Rule: Q[2, 3, 4, 8, 56]

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)
-8.1998667435546562883 - 1.3732104208124001032e-1772j 6.05e-495, 6.05e-495j
10.008190773970554619 - 1.0981038862464550291e-1791j 8.05e-498, 8.05e-498j
-7.8195443062116782141 + 1.925129140580773931e-1786j 1.33e-494, 1.33e-494j
-10.008190773970554619 - 4.0362303849260716967e-1804j 7.91e-498, 7.91e-498j
9.0265841868074020033 + 6.1674069748936524982e-1803j 4.7e-496, 4.7e-496j
-8.6001845953456881047 - 6.9240414155985943414e-1803j 1.98e-495, 1.98e-495j
10.635551308412488729 - 2.3445753565890837399e-1815j 3.44e-499, 3.44e-499j
8.6001845953456881047 + 4.5585652390408990279e-1811j 1.9e-495, 1.9e-495j
-10.635551308412488729 + 2.1665168470068968729e-1818j 3.2e-499, 3.2e-499j
4.9350521114651709022 + 9.4885731014577258978e-1814j 2.4e-494, 2.4e-494j
-1.9787768421383740114 + 3.61403871635382704e-1837j 1.06e-500, 1.06e-500j
-7.4552605505331661822 + 1.3568079626588043249e-1829j 2.64e-494, 2.64e-494j
-1.3029123464880376825 + 8.2873029666914819077e-1860j 2.64e-503, 2.64e-503j
9.4892837347736886648 + 6.8982235056508921752e-1850j 8.26e-497, 8.26e-497j
-6.4354499953554275511 + 5.1056613914263616752e-1865j 6.58e-494, 6.58e-494j
7.4552605505331661822 - 2.1288812555291328643e-1878j 2.71e-494, 2.71e-494j
-9.0265841868074020033 + 4.174306826653554019e-1880j 4.77e-496, 4.77e-496j
-2.6822224032999497583 - 3.3726787511289861212e-1887j 1.25e-496, 1.25e-496j
-2.561096596199577157 + 1.5430812355990259514e-1887j 7.47e-498, 7.47e-498j
7.1043081027331607331 + 1.8901408604076977486e-1882j 4.42e-494, 4.42e-494j
-6.1155075281862551992 + 5.704408593073396404e-1892j 6.47e-494, 6.47e-494j
-7.1043081027331607331 + 2.4345538489532606122e-1905j 4.25e-494, 4.25e-494j
-4.9350521114651709022 - 8.040763405665762949e-1922j 2.37e-494, 2.37e-494j
-4.1373536398299946158 - 1.2337674682915222811e-1933j 2.41e-495, 2.41e-495j
-3.5903080492061445799 - 2.3054141570480099284e-1941j 2.19e-496, 2.19e-496j
2.2456732555148681046 + 3.4126163205471938887e-1946j 1.18e-499, 1.18e-499j
1.7320508075688772935 + 1.9769122314547853558e-1948j 1.58e-501, 1.58e-501j
2.561096596199577157 + 1.5150257398778580868e-1944j 7.97e-498, 7.97e-498j
6.1155075281862551992 - 1.8631256140490837351e-1939j 7.1e-494, 7.1e-494j
-1.7320508075688772935 - 1.2247586118283763911e-1948j 1.54e-501, 1.54e-501j
6.7647984970465292384 + 6.8966243390587546766e-1940j 5.53e-494, 5.53e-494j
-4.6672580226420521533 - 3.8404413435510342986e-1945j 1.27e-494, 1.27e-494j
-5.5036789245151144042 - 8.3882922341492476141e-1968j 4.83e-494, 4.83e-494j
2.6822224032999497583 - 7.0172232551416605121e-1984j 1.3e-496, 1.3e-496j
-3.3121706201791718697 - 1.2043560026765455133e-1984j 6.66e-497, 6.66e-497j
4.4037260958758466183 + 1.0951725389360034126e-1981j 5.78e-495, 5.78e-495j
3.0303564215132354582 + 4.3673891169573031378e-1985j 2.08e-497, 2.08e-497j
