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

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)
9.1075488611794088727 + 1.0082408410287642091e-1164j 1.16e-488, 1.16e-488j
-10.829272813534168045 + 1.2425878549550138239e-1181j 5.41e-491, 5.41e-491j
8.2013547208892816223 - 2.9442086003083652202e-1178j 4.23e-488, 4.23e-488j
-9.7551273750832167558 - 2.560650431570647592e-1190j 2.46e-489, 2.46e-489j
-10.453442084393811949 + 9.4650004135364397168e-1192j 2.45e-490, 2.45e-490j
10.829272813534168045 + 2.1982285174328570455e-1188j 5.36e-491, 5.36e-491j
11.665452867466220216 - 8.959706558338862814e-1196j 9.26e-493, 9.26e-493j
-10.096678365124808763 - 3.8819716062804604085e-1201j 8.64e-490, 8.64e-490j
-12.746861799264025423 - 1.0856131413285476778e-1205j 1.21e-495, 1.21e-495j
-6.5392302982875889753 - 6.666581927221840043e-1197j 2.19e-488, 2.19e-488j
1.7320508075688772935 + 2.9925496879562103176e-1210j 1.19e-499, 1.19e-499j
-11.230022720039603436 - 1.4277034598970303173e-1202j 9.07e-492, 9.07e-492j
-1.593468243410359475 + 4.1578323406707789533e-1211j 1.91e-500, 1.91e-500j
7.912498089705349314 - 2.6877400559954396134e-1198j 5.01e-488, 5.01e-488j
-9.1075488611794088727 + 1.0089260187391677113e-1209j 1.22e-488, 1.22e-488j
-12.154516367570292414 + 9.0392954418436203005e-1217j 5.55e-494, 5.55e-494j
5.5029020033299973065 + 6.4988568490387050621e-1209j 2.49e-489, 2.49e-489j
-8.4962623493192049594 + 1.823459611325768505e-1214j 3.25e-488, 3.25e-488j
-0.7071067811865475244 + 7.3325407306481450376e-1232j 1.04e-505, 1.04e-505j
-5.7579306028387025229 - 2.2345748144496088815e-1215j 4.8e-489, 4.8e-489j
3.1427583224059007146 + 3.3026989516058636423e-1219j 1.17e-493, 1.17e-493j
0.93157368698506392525 + 5.3142770196326690161e-1230j 1.7e-504, 1.7e-504j
12.746861799264025423 - 4.7125288615345657244e-1221j 1.26e-495, 1.26e-495j
-4.0108807365021198406 + 3.8796742777869246568e-1218j 1.94e-491, 1.94e-491j
10.096678365124808763 - 6.6618175090541471749e-1213j 8.77e-490, 8.77e-490j
-2.7800675753189594109 - 6.6086236911583593019e-1228j 3.07e-495, 3.07e-495j
1.3838640636386493628 - 5.064814859000945364e-1234j 7.71e-502, 7.71e-502j
-3.6749852443872490457 + 2.8232739902692804649e-1222j 1.91e-490, 1.91e-490j
10.453442084393811949 - 3.1435382467166972525e-1219j 2.51e-490, 2.51e-490j
9.7551273750832167558 + 2.0810481610524387591e-1235j 2.54e-489, 2.54e-489j
4.9996676859815507075 - 1.5990262074210012054e-1272j 5.01e-490, 5.01e-490j
6.2758587585066707489 - 8.9455699079074547027e-1320j 1.5e-488, 1.5e-488j
12.154516367570292414 + 1.0495198560463314832e-1362j 5.45e-494, 5.45e-494j
5.2502146713148472345 - 3.6830431706376237481e-1429j 1.09e-489, 1.09e-489j
7.3504220508398602665 - 6.7975402275576715478e-1480j 5.03e-488, 5.03e-488j
2.5607755625350494119 - 1.8018345669240557187e-1513j 2.79e-496, 2.79e-496j
-4.2580918938723234208 + 3.342653230739996052e-1512j 3.73e-491, 3.73e-491j
1.1587454282597314315 - 4.162039960250336616e-1536j 3.36e-503, 3.36e-503j
6.8058994400492685949 - 8.8463446745555620922e-1533j 3.14e-488, 3.14e-488j
-1.1587454282597314315 - 3.2034444356897401e-1570j 3.25e-503, 3.25e-503j
