Rule: Q[18, 20, 38]

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)
0.99859651643055150742 - 8.9465750487514764967e-1767j 9.16e-483, 9.16e-483j
-0.95582394957139775518 - 1.7162462551366147159e-1768j 1.29e-483, 1.29e-483j
-0.9424907948868813648 - 4.5591940180796661278e-1788j 5.99e-484, 5.99e-484j
-0.99593963718398972922 - 4.7251911150276457307e-1817j 1.07e-482, 1.07e-482j
0.9424907948868813648 + 2.9971262343274922496e-1847j 6.13e-484, 6.13e-484j
0.9274951474090583421 - 1.0266407766039290161e-1847j 2.72e-484, 2.72e-484j
0.95582394957139775518 + 2.1698564081334347515e-1846j 1.24e-483, 1.24e-483j
-0.89260246649755573921 + 3.5258518275759360333e-1846j 3.82e-485, 3.82e-485j
0.98536494677783579131 - 1.0958597824362854354e-1847j 5.59e-483, 5.59e-483j
0.99978525666522903894 - 9.3260081929052871333e-1848j 4.16e-483, 4.16e-483j
0.87272892725987417466 - 3.7118807449066579e-1857j 1.19e-485, 1.19e-485j
-0.97731704962157666034 + 1.8557091235838677266e-1852j 3.56e-483, 3.56e-483j
-0.9274951474090583421 - 6.4233050112607331125e-1873j 2.46e-484, 2.46e-484j
-0.87272892725987417466 + 6.0631747045883936156e-1884j 1.27e-485, 1.27e-485j
0.99593963718398972922 - 6.1792858416731173056e-1881j 9.9e-483, 9.9e-483j
0.8512614839549979707 - 8.7295081773066196961e-1892j 3.75e-486, 3.75e-486j
0.9108615913643796364 - 9.7644200045206375376e-1890j 1.04e-484, 1.04e-484j
-0.99859651643055150742 - 9.7767012974762272875e-1890j 9.57e-483, 9.57e-483j
-0.75036505635902229785 + 1.8267230651556170556e-1919j 1.15e-488, 1.15e-488j
-0.96745167461104915612 + 1.9051141400701880912e-1913j 2.06e-483, 2.06e-483j
0.96745167461104915612 - 5.6431912786735659755e-1927j 2.32e-483, 2.32e-483j
0.77772782320499623144 + 6.5947770221688603308e-1932j 5.3e-488, 5.3e-488j
-0.99978525666522903894 + 6.6792645726557691506e-1923j 3.99e-483, 3.99e-483j
-0.8512614839549979707 - 1.1698499836519088687e-1958j 3.65e-486, 3.65e-486j
-0.82823630980177935264 + 1.6793567521632389656e-1959j 1.11e-486, 1.11e-486j
-0.98536494677783579131 - 1.0112674530942174867e-1954j 5.96e-483, 5.96e-483j
0.75036505635902229785 - 6.3704419680752692578e-1984j 1.08e-488, 1.08e-488j
0.82823630980177935264 - 1.193181909248898138e-1980j 1.01e-486, 1.01e-486j
0.99156516842093094673 + 1.3043584731064685661e-1975j 8.19e-483, 8.19e-483j
-0.66045720217883859543 - 7.9249061055812181971e-1986j 4.7e-491, 4.7e-491j
-0.9108615913643796364 - 5.5568788070165445608e-1979j 9.75e-485, 9.75e-485j
0.80370495897252311568 + 8.4316235944433888887e-1990j 2.46e-487, 2.46e-487j
0.66045720217883859543 + 1.2115131199040054773e-1994j 4.45e-491, 4.45e-491j
0.042667458043371127104 - 3.6438932714193625397e-2013j 1.22e-509, 1.22e-509j
-0.72166984583245439119 + 1.1802901407594855574e-1992j 2.05e-489, 2.05e-489j
-0.77772782320499623144 + 1.1063223365523556776e-1990j 5.55e-488, 5.55e-488j
0.89260246649755573921 - 7.5423226970022661532e-1987j 4.1e-485, 4.1e-485j
0.62802376237896618507 + 6.7449140731777102603e-1995j 5.86e-492, 5.86e-492j
