Rule: Q[1, 2, 40, 42]

Type

This rule is of Legendre 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)
-0.99622490528349537804 + 1.8758820966452393361e-1711j 2.3e-479, 2.3e-479j
-0.97052802277417746491 - 4.1443707771655581202e-1714j 9.14e-480, 9.14e-480j
-0.92491477735576524157 - 3.6002579419473899008e-1716j 7.41e-481, 7.41e-481j
-0.98544547375341350032 + 3.2595173318929282849e-1714j 8.75e-480, 8.75e-480j
0.9111298583517110982 - 1.2691197231067530902e-1714j 2.99e-481, 2.99e-481j
0.92491477735576524157 - 1.1523251054520686193e-1728j 7.09e-481, 7.09e-481j
0.9924318645134292499 + 1.7353473403054837481e-1753j 2.05e-479, 2.05e-479j
-0.9111298583517110982 + 6.6342621993616258996e-1791j 3e-481, 3e-481j
-0.9775397615886434099 + 2.6059656101366290892e-1788j 9.76e-480, 9.76e-480j
0.9775397615886434099 - 3.5519559851324542802e-1787j 9.07e-480, 9.07e-480j
0.89446325430860287476 - 3.7241178832256188211e-1818j 1.09e-481, 1.09e-481j
0.99500543531732884972 - 3.3741396160108317318e-1825j 3.95e-479, 3.95e-479j
-0.93737166642545683026 - 1.4206190878276426278e-1862j 1.32e-480, 1.32e-480j
-0.85912690287453322245 - 4.4519800174314050709e-1863j 1.52e-482, 1.52e-482j
0.99933888183715517208 - 5.1202917970759222772e-1857j 1.31e-480, 1.31e-480j
0.84040142237434780595 + 2.924106381489585843e-1884j 4.26e-483, 4.26e-483j
0.85912690287453322245 + 1.72819154021144686e-1883j 1.44e-482, 1.44e-482j
0.99622490528349537804 + 1.0197489256201276845e-1879j 2.29e-479, 2.29e-479j
-0.79671554390300336293 + 3.8381469665894207857e-1902j 2.67e-484, 2.67e-484j
-0.95031557461792315437 + 7.290282245201928421e-1898j 2.18e-480, 2.18e-480j
0.87663768802434542877 + 1.1405610303769669067e-1900j 3.73e-482, 3.73e-482j
0.77459666924148337704 - 3.0280569560192592559e-1903j 6.61e-485, 6.61e-485j
-0.9924318645134292499 + 9.7046766452219813696e-1895j 2.07e-479, 2.07e-479j
-0.84040142237434780595 - 2.6376559309934856384e-1904j 4.18e-483, 4.18e-483j
-0.77459666924148337704 + 2.368741331948436301e-1906j 6.78e-485, 6.78e-485j
-0.9620826879393802647 + 6.6786405508600748259e-1901j 4.72e-480, 4.72e-480j
0.75139187580841733814 + 1.6219139696880016875e-1906j 1.38e-485, 1.38e-485j
0.95031557461792315437 + 1.0767034308270113695e-1899j 2.27e-480, 2.27e-480j
-0.87663768802434542877 - 4.4318548241355380321e-1915j 4.1e-482, 4.1e-482j
-0.64602510655924630182 - 2.4875926449754572436e-1922j 9.45e-489, 9.45e-489j
-0.89446325430860287476 + 2.1094659632610678089e-1914j 1.06e-481, 1.06e-481j
0.79671554390300336293 - 1.8195789902845250056e-1916j 2.67e-484, 2.67e-484j
0.72586431796511602165 - 1.1991037915461894382e-1918j 2.48e-486, 2.48e-486j
-0.99500543531732884972 - 1.0183808036910306073e-1910j 3.76e-479, 3.76e-479j
-0.75139187580841733814 + 1.1514806013151277932e-1920j 1.38e-485, 1.38e-485j
-0.72586431796511602165 - 5.4035472161806807163e-1920j 2.24e-486, 2.24e-486j
-0.36095839141911379512 + 5.7187271762093652501e-1931j 4.95e-498, 4.95e-498j
0.67316760748285544209 + 1.9235570401081518572e-1920j 6.65e-488, 6.65e-488j
0.81909853184898019962 + 1.4216432862105114239e-1916j 1.01e-483, 1.01e-483j
0.98544547375341350032 + 5.6742440526001877876e-1910j 8.99e-480, 8.99e-480j
-0.5867388869148402037 - 3.5714902371509205329e-1935j 1.4e-490, 1.4e-490j
-0.14718617801380025857 + 1.1519379330974146175e-1950j 1.32e-505, 1.32e-505j
