Rule: Q[8, 13, 86]

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)
14.503502877372048227 + 1.8456478902065081005e-2431j 2.32e-998, 2.32e-998j
13.60228813497460944 - 1.5811349594207987744e-2439j 1.66e-997, 1.66e-997j
15.47626206686341194 + 1.8797106196937521767e-2450j 1.78e-999, 1.78e-999j
-13.60228813497460944 - 2.1676269091795463143e-2459j 1.64e-997, 1.64e-997j
-16.0001557075749526 - 6.4652852230391750149e-2462j 3.43e-1000, 3.43e-1000j
17.166559689771098658 - 3.9686054286638671588e-2457j 4.87e-1002, 4.87e-1002j
-16.559037674723682729 + 2.1797213541893842533e-2465j 4.61e-1001, 4.61e-1001j
-1.3466427403538444606 - 1.4229785785192843775e-2478j 2.12e-1014, 2.12e-1014j
-4.6711908944959916365 - 5.4387737450231637521e-2466j 8.96e-1003, 8.96e-1003j
-12.342843723355831991 - 7.3097312396212325721e-2459j 8.2e-997, 8.2e-997j
1.6365190424351079992 + 1.5591438088982518684e-2478j 3.62e-1013, 3.62e-1013j
-14.979173598718515576 - 3.465616531444382135e-2461j 7.47e-999, 7.47e-999j
-1.0668273675044222989 + 9.9314567876465182965e-2480j 1.33e-1015, 1.33e-1015j
-4.1445471861258943321 + 1.649796101331730195e-2468j 9.37e-1004, 9.37e-1004j
-13.171842934064371624 + 6.8289228654270260496e-2460j 3.18e-997, 3.18e-997j
-14.503502877372048227 + 9.7561589406431577079e-2462j 2.47e-998, 2.47e-998j
-11.162466879528571 + 2.4374049879945269173e-2460j 1.47e-996, 1.47e-996j
-14.045468209671704213 - 2.1793641535765029674e-2462j 6.48e-998, 6.48e-998j
14.045468209671704213 - 1.0454368607893927124e-2465j 6.73e-998, 6.73e-998j
3.680294992838983705 + 1.7147345747711510921e-2493j 5.47e-1005, 5.47e-1005j
6.513221939183309358 + 2.915514006711177538e-2487j 3.56e-999, 3.56e-999j
-17.166559689771098658 + 5.6210284783056819189e-2487j 4.55e-1002, 4.55e-1002j
16.559037674723682729 - 2.0650301578950118233e-2484j 4.73e-1001, 4.73e-1001j
-7.9291908103075588996 + 1.7044172744931909817e-2496j 1.02e-997, 1.02e-997j
17.849230937530585313 - 4.0279117054455125309e-2494j 2.23e-1003, 2.23e-1003j
7.5972556691033815166 - 4.7706583362569341865e-2523j 6.06e-998, 6.06e-998j
2.5232023333578161114 + 4.7166227559857148106e-2567j 2.95e-1009, 2.95e-1009j
10.782647984952613138 + 5.0414793087218941626e-2587j 1.5e-996, 1.5e-996j
4.6711908944959916365 - 2.6863940532991371307e-2629j 8.93e-1003, 8.93e-1003j
-17.849230937530585313 + 2.6040888736793929869e-2656j 2.33e-1003, 2.33e-1003j
11.162466879528571 + 2.9425904244624927331e-2714j 1.52e-996, 1.52e-996j
3.9167718690613327338 - 1.2639361980221632251e-2758j 2.52e-1004, 2.52e-1004j
5.2674121990513315738 - 2.0661738148440473331e-2750j 1.16e-1001, 1.16e-1001j
-10.408676026690916679 + 2.5379360564813622369e-2762j 1.29e-996, 1.29e-996j
18.676489708558864251 - 5.9705875631513447998e-2816j 5.08e-1005, 5.08e-1005j
11.941874010691770221 - 4.1292804030700218786e-2873j 1.11e-996, 1.11e-996j