0.72166984583245439119 - 3.3379285476471683978e-1992j 1.91e-489, 1.91e-489j
-0.62802376237896618507 - 3.3753441616419334878e-1996j 5.94e-492, 5.94e-492j
-0.52409209187493722694 + 4.5211756177837881548e-1998j 6.93e-495, 6.93e-495j
0.16885305857357954056 - 7.6877542160240666555e-2009j 1.42e-505, 1.42e-505j
0.69168704306035320787 + 5.8977143593371986786e-1993j 3.48e-490, 3.48e-490j
0.5944398006694373254 - 7.263908982486181823e-1996j 7.01e-493, 7.01e-493j
-0.80370495897252311568 - 4.3142653484818405106e-1990j 2.52e-487, 2.52e-487j
-0.55977083107394753461 - 4.1637482565408670479e-1997j 6.99e-494, 6.99e-494j
-0.69168704306035320787 - 4.296317309445171114e-1993j 3.11e-490, 3.11e-490j
-0.1268906073935182337 - 1.1862750978937831542e-2010j 5.25e-507, 5.25e-507j
0.48748173749781995875 - 2.0937341957237781277e-2000j 5.71e-496, 5.71e-496j
0.97731704962157666034 + 4.8293393939046615625e-1995j 3.71e-483, 3.71e-483j
-0.41175116146284264604 - 5.4866176239032534241e-2011j 3.89e-498, 3.89e-498j
-0.45001333040213137396 + 3.2121659371330051139e-2010j 5.17e-497, 5.17e-497j
-0.99156516842093094673 + 2.9725060783440091753e-2015j 8.2e-483, 8.2e-483j
0.55977083107394753461 + 1.7239791339186991002e-2033j 7.36e-494, 7.36e-494j
0.52409209187493722694 - 1.1157066130199879889e-2034j 6.54e-495, 6.54e-495j
-0.084775013041735301242 + 1.5562283085370388085e-2047j 2.98e-508, 2.98e-508j
-0.48748173749781995875 - 6.6622168510703229924e-2034j 5.54e-496, 5.54e-496j
-0.5944398006694373254 - 6.2465964222501814335e-2038j 6.88e-493, 6.88e-493j
0.33306496018034047177 - 1.172670406644128257e-2047j 1.59e-500, 1.59e-500j
0.41175116146284264604 + 5.1581942873556151347e-2045j 3.99e-498, 3.99e-498j
0.21055324069300071552 - 1.3333909356181729968e-2050j 2.73e-504, 2.73e-504j
-0.16885305857357954056 - 1.2569423127342563242e-2051j 1.3e-505, 1.3e-505j
-0.37275077275742226144 + 1.2569194113403270997e-2045j 2.8e-499, 2.8e-499j
-0.042667458043371127104 + 6.482806416155600613e-2056j 1.22e-509, 1.22e-509j
0.45001333040213137396 - 5.1406539264782767899e-2044j 5.06e-497, 5.06e-497j
0.29275267207064298679 - 5.5818188405305046678e-2049j 9.72e-502, 9.72e-502j
-0.0037762088409933203275 - 3.7070369376773591115e-2057j 4.33e-511, 4.33e-511j
-0.33306496018034047177 + 1.492342800959520421e-2046j 1.75e-500, 1.75e-500j
-0.21055324069300071552 + 4.1194117910445943919e-2050j 2.57e-504, 2.57e-504j
0.084775013041735301242 + 2.0255113435265507992e-2054j 2.8e-508, 2.8e-508j
0.37275077275742226144 - 2.2413354560821225856e-2046j 2.53e-499, 2.53e-499j
0.1268906073935182337 - 3.9737362580187151417e-2053j 6e-507, 6e-507j
-0.25188622569150550959 + 7.3296492180176932875e-2049j 5.01e-503, 5.01e-503j
-0.29275267207064298679 - 6.1155055310153151918e-2048j 9.47e-502, 9.47e-502j
0.0037762088409933203275 + 1.2775143972594591463e-2056j 4.33e-511, 4.33e-511j
0.25188622569150550959 - 2.1449149761618234861e-2049j 5.47e-503, 5.47e-503j

Weights

Weight (ball center) Error (ball radius)