0.5867388869148402037 + 7.1920433668087218467e-1935j 1.46e-490, 1.46e-490j
0.61679433055058544069 - 1.4324193022675913206e-1933j 1.28e-489, 1.28e-489j
0.97052802277417746491 - 4.8387198950789123118e-1922j 8.65e-480, 8.65e-480j
-0.61679433055058544069 - 9.1723518131440690209e-1944j 1.19e-489, 1.19e-489j
-0.81909853184898019962 + 4.8551901362137928258e-1936j 1.11e-483, 1.11e-483j
0.9620826879393802647 + 1.763737565709679538e-1932j 4.78e-480, 4.78e-480j
0.46128184088785847193 + 3.6479326100405162919e-1958j 1.19e-494, 1.19e-494j
-0.11109493391388428188 + 1.9773281543017721854e-1970j 5.65e-507, 5.65e-507j
-0.6993711330167845415 - 1.354535321199584736e-1949j 3.85e-487, 3.85e-487j
-0.49426799723683859419 + 2.6606483020455671046e-1957j 1.44e-493, 1.44e-493j
0.93737166642545683026 - 1.3335283349502463141e-1947j 1.26e-480, 1.26e-480j
0.52659128058949802663 - 2.1765443604110755267e-1959j 1.7e-492, 1.7e-492j
0.5570523665246407664 + 1.1028925229590823189e-1958j 1.6e-491, 1.6e-491j
-0.99933888183715517208 - 3.289347206268019021e-1958j 1.29e-480, 1.29e-480j
-0.46128184088785847193 - 2.3506194365847624944e-1976j 1.24e-494, 1.24e-494j
-0.67316760748285544209 - 5.3018053251750713346e-1971j 6.7e-488, 6.7e-488j
0.6993711330167845415 + 3.3576252227395250359e-1969j 3.8e-487, 3.8e-487j
0.39569138851584868174 + 4.6864713079521439324e-1981j 7.41e-497, 7.41e-497j
0.64602510655924630182 - 1.8215575458233728404e-1979j 9.85e-489, 9.85e-489j
-0.5570523665246407664 + 8.5632228707741759971e-1981j 1.63e-491, 1.63e-491j
-0.52659128058949802663 + 2.6041203860652656371e-1987j 1.74e-492, 1.74e-492j
0.073794608601719502957 + 6.9552299644244442685e-2003j 2.65e-508, 2.65e-508j
0.42878412349565830447 + 3.1011069040948902065e-1989j 1.07e-495, 1.07e-495j
0.49426799723683859419 + 1.7730863778243146255e-1988j 1.65e-493, 1.65e-493j
0.036394968383913911559 - 5.5266492937140396334e-2006j 1.15e-509, 1.15e-509j
-0.42878412349565830447 + 8.846794557665095164e-1992j 1.12e-495, 1.12e-495j
-0.39569138851584868174 - 3.9855863997611306324e-1993j 7.83e-497, 7.83e-497j
0.21977387039115541258 - 1.1236853151228379937e-2000j 4.77e-503, 4.77e-503j
0.36095839141911379512 - 2.7054269423159718423e-1993j 5.29e-498, 5.29e-498j
0.18307356617022244873 + 6.7003595894632125294e-2001j 2.46e-504, 2.46e-504j
-0.32574039046727197686 + 4.0080467125919700121e-1995j 2.92e-499, 2.92e-499j
-0.29116123856529143669 + 1.2942381829656744647e-1995j 1.78e-500, 1.78e-500j
2.6075263210175953176e-2058 + 4.2620668357627492123e-2058j 4.25e-2056, 4.25e-2056j
0.32574039046727197686 + 3.629629302455080936e-1995j 2.81e-499, 2.81e-499j
0.25617854793502086118 - 9.6674183783655631644e-1998j 9.85e-502, 9.85e-502j
-0.036394968383913911559 + 7.0234376467510844777e-2006j 1.15e-509, 1.15e-509j
-0.25617854793502086118 - 9.9059710546716280207e-1997j 1.07e-501, 1.07e-501j
-0.073794608601719502957 + 2.1813839175225722051e-2003j 2.7e-508, 2.7e-508j
0.14718617801380025857 + 2.8436712958694046864e-2001j 1.25e-505, 1.25e-505j
0.29116123856529143669 + 4.8678513363788064994e-1996j 1.65e-500, 1.65e-500j
0.11109493391388428188 - 2.8693262356686580592e-2002j 5.83e-507, 5.83e-507j
-0.18307356617022244873 + 6.9720105632719567677e-2000j 2.3e-504, 2.3e-504j
-0.21977387039115541258 + 3.3830421855236210803e-1998j 4.63e-503, 4.63e-503j

Weights

Weight (ball center) Error (ball radius)