14.979173598718515576 + 8.6664671021117669159e-2908j 7.5e-999, 7.5e-999j
7.2809193681320787271 - 1.390578297167977593e-2917j 3.92e-998, 3.92e-998j
3.1611809891611667576 - 3.2249423321058885816e-2930j 1.77e-1006, 1.77e-1006j
-15.47626206686341194 - 6.5597593936685590277e-2920j 1.78e-999, 1.78e-999j
-5.5776695595810094094 - 2.5718548126106994151e-2927j 4.45e-1001, 4.45e-1001j
-9.31777821322528483 - 2.6016827370370126646e-2922j 5.94e-997, 5.94e-997j
8.6138254939156561728 + 1.3193483916280512915e-2944j 2.57e-997, 2.57e-997j
-18.676489708558864251 - 2.0217407317461485903e-2965j 5.09e-1005, 5.09e-1005j
-11.941874010691770221 + 8.6907764693380253158e-2984j 1.12e-996, 1.12e-996j
1.3466427403538444606 - 2.7998084021722095918e-3017j 2.08e-1014, 2.08e-1014j
-6.513221939183309358 - 2.1552951140763099915e-2999j 3.33e-999, 3.33e-999j
8.963548577965232379 - 5.4710283540322721431e-3030j 4.09e-997, 4.09e-997j
1.931843291151814829 - 5.9366978107871830917e-3077j 7.1e-1012, 7.1e-1012j
11.548665108078403759 + 5.2818219470585635443e-3078j 1.38e-996, 1.38e-996j
-4.3951104890775863417 - 2.2289830279745144433e-3109j 2.83e-1003, 2.83e-1003j
-3.1611809891611667576 - 3.1609889672125330204e-3112j 1.73e-1006, 1.73e-1006j
5.8918730094612552215 - 1.1002284749546250777e-3105j 1.85e-1000, 1.85e-1000j
8.268797774828517184 - 1.3224980287137755533e-3102j 1.56e-997, 1.56e-997j
-10.040104630681630643 + 5.5874563066965972371e-3117j 1.08e-996, 1.08e-996j
9.676567120136705075 + 2.0367160979618894511e-3138j 8.67e-997, 8.67e-997j
-4.9639840200927492835 + 1.5183007952213602941e-3175j 3.29e-1002, 3.29e-1002j
-1.6365190424351079992 + 3.9647136471173898968e-3199j 3.82e-1013, 3.82e-1013j
-11.548665108078403759 - 1.3285442556112183461e-3181j 1.38e-996, 1.38e-996j
-9.676567120136705075 - 3.5108669550093172458e-3231j 8.17e-997, 8.17e-997j
5.5776695595810094094 - 4.37765114507939413e-3266j 4.63e-1001, 4.63e-1001j
12.752471667426736412 + 3.8975316213786000943e-3298j 5.48e-997, 5.48e-997j
-2.8024858612875416991 + 1.8534453952780540111e-3332j 6.91e-1008, 6.91e-1008j
-5.2674121990513315738 - 7.0015284232116336107e-3326j 1.23e-1001, 1.23e-1001j
3.4215630331028556842 + 2.9999399354416553147e-3330j 8.62e-1006, 8.62e-1006j
-8.963548577965232379 + 4.2441958428460673582e-3321j 3.95e-997, 3.95e-997j
-2.5232023333578161114 + 6.7562316526434894705e-3345j 2.72e-1009, 2.72e-1009j
-6.2065301836485300518 + 3.3513490038554224258e-3335j 7.79e-1000, 7.79e-1000j
-5.8918730094612552215 + 2.0215965707272578263e-3335j 1.88e-1000, 1.88e-1000j
10.040104630681630643 - 4.9915760256025354271e-3330j 1.07e-996, 1.07e-996j
1.0668273675044222989 - 1.1465590753996404037e-3364j 1.37e-1015, 1.37e-1015j
-7.5972556691033815166 - 5.085659534115917689e-3345j 5.76e-998, 5.76e-998j
