Rule: Q[1, 2, 40, 44]

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.

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)
-0.99988876134358866069 + 1.2425524665753439084e-1697j 2.04e-477, 2.04e-477j
-0.97052802277417746491 - 1.1555023762991586812e-1712j 2.22e-478, 2.22e-478j
-0.92491477735576524157 + 1.0107064511650669239e-1716j 8.22e-480, 8.22e-480j
0.99933888183715517208 + 4.6168340273213074607e-1711j 4.29e-477, 4.29e-477j
0.93825582835274084978 + 1.053755984772398951e-1737j 2.1e-479, 2.1e-479j
0.95031557461792315437 + 5.3643284463154914964e-1737j 4.95e-479, 4.95e-479j
-0.99933888183715517208 + 1.1812092610029604166e-1736j 4.35e-477, 4.35e-477j
-0.93825582835274084978 + 6.4059817340134221011e-1741j 2.13e-479, 2.13e-479j
-0.95031557461792315437 - 2.9816044652210164861e-1740j 5.37e-479, 5.37e-479j
0.96107771692069450679 - 1.5663023800666331575e-1735j 1.1e-478, 1.1e-478j
0.85912690287453322245 + 6.1882435955044272568e-1748j 8.55e-482, 8.55e-482j
0.98544547375341350032 - 7.4532987369688820925e-1742j 7.76e-478, 7.76e-478j
-0.91031061962910991036 - 3.1243988788581923589e-1785j 3.18e-480, 3.18e-480j
-0.87739425538707176377 + 1.141136111220703041e-1786j 3.04e-481, 3.04e-481j
0.99988876134358866069 - 1.7341440421542950183e-1778j 2.18e-477, 2.18e-477j
0.87739425538707176377 - 1.0102728573727118207e-1812j 3.09e-481, 3.09e-481j
0.92491477735576524157 - 4.3444006316351072106e-1809j 8.13e-480, 8.13e-480j
-0.99779971295100584467 - 3.6700712102152499827e-1814j 3.36e-477, 3.36e-477j
-0.77459666924148337704 + 1.9476340020379751861e-1826j 2.26e-484, 2.26e-484j
-0.97865413764512959743 + 1.1108137525326507414e-1817j 4.39e-478, 4.39e-478j
0.91031061962910991036 + 1.8145722125073332595e-1817j 2.82e-480, 2.82e-480j
0.797392160283688017 - 4.3054252328962335863e-1826j 1.13e-483, 1.13e-483j
0.99500543531732884972 - 1.1951939590232364918e-1817j 2.04e-477, 2.04e-477j
-0.85912690287453322245 + 4.4306827954362189364e-1856j 8.92e-482, 8.92e-482j
-0.83968614637524280884 - 2.6239275858392288478e-1857j 2.29e-482, 2.29e-482j
0.99779971295100584467 - 1.7910429154225484255e-1848j 3.32e-477, 3.32e-477j
0.72586431796511602165 - 5.8168249331071788629e-1880j 6.96e-486, 6.96e-486j
0.89446325430860287476 + 2.584295880241646192e-1872j 9.34e-481, 9.34e-481j
-0.96107771692069450679 - 5.753235635333452707e-1871j 1.08e-478, 1.08e-478j
-0.64542174799487529885 - 1.5835603257345962207e-1882j 2.14e-488, 2.14e-488j
0.97865413764512959743 - 1.5025499558073064749e-1869j 4.22e-478, 4.22e-478j
0.83968614637524280884 - 7.0818744646610871681e-1892j 2.24e-482, 2.24e-482j
0.69999401161208205264 - 3.9512829030155150559e-1896j 1.12e-486, 1.12e-486j
-0.98544547375341350032 + 1.9736211442810026454e-1884j 7.79e-478, 7.79e-478j
-0.75074319529946827624 - 3.6744006925570275099e-1894j 4.69e-485, 4.69e-485j
-0.797392160283688017 - 1.0307464810614102924e-1892j 1.2e-483, 1.2e-483j
-0.99089501825791414073 + 2.7293795436206074522e-1885j 1.32e-477, 1.32e-477j
0.61679433055058544069 + 2.9201595157815076937e-1899j 2.51e-489, 2.51e-489j
0.81909853184898019962 + 6.0885152585132434367e-1892j 5.52e-483, 5.52e-483j
-0.81909853184898019962 + 6.6344458030467954945e-1893j 5.19e-483, 5.19e-483j
-0.58732445339233239614 - 1.5451924363799074252e-1900j 3e-490, 3e-490j
-0.52601941863378848131 - 6.3707252949387724081e-1903j 2.87e-492, 2.87e-492j
0.77459666924148337704 + 1.6206184149545609233e-1893j 2.3e-484, 2.3e-484j