0.7071067811865475244 + 1.2929647236139456129e-1993j 7.22e-506, 7.22e-506j
4.6672580226420521533 + 3.9298157134578967742e-1981j 1.23e-494, 1.23e-494j
8.1998667435546562883 - 5.4187401984021795198e-1983j 5.62e-495, 5.62e-495j
-9.4892837347736886648 - 2.491418260359811396e-1984j 8.09e-497, 8.09e-497j
-3.8658170798981634021 + 6.2852393532618141302e-1984j 7.24e-496, 7.24e-496j
-2.6683216593498730133 - 5.7136732041243874725e-1985j 1.23e-496, 1.23e-496j
7.8195443062116782141 - 1.7700583926937926108e-1981j 1.39e-494, 1.39e-494j
5.804766556154487229 + 9.2610773800395915433e-1981j 6.35e-494, 6.35e-494j
-2.2456732555148681046 + 6.5301843232068485585e-1990j 1.09e-499, 1.09e-499j
4.1373536398299946158 + 7.6627794616402751097e-1984j 2.21e-495, 2.21e-495j
1.514779164673178832 - 2.3471533393008211281e-1992j 2.29e-502, 2.29e-502j
0.83127776868067864513 + 4.4366519294586244138e-1995j 2.86e-505, 2.86e-505j
5.2133911127295897702 - 9.4904765176063213698e-1985j 3.73e-494, 3.73e-494j
-5.804766556154487229 - 5.1820259287372899895e-1990j 5.72e-494, 5.72e-494j
-0.52082470896936865978 - 4.2816961287114258468e-2018j 4.37e-507, 4.37e-507j
-6.7647984970465292384 + 2.5999648543129521062e-2005j 5.81e-494, 5.81e-494j
0.52082470896936865978 + 3.5399585512177356416e-2021j 4.18e-507, 4.18e-507j
3.5903080492061445799 + 2.5562045692660940022e-2010j 2.22e-496, 2.22e-496j
3.3121706201791718697 - 4.3515880139364063004e-2012j 6.06e-497, 6.06e-497j
-1.0672470719421885588 + 1.8938316024454315246e-2019j 2.54e-504, 2.54e-504j
-1.514779164673178832 + 9.7758622624656512979e-2017j 2.21e-502, 2.21e-502j
2.6683216593498730133 - 5.077307999239625423e-2011j 1.25e-496, 1.25e-496j
1.9787768421383740114 - 9.9066068200699969185e-2015j 1.1e-500, 1.1e-500j
6.4354499953554275511 - 4.1117329673229735691e-2008j 6.91e-494, 6.91e-494j
-3.0303564215132354582 - 5.8913803232228971912e-2012j 2.19e-497, 2.19e-497j
-0.83127776868067864513 - 6.0762978396507728162e-2021j 2.86e-505, 2.86e-505j
5.5036789245151144042 - 2.9961326795348935022e-2009j 4.79e-494, 4.79e-494j
-6.6703044967102877328e-2057 - 1.2493371217538000514e-2056j 1.03e-2054, 1.03e-2054j
1.3029123464880376825 + 7.9457871539539199921e-2019j 2.65e-503, 2.65e-503j
3.8658170798981634021 - 5.1598816445856363634e-2011j 7.34e-496, 7.34e-496j
-0.7071067811865475244 + 1.3972035308345573808e-2020j 7.7e-506, 7.7e-506j
-5.2133911127295897702 - 5.1924781825731057084e-2016j 3.76e-494, 3.76e-494j
1.0672470719421885588 - 9.1997896276116188439e-2028j 2.46e-504, 2.46e-504j
0.26376891207373976444 - 9.4218982043761542307e-2033j 9.62e-509, 9.62e-509j
-4.4037260958758466183 - 4.6147479652510006309e-2029j 5.72e-495, 5.72e-495j
-0.26376891207373976444 - 6.5918278388401757508e-2045j 9.62e-509, 9.62e-509j

Weights

Weight (ball center) Error (ball radius)
1.3833981854941450512e-30 - 1.7694530671425065414e-1801j 6.88e-179, 1.39e-425j
