Rule: Q[4, 37, 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.

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)
0.99969582504707330934 + 5.7705396525439528368e-1750j 4.24e-482, 4.24e-482j
-0.93030417494014471 - 5.0585164599323137762e-1764j 1.43e-482, 1.43e-482j
-0.9151199048052934817 - 1.4610023201299849725e-1764j 6.74e-483, 6.74e-483j
-0.99506899473048301953 - 1.7657198965125361675e-1763j 1.41e-481, 1.41e-481j
0.94404214480607462039 - 2.6296995209994334978e-1762j 2.9e-482, 2.9e-482j
0.93030417494014471 + 1.8153603637864228047e-1770j 1.39e-482, 1.39e-482j
-0.8804930303043403715 - 1.1449957203672009527e-1784j 1.07e-483, 1.07e-483j
-0.89850237540524299387 - 1.0463285553674167751e-1784j 2.66e-483, 2.66e-483j
-0.94404214480607462039 - 4.0954425529799184503e-1782j 2.68e-482, 2.68e-482j
0.79522108427446577387 - 3.3013549457830076672e-1786j 8.21e-486, 8.21e-486j
0.8804930303043403715 + 1.386874358808212903e-1781j 1.11e-483, 1.11e-483j
0.95632572175562874702 - 1.8347567633597141152e-1805j 5.02e-482, 5.02e-482j
-0.86113631159405257522 + 1.4033038539644545171e-1826j 3.52e-484, 3.52e-484j
-0.84045306985318616804 - 3.8670449324721671546e-1831j 1.19e-484, 1.19e-484j
0.99037016045108105218 + 4.2032283621797673283e-1834j 1.59e-481, 1.59e-481j
0.86113631159405257522 + 5.4766395329581025898e-1858j 3.61e-484, 3.61e-484j
0.9151199048052934817 + 4.06853237133202152e-1866j 6.78e-483, 6.78e-483j
0.24965614963949324715 - 3.0536459416537986591e-1905j 1.41e-501, 1.41e-501j
-0.77078666167911538462 - 8.6049873467836943363e-1884j 1.96e-486, 1.96e-486j
0.99506899473048301953 + 2.6338675500776909861e-1886j 1.41e-481, 1.41e-481j
-0.99816981403777082918 - 6.5753998963384856804e-1907j 1.09e-481, 1.09e-481j
0.77078666167911538462 + 4.4369310281689577586e-1918j 2.03e-486, 2.03e-486j
0.27723606894235341918 + 1.2879836727427402518e-1932j 1.43e-500, 1.43e-500j
-0.81846385055857360972 - 1.6937838075201363556e-1919j 3.24e-485, 3.24e-485j
-0.79522108427446577387 + 3.0012409783322401303e-1918j 8.26e-486, 8.26e-486j
0.99816981403777082918 - 2.3834452775699352773e-1911j 1.08e-481, 1.08e-481j
0.74519068266733664373 + 1.5336512389964110895e-1935j 4.36e-487, 4.36e-487j
0.97639222378840376842 - 6.7139494393621789902e-1928j 1.15e-481, 1.15e-481j
-0.97639222378840376842 - 2.6176465910395102446e-1947j 1.19e-481, 1.19e-481j
-0.6321709602805231133 + 1.4680613009120137212e-1957j 3.91e-490, 3.91e-490j
-0.74519068266733664373 + 2.5076274465673566279e-1954j 4.27e-487, 4.27e-487j
0.98413624859999195727 - 6.2103491320012666827e-1946j 1.44e-481, 1.44e-481j
0.6618756469678823958 - 8.3044000430567929209e-1971j 2.56e-489, 2.56e-489j
-0.99037016045108105218 + 1.2730947281111043291e-1961j 1.51e-481, 1.51e-481j
-0.71845464895300619817 + 1.6482047782463391117e-1975j 8.97e-488, 8.97e-488j
-0.69064994834708266285 - 3.0069275471380611406e-1975j 1.55e-488, 1.55e-488j
0.96712154238710687702 + 1.7173665590183021067e-1966j 8.28e-482, 8.28e-482j
0.6321709602805231133 - 5.2237729347322421737e-1987j 3.81e-490, 3.81e-490j
0.84045306985318616804 - 6.3697401287595217822e-1982j 1.17e-484, 1.17e-484j
-0.95632572175562874702 - 2.7250820660412506711e-1975j 5.16e-482, 5.16e-482j
-0.5380518436120006096 + 9.0484266391230741981e-1989j 9.54e-493, 9.54e-493j
