 1.9179703550994613581e-1646j 1.8e-491, 1.8e-491j
0.73476434732331140816 - 1.7211956851108373517e-1660j 3.1e-505, 3.1e-505j
7.3463139400695203399 - 2.3255767705788230886e-1641j 7.2e-487, 7.2e-487j
8.1550137567230136803 + 3.174858072418751488e-1641j 9.39e-487, 9.39e-487j
-10.247990209125360078 + 1.4735464294043684383e-1644j 2.88e-488, 2.88e-488j
0.90837151181107858892 + 2.5022070492621107408e-1674j 5.09e-504, 5.09e-504j
-5.0917452971365012878 + 1.0105639413687180105e-1656j 2.05e-489, 2.05e-489j
7.6117389538933716533 - 3.3870739569330379307e-1659j 9.88e-487, 9.88e-487j
-4.1445529067850734025 + 2.2791613040652246621e-1665j 1.89e-491, 1.89e-491j
-3.2186621749283620582 + 2.6269683364951024975e-1668j 1.66e-493, 1.66e-493j
3.4490818589268770537 + 5.2265846707240786762e-1668j 3.99e-493, 3.99e-493j
9.3050981178737809835 - 4.4916110051450481117e-1660j 3.13e-487, 3.13e-487j
4.8524867552345506937 + 5.3537563675989662682e-1668j 7.14e-490, 7.14e-490j
-7.8811719756347078181 + 8.8907811384286738592e-1667j 9.91e-487, 9.91e-487j
-0.29331657200893332007 - 2.5471391385071547717e-1710j 8.53e-508, 8.53e-508j
-6.0684659147569613041 + 1.3924248866511127022e-1690j 6.94e-488, 6.94e-488j
1.0363502036603261655 + 1.7315572713729884106e-1733j 2.47e-503, 2.47e-503j
1.4327708969072488925 + 3.4918925946503473944e-1732j 1.24e-501, 1.24e-501j
-7.3463139400695203399 - 3.4566908206972452403e-1715j 7.11e-487, 7.11e-487j
-1.4327708969072488925 + 3.3484724467657964354e-1760j 1.17e-501, 1.17e-501j
-7.0845482630659029811 + 1.5162933161475915369e-1744j 5.47e-487, 5.47e-487j
6.3183222432279279225 - 1.1399107027854067597e-1764j 1.28e-487, 1.28e-487j
0.29331657200893332007 + 9.2350071143628976514e-1786j 8.53e-508, 8.53e-508j
-5.821045823352423588 - 1.8876031988700047812e-1764j 3.09e-488, 3.09e-488j
3.6797457980591619091 + 4.9590541736071436578e-1786j 1.41e-492, 1.41e-492j
-2.0932373761320241725 - 1.7324708399697752444e-1791j 1.25e-498, 1.25e-498j
-8.7178626297942803174 - 1.5762275612350040479e-1778j 7.33e-487, 7.33e-487j
3.0000019506766675549 - 4.4514402415357035328e-1799j 9.83e-494, 9.83e-494j
3.0486879132310133212 - 1.081599019081550554e-1797j 1.62e-493, 1.62e-493j
-2.3166778458612639611 + 1.0667921102011840608e-1801j 1.44e-497, 1.44e-497j
-0.73476434732331140816 + 4.1221282810379984298e-1809j 3.25e-505, 3.25e-505j
-4.3789971524607627847 + 6.5991898874673882243e-1795j 6.47e-491, 6.47e-491j
0.18980675578014987883 - 7.6988669559510243424e-1813j 2.99e-508, 2.99e-508j
0.52464762327529031788 - 1.4563931700676419576e-1810j 1.26e-506, 1.26e-506j
-0.1328720289509441177 - 6.7245112728962206801e-1813j 9.14e-509, 9.14e-509j
-3.0486879132310133212 + 2.0465770258723616945e-1797j 1.63e-493, 1.63e-493j
-0.18980675578014987883 + 1.1261363698519461408e-1855j 2.99e-508, 2.99e-508j