0.0018650207357514555634 + 7.2066201404627637756e-1769j 1.18e-141, 4.46e-376j
0.012489114154723196883 - 1.6809084449919260712e-1770j 5.72e-142, 2.16e-376j
0.014169968772078703794 + 1.749135169595562082e-1768j 5.17e-142, 1.95e-376j
0.0034883268912083345418 - 4.3496051146344939176e-1769j 5.67e-143, 2.14e-377j
0.014169968772078703794 - 3.7327639450456981495e-1768j 1.78e-143, 6.73e-378j
0.015817346515839589325 + 3.0215262355383764124e-1768j 1.02e-143, 3.83e-378j
0.012489114154723196883 + 4.7914146003376112319e-1768j 7.15e-144, 2.69e-378j
0.019068667825101818068 - 4.9000367597884811254e-1769j 1.81e-145, 6.83e-380j
0.0071257224493184157921 - 1.5656418059362215231e-1767j 6.84e-145, 2.58e-379j
0.00059541477811261148086 + 5.6640723308104663486e-1767j 1.31e-144, 4.95e-379j
0.020675103874874755844 - 1.8304026350971291982e-1768j 3.2e-146, 1.2e-380j
0.0089645924372801437705 + 8.8766051508783111208e-1769j 2.04e-147, 7.7e-382j
0.015817346515839589325 - 9.029811977556454657e-1769j 6.62e-147, 2.49e-381j
0.020675103874874755844 + 4.1081055518536757687e-1769j 7.71e-148, 2.91e-382j
0.0034883268912083345418 - 7.7529174470352874373e-1767j 2.4e-145, 9.03e-380j
0.022253748917448557547 + 1.5999289593623455948e-1768j 2.65e-148, 9.97e-383j
0.01744790672141117511 - 2.5106967964033116816e-1768j 5.24e-147, 1.97e-381j
0.0018650207357514555634 + 3.2540935221655292849e-1769j 6.08e-149, 2.29e-383j
0.028037104525957216602 - 2.8695800180838415903e-1769j 1.46e-152, 5.5e-387j
0.010756573644166783559 - 1.7202669402854288235e-1768j 8.78e-149, 3.31e-383j
0.010756573644166783559 - 6.503704464607011031e-1768j 5.18e-148, 1.95e-382j
0.026679542075135669866 - 1.1712792100314275463e-1768j 2.85e-152, 1.07e-386j
0.00059541477811261148086 - 1.2687803019393599694e-1769j 2.85e-149, 1.07e-383j
0.022253748917448557547 - 3.6068254573659708962e-1769j 7.5e-152, 2.83e-386j
0.023787887935139790106 + 3.2800679128606334662e-1769j 1.1e-152, 4.16e-387j
0.0071257224493184157921 - 6.3028672415002501909e-1769j 1.12e-149, 4.23e-384j
0.028037104525957216602 + 1.0846881444124144968e-1768j 2.71e-155, 1.02e-389j
0.023787887935139790106 - 1.4204311827101383374e-1768j 9.72e-154, 3.66e-388j
0.0052778301509698257492 + 3.005832970447937773e-1767j 6.99e-151, 2.63e-385j
0.031839502902372755754 + 2.8914821334926453787e-1769j 2.4e-158, 9.05e-393j
0.01744790672141117511 + 6.2586754268622103963e-1769j 3.15e-152, 1.19e-386j
0.025264584115077721991 + 1.2806496328103565728e-1768j 1.62e-154, 6.1e-389j
0.031839502902372755754 - 9.1248699309087899508e-1769j 1.23e-158, 4.64e-393j
0.041972126566202637983 - 5.4131962243581976752e-1768j 1e-162, 3.78e-397j
0.029345914359214064208 + 2.8432835710716807744e-1769j 6.51e-158, 2.45e-392j
0.026679542075135669866 + 2.9398941501332105486e-1769j 9.39e-157, 3.54e-391j
0.019068667825101818068 + 2.1265734187227450071e-1768j 2.09e-154, 7.88e-389j
0.03301858141596863835 + 8.7640096333980295912e-1769j 2.09e-160, 7.88e-395j