-2.2288115845706893975 - 6.6422190989167978954e-3363j 1.38e-1010, 1.38e-1010j
-12.752471667426736412 - 6.3410242373424756808e-3349j 5.26e-997, 5.26e-997j
16.0001557075749526 - 2.5856896178019516305e-3363j 3.49e-1000, 3.49e-1000j
4.1445471861258943321 - 3.9909409529974412244e-3386j 8.94e-1004, 8.94e-1004j
2.8024858612875416991 - 5.760803834261752163e-3390j 6.94e-1008, 6.94e-1008j
-3.9167718690613327338 - 5.4602107132154870707e-3387j 2.73e-1004, 2.73e-1004j
7.9291908103075588996 + 3.1024775629069294912e-3379j 9.13e-998, 9.13e-998j
-0.53907981135137510807 + 4.8326731136743268733e-3410j 4.89e-1018, 4.89e-1018j
6.7837355311414818919 + 2.7855805909669495673e-3389j 1.19e-998, 1.19e-998j
2.9897145805308285518 - 1.9804180707693857885e-3414j 5.33e-1007, 5.33e-1007j
12.342843723355831991 - 7.498016019109102942e-3403j 8.84e-997, 8.84e-997j
-7.2809193681320787271 - 1.7191310006740889243e-3422j 3.87e-998, 3.87e-998j
-2.9897145805308285518 + 6.4652403765135012584e-3431j 5.02e-1007, 5.02e-1007j
13.171842934064371624 + 1.5514102199889101552e-3418j 3e-997, 3e-997j
6.2065301836485300518 - 2.7577582077969386943e-3430j 7.51e-1000, 7.51e-1000j
2.2288115845706893975 - 8.9527426700099822138e-3441j 1.26e-1010, 1.26e-1010j
-8.6138254939156561728 - 4.0040945610093531338e-3428j 2.58e-997, 2.58e-997j
-3.4215630331028556842 - 3.3979148414651842506e-3436j 8.47e-1006, 8.47e-1006j
4.9639840200927492835 - 1.842431277912352251e-3432j 3.18e-1002, 3.18e-1002j
-6.7837355311414818919 - 3.8829208541898056351e-3429j 1.29e-998, 1.29e-998j
0.79980551632613013884 - 1.2367798801438550481e-3446j 1.01e-1016, 1.01e-1016j
2.7228084942927257103e-3531 - 6.5906106762508337677e-3532j 3e-3529, 3e-3529j
-3.680294992838983705 - 2.067464051068902959e-3435j 5.14e-1005, 5.14e-1005j
-0.79980551632613013884 - 1.182698627710988714e-3447j 9.41e-1017, 9.41e-1017j
-8.268797774828517184 - 7.5346919693057771971e-3429j 1.57e-997, 1.57e-997j
10.408676026690916679 + 2.4502254877582848643e-3426j 1.34e-996, 1.34e-996j
7.0083416616804912679 + 2.4323541689246772433e-3433j 2.59e-998, 2.59e-998j
-1.931843291151814829 - 7.8327081613659525801e-3447j 7.08e-1012, 7.08e-1012j
-0.27303081997926625885 + 4.9227934170496859996e-3454j 3.72e-1019, 3.72e-1019j
9.31777821322528483 + 1.2039841064264645666e-3432j 6e-997, 6e-997j
4.3951104890775863417 + 3.4466050542949928235e-3438j 2.83e-1003, 2.83e-1003j
0.53907981135137510807 - 3.1300291739114787617e-3454j 4.89e-1018, 4.89e-1018j
-10.782647984952613138 - 1.3645015088257940186e-3431j 1.57e-996, 1.57e-996j
0.27303081997926625885 - 3.999726650975171292e-3454j 3.72e-1019, 3.72e-1019j
-7.0083416616804912679 - 1.2727359777108217648e-3433j 2.4e-998, 2.4e-998j

Weights

Weight (ball center) Error (ball radius)
3.9113405800762594167e-47 - 5.7816453080135554651e-2477j 2.74e-371, 2.97e-866j