0.1328720289509441177 + 6.5052597895780902747e-1813j 9.14e-509, 9.14e-509j
3.2186621749283620582 + 7.3243592195575845573e-1798j 1.74e-493, 1.74e-493j
5.821045823352423588 - 6.1905225895384332538e-1793j 3.16e-488, 3.16e-488j

Weights

Weight (ball center) Error (ball radius)
4.2762919934626482057e-50 + 9.2921531908019208752e-1349j 7.21e-140, 2.55e-380j
4.5700133005207940038e-47 + 2.1777122179534016064e-1350j 8.58e-140, 3.03e-380j
2.2242320373546175114e-70 + 6.7922586689764342554e-1361j 1.81e-148, 6.41e-389j
2.3083664669026099558e-53 + 4.0924415409521938645e-1351j 1.83e-141, 6.46e-382j
6.4049423601347529267e-57 - 1.3386343811348072109e-1355j 6.95e-145, 2.46e-385j
7.6843784760767863974e-61 + 1.1629568275111354291e-1357j 1.28e-146, 4.53e-387j
6.4049423601347529267e-57 - 2.5131562321138535033e-1353j 2.09e-144, 7.38e-385j
0.026052658872880845959 - 2.5558320863037624462e-1321j 5.43e-54, 1.92e-294j
1.6218944314862408201e-28 + 2.1979346577614009317e-1340j 2.21e-131, 7.8e-372j
1.5940758932912421886e-34 - 1.1598492393176571703e-1343j 2.32e-135, 8.2e-376j
0.00059015327211331673684 - 3.1529861466348616456e-1324j 7.84e-84, 2.77e-324j
4.2762919934626482057e-50 - 5.3774609843134544009e-1352j 1.51e-142, 5.35e-383j
0.0037668650367335293651 + 1.6173029957908681079e-1322j 2.31e-76, 8.16e-317j
6.0865042745350803332e-14 + 6.6051976566929769149e-1329j 5.77e-119, 2.04e-359j
3.0014325571500269395e-65 - 5.6912422731886432158e-1360j 2.31e-149, 8.18e-390j
3.0581057469775841934e-44 + 1.8126999711943312078e-1346j 1.4e-142, 4.96e-383j
6.2367147880162702488e-10 + 1.6566071822330399476e-1330j 4.87e-112, 1.72e-352j
6.5129923776533568867e-19 - 1.2337330274877010238e-1332j 3.36e-126, 1.19e-366j
0.033587282258781753836 - 1.2098408252055873142e-1320j 5.18e-65, 1.83e-305j
7.5198494941384824527e-11 - 1.3135386321620240949e-1327j 2.42e-114, 8.56e-355j
1.8146394146813614011e-05 - 4.113315223773206339e-1323j 4.47e-99, 1.58e-339j
2.9714738043572782085e-08 + 4.9044729019412172962e-1326j 1.14e-108, 4.03e-349j
1.3618607379419571076e-41 - 3.4304762176433028242e-1345j 3.8e-142, 1.34e-382j
9.5083891347958357506e-37 + 7.2693380244877755648e-1345j 5.02e-140, 1.77e-380j
1.4239350524166428046e-17 - 2.8992643459622119269e-1334j 5.66e-128, 2e-368j
8.4066587431759161919e-22 - 1.9481629774022716902e-1334j 3.63e-131, 1.28e-371j
2.2242320373546175114e-70 + 1.1871806478458969367e-1362j 5.12e-155, 1.81e-395j
7.9238689112195113766e-77 - 2.1838533404865843411e-1364j 1.74e-158, 6.15e-399j
2.0428189686462970925e-32 + 1.6187680991758609037e-1342j 3.55e-138, 1.25e-378j
1.3618607379419571076e-41 + 1.8117109261144974903e-1347j 5.96e-143, 2.11e-383j
2.0428189686462970925e-32 - 1.9541732058744274204e-1340j 5.53e-141, 1.95e-381j
0.026052658872880845959 + 7.08736559734535367e-1322j 3.45e-80, 1.22e-320j
6.0443913019392748447e-05 - 7.0437618612575767553e-1325j 9.8e-106, 3.46e-346j