-0.99969582504707330934 - 2.4417276090963142699e-1975j 3.79e-482, 3.79e-482j
0.69064994834708266285 + 5.5538018872836392475e-1988j 1.5e-488, 1.5e-488j
0.60153666609275201815 + 6.0610262947434080957e-1990j 5.54e-491, 5.54e-491j
-0.50550034841465622286 + 4.8896458937847120959e-1994j 1e-493, 1e-493j
-0.57008693010920404008 - 1.5910558973419318938e-1992j 7.07e-492, 7.07e-492j
-0.60153666609275201815 + 1.0546308329047875394e-1991j 5.32e-491, 5.32e-491j
0.89850237540524299387 - 1.0416081619965131321e-1980j 2.65e-483, 2.65e-483j
0.50550034841465622286 + 5.192911473534700393e-1996j 1.03e-493, 1.03e-493j
0.71845464895300619817 + 5.1813689294350291866e-1990j 8.71e-488, 8.71e-488j
-0.6618756469678823958 + 1.8517304128050806081e-1990j 2.52e-489, 2.52e-489j
-0.47227408324693184291 - 1.3423247979437603403e-1996j 1.08e-494, 1.08e-494j
0.81846385055857360972 + 9.1054022110805344462e-1988j 3.07e-485, 3.07e-485j
0.57008693010920404008 + 2.6936444159678044034e-1994j 7.13e-492, 7.13e-492j
0.5380518436120006096 - 9.0189765567601935611e-1995j 9.14e-493, 9.14e-493j
-0.96712154238710687702 - 5.9161293045567311447e-1982j 8.24e-482, 8.24e-482j
-0.43860664448184905982 - 2.0674563000615314885e-2000j 1.17e-495, 1.17e-495j
-0.3399810435848562648 + 1.9462240268854080603e-2003j 1.25e-498, 1.25e-498j
0.035902643446137412127 + 2.7202687353164726474e-2014j 9.55e-510, 9.55e-510j
0.40554253094583655843 + 7.107325502715499048e-2001j 1.29e-496, 1.29e-496j
0.14636780220938547625 - 1.5183485211473640292e-2009j 1.81e-505, 1.81e-505j
-0.30583432046023013023 - 1.4416640764726539308e-2003j 1.13e-499, 1.13e-499j
-0.40554253094583655843 + 3.1480630569564117929e-2001j 1.35e-496, 1.35e-496j
-0.98413624859999195727 - 4.8269485010792995954e-2009j 1.38e-481, 1.38e-481j
0.43860664448184905982 + 2.3138466552314691636e-2022j 1.18e-495, 1.18e-495j
0.47227408324693184291 - 8.1520368945597318132e-2024j 1.12e-494, 1.12e-494j
-0.074184708318281580068 + 6.6164790917918650113e-2038j 2.44e-508, 2.44e-508j
-0.37334725534840493358 + 1.0549603737307076058e-2027j 1.38e-497, 1.38e-497j
-0.17836429603140252798 - 6.2392286692154543369e-2034j 2.75e-504, 2.75e-504j
0.21421425536824183715 - 4.4568387039438840489e-2032j 5.81e-503, 5.81e-503j
0.3399810435848562648 + 4.5697775743647168574e-2028j 1.29e-498, 1.29e-498j
0.074184708318281580068 - 1.4218929699862724726e-2038j 2.23e-508, 2.23e-508j
-0.21421425536824183715 + 5.3896172692149106941e-2033j 5.63e-503, 5.63e-503j
-0.24965614963949324715 - 9.1889567174171441995e-2032j 1.38e-501, 1.38e-501j
-0.035902643446137412127 - 8.9595221831200864234e-2040j 9.55e-510, 9.55e-510j
0.37334725534840493358 + 6.1266395668137802653e-2027j 1.35e-497, 1.35e-497j
0.17836429603140252798 + 2.3082266651224880229e-2033j 2.79e-504, 2.79e-504j
-0.14636780220938547625 + 3.1497483925942973052e-2035j 1.68e-505, 1.68e-505j
-0.27723606894235341918 + 1.4585397712479550082e-2030j 1.39e-500, 1.39e-500j
-0.11238989210078422628 - 2.3929279556052275299e-2036j 7.8e-507, 7.8e-507j
0.11238989210078422628 + 2.3907785167852380092e-2037j 7.15e-507, 7.15e-507j
0.30583432046023013023 + 1.235410982652124229e-2028j 1.13e-499, 1.13e-499j
2.3218031939416673276e-2281 + 1.1400540662808429051e-2281j 2.15e-2279, 2.15e-2279j

Weights