1.1583175568807071943e-41 - 8.8634541421349448051e-2475j 1.24e-368, 1.34e-863j
1.9862224808159359049e-53 + 2.60445987572836364e-2481j 1.39e-374, 1.51e-869j
1.1583175568807071943e-41 - 2.8421126446982542482e-2476j 1.07e-371, 1.16e-866j
5.5243955275243011189e-57 + 9.6647576563839691046e-2485j 2.22e-379, 2.4e-874j
2.6004309096671793304e-65 - 3.5721408063405577611e-2488j 5.92e-382, 6.4e-877j
6.6430926429996698e-61 - 7.1953555095096587131e-2487j 4.05e-381, 4.38e-876j
0.046021852345968517843 + 3.0421571304528650568e-2450j 1.78e-274, 1.93e-769j
2.0830245480031517638e-06 - 4.4669213706990239815e-2453j 3.61e-328, 3.9e-823j
1.3398057129951371991e-34 + 3.0337648698652522255e-2472j 4.19e-369, 4.54e-864j
0.030666214643099631334 - 2.7891251498865276195e-2450j 2.59e-286, 2.81e-781j
3.6700743390969050328e-50 + 5.2684082699090938086e-2481j 4.01e-377, 4.33e-872j
0.061729223522720115293 - 4.2318098944879085495e-2450j 7.13e-271, 7.72e-766j
1.7529618458764006642e-05 - 5.9527907362168322334e-2452j 9.3e-326, 1.01e-820j
3.5847761392900077755e-39 + 7.2778295870471696346e-2475j 7.09e-372, 7.67e-867j
3.9113405800762594167e-47 - 2.488688583882797121e-2479j 4.35e-376, 4.7e-871j
1.3406584721024739022e-28 - 1.0275082907425967716e-2468j 5.74e-368, 6.21e-863j
2.6095295746700699351e-44 + 9.3025201700209069377e-2478j 1.14e-374, 1.23e-869j
2.6095295746700699351e-44 + 5.7982213174903098178e-2476j 1.46e-380, 1.58e-875j
0.00011364047725409753646 - 4.9102930466523392698e-2451j 4.36e-330, 4.72e-825j
7.2532696546917221601e-11 - 4.8260282188096671777e-2456j 7.46e-353, 8.07e-848j
2.6004309096671793304e-65 + 3.003722438320935277e-2489j 1.05e-386, 1.13e-881j
6.6430926429996698e-61 + 1.0873384099066947164e-2485j 1.36e-388, 1.47e-883j
2.9870363017598333527e-15 + 6.0437390735304367438e-2460j 1.57e-359, 1.7e-854j
1.9312626309106388251e-70 + 5.1177230935643933147e-2491j 9.38e-393, 1.01e-887j
3.8132875557325393184e-14 - 1.5486856554533657357e-2458j 7.63e-361, 8.26e-856j
0.0048078337114488667693 + 2.2431975303207902148e-2450j 4.94e-321, 5.35e-816j
8.5173657112380900139e-27 + 8.5676049963491921195e-2467j 4.67e-374, 5.05e-869j
2.0830245480031517638e-06 - 8.7112613395760907766e-2453j 2.62e-344, 2.83e-839j
1.9312626309106388251e-70 - 5.292443105636702372e-2492j 9.91e-393, 1.07e-887j
1.3406584721024739022e-28 - 7.8933530193148254592e-2468j 2.48e-375, 2.68e-870j
4.2104353085066137201e-05 + 2.5490095194948110541e-2451j 1.4e-338, 1.52e-833j
1.1578986904105689807e-07 - 5.7561176163835458277e-2454j 3.51e-349, 3.8e-844j
4.4123923467438657753e-25 - 1.4193140144283260369e-2466j 3.3e-373, 3.57e-868j
6.8959105154243180667e-77 - 1.5182413787207223652e-2494j 2.03e-397, 2.19e-892j
1.7085924986813539886e-32 - 5.2551037930858015391e-2470j 1.65e-378, 1.79e-873j
3.6700743390969050328e-50 - 3.2654316089082745741e-2479j 1.27e-385, 1.37e-880j