9.9584637000605484907e-45 - 1.5015502376346119634e-1811j 2.04e-186, 4.13e-433j
5.8430732260756295018e-28 + 1.5897569379395679814e-1800j 1.27e-177, 2.58e-424j
9.9584637000605484907e-45 + 1.5119144927812325586e-1810j 4.31e-186, 8.73e-433j
1.0261126079764651203e-36 - 3.9954445689798066749e-1807j 5.06e-184, 1.02e-430j
1.7564093555621284979e-33 - 1.0971628246882410185e-1803j 4.45e-181, 9.02e-428j
3.0665148421727770414e-50 + 1.5046532891718762649e-1814j 2.3e-189, 4.65e-436j
1.7564093555621284979e-33 + 2.6143602551215046094e-1805j 5.3e-183, 1.07e-429j
3.0665148421727770414e-50 - 1.1635650253616929332e-1813j 3.03e-189, 6.14e-436j
4.069638243244803598e-12 - 4.213084990798505308e-1792j 1.09e-169, 2.21e-416j
0.0029045751721469969459 - 4.5822890891199160278e-1785j 2.57e-145, 5.2e-392j
1.4652395848419301853e-25 - 2.0101084262323826443e-1799j 1.95e-178, 3.95e-425j
0.023107126434161628982 + 3.4180882889447244265e-1784j 6.63e-136, 1.34e-382j
2.1452672501731592781e-40 + 3.6148587505321997298e-1809j 4.42e-186, 8.95e-433j
1.8897789063406625602e-19 + 4.0535934787449378593e-1796j 3.6e-176, 7.29e-423j
1.4652395848419301853e-25 - 9.560696343761284971e-1801j 2.55e-181, 5.15e-428j
1.0261126079764651203e-36 + 8.3253752989297784279e-1806j 7.12e-184, 1.44e-430j
0.0014238623723496302457 - 2.1326497358349886327e-1784j 1.9e-156, 3.85e-403j
0.00047131424683225061292 - 4.8051985335297169078e-1785j 2.68e-155, 5.43e-402j
2.3434044400319940545e-23 + 1.9596944014106827438e-1799j 3.09e-181, 6.26e-428j
1.018267074134582642e-17 - 4.2534191852797675393e-1795j 4.7e-176, 9.51e-423j
2.3434044400319940545e-23 + 2.7375536686922001978e-1798j 1.14e-178, 2.3e-425j
4.069638243244803598e-12 - 1.6949975946421388693e-1791j 8.73e-173, 1.77e-419j
5.5853005631861127259e-09 + 1.8463295272564621055e-1789j 1.76e-169, 3.57e-416j
3.9412330842858124892e-07 + 3.2764636960557779012e-1788j 1.24e-166, 2.5e-413j
0.0010043012758969609758 + 1.3315350533989211726e-1785j 2.52e-161, 5.1e-408j
0.0065338157402290721841 + 6.8854312903296285884e-1785j 3.64e-156, 7.37e-403j
0.00047131424683225061292 - 2.5179353546241535984e-1785j 2.27e-164, 4.61e-411j
1.018267074134582642e-17 - 6.1930993262002595154e-1796j 2.31e-183, 4.69e-430j
0.0065338157402290721841 + 1.0573203776408557123e-1784j 2.41e-156, 4.87e-403j
2.5183090654699765732e-21 - 3.3497472040099997701e-1798j 6.67e-185, 1.35e-431j
5.1657215081139689369e-11 + 9.2579300131089613837e-1791j 2.67e-174, 5.41e-421j
1.1684738350930622381e-14 - 3.4413400493875027369e-1793j 9.22e-178, 1.87e-424j
0.0014238623723496302457 - 1.0813367346909455576e-1784j 1.01e-167, 2.04e-414j
2.7116343346449019217e-06 - 1.5107687948943794528e-1787j 1.25e-168, 2.52e-415j
5.6362673913019474931e-10 - 1.3033539114812098745e-1790j 1.38e-178, 2.8e-425j
1.6582792173891262534e-05 + 3.9535122545441129649e-1787j 9.03e-172, 1.83e-418j