-2.9734984797723849738 - 6.8932116184771536542e-1562j 2.56e-494, 2.56e-494j
-8.7980071385385987822 + 5.7170010636505896402e-1554j 2.01e-488, 2.01e-488j
6.5392302982875889753 - 3.3385413327184988742e-1563j 2.31e-488, 2.31e-488j
3.3458138508944375668 + 1.1525527234997167787e-1582j 5.99e-493, 5.99e-493j
-6.8058994400492685949 - 4.4545290875413665265e-1578j 3.11e-488, 3.11e-488j
2.7800675753189594109 - 4.044296636462361623e-1585j 3.3e-495, 3.3e-495j
-5.5029020033299973065 + 1.0476923293143501419e-1580j 2.34e-489, 2.34e-489j
-7.6290380015512510669 - 6.0900351020316430241e-1578j 5.24e-488, 5.24e-488j
-8.2013547208892816223 + 1.8112626874316395531e-1579j 4.27e-488, 4.27e-488j
4.2580918938723234208 + 1.1785314183323958956e-1580j 4e-491, 4e-491j
-0.93157368698506392525 - 1.2996088461080043285e-1594j 1.63e-504, 1.63e-504j
-3.7045500301318508604 - 2.0853599246866653196e-1581j 1.34e-490, 1.34e-490j
-6.0155080728209515942 + 6.8085350836355183156e-1579j 8.8e-489, 8.8e-489j
-11.665452867466220216 + 1.0968155185919008786e-1584j 9.83e-493, 9.83e-493j
-9.4260819061892751299 + 2.0958778373333529339e-1579j 6.05e-489, 6.05e-489j
-1.8904601289672687258 - 1.4428382382859349324e-1590j 5.52e-499, 5.52e-499j
-4.7510451839247759756 + 7.2290763902834866657e-1581j 2.15e-490, 2.15e-490j
5.7579306028387025229 - 3.8022108993452305876e-1577j 4.92e-489, 4.92e-489j
-7.0761783342754597201 - 5.9330347042890991153e-1583j 4.25e-488, 4.25e-488j
0.51198835281687566252 - 1.5341441838790325269e-1600j 1.71e-506, 1.71e-506j
8.7980071385385987822 - 8.8885503462088282933e-1581j 2.09e-488, 2.09e-488j
9.4260819061892751299 - 2.4681135475969739895e-1585j 6.1e-489, 6.1e-489j
-0.51198835281687566252 - 1.4870797961303053088e-1610j 1.77e-506, 1.77e-506j
3.7045500301318508604 - 5.0293281684754108762e-1595j 1.38e-490, 1.38e-490j
8.4962623493192049594 - 2.0725810557423412352e-1590j 3.2e-488, 3.2e-488j
-7.3504220508398602665 - 1.0169986236643673709e-1603j 4.77e-488, 4.77e-488j
-4.5040545022880217428 + 6.5522797134327770621e-1614j 9.06e-491, 9.06e-491j
6.0155080728209515942 - 7.0308723165221904936e-1610j 8.54e-489, 8.54e-489j
-2.5607755625350494119 + 1.0400572000111671895e-1638j 2.77e-496, 2.77e-496j
3.6749852443872490457 + 3.5307291732040175263e-1630j 1.92e-490, 1.92e-490j
2.1073815878368520694 - 5.2348893930628494614e-1642j 3.43e-498, 3.43e-498j
7.6290380015512510669 + 2.5372081666998387207e-1631j 5.13e-488, 5.13e-488j
-4.9996676859815507075 - 1.0542772667414848863e-1647j 4.76e-490, 4.76e-490j
-3.1427583224059007146 - 1.9734135977465358813e-1650j 1.15e-493, 1.15e-493j
11.230022720039603436 - 1.5663265131740195762e-1643j 8.51e-492, 8.51e-492j
2.3340614410930684954 + 4.9235675669082929599e-1657j 2.93e-497, 2.93e-497j
7.0761783342754597201 + 3.1221442609788511551e-1646j 4.23e-488, 4.23e-488j
-3.3458138508944375668 + 1.6017584467852401743e-1661j 5.9e-493, 5.9e-493j
-1.3838640636386493628 - 3.0634935126610779015e-1670j 8.2e-502, 8.2e-502j
-3.6336683685012485862 + 1.8990498470453463639e-1659j 6.89e-491, 6.89e-491j