0.64542174799487529885 - 3.6523636468930143468e-1898j 2.3e-488, 2.3e-488j
-0.14718617801380025857 + 1.6328424411761443724e-1916j 1.4e-505, 1.4e-505j
-0.61679433055058544069 + 1.2681438435694409341e-1899j 2.81e-489, 2.81e-489j
-0.72586431796511602165 + 9.1004625136895508509e-1896j 6.77e-486, 6.77e-486j
0.99089501825791414073 - 7.4457384668335065412e-1889j 1.33e-477, 1.33e-477j
0.49426799723683859419 + 6.9236098386608250313e-1933j 2.54e-493, 2.54e-493j
0.75074319529946827624 - 8.348921737027383589e-1924j 4.46e-485, 4.46e-485j
-0.69999401161208205264 - 4.3845020773941410543e-1926j 1.14e-486, 1.14e-486j
-0.49426799723683859419 + 3.8086670211483181911e-1932j 2.66e-493, 2.66e-493j
-0.99500543531732884972 - 3.9088893107716407626e-1919j 2.09e-477, 2.09e-477j
0.5570523665246407664 + 2.7698190272563508422e-1934j 2.91e-491, 2.91e-491j
0.58732445339233239614 - 3.6048394461607518152e-1932j 2.86e-490, 2.86e-490j
0.97052802277417746491 - 9.7165183875326407816e-1932j 2.17e-478, 2.17e-478j
-0.46184146907720902976 - 2.4164651318165754419e-1952j 2.13e-494, 2.13e-494j
-0.67316760748285544209 - 8.3830047993916760901e-1946j 1.81e-487, 1.81e-487j
0.67316760748285544209 + 1.6085344649179825492e-1946j 1.54e-487, 1.54e-487j
0.42878412349565830447 + 3.5667675986828425971e-1956j 1.41e-495, 1.41e-495j
0.18360541550797299565 + 4.0843782407361018669e-1965j 2.59e-504, 2.59e-504j
-0.5570523665246407664 + 1.0348524941476574514e-1948j 2.94e-491, 2.94e-491j
-0.42878412349565830447 + 1.5168760088703013432e-1953j 1.42e-495, 1.42e-495j
-0.89446325430860287476 - 2.1184734130585414204e-1961j 9.77e-481, 9.77e-481j
0.46184146907720902976 - 3.7443264518190032397e-1975j 1.93e-494, 1.93e-494j
0.52601941863378848131 + 9.5046972291428905563e-1975j 2.74e-492, 2.74e-492j
-7.2414934699791277306e-2044 - 3.4617607697203470818e-2044j 6.98e-2042, 6.98e-2042j
-0.39514111339613461763 - 2.0760850958204483658e-1979j 9.89e-497, 9.89e-497j
-0.29116123856529143669 - 1.3360391056176522039e-1983j 2.09e-500, 2.09e-500j
0.21977387039115541258 + 4.5142504445502348518e-1987j 5e-503, 5e-503j
0.39514111339613461763 + 3.0792017090921347142e-1979j 9.61e-497, 9.61e-497j
0.14718617801380025857 + 1.7420086911556261112e-1988j 1.3e-505, 1.3e-505j
-0.25564214169224484281 - 3.1173716998370392518e-1984j 1.01e-501, 1.01e-501j
-0.36095839141911379512 + 1.0715653968389855732e-1980j 6.46e-498, 6.46e-498j
-0.036922520355545183064 - 5.864460447634878967e-1993j 1.2e-509, 1.2e-509j
0.36095839141911379512 - 1.2050621393472144472e-1981j 6.42e-498, 6.42e-498j
0.25564214169224484281 - 1.0107245056945858865e-1985j 9.64e-502, 9.64e-502j
-0.073794608601719502957 + 1.1299463020359270274e-1991j 2.69e-508, 2.69e-508j
-0.32628264583620278827 - 1.5991401584114411357e-1982j 3.65e-499, 3.65e-499j
-0.11056590262412224025 + 2.7098166805345075724e-1990j 5.86e-507, 5.86e-507j
0.036922520355545183064 + 4.2800453405916569472e-1993j 1.2e-509, 1.2e-509j
0.32628264583620278827 + 3.3759573200520735886e-1982j 3.89e-499, 3.89e-499j
0.11056590262412224025 + 1.2246776399304124864e-1990j 5.55e-507, 5.55e-507j
-0.18360541550797299565 + 7.0583312333511679433e-1987j 2.61e-504, 2.61e-504j
-0.21977387039115541258 + 2.1461941919002214974e-1985j 5.07e-503, 5.07e-503j
0.073794608601719502957 - 5.8814034985157963976e-1992j 2.69e-508, 2.69e-508j
0.29116123856529143669 + 1.8805246143080801019e-1984j 1.97e-500, 1.97e-500j

Weights