8.4066587431759161919e-22 + 1.1756281073089049653e-1336j 1.48e-133, 5.22e-374j
3.0014325571500269395e-65 - 4.4962972293046023724e-1358j 4.87e-155, 1.72e-395j
0.0037668650367335293651 - 2.1370381056427616779e-1323j 3.43e-95, 1.21e-335j
4.2297840119909140988e-39 + 6.1183216968690722893e-1344j 4.58e-144, 1.62e-384j
3.0581057469775841934e-44 - 6.9597821695375317889e-1349j 7.91e-145, 2.8e-385j
2.0427368147150793241e-30 + 2.3744703760746692304e-1339j 1.82e-140, 6.42e-381j
2.3381326913891779843e-23 - 1.5818126329659577596e-1337j 1.99e-135, 7.02e-376j
0.0081168249556428289939 - 3.5721531107885679931e-1322j 1.05e-95, 3.73e-336j
2.5418443644989321771e-20 + 1.5720365815312103108e-1333j 8.62e-134, 3.05e-374j
4.3302427451911070056e-15 - 7.5485306826665539541e-1333j 2.93e-130, 1.03e-370j
7.9238689112195113766e-77 - 5.191105113779602576e-1366j 5.58e-161, 1.97e-401j
2.5418443644989321771e-20 - 8.0122978830956980307e-1336j 9.23e-134, 3.26e-374j
0.00019924034777574045714 - 2.4332589433285518562e-1323j 5.23e-111, 1.85e-351j
0.00019924034777574045714 + 1.3557779864544068341e-1324j 2.11e-110, 7.46e-351j
9.5083891347958357506e-37 - 9.9490359583073551096e-1343j 6.19e-145, 2.19e-385j
2.9714738043572782085e-08 - 1.6216777123046404167e-1328j 9.72e-123, 3.43e-363j
2.3083664669026099558e-53 + 1.0048317077296340985e-1353j 6.91e-150, 2.44e-390j
7.9896967960131826001e-12 + 4.4997414338518833418e-1328j 1.4e-128, 4.96e-369j
6.0443913019392748447e-05 + 1.7568699342704426416e-1323j 8.11e-115, 2.87e-355j
7.6843784760767863974e-61 + 1.338819140503673387e-1355j 4.49e-156, 1.59e-396j
0.0015718291405253260663 + 7.9694246232032197646e-1324j 2.15e-109, 7.58e-350j
1.7170159245967418876e-07 - 1.820509691170352289e-1325j 2.05e-122, 7.26e-363j
4.3302427451911070056e-15 + 8.4228380156855212237e-1330j 6.06e-133, 2.14e-373j
0.092995740813785467044 - 1.8770227891863723183e-1320j 2.78e-99, 9.84e-340j
0.029228653625810267497 + 7.7882856580667580596e-1321j 4.63e-99, 1.64e-339j
8.85476631758619514e-07 + 7.68111390126448684e-1325j 4.17e-121, 1.47e-361j
7.5198494941384824527e-11 - 1.3373950661214218908e-1330j 3.81e-130, 1.35e-370j
6.0865042745350803332e-14 + 3.3780809895002613967e-1332j 1.32e-133, 4.65e-374j
7.4550746970292416281e-13 - 1.3636806627309766428e-1331j 8.38e-133, 2.96e-373j
0.0081168249556428289939 + 6.1163107309349574657e-1323j 5.45e-112, 1.93e-352j
1.0378426794384682475e-26 + 2.7128403781618422698e-1337j 4.33e-145, 1.53e-385j
4.572156800451468725e-09 + 1.3716711655302591136e-1329j 7.41e-129, 2.62e-369j
0.065910651523772700483 - 6.0730778801176560313e-1321j 9.81e-108, 3.47e-348j
5.4215829909174734519e-25 + 1.9444442313187354457e-1338j 3.85e-142, 1.36e-382j
2.0427368147150793241e-30 - 1.9972558558129654785e-1341j 3.1e-145, 1.1e-385j
4.5700133005207940038e-47 - 1.0433019170439553075e-1347j 1.27e-155, 4.49e-396j