1.8904601289672687258 - 5.4219816336396327447e-1667j 5.15e-499, 5.15e-499j
2.9734984797723849738 - 1.9243926014592590668e-1662j 2.59e-494, 2.59e-494j
-7.912498089705349314 + 4.3885706844133971571e-1658j 5e-488, 5e-488j
-0.22276080049681409405 + 1.2077482688620842852e-1676j 1.02e-508, 1.02e-508j
-2.1073815878368520694 - 8.2978377665387901341e-1667j 3.5e-498, 3.5e-498j
-5.2502146713148472345 - 3.5286456368144275594e-1658j 1.11e-489, 1.11e-489j
0.7071067811865475244 - 1.0840011882254167624e-1673j 1.2e-505, 1.2e-505j
1.593468243410359475 + 8.3699587161817705525e-1669j 1.79e-500, 1.79e-500j
-6.2758587585066707489 + 2.4631976324282674958e-1658j 1.45e-488, 1.45e-488j
-0.41183732575728475763 + 3.2713824189672959778e-1675j 3.17e-507, 3.17e-507j
4.5040545022880217428 - 1.7930882362698667926e-1658j 8.98e-491, 8.98e-491j
3.6336683685012485862 - 3.4515667909972843515e-1660j 6.83e-491, 6.83e-491j
0.41183732575728475763 - 1.6111060754274501104e-1675j 3.17e-507, 3.17e-507j
-1.29899989308548045e-1729 - 2.0923007913801351608e-1729j 2.44e-1727, 2.44e-1727j
-2.3340614410930684954 + 1.057389660019447795e-1665j 2.85e-497, 2.85e-497j
-1.7320508075688772935 - 9.8105607784718031523e-1669j 1.16e-499, 1.16e-499j
0.22276080049681409405 + 2.5468600056916541146e-1723j 1.02e-508, 1.02e-508j
4.0108807365021198406 + 3.180734952774160371e-1659j 1.87e-491, 1.87e-491j
4.7510451839247759756 - 2.1145112566419162062e-1659j 2.29e-490, 2.29e-490j

Weights

Weight (ball center) Error (ball radius)
1.6767964394761047928e-37 - 1.7015457414972137293e-1200j 6.68e-139, 9.79e-379j
2.5594450394246330659e-52 + 7.4825782646452448625e-1211j 1.18e-146, 1.73e-386j
1.0112184014124114968e-30 + 1.153704379474322691e-1197j 3.65e-135, 5.34e-375j
8.8676599600525200927e-43 - 1.1367453049795136102e-1205j 8.6e-143, 1.26e-382j
7.1953776462791016264e-49 - 5.4605722798593989264e-1209j 1.99e-145, 2.91e-385j
2.5594450394246330659e-52 - 8.6645032918096204643e-1210j 1.06e-148, 1.56e-388j
2.0516429766997323341e-60 - 2.7855268716235879602e-1214j 4.61e-152, 6.76e-392j
1.0484416721113395817e-45 + 2.8566495686790883705e-1207j 1.57e-144, 2.31e-384j
1.0521637729265030645e-71 + 3.5137343516277547209e-1221j 1.6e-156, 2.35e-396j
4.0135293218232484271e-20 + 1.0767624022507220068e-1192j 1.7e-131, 2.49e-371j
0.003196786906909953437 + 1.6720673465243722491e-1179j 1.89e-82, 2.77e-322j
3.9822883766972640697e-56 - 6.7264318470333244821e-1213j 5.01e-150, 7.34e-390j
0.0082709911224594815656 - 1.2926685789910414879e-1179j 7.65e-81, 1.12e-320j
1.0416782486547731039e-28 - 1.2242101000407735176e-1196j 5.2e-140, 7.62e-380j
1.6767964394761047928e-37 - 9.2813921626502185989e-1203j 7.24e-143, 1.06e-382j
2.0612648742488581491e-65 - 7.5243693591278876226e-1218j 1.5e-154, 2.2e-394j
1.010841502193585849e-14 + 1.3408032636705126923e-1188j 1.04e-129, 1.53e-369j
7.5113800283787913903e-33 - 3.7404153180771761584e-1200j 3.13e-141, 4.58e-381j
0.075262154453923946963 - 4.5804655192728680289e-1179j 3.26e-86, 4.78e-326j
5.7759192865390872026e-16 - 4.826077103297591238e-1190j 1.41e-129, 2.06e-369j
5.2022534168331593179e-06 - 1.6012769484099145433e-1181j 4.64e-111, 6.81e-351j