Weight (ball center) Error (ball radius)
0.00027163132040367754158 + 1.0025946383278729533e-1697j 1.62e-135, 2.16e-363j
0.0087901324843338179587 - 7.4965379163670436626e-1700j 5.45e-136, 7.28e-364j
0.013975782819376060054 - 1.894168291204741658e-1700j 2.03e-136, 2.71e-364j
0.0009583465259321214943 - 3.3526082426740428505e-1701j 9.75e-137, 1.3e-364j
0.012703282357584955802 + 8.0323793330973098473e-1702j 4.99e-137, 6.67e-365j
0.011413489621696297354 - 8.8414217466235144885e-1702j 3.49e-137, 4.66e-365j
0.0009583465259321214943 - 1.2189265098069024847e-1697j 4.92e-136, 6.57e-364j
0.012703282357584955802 + 2.5259081129744368855e-1700j 1.03e-136, 1.37e-364j
0.011413489621696297354 - 3.4782067010750553685e-1700j 6.77e-137, 9.03e-365j
0.01010842670578872552 + 9.8598113379956210479e-1702j 1.24e-137, 1.65e-365j
0.018858347971148227756 - 5.7055815985606467671e-1702j 3.34e-139, 4.45e-367j
0.0061210945353003479013 - 1.526720218926109778e-1701j 1.01e-138, 1.35e-366j
0.015229216127214864656 + 1.4588252191064162097e-1700j 5.83e-138, 7.78e-366j
0.017672197342870485295 + 9.2279287918548391613e-1701j 2.52e-139, 3.37e-367j
0.00027163132040367754158 + 1.6877685805377109609e-1701j 3.02e-138, 4.03e-366j
0.017672197342870485295 + 6.0213114820048593411e-1702j 8.87e-140, 1.18e-367j
0.013975782819376060054 - 7.378069516186970849e-1702j 6.38e-139, 8.51e-367j
0.0021514277963663663595 + 2.8390727708477005374e-1698j 8.81e-139, 1.18e-366j
0.023329794054048979416 - 3.8183330834601065381e-1701j 5.78e-143, 7.72e-371j
0.0074603092711380646678 + 1.2022961205402627306e-1699j 1.8e-139, 2.4e-367j
0.015229216127214864656 + 6.8411105937752175073e-1702j 8.35e-140, 1.11e-367j
0.022255997573641829458 + 5.0141565050282547415e-1702j 4.77e-143, 6.36e-371j
0.0034462136115757381563 - 2.3306106493827880011e-1701j 5.85e-140, 7.81e-368j
0.018858347971148227756 - 7.5352554168795355317e-1701j 2.31e-142, 3.08e-370j
0.0200187400672483247 + 6.243788323714445377e-1701j 4.82e-143, 6.43e-371j
0.0021514277963663663595 + 2.9689115621332104542e-1701j 1.23e-139, 1.64e-367j
0.025380368591159015903 - 4.5864648368808392651e-1702j 1.59e-146, 2.12e-374j
0.016461919739327930432 - 6.3951749418164111373e-1702j 5.54e-142, 7.4e-370j
0.01010842670578872552 + 4.9817673817485719893e-1700j 3.36e-142, 4.48e-370j
0.02819305644286810178 + 2.0122203457223701633e-1701j 2.81e-149, 3.76e-377j
0.0074603092711380646678 + 1.2903589660352717747e-1701j 2.4e-141, 3.21e-369j
0.0200187400672483247 + 5.4375128328370854041e-1702j 4.83e-144, 6.45e-372j
0.026354354818097008742 + 4.485916054012866282e-1702j 6.15e-148, 8.21e-376j
0.0061210945353003479013 - 2.0985872496797450533e-1699j 1.1e-143, 1.47e-371j
0.024371723111711210516 + 3.3068914866965248801e-1701j 7.26e-149, 9.68e-377j
0.022255997573641829458 + 4.4503699209596225544e-1701j 8.8e-148, 1.17e-375j
0.0047782497874285984756 + 4.1137569866407857765e-1699j 4.5e-144, 6.01e-372j
0.02905526128699017471 - 4.2820692233114004959e-1702j 5.21e-153, 6.95e-381j
0.021151805321517314918 - 5.2091020990955651477e-1702j 1.58e-147, 2.11e-375j
0.021151805321517314918 - 5.2410412629483225453e-1701j 1.71e-147, 2.28e-375j
0.029877783818751233406 + 1.6321475793112252394e-1701j 4.65e-154, 6.2e-382j
0.031399332747848111607 + 1.3526292545003938537e-1701j 1.65e-155, 2.21e-383j
0.023329794054048979416 - 4.8478293137194872279e-1702j 3.03e-150, 4.04e-378j
0.02819305644286810178 + 4.3351436224321920962e-1702j 2.18e-153, 2.92e-381j