3.7100233945684234342e-13 + 1.166301352516009677e-2457j 4.41e-362, 4.77e-857j
0.00064824284002771507636 - 2.5955665994405276775e-2450j 2.68e-336, 2.9e-831j
1.9862224808159359049e-53 - 8.4507310480065965871e-2483j 2.01e-387, 2.18e-882j
2.1902536678021352574e-08 + 6.8758199577192642257e-2455j 2.9e-358, 3.14e-853j
1.9953324336500906962e-20 + 1.3106097916117284655e-2463j 2.24e-373, 2.42e-868j
1.0712640358450014033e-17 + 3.6994924728430613216e-2461j 4.73e-370, 5.12e-865j
6.8959105154243180667e-77 + 1.9094634448952169793e-2495j 2.2e-399, 2.38e-894j
1.7085924986813539886e-32 - 5.0903571138115704514e-2471j 1.91e-379, 2.07e-874j
0.046021852345968517843 + 3.664904205323756138e-2450j 3.67e-321, 3.97e-816j
7.2532696546917221601e-11 - 1.8347924759476202545e-2456j 2.19e-364, 2.37e-859j
5.0198638641163305583e-19 - 4.8088215330690507266e-2462j 5.35e-372, 5.79e-867j
0.018318149936272770612 + 2.2721034672171950271e-2450j 3.02e-331, 3.26e-826j
1.6991345264205265607e-30 + 6.7166827227251951476e-2469j 1.61e-379, 1.74e-874j
6.7489864688733630258e-06 + 1.7717206702088304769e-2452j 1.87e-354, 2.03e-849j
0.00064824284002771507636 - 1.6665882204790771339e-2450j 1.26e-347, 1.37e-842j
3.6423846737370645884e-09 - 4.3862447816843577714e-2455j 8.27e-361, 8.95e-856j
1.9434467459691796271e-16 - 2.7747936219576878318e-2460j 1.26e-369, 1.36e-864j
1.884280221135955018e-23 + 1.4780933310491837375e-2465j 2.43e-377, 2.63e-872j
6.6954012123941596738e-22 - 7.1904973032307419938e-2464j 1.34e-375, 1.45e-870j
5.3168655710335509311e-07 + 1.0882221110482886621e-2453j 5.99e-360, 6.48e-855j
0.030666214643099631334 - 2.223518017915904776e-2450j 1.14e-336, 1.24e-831j
1.6991345264205265607e-30 + 7.6180708427149130828e-2470j 6.96e-381, 7.54e-876j
6.6954012123941596738e-22 - 1.4354004656077418719e-2464j 1.04e-376, 1.12e-871j
2.1902536678021352574e-08 + 1.5469088501016910237e-2454j 1.68e-362, 1.82e-857j
8.0267489477653620155e-37 - 2.4806544093032191539e-2472j 3.32e-386, 3.6e-881j
0.0020208508657010895863 - 2.2816812529044269055e-2450j 2.19e-351, 2.36e-846j
1.1578986904105689807e-07 - 2.6890020726227736545e-2454j 5.07e-363, 5.49e-858j
0.00030411362300223951226 + 9.8185442347213113934e-2451j 6.31e-354, 6.82e-849j
5.0198638641163305583e-19 - 1.1352368236501808365e-2462j 3.23e-376, 3.49e-871j
0.0048078337114488667693 + 1.5783548154936937677e-2450j 3.46e-351, 3.74e-846j
5.3909146513072089309e-10 + 5.4250496828149908739e-2456j 3.62e-368, 3.92e-863j
3.6423846737370645884e-09 - 1.8520238557171297175e-2455j 8.69e-367, 9.41e-862j
1.884280221135955018e-23 + 8.1278260867286865228e-2465j 1.11e-381, 1.2e-876j
0.061729223522720115293 - 4.9037930522741072962e-2450j 4.02e-342, 4.35e-837j
3.8132875557325393184e-14 - 4.8394758033966408921e-2459j 5.78e-374, 6.26e-869j