0.053724040181747627377 + 2.6506258154960616089e-1179j 3.14e-87, 4.6e-327j
1.0521637729265030645e-71 - 2.1100299638443433925e-1220j 3.21e-162, 4.71e-402j
1.456802580685757253e-08 + 3.4069001793081405072e-1184j 1.32e-119, 1.94e-359j
1.0484416721113395817e-45 - 5.5462653983316122274e-1206j 5.81e-152, 8.52e-392j
5.2309664489915798394e-05 - 2.0969717464162006868e-1181j 5.41e-109, 7.94e-349j
0.018437380681619400988 + 1.5433607071455772499e-1179j 2.02e-87, 2.97e-327j
-2.2161585318034032876e-06 + 1.8225519238024169141e-1181j 1.44e-117, 2.11e-357j
7.1953776462791016264e-49 + 7.9363060219220993957e-1208j 2.11e-153, 3.1e-393j
8.8676599600525200927e-43 + 3.3111133538501365147e-1204j 5.65e-151, 8.28e-391j
1.9618065064568072595e-12 + 4.796677843890699041e-1187j 1.82e-131, 2.67e-371j
1.1591145678454884507e-18 - 4.7683268992780232735e-1191j 8.46e-139, 1.24e-378j
2.0612648742488581491e-65 + 5.2505854355905258636e-1217j 7.21e-162, 1.06e-401j
1.516654245622667485e-13 - 8.1108410276208244919e-1188j 3.44e-133, 5.04e-373j
5.3521392863127632101e-25 - 1.1665469439735375516e-1194j 6.18e-144, 9.06e-384j
0.00017999514162850277089 + 6.0715366500851558359e-1181j 1.52e-115, 2.22e-355j
1.8526994105183936343e-09 - 3.968110321031683273e-1185j 1.22e-130, 1.79e-370j
0.033437083827080095737 - 1.7774106929733648095e-1179j 3.84e-95, 5.62e-335j
1.1577142575324093617e-21 - 8.4898884332251125678e-1193j 6.72e-142, 9.85e-382j
0.033437083827080095737 - 1.3761818841035631546e-1179j 1.76e-96, 2.59e-336j
1.4163101883845628971e-05 + 1.4201980612703766593e-1181j 4.72e-121, 6.92e-361j
4.1672964960251239027e-35 + 2.0141390031607323834e-1201j 3.65e-152, 5.35e-392j
4.0135293218232484271e-20 + 6.5598807557954595094e-1192j 8.41e-141, 1.23e-380j
1.7452665279993149552e-06 + 6.7361069898808494783e-1182j 6.16e-125, 9.03e-365j
1.1577142575324093617e-21 - 1.2279391887649863287e-1193j 6.24e-146, 9.15e-386j
5.2309664489915798394e-05 - 3.9396395932983205863e-1181j 7.41e-121, 1.09e-360j
1.010841502193585849e-14 + 3.3079898191829814052e-1189j 2.77e-140, 4.05e-380j
8.3782982660139833767e-27 + 1.090426465348098259e-1196j 7.67e-150, 1.12e-389j
1.0112184014124114968e-30 + 6.0401292498421369839e-1199j 1.86e-151, 2.73e-391j
1.8526994105183936343e-09 - 1.0936551738701357254e-1184j 2.08e-135, 3.06e-375j
0.053724040181747627377 + 2.1586997030453720641e-1179j 5.77e-107, 8.46e-347j
1.1819458498132040614e-06 - 8.903439496464289925e-1182j 1.07e-130, 1.57e-370j
2.8123690297000741468e-17 + 6.700031881050833346e-1191j 1.88e-143, 2.75e-383j
2.0516429766997323341e-60 + 3.4299859200331493122e-1215j 6.44e-166, 9.43e-406j
4.7184013763808136689e-40 + 3.589317078571498206e-1204j 7.22e-156, 1.06e-395j
0.0032003916747687682481 - 5.190082557425104516e-1180j 6.06e-122, 8.89e-362j
2.1996246369491829962e-11 - 8.8646981286157384141e-1187j 1.95e-139, 2.86e-379j
5.7759192865390872026e-16 - 2.1417948117766138675e-1189j 3.54e-145, 5.19e-385j
2.7554128191962999323e-23 + 1.2922508739417194061e-1194j 2.15e-148, 3.15e-388j
0.060813884128018828909 + 1.4623194371871227125e-1178j 1.56e-115, 2.28e-355j