0.033496349636703024431 - 1.4931116588556848825e-1783j 2.44e-158, 4.95e-405j
5.1657215081139689369e-11 + 2.541721238906865334e-1791j 1.82e-179, 3.69e-426j
1.3833981854941450512e-30 - 1.1583705344645428112e-1803j 1.76e-191, 3.57e-438j
2.1452672501731592781e-40 - 4.959123451259144079e-1808j 1.24e-194, 2.52e-441j
4.9952308516800562231e-08 - 7.6830062109438815153e-1789j 2.22e-174, 4.5e-421j
-0.0015742361098040494989 + 2.5160614452254216522e-1784j 7.15e-168, 1.45e-414j
5.8430732260756295018e-28 + 3.7742975165659111999e-1802j 2.25e-190, 4.55e-437j
4.013554210456157145e-16 + 6.8971075593917599592e-1795j 2.92e-185, 5.92e-432j
0.0010043012758969609758 + 2.3359339410753025894e-1785j 3.45e-169, 6.99e-416j
5.5853005631861127259e-09 + 6.0797632409766922116e-1790j 2.11e-180, 4.28e-427j
0.011735264419510558449 - 1.4611825757389373885e-1784j 1.18e-169, 2.39e-416j
0.058244221559463620652 + 1.1813029685300577437e-1783j 5.26e-167, 1.07e-413j
2.5205345026561724788e-13 + 5.7966597636821019194e-1793j 1.66e-183, 3.36e-430j
4.013554210456157145e-16 + 4.0328777353896161877e-1794j 2.58e-184, 5.22e-431j
0.10595247960151482044 + 1.0289424578401833253e-1783j 2.35e-169, 4.75e-416j
2.5183090654699765732e-21 - 3.4930635299489004684e-1797j 2.98e-187, 6.04e-434j
0.10595247960151482044 + 9.0603966760426910973e-1784j 4.02e-169, 8.14e-416j
3.9412330842858124892e-07 + 1.2809862434053417065e-1788j 4.45e-180, 9.01e-427j
2.7116343346449019217e-06 - 6.4143109936248923948e-1788j 4.35e-179, 8.81e-426j
0.044029452635040884474 - 5.6910945399496023254e-1784j 4.99e-172, 1.01e-418j
0.011735264419510558449 - 2.1233635555379154203e-1784j 1.41e-172, 2.85e-419j
-0.0015742361098040494989 + 1.2805912819080548453e-1784j 2.44e-175, 4.94e-422j
0.0029045751721469969459 - 2.800651396986001112e-1785j 1.34e-175, 2.71e-422j
1.8897789063406625602e-19 + 4.8869651076790093314e-1797j 2.38e-189, 4.82e-436j
1.6582792173891262534e-05 + 8.588593916730688852e-1787j 3.74e-179, 7.59e-426j
0.058244221559463620652 + 1.4478372803675105491e-1783j 5.78e-175, 1.16e-421j
1.1684738350930622381e-14 - 6.7708747407985397944e-1794j 2.39e-186, 4.83e-433j
0.14917551650952303489 + 4.9528245316328068765e-1784j 2.85e-176, 5.57e-423j
0.023107126434161628982 + 2.4807899700310979246e-1784j 1.04e-176, 2.08e-423j
4.9952308516800562231e-08 - 2.7597715116392539289e-1789j 8.78e-182, 1.79e-428j
0.033496349636703024431 - 1.774927540946714903e-1783j 7.18e-176, 1.47e-422j
2.5205345026561724788e-13 + 2.6034731813276855056e-1792j 5.4e-187, 1.07e-433j
0.044029452635040884474 - 4.3802648458650042447e-1784j 2.98e-177, 4.96e-424j
0.13806397005414929655 - 5.5128028626759664149e-1784j 2.69e-177, 4.46e-424j
5.6362673913019474931e-10 - 4.3272743427019720278e-1790j 7.41e-185, 1.69e-431j
0.13806397005414929655 - 5.8792565139394132211e-1784j 1.84e-177, 2.69e-424j

Download

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