Rule: Q[4, 8, 12, 39]

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)
0.99546177155001729373 + 6.9408638768627540293e-879j 1.98e-233, 1.98e-233j
0.98120943726842291166 + 4.6970751105139665596e-888j 6e-233, 6e-233j
-0.94686579301201898279 - 1.0528078153918122364e-897j 1.48e-233, 1.48e-233j
-0.99546177155001729373 - 5.1405800256330411339e-897j 1.91e-233, 1.91e-233j
0.94686579301201898279 - 5.0516648460890980091e-897j 1.5e-233, 1.5e-233j
0.92849639635025196002 - 1.9263841742112883325e-901j 6.83e-234, 6.83e-234j
0.98912453441893804588 + 1.5165495280588278008e-907j 3.78e-233, 3.78e-233j
-0.88518453566095265907 + 1.0728130460000455777e-931j 1.13e-234, 1.13e-234j
0.99913089292685398679 - 7.3439289968183773943e-928j 5.29e-234, 5.29e-234j
-0.99913089292685398679 + 3.9795895400711445043e-939j 5.7e-234, 5.7e-234j
0.88518453566095265907 - 1.1476920378990999579e-947j 1.12e-234, 1.12e-234j
0.96212175385237495876 + 1.1692434507974446251e-965j 3.3e-233, 3.3e-233j
-0.92849639635025196002 + 6.276769448279505224e-982j 6.37e-234, 6.37e-234j
-0.86113631159405257522 + 1.2692005394270691924e-980j 4.31e-235, 4.31e-235j
-0.98120943726842291166 - 3.0165330472108552066e-977j 6.06e-233, 6.06e-233j
0.86113631159405257522 - 8.6549587760492830204e-983j 4.35e-235, 4.35e-235j
0.73850671499704622703 + 9.6464385834829613226e-986j 1.43e-237, 1.43e-237j
-0.9732162594524303742 + 7.9222149027915068244e-979j 5.64e-233, 5.64e-233j
-0.77313202939632075478 + 3.3898954610188074381e-985j 6.48e-237, 6.48e-237j
-0.90772042897585588602 - 3.648305478268023745e-981j 2.82e-234, 2.82e-234j
0.9732162594524303742 + 2.7387510545115789413e-980j 6.28e-233, 6.28e-233j
0.77313202939632075478 - 2.5097358562514287214e-994j 6.35e-237, 6.35e-237j
-0.98912453441893804588 + 7.3454446321263509235e-987j 3.99e-233, 3.99e-233j
-0.8348561121830108777 - 2.3686068067352930023e-994j 1.21e-235, 1.21e-235j
-0.80550083680227534102 - 3.9944210443314702937e-994j 2.93e-236, 2.93e-236j
0.90772042897585588602 - 5.7456560280038810802e-992j 2.76e-234, 2.76e-234j
0.70286882590834304222 - 5.127065940202490376e-997j 2.76e-238, 2.76e-238j
0.8348561121830108777 - 1.0680993975278577956e-993j 1.27e-235, 1.27e-235j
-0.6674742407002819824 - 2.4492740923106059275e-997j 4.84e-239, 4.84e-239j
-0.63126568760916818812 + 2.2320554579188017435e-998j 7.09e-240, 7.09e-240j
-0.96212175385237495876 - 1.1459977934099620324e-996j 3.39e-233, 3.39e-233j