0.036528072446633151231 - 6.4674019455281283978e-1702j 2.56e-159, 3.41e-387j
0.02905526128699017471 - 1.8070607023224294072e-1701j 1.13e-153, 1.51e-381j
0.025380368591159015903 - 2.8884670716958523468e-1701j 4.55e-152, 6.07e-380j
0.0047782497874285984756 + 1.8584676967144090758e-1701j 4.29e-148, 5.72e-376j
0.032096281331450609599 - 4.1973201058328507555e-1702j 1.62e-158, 2.16e-386j
0.024371723111711210516 + 4.7062853398365496842e-1702j 4.94e-152, 6.59e-380j
0.026354354818097008742 + 2.5427358858286420804e-1701j 2.96e-153, 3.95e-381j
0.032096281331450609599 - 1.2403474400688425666e-1701j 1.26e-157, 1.68e-385j
0.0034462136115757381563 - 9.5208094523831609414e-1699j 1.73e-148, 2.31e-376j
0.030659497445912583545 - 4.2154749999296886791e-1702j 9.95e-158, 1.33e-385j
0.029877783818751233406 + 4.2424409668555185557e-1702j 7.77e-157, 1.04e-384j
0.0087901324843338179587 - 1.1170423014727393551e-1701j 3.08e-150, 4.11e-378j
0.032749392454875851171 + 1.1425010834801551023e-1701j 7.11e-160, 9.48e-388j
0.027292345389274912951 - 2.2544958671043390289e-1701j 3.36e-155, 4.48e-383j
0.027292345389274912951 - 4.4026698968397519618e-1702j 1.73e-155, 2.31e-383j
0.033357772144022826989 - 4.2247489282892461096e-1702j 3.12e-162, 4.16e-390j
0.03630211059311798267 + 4.6885890173493418026e-1702j 6.6e-166, 8.8e-394j
0.030659497445912583545 - 1.4820928175004287806e-1701j 3.88e-158, 5.18e-386j
0.033357772144022826989 - 1.0568613593901246423e-1701j 1.41e-161, 1.88e-389j
0.016461919739327930432 - 1.1491253760291431014e-1700j 2.95e-154, 3.93e-382j
0.032749392454875851171 + 4.205424513940807465e-1702j 7.8e-162, 1.04e-389j
0.031399332747848111607 + 4.2005773621960232038e-1702j 2.48e-160, 3.3e-388j
0.0369309280688936748 - 5.4555261523235889891e-1702j 1.23e-167, 1.64e-395j
0.033920588205293049642 + 9.8160654696123143789e-1702j 1.36e-163, 1.82e-391j
0.035328294952639725635 - 8.0440482789076236131e-1702j 4.09e-166, 5.46e-394j
0.036026564578775565295 - 4.5841639729679424629e-1702j 1.7e-167, 2.27e-395j
0.033920588205293049642 + 4.2552798419134536968e-1702j 2.15e-165, 2.86e-393j
0.036528072446633151231 - 4.8076809606472794977e-1702j 4.45e-168, 5.93e-396j
0.03570181179996045511 + 7.5804413944151996841e-1702j 1.22e-167, 1.62e-395j
0.034437073350424121464 - 9.1523785874298828601e-1702j 5.56e-166, 7.42e-394j
0.036905706115932714683 + 5.6685563375761286006e-1702j 1.86e-169, 2.49e-397j
0.034437073350424121464 - 4.2971267013230470592e-1702j 8.92e-168, 1.19e-395j
0.03570181179996045511 + 4.4934918524247724926e-1702j 8.3e-169, 1.11e-396j
0.036830075569497238946 - 5.9062900934361682439e-1702j 5.27e-170, 7.03e-398j
0.034906523111703232295 + 8.565147793213019079e-1702j 2.09e-168, 2.8e-396j
0.036704140624671556791 + 6.1714955180514412729e-1702j 3.74e-170, 4.99e-398j
0.036905706115932714683 + 5.2648234877219395028e-1702j 4.31e-171, 5.75e-399j
0.034906523111703232295 + 4.350520530376089268e-1702j 1.02e-170, 1.36e-398j
0.036704140624671556791 + 4.9425268899114934752e-1702j 1.92e-171, 2.56e-399j
0.03630211059311798267 + 6.7977839165670117242e-1702j 1.52e-171, 2.02e-399j
0.036026564578775565295 - 7.1670641855678374869e-1702j 1.66e-171, 2.21e-399j
0.036830075569497238946 - 5.0944076000797069785e-1702j 4.62e-172, 6.2e-400j
0.035328294952639725635 - 4.415815352017403059e-1702j 4.23e-172, 5.5e-400j

Download

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