0.029345914359214064208 - 1.0150438206234278089e-1768j 3.55e-158, 1.34e-392j
0.03301858141596863835 - 2.9625743636752543731e-1769j 6.31e-162, 2.38e-396j
0.036157415502460920747 + 3.411930755616550121e-1769j 3.46e-164, 1.3e-398j
0.041848996550780574355 + 1.5319173181490192951e-1768j 2.12e-166, 8e-401j
0.030613060848870384532 + 9.5839019365384675784e-1769j 8.25e-160, 3.11e-394j
0.03413837388685956351 - 8.4995803527124369847e-1769j 2.1e-162, 7.92e-397j
0.025264584115077721991 - 3.0692333287196390195e-1769j 2.02e-159, 7.6e-394j
0.035186903163667422993 - 3.2165222947096736929e-1769j 2.7e-164, 1.02e-398j
0.030613060848870384532 - 2.8518904090968380673e-1769j 1.63e-161, 6.16e-396j
0.04205804704741282279 + 1.6009816145681452243e-1768j 4.28e-169, 1.61e-403j
0.037050948877345393226 + 8.2877858854269616245e-1769j 6.82e-167, 2.57e-401j
0.0089645924372801437705 + 9.553494954443395049e-1768j 3.42e-159, 1.29e-393j
0.038640029553349914322 - 4.3710429619618789962e-1769j 7.01e-168, 2.64e-402j
0.037875157757609286949 + 3.9779070517541965944e-1769j 2.72e-167, 1.02e-401j
0.0052778301509698257492 + 5.1352521588268598838e-1769j 1.58e-160, 5.95e-395j
0.035186903163667422993 + 8.3322915415880981108e-1769j 2.32e-166, 8.76e-401j
0.036157415502460920747 - 8.2621040268377624608e-1769j 3.57e-167, 1.35e-401j
0.042150825651980189516 - 2.4742675859447376646e-1768j 2.46e-171, 9.26e-406j
0.037050948877345393226 - 3.6626730638665959817e-1769j 1.37e-167, 5.15e-402j
0.03413837388685956351 + 3.068986719929606788e-1769j 3.99e-166, 1.5e-400j
0.040009821255218796344 + 9.4681643943253157916e-1769j 3.71e-172, 1.4e-406j
0.038640029553349914322 + 8.6316981036133347784e-1769j 2.97e-171, 1.12e-405j
0.041533810036132101216 - 1.2817592963992531471e-1768j 7.13e-173, 2.69e-407j
0.041848996550780574355 - 1.1677877852748956606e-1768j 2.14e-173, 8.07e-408j
0.039352109205887489689 + 4.864221739814658361e-1769j 1.39e-172, 5.25e-407j
0.041972126566202637983 + 5.0561712415541589504e-1768j 1.81e-173, 6.82e-408j
0.037875157757609286949 - 8.4091789828008152481e-1769j 5.59e-172, 2.11e-406j
0.040602734526903800048 - 1.0184627969624890706e-1768j 1.07e-173, 4.02e-408j
0.021725696555975424463 - 2.9321112441851591784e-1767j 4.34e-174, 1.63e-408j
0.040009821255218796344 - 5.4952401124077079944e-1769j 1.82e-174, 6.85e-409j
0.041533810036132101216 + 9.1279598227925328827e-1769j 5.56e-175, 2.09e-409j
0.042150825651980189516 + 2.83388741325545834e-1768j 2.52e-175, 9.47e-410j
0.039352109205887489689 - 8.9725158177285435506e-1769j 4.11e-175, 1.55e-409j
0.04205804704741282279 - 1.9622432062676823999e-1768j 1.48e-175, 5.57e-410j
0.041115487816192053411 - 7.4736619740987645087e-1769j 2.57e-176, 9.65e-411j
0.040602734526903800048 + 6.3283732891991959678e-1769j 2.41e-176, 9.04e-411j
0.021725696555975424463 + 2.9498644612611980521e-1767j 3e-175, 1.16e-409j
0.041115487816192053411 + 1.1234820078607273097e-1768j 8.13e-177, 2.9e-411j

Download

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