0.80550083680227534102 + 4.7757553044706025752e-1003j 3.02e-236, 3.02e-236j
0.6674742407002819824 + 1.9815378507655053964e-1006j 5.18e-239, 5.18e-239j
-0.15533674791170364347 + 3.016278065995971474e-1019j 5.51e-251, 5.51e-251j
-0.5471825721852412894 + 3.0244440069157089191e-1010j 7.26e-242, 7.26e-242j
-0.73850671499704622703 - 4.7308160080380179642e-1006j 1.33e-237, 1.33e-237j
0.28862954642910963825 + 3.2305994853569520341e-1018j 9.51e-248, 9.51e-248j
0.59134073391232386307 + 1.4076868866146982599e-1010j 7.9e-241, 7.9e-241j
0.3399810435848562648 - 5.9123663418847651037e-1016j 6.93e-246, 6.93e-246j
-0.70286882590834304222 + 2.6251988290155690477e-1011j 2.79e-238, 2.79e-238j
-0.59134073391232386307 - 4.6042853430549004203e-1014j 7.99e-241, 7.99e-241j
-0.11438138061042355922 - 4.807317803716731861e-1024j 2.17e-252, 2.17e-252j
0.63126568760916818812 - 1.7215059042953876714e-1014j 7.13e-240, 7.13e-240j
0.5471825721852412894 - 2.4021522361875815384e-1018j 6.64e-242, 6.64e-242j
0.057798702582244333721 - 9.4979005760274574825e-1032j 3.58e-254, 3.58e-254j
-0.49975742537940093891 + 1.6968604616157452738e-1021j 5.48e-243, 5.48e-243j
-0.35241611849142175407 - 1.7797787783207347432e-1024j 1.44e-245, 1.44e-245j
0.11438138061042355922 + 3.6366623439292454026e-1030j 2.04e-252, 2.04e-252j
0.49975742537940093891 - 1.4248541209803031252e-1021j 5.75e-243, 5.75e-243j
0.23340655143757361825 + 7.0403697877739251469e-1028j 3.02e-249, 3.02e-249j
-0.3399810435848562648 + 6.1259941405432748806e-1026j 7.21e-246, 7.21e-246j
-0.44989474445373993198 + 1.3972647244939739497e-1023j 4.77e-244, 4.77e-244j
1.3581355343748206371e-1068 - 6.0310399285399531619e-1068j 3.89e-1066, 3.89e-1066j
0.39829549941770898999 - 1.7028105672371201726e-1023j 4.49e-245, 4.49e-245j
0.44989474445373993198 - 1.8845283736142467193e-1022j 4.86e-244, 4.86e-244j
-0.057798702582244333721 - 3.7604496261618147195e-1033j 3.28e-254, 3.28e-254j
-0.39829549941770898999 - 2.2155236253019200062e-1024j 4.75e-245, 4.75e-245j
-0.18035397587349104032 + 2.0910569246002046714e-1029j 2.33e-250, 2.33e-250j
0.18035397587349104032 - 2.4829659294288239233e-1029j 2.47e-250, 2.47e-250j
0.35241611849142175407 - 5.676974936049210813e-1024j 1.41e-245, 1.41e-245j
0.15533674791170364347 - 9.2669495216296385265e-1031j 5.72e-251, 5.72e-251j
-0.23340655143757361825 - 2.3332832835905127371e-1028j 3.01e-249, 3.01e-249j
-0.28862954642910963825 + 4.8958422365116457773e-1027j 9.47e-248, 9.47e-248j