0.0098703466395774903168 - 1.5145344826292064298e-2450j 7.19e-353, 7.78e-848j
8.0267489477653620155e-37 - 1.5936738954382387685e-2473j 1.76e-386, 1.9e-881j
5.5243955275243011189e-57 - 1.9698004542348508556e-2483j 2.2e-399, 2.38e-894j
1.7529618458764006642e-05 - 1.0716131205449202844e-2451j 1.73e-363, 1.87e-858j
0.0020208508657010895863 - 3.3746423436925328443e-2450j 4.74e-358, 5.12e-853j
4.2104353085066137201e-05 + 1.4649988160443263475e-2451j 1.01e-362, 1.09e-857j
2.9870363017598333527e-15 + 2.0622281025307204109e-2459j 2.96e-376, 3.21e-871j
0.090377086877199247019 - 6.4610053210514157864e-2450j 2.6e-351, 2.81e-846j
9.6382152267366238915e-12 + 1.9616034027330024555e-2456j 2.2e-373, 2.38e-868j
0.00056266851824330161514 + 4.0799653739830911994e-2450j 6.24e-360, 6.75e-855j
1.3398057129951371991e-34 + 3.769468956503264464e-2471j 1.92e-389, 2.08e-884j
3.7100233945684234342e-13 + 3.8668506975049791225e-2458j 6.27e-377, 6.78e-872j
0.00056266851824330161514 + 2.6853756882430818741e-2450j 3.67e-363, 3.97e-858j
3.5847761392900077755e-39 + 1.5125145144738423481e-2473j 5.63e-392, 6.1e-887j
5.3909146513072089309e-10 + 1.3541437674432655371e-2455j 8.02e-373, 8.68e-868j
0.0098703466395774903168 - 2.0645460912992859712e-2450j 1.41e-361, 1.52e-856j
1.0712640358450014033e-17 + 9.4253210389396834782e-2462j 6.54e-381, 7.08e-876j
0.00030411362300223951226 + 6.0701883043534202473e-2451j 5.76e-367, 6.23e-862j
5.3168655710335509311e-07 + 2.2207563395601595858e-2453j 5.79e-371, 6.27e-866j
9.6382152267366238915e-12 + 7.113710410964369356e-2457j 2.42e-376, 2.61e-871j
0.075856158466339825262 + 6.1928702364190889589e-2450j 5.78e-363, 6.26e-858j
0.10948974611226694216 - 6.9408425575722252321e-2450j 1.22e-363, 1.32e-858j
0.00011364047725409753646 - 2.9226637642402128065e-2451j 4.2e-368, 4.55e-863j
0.075856158466339825262 + 5.5455473322587864425e-2450j 1e-363, 1.08e-858j
1.9434467459691796271e-16 - 7.5969596947675097896e-2461j 3.14e-380, 3.39e-875j
4.4123923467438657753e-25 - 8.6348424741176339301e-2466j 9.45e-387, 1.02e-881j
2.042400948365774963e-12 - 6.6218739605634994477e-2457j 1.62e-377, 1.75e-872j
0.018318149936272770612 + 1.7379684187669016322e-2450j 6.88e-369, 7.43e-864j
0.10388960484938631263 + 6.8115417776814670317e-2450j 9.77e-368, 1.05e-862j
1.9953324336500906962e-20 + 6.0204492189771577114e-2463j 5.19e-384, 5.62e-879j
6.7489864688733630258e-06 + 3.3124021093906950288e-2452j 1.76e-372, 1.89e-867j
0.090377086877199247019 - 6.9598439665311126202e-2450j 1.21e-369, 1.24e-864j
8.5173657112380900139e-27 + 1.2607231444815197435e-2467j 4.92e-389, 5.37e-884j
0.10388960484938631263 + 7.0729189860216392823e-2450j 3.19e-370, 2.31e-865j
2.042400948365774963e-12 - 2.3071951534053797447e-2457j 1.09e-378, 1.22e-873j

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