0.006135302381468397343 + 1.338688125056416432e-1710j 1.67e-129, 1.04e-359j
0.0067528979026245891915 - 7.8659725196590891188e-1712j 7.85e-130, 4.89e-360j
0.012501542826540675337 - 4.0595745322279423576e-1712j 1.94e-130, 1.21e-360j
0.0075881460235888154626 - 8.0862427421978705453e-1712j 2.88e-130, 1.8e-360j
0.015372394574014444213 + 1.4890992033811212573e-1713j 2.19e-131, 1.37e-361j
0.012501542826540675337 - 1.3641930755586180782e-1713j 1.92e-131, 1.2e-361j
0.0068584913419369783957 + 6.3532057393672537993e-1714j 8.31e-133, 5.18e-363j
0.015372394574014444213 + 3.3849073684744013088e-1712j 2.59e-131, 1.61e-361j
0.0077910221644058569791 + 7.7406732268857370757e-1712j 3.29e-130, 2.05e-360j
0.0077910221644058569791 + 7.2451325937437904498e-1714j 3.51e-133, 2.18e-363j
0.017608650368993399455 - 1.6339689903271956991e-1713j 5.32e-133, 3.31e-363j
-0.0038340088414281268223 - 9.9202930171896780429e-1714j 2.87e-133, 1.79e-363j
0.012813564929980027029 + 4.0797996799502717443e-1712j 9.35e-133, 5.83e-363j
0.017629308636108387913 - 2.7449565890240446149e-1712j 1.04e-134, 6.47e-365j
0.0017051383043378171418 - 2.6897542493646678548e-1715j 2.66e-134, 1.66e-364j
0.02010247052903410573 + 2.086337953620558683e-1713j 3.81e-136, 2.37e-366j
0.017629308636108387913 - 2.0770724410174223582e-1713j 7.85e-136, 4.89e-366j
0.006135302381468397343 + 5.6765635313103852371e-1714j 9.37e-134, 5.84e-364j
0.022290566860453298341 + 2.093121759866716855e-1712j 9.62e-139, 5.99e-369j
0.012713887138843940361 - 4.1299163282855203755e-1712j 9.04e-136, 5.63e-366j
0.017727367303569019492 + 1.8382731654119858414e-1713j 9.14e-136, 5.69e-366j
0.022241672786748671461 - 2.6056443018758825131e-1713j 6.35e-139, 3.95e-369j
0.0068584913419369783957 + 3.3963357513583210723e-1711j 1.63e-136, 1.01e-366j
0.02010247052903410573 + 2.416882216732033886e-1712j 1.6e-138, 9.97e-369j
0.022241672786748671461 - 2.0640545484896472126e-1712j 3.14e-140, 1.95e-370j
0.010429617463937363602 + 5.7385575533348809327e-1712j 3.48e-136, 2.17e-366j
0.024428010969977580104 + 2.6377343788148677692e-1713j 7.52e-142, 4.68e-372j
0.012713887138843940361 - 9.4045941762802840276e-1714j 2.11e-139, 1.31e-369j
0.017727367303569019492 + 2.7772272155731050947e-1712j 3.48e-138, 2.17e-368j
0.028234360350637892946 + 1.5104925132847813551e-1712j 2.19e-144, 1.37e-374j
0.017608650368993399455 - 2.8197347843005176775e-1712j 6.03e-138, 3.76e-368j
0.022290566860453298341 + 2.3535515752480585275e-1713j 3.29e-142, 2.05e-372j
0.026340037898502226051 - 2.6730734967074485873e-1713j 4.93e-144, 3.07e-374j
-0.0038340088414281268223 - 1.6491801153547387e-1710j 2.23e-137, 1.39e-367j
0.024428010969977580104 + 1.869209730496506281e-1712j 2.58e-142, 1.61e-372j
0.026340037898502226051 - 1.692224129364490025e-1712j 2.45e-143, 1.53e-373j
0.035273834943073891508 - 9.911044192764578092e-1713j 8.55e-150, 5.32e-380j
0.026294917356826307861 - 3.1894238018788800248e-1713j 2.21e-147, 1.38e-377j
0.022189186644821545055 - 2.1032994934934636138e-1713j 1.46e-143, 9.08e-374j
0.0075881460235888154626 - 4.321140636753627347e-1714j 1.2e-143, 7.49e-374j
0.029898341398616813225 + 1.3679712842961354296e-1712j 1.33e-148, 8.26e-379j
0.035675772579771702052 - 8.2515563551449932154e-1713j 7.11e-153, 4.43e-383j
0.029898341398616813225 + 3.5496998738909190871e-1713j 5.84e-150, 3.64e-380j