Weight (ball center) Error (ball radius)
0.00081887236617353783478 - 6.1548004500699228268e-1750j 1.41e-136, 2.11e-370j
0.014461383364686907629 + 5.88468731984677547e-1752j 6.93e-137, 1.04e-370j
0.015905516337306301669 - 6.9902068353206897544e-1752j 5.43e-137, 8.14e-371j
0.0039095699631693928854 - 1.1534775469561030906e-1752j 2.32e-138, 3.48e-372j
0.013013914451881369839 - 1.7222659958535025166e-1750j 8.39e-139, 1.26e-372j
0.014461383364686907629 + 1.6367165948215770713e-1750j 7.33e-139, 1.1e-372j
0.018688612730499705386 - 9.875005038877205325e-1752j 4.04e-138, 6.07e-372j
0.017322048485117784116 + 8.2986962594159549099e-1752j 7.56e-138, 1.13e-371j
0.013013914451881369839 - 4.931242920082753242e-1752j 1.36e-138, 2.05e-372j
0.023846969568607435326 - 1.8043759608973561304e-1750j 6.3e-141, 9.45e-375j
0.018688612730499705386 - 1.5575871740839308089e-1750j 1.61e-140, 2.42e-374j
0.011547217605882230287 + 1.8516624293870716255e-1750j 2.93e-140, 4.39e-374j
0.020021874220034327682 + 1.178046809647286649e-1751j 2.47e-139, 3.7e-373j
0.021341946374785955712 - 1.4094847251905779564e-1751j 4.09e-140, 6.14e-374j
0.005475676641872702246 + 3.4925763920427247996e-1750j 6.17e-142, 9.26e-376j
0.020021874220034327682 + 1.5820980510308430781e-1750j 1.07e-141, 1.6e-375j
0.015905516337306301669 - 1.5825967917252899324e-1750j 4.65e-141, 6.98e-375j
0.032321422936823179143 - 6.3617504397957774197e-1749j 1.27e-145, 1.91e-379j
0.025017319598993183756 + 2.5108995570832862826e-1751j 1.12e-143, 1.68e-377j
0.0039095699631693928854 - 4.9729864153755672418e-1750j 4.3e-142, 6.45e-376j
0.0022910266006860116145 + 6.9896643425651570916e-1753j 4.83e-145, 7.25e-379j
0.025017319598993183756 + 1.9420383292377618627e-1750j 4.95e-144, 7.43e-378j
0.02497268342106962748 + 7.0582431626963570157e-1749j 7.47e-146, 1.12e-379j
0.022627268019854244596 + 1.6955888465499890374e-1751j 5.6e-143, 8.41e-377j
0.023846969568607435326 - 2.0555230436844686504e-1751j 1.52e-143, 2.27e-377j
0.0022910266006860116145 + 9.1509236394064924178e-1750j 1.81e-142, 2.72e-376j
0.026172435302161467311 - 2.1188829113469734943e-1750j 6.41e-146, 9.62e-380j
0.0085041571509371185766 + 2.3368905356750248905e-1750j 2.53e-143, 3.8e-377j
0.0085041571509371185766 + 2.7558471020951089576e-1752j 1.51e-146, 2.26e-380j
0.030173944398837545709 - 7.9980037048515526948e-1751j 9.06e-149, 1.36e-382j
0.026172435302161467311 - 3.0905539968032751319e-1751j 1.76e-146, 2.64e-380j
0.0069880033928710094563 - 2.7702593818549446544e-1750j 6.1e-144, 9.15e-378j
0.029238749367941732706 + 3.0431634672128475093e-1750j 1.22e-149, 1.83e-383j
0.005475676641872702246 + 1.6366590980310198055e-1752j 3.23e-147, 4.85e-381j
0.027287353934700446759 + 3.8437118214733635539e-1751j 1.48e-147, 2.22e-381j
0.028303498020392405091 - 4.8450341096470761493e-1751j 1.99e-148, 2.99e-382j
0.010037274911267266344 - 2.0471679307025070183e-1750j 3.93e-145, 5.89e-379j
0.030173944398837545709 - 3.5512363505426060183e-1750j 1.06e-151, 1.59e-385j
0.021341946374785955712 - 1.628747101417046458e-1750j 4.03e-148, 6.05e-382j
0.011547217605882230287 + 4.1056189260643089073e-1752j 3.41e-148, 5.11e-382j
0.032276470095835826694 + 1.9437913061545625502e-1750j 5.38e-154, 8.08e-388j
0.00081887236617353783478 - 2.3633866126689442904e-1753j 8.03e-149, 1.21e-382j