4.1672964960251239027e-35 + 1.1650861926792612255e-1199j 3.87e-158, 5.68e-398j
4.7184013763808136689e-40 - 2.0884199765740170923e-1202j 1.38e-160, 2.02e-400j
0.060813884128018828909 + 1.3066590341376536248e-1178j 1.79e-119, 2.62e-359j
1.1819458498132040614e-06 - 2.1112672956066073495e-1181j 1.91e-139, 2.79e-379j
7.5113800283787913903e-33 - 1.0771637166023790537e-1198j 3.97e-157, 5.82e-397j
5.3521392863127632101e-25 - 1.2454578524135436038e-1195j 2.5e-152, 3.66e-392j
2.1508653235661400562e-10 + 5.726078919039658182e-1186j 2.18e-142, 3.2e-382j
2.8123690297000741468e-17 + 3.2769607690574582096e-1190j 1.74e-148, 2.55e-388j
0.00017999514162850277089 + 3.4065708647632898819e-1181j 2.1e-135, 3.08e-375j
-2.2161585318034032876e-06 + 4.2883680281703019686e-1181j 6.45e-140, 9.46e-380j
0.001493058526709152699 + 2.6477765522448028313e-1180j 6.06e-134, 8.89e-374j
8.3782982660139833767e-27 + 1.234351248532716001e-1195j 9.94e-155, 1.46e-394j
1.9618065064568072595e-12 + 1.3967448895687625492e-1187j 3.62e-146, 5.3e-386j
5.2022534168331593179e-06 - 7.7967690513122241303e-1182j 2.77e-139, 4.07e-379j
3.9822883766972640697e-56 + 6.4452755732571316959e-1212j 9.47e-171, 1.39e-410j
0.0005529014595031520792 - 1.1427004036104297045e-1180j 1.29e-137, 1.89e-377j
2.7554128191962999323e-23 + 1.0295234343229257181e-1193j 5.14e-153, 7.54e-393j
1.7452665279993149552e-06 + 3.1165609140150867967e-1182j 8.55e-141, 1.25e-380j
0.018437380681619400988 + 1.1362084131385712389e-1179j 5.6e-135, 8.21e-375j
1.449342730595595672e-06 - 1.0279717982272717182e-1181j 1.57e-140, 2.3e-380j
0.0032003916747687682481 - 7.9090858854878177588e-1180j 1.56e-137, 2.29e-377j
1.4163101883845628971e-05 + 2.7970882116955939575e-1181j 3.36e-141, 4.93e-381j
1.0416782486547731039e-28 - 8.5957061610552885977e-1198j 2.99e-158, 4.38e-398j
0.11708217460698662656 + 8.4631197023284904136e-1179j 3.73e-136, 5.46e-376j
0.001493058526709152699 + 1.6526967199998631457e-1180j 2.44e-140, 3.58e-380j
1.516654245622667485e-13 - 2.1790458056631167436e-1188j 3.71e-149, 5.44e-389j
0.075262154453923946963 - 5.3515864195395943245e-1179j 5.28e-138, 7.74e-378j
0.0082709911224594815656 - 1.8409262926531493202e-1179j 3.49e-139, 5.12e-379j
1.1591145678454884507e-18 - 8.7772647131561914477e-1192j 9.3e-153, 1.36e-392j
0.06091103778811356219 - 1.4919817729085410668e-1178j 3.93e-138, 5.77e-378j
2.1508653235661400562e-10 + 1.6930859255584191866e-1185j 3.49e-150, 5.12e-390j
1.449342730595595672e-06 - 2.3927471571152991181e-1181j 2.02e-145, 2.96e-385j
0.06091103778811356219 - 1.6333051783440889841e-1178j 2.97e-140, 4.36e-380j
0.12672853484846303562 - 6.8648030127295811408e-1179j 8.36e-141, 1.23e-380j
0.0005529014595031520792 - 6.7648404414109880048e-1181j 9.41e-144, 1.38e-383j
0.003196786906909953437 + 1.1377107860666192629e-1179j 1.99e-142, 2.92e-382j
0.11708217460698662656 + 8.8874970328222845973e-1179j 2.25e-141, 3.31e-381j
1.456802580685757253e-08 + 8.7690975860024416531e-1184j 9.65e-149, 1.41e-388j
2.1996246369491829962e-11 - 2.8199735797072250622e-1186j 2.22e-151, 3.26e-391j

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