0.029959178124191275524 - 3.2813394815367319121e-1713j 2.97e-149, 1.85e-379j
0.0067528979026245891915 - 1.0080581672359428893e-1713j 1.26e-144, 7.83e-375j
0.029959178124191275524 - 1.3900463519203577171e-1712j 5.13e-150, 3.2e-380j
0.022189186644821545055 - 2.1274600839856333195e-1712j 8.2e-146, 5.11e-376j
0.010429617463937363602 + 9.7165140034056749607e-1714j 5.67e-145, 3.53e-375j
0.032767660488310223657 + 4.2205530931077247922e-1713j 2e-154, 1.24e-384j
0.036724055487141997864 + 7.776459238027922688e-1713j 5.27e-156, 3.28e-386j
0.026379562916778022664 + 1.6648382785847106854e-1712j 6.32e-149, 3.94e-379j
0.032960439789155110533 - 1.1658075786640534565e-1712j 9.08e-154, 5.65e-384j
0.012813564929980027029 + 1.1793562737795108818e-1713j 1e-145, 6.24e-376j
0.031431578045524178361 + 3.8767205108631809676e-1713j 9.33e-154, 5.81e-384j
0.02972330787490541756 - 3.8216397373464659543e-1713j 5.79e-153, 3.61e-383j
0.0017051383043378171418 + 1.7522002300490511413e-1712j 1.44e-148, 9e-379j
0.032767660488310223657 + 1.1477274777569082591e-1712j 3.31e-155, 2.06e-385j
0.026294917356826307861 - 1.6405947667604799589e-1712j 4.29e-151, 2.67e-381j
0.026379562916778022664 + 2.9300620193217408942e-1713j 4.38e-151, 2.73e-381j
0.033946997058123631984 + 4.5768260781643311229e-1713j 4.41e-158, 2.75e-388j
0.028234360350637892946 + 3.2339968324472357305e-1713j 5.08e-152, 3.16e-382j
0.02972330787490541756 - 1.3479334415176741691e-1712j 3.59e-154, 2.23e-384j
0.031431578045524178361 + 1.2539792474033836311e-1712j 7.83e-155, 4.88e-385j
0.037641274008861946682 - 6.3109872521107350544e-1713j 2.5e-162, 1.56e-392j
0.032463150809657226021 - 4.5115312009437315361e-1713j 2.43e-158, 1.52e-388j
0.032960439789155110533 - 3.9346360734713579844e-1713j 6.79e-157, 4.23e-387j
0.036923970834279764574 + 6.7034475431068045095e-1713j 9.25e-163, 5.76e-393j
0.032463150809657226021 - 1.1310473306675032341e-1712j 3.81e-159, 2.37e-389j
0.033946997058123631984 + 1.0592165883332824355e-1712j 6.4e-160, 3.99e-390j
0.036846092423823773469 - 5.4299545433569462173e-1713j 4.03e-164, 2.51e-394j
0.035273834943073891508 - 4.6452855812112378427e-1713j 2.63e-162, 1.64e-392j
0.036319526061966122282 + 5.7753125063454768079e-1713j 2.22e-164, 1.38e-394j
0.034923886025430438987 + 9.7606362098603592044e-1713j 3.11e-163, 1.94e-393j
0.034460706509963887248 - 9.6202849704655171012e-1713j 4.08e-164, 2.54e-394j
0.036083580584408554961 - 7.1086891713375660358e-1713j 6.76e-166, 4.21e-396j
0.034923886025430438987 + 4.95620085028060084e-1713j 9.01e-165, 5.61e-395j
0.035724328412297085681 + 5.3501339368392156546e-1713j 1.65e-165, 1.03e-395j
0.036923970834279764574 + 7.2109982115537775891e-1713j 1.05e-166, 6.57e-397j
0.035724328412297085681 + 9.0464045082952280536e-1713j 7.04e-167, 4.39e-397j
0.037641274008861946682 - 7.319064306176488732e-1713j 6.11e-167, 3.8e-397j
0.035675772579771702052 - 6.1300522446239575929e-1713j 1.73e-167, 1.08e-397j
0.034460706509963887248 - 5.2740396986095821648e-1713j 3.91e-167, 2.44e-397j
0.036724055487141997864 + 6.2182402280170927279e-1713j 1.23e-167, 7.66e-398j
0.036319526061966122282 + 8.3708843111397765256e-1713j 7.05e-169, 4.4e-399j
0.036846092423823773469 - 8.4977024626701143281e-1713j 6.71e-169, 4.17e-399j

Download

All rules of Legendre type can be downloaded here packed into a single file.