0.033587282258781753836 + 4.6997247372612814962e-1321j 3.39e-113, 1.2e-353j
7.4550746970292416281e-13 - 1.9822535902710265017e-1328j 2.38e-136, 8.42e-377j
1.0378426794384682475e-26 - 2.1737263724366456524e-1339j 1.58e-143, 5.59e-384j
4.572156800451468725e-09 - 1.4046503920242411387e-1326j 1.2e-132, 4.23e-373j
4.1416161202311286863e-06 - 4.8809425374171506006e-1324j 4.58e-128, 1.62e-368j
8.85476631758619514e-07 - 9.150010168542431679e-1327j 4.28e-131, 1.51e-371j
4.2297840119909140988e-39 - 3.9351159700459384062e-1346j 1.29e-150, 4.56e-391j
7.9896967960131826001e-12 + 4.8034479583717956834e-1331j 4.33e-135, 1.53e-375j
1.6218944314862408201e-28 - 2.6429605693328203855e-1338j 3.91e-149, 1.38e-389j
0.14677663487395646589 + 8.9726915957328165895e-1320j 1.72e-120, 6.06e-361j
1.4239350524166428046e-17 + 9.6816860256724849598e-1332j 5.95e-143, 2.1e-383j
0.029228653625810267497 - 2.6384495900199248888e-1321j 6.52e-122, 2.3e-362j
0.015638133193231573865 - 1.9269919013853543913e-1322j 2.89e-124, 1.02e-364j
5.4215829909174734519e-25 - 2.586471012576876481e-1336j 6.17e-148, 2.18e-388j
0.015638133193231573865 + 8.7842596201412122868e-1322j 1.57e-126, 5.56e-367j
2.3381326913891779843e-23 + 2.3089832205573320789e-1335j 4.77e-147, 1.69e-387j
6.5129923776533568867e-19 + 5.0216002994249459842e-1335j 7.86e-145, 2.78e-385j
0.14677663487395646589 - 6.6315060526712009507e-1320j 3.33e-127, 1.18e-367j
2.6726223190772580483e-16 - 8.0447613749131311316e-1331j 5.95e-143, 2.1e-383j
1.7170159245967418876e-07 + 1.2468341524180013068e-1327j 1.71e-137, 6.06e-378j
0.0015718291405253260663 - 7.9138261834399154697e-1323j 5.86e-133, 2.07e-373j
1.5940758932912421886e-34 + 1.4645258441046704021e-1341j 1.66e-153, 5.86e-394j
1.8146394146813614011e-05 + 1.1412559580388269822e-1324j 3.21e-136, 1.14e-376j
-2.0628224419483233797e-06 - 9.5253773437205427906e-1325j 2e-136, 7.07e-377j
0.00059015327211331673684 + 4.1703125663074416723e-1323j 1.6e-135, 5.65e-376j
0.065910651523772700483 + 1.3011145527217001987e-1320j 1.1e-131, 3.88e-372j
6.2367147880162702488e-10 + 4.1941347580043948695e-1327j 7.46e-141, 2.64e-381j
-0.13622621350934075325 + 2.1170425173842714753e-1319j 3.09e-132, 1.09e-372j
0.092995740813785467044 + 1.0914790628205804949e-1320j 1.21e-132, 4.28e-373j
0.2117097843206967301 + 2.09830359484245827e-1319j 7.46e-133, 2.64e-373j
-2.0628224419483233797e-06 + 3.7261799322038165682e-1323j 1.34e-137, 4.73e-378j
-0.13622621350934075325 - 2.5742223353072282703e-1319j 8.39e-133, 2.97e-373j
0.2117097843206967301 - 1.8299436870759310147e-1319j 6.47e-133, 2.28e-373j
4.1416161202311286863e-06 + 9.2237677194759371722e-1326j 8.34e-140, 2.76e-380j
2.6726223190772580483e-16 + 1.5425308414621936387e-1333j 6.95e-148, 2.65e-388j

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