Weights

Weight (ball center) Error (ball radius)
0.0050793026072318104849 - 5.790900430241523471e-879j 3.39e-36, 5.18e-146j
0.0079369513161243331483 - 1.512277481287314861e-878j 1.67e-36, 2.55e-146j
0.016946929514198388697 + 1.3138096502891249349e-880j 1.41e-38, 2.15e-148j
0.0050793026072318104849 - 1.7707735746314685853e-881j 2.81e-39, 4.29e-149j
0.016946929514198388697 + 5.2511519991900901918e-879j 4.12e-37, 6.29e-147j
0.01968233787982999276 - 3.9816611241012547252e-879j 1.73e-37, 2.65e-147j
0.0074412337186628722134 + 1.5166109227543090739e-878j 7.63e-37, 1.17e-146j
0.023257462249903164141 - 1.9551036782092526269e-880j 4.56e-41, 6.97e-151j
0.0022263577342987368859 - 2.3061038776931574264e-879j 2.45e-37, 3.74e-147j
0.0022263577342987368859 + 4.2421569008645596924e-882j 3.15e-41, 4.82e-151j
0.023257462249903164141 - 3.3341953877291179164e-879j 8.39e-40, 1.28e-149j
0.013377600310240439592 - 8.4044712178335908213e-879j 1.09e-37, 1.67e-147j
0.01968233787982999276 - 1.3858587755711327378e-880j 1.43e-41, 2.19e-151j
0.024978101497350719179 + 2.2581626100554272428e-880j 2.67e-42, 4.08e-152j
0.0079369513161243331483 - 1.090392984591773611e-880j 4.38e-42, 6.7e-152j
0.024978101497350719179 + 3.1211509528509253943e-879j 3.53e-42, 5.39e-152j
0.035379214637404163199 + 2.2914373980771005061e-879j 2.65e-44, 4.04e-154j
0.0088550746136729907008 + 1.5113594635703588384e-880j 2.23e-42, 3.4e-152j
0.033663484195327567667 - 2.8199845380354733052e-880j 1e-45, 1.53e-155j
0.021756127646768193258 + 1.6192156497482373558e-880j 3.98e-43, 6.09e-153j
0.0088550746136729907008 + 1.3375192986073138788e-878j 2.73e-41, 4.18e-151j
0.033663484195327567667 - 2.2432478554031454669e-879j 4.03e-46, 6.16e-156j
0.0074412337186628722134 + 4.842884901088681167e-881j 1.85e-42, 2.83e-152j
0.027745351766174780149 - 2.4284979054384227148e-880j 5.48e-45, 8.38e-155j
0.030945281764615104209 + 2.5529222352953752745e-880j 1.06e-45, 1.62e-155j
0.021756127646768193258 + 3.5122125022288416676e-879j 2.09e-44, 3.19e-154j
0.035638757857265603346 - 2.5258353174818555353e-879j 1.32e-48, 2.01e-158j
0.027745351766174780149 - 2.7676005718103062987e-879j 2.04e-46, 3.12e-156j
0.035312389644374428377 + 5.3603564487731353013e-880j 2.9e-50, 4.43e-160j
0.037713412838650731961 - 5.8008865232131923344e-880j 7.17e-51, 1.1e-160j
0.013377600310240439592 - 1.4313832103203186722e-880j 2.63e-45, 4.02e-155j
0.030945281764615104209 + 2.420348949094092203e-879j 2.14e-47, 3.27e-157j
0.035312389644374428377 + 2.7177648351831786929e-879j 1.78e-50, 2.72e-160j
0.018606939786515346631 + 1.0771023643947529212e-878j 1.64e-55, 2.5e-165j
0.045957132020875180462 - 5.9877775813247991004e-880j 2.71e-53, 4.14e-163j
0.035379214637404163199 + 3.3956581724149838939e-880j 4.1e-50, 6.27e-160j
0.054748222515463814368 - 7.40619059128575608e-879j 2.21e-56, 3.37e-166j
0.042172695098393654991 + 2.2604649340309226103e-879j 3.66e-54, 5.59e-164j
0.037289239901663013786 + 2.6468583387685150327e-878j 2.74e-56, 4.19e-166j
0.035638757857265603346 - 4.3515767592501714861e-880j 7.68e-52, 1.17e-161j
0.042172695098393654991 + 5.756869122245863751e-880j 1.15e-53, 1.75e-163j
0.054650967099603138345 - 5.4338370101387755866e-879j 5.39e-58, 8.23e-168j
0.037713412838650731961 - 2.5910386769458040123e-879j 2.73e-54, 4.18e-164j
0.045957132020875180462 - 2.0605487005647717226e-879j 4.96e-56, 7.58e-166j
0.0575319593728533597 + 3.3268151400262119243e-879j 1.06e-59, 1.62e-169j
0.04875758690686564741 + 7.2263484316702255283e-880j 1.69e-57, 2.59e-167j
0.019078914122264293309 - 1.2474674225624851023e-878j 1.58e-59, 2.41e-169j
0.054650967099603138345 - 6.8446725834300474766e-879j 2.69e-60, 4.12e-170j
0.04875758690686564741 + 2.179721638965112697e-879j 9.21e-59, 1.41e-168j
0.055319759710183459902 + 6.7239236507092618781e-879j 3.22e-60, 4.92e-170j
0.037289239901663013786 + 1.2991680445246072429e-878j 4.28e-60, 6.55e-170j
0.050864427017797216006 - 1.093999158329947434e-879j 8.16e-60, 1.25e-169j
0.057920033886296448696 - 2.6232750120626405531e-879j 2e-61, 3.06e-171j
0.052077905476356592646 + 5.8289122478816244402e-879j 5.94e-61, 9.09e-171j
0.050864427017797216006 - 2.8983034777876840691e-879j 1e-60, 1.53e-170j
0.0575319593728533597 + 2.9616906460431597422e-879j 3.54e-62, 5.41e-172j
0.052077905476356592646 + 2.4974433282981053746e-879j 7.03e-62, 1.07e-171j
0.046048862235923038126 - 9.4453918636869463667e-879j 1.72e-62, 2.63e-172j
0.046048862235923038126 - 1.3625241413255990053e-878j 6.74e-63, 1.03e-172j
0.019078914122264293309 - 2.614796864615805438e-878j 2.28e-62, 3.49e-172j
0.018606939786515346631 + 1.4754087443880820276e-878j 6.34e-63, 9.69e-173j
0.055319759710183459902 + 4.1696909439383136499e-879j 1.73e-64, 2.65e-174j
0.054748222515463814368 - 4.0767615994306299535e-879j 1.7e-64, 2.59e-174j

Download

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