0.028303498020392405091 - 2.6500217432576075443e-1750j 9.73e-152, 1.46e-385j
0.031076221153336405546 + 4.2288039671496556422e-1750j 2.13e-153, 3.2e-387j
0.032865870729444497434 - 2.7063232748562837641e-1750j 2.95e-155, 4.43e-389j
0.031778968668081289106 - 1.4150877588192507684e-1750j 3.26e-154, 4.89e-388j
0.031076221153336405546 + 1.0515256468126510388e-1750j 9.3e-154, 1.4e-387j
0.017322048485117784116 + 1.5566788523828579173e-1750j 2.6e-151, 3.9e-385j
0.032865870729444497434 - 8.2427857600436885481e-1750j 2.43e-158, 3.64e-392j
0.027287353934700446759 + 2.3481893297778819158e-1750j 5.6e-154, 8.41e-388j
0.029238749367941732706 + 6.187167038751720954e-1751j 5.22e-154, 7.84e-388j
0.033559526806435475882 + 3.8621340512581400177e-1750j 5.64e-158, 8.46e-392j
0.022627268019854244596 + 1.7010526293186370767e-1750j 5.6e-153, 8.41e-387j
0.031778968668081289106 - 5.1707038355157656819e-1750j 2.26e-157, 3.39e-391j
0.032276470095835826694 + 6.4748175425287465434e-1750j 3.54e-158, 5.32e-392j
0.010037274911267266344 - 3.3905042700643577312e-1752j 3.63e-156, 5.44e-390j
0.033577128528850250958 - 5.7775029488192874519e-1750j 4.3e-161, 6.46e-395j
0.034324734352574759815 + 1.7793648819626658321e-1749j 8.63e-164, 1.29e-397j
0.037160873916739989957 - 2.7791117903153256593e-1749j 1.36e-166, 2.04e-400j
0.032443609212588761425 + 2.0810931444653762786e-1749j 7.99e-164, 1.2e-397j
0.031556517060216867237 + 4.5907650425081292653e-1749j 4.33e-166, 6.5e-400j
0.032731916569104438921 - 2.8515978676136622317e-1749j 3.88e-165, 5.82e-399j
0.032443609212588761425 + 8.7991360457992670804e-1750j 4.25e-163, 6.37e-397j
0.0069880033928710094563 - 2.1727677056453177363e-1752j 2.74e-159, 4.12e-393j
0.033577128528850250958 - 1.481011761913804446e-1749j 2.47e-164, 3.7e-398j
0.033559526806435475882 + 1.0778746219004599953e-1749j 8.83e-164, 1.32e-397j
0.038917353386098962915 + 2.497360698831846525e-1749j 4.64e-169, 6.96e-403j
0.032408983905305447007 - 1.2628316032152958269e-1749j 4.74e-165, 7.12e-399j
0.033860812872015875691 - 3.3832838922402512328e-1749j 7.16e-169, 1.07e-402j
0.036914646767932656562 + 4.9504608269821142109e-1749j 3.12e-168, 4.68e-402j
0.034324734352574759815 + 3.6133402497633089415e-1749j 1.12e-167, 1.68e-401j
0.038917353386098962915 + 2.8977145609511614015e-1749j 1.7e-169, 2.54e-403j
0.036914646767932656562 + 3.2032815311009134648e-1749j 2.2e-169, 3.3e-403j
0.032321422936823179143 - 3.8192321552024711212e-1749j 3.46e-169, 5.19e-403j
0.037160873916739989957 - 2.5864165271602606487e-1749j 8.15e-171, 1.22e-404j
0.032408983905305447007 - 2.7683023134124148282e-1749j 2.31e-169, 3.47e-403j
0.033860812872015875691 - 4.8527442219489600786e-1749j 3.35e-170, 5.02e-404j
0.031556517060216867237 + 3.418159658738828151e-1749j 6.16e-171, 9.25e-405j
0.02497268342106962748 + 3.9933975005654155666e-1749j 3.36e-170, 5.04e-404j
0.036683864308436888983 - 2.8813147322077474053e-1749j 2.71e-171, 4.08e-405j
0.036683864308436888983 - 3.6112335569042675807e-1749j 4.87e-172, 7.42e-406j
0.032731916569104438921 - 5.3654035487248302016e-1749j 5.83e-172, 9.29e-406j
0.035147525000897433422 + 2.7754927506121874932e-1749j 4.03e-172, 5.92e-406j

Download

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