Rule: Q[3, 42, 46]

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)
0.99610148483727753794 + 2.0209107564575310131e-1777j 5.57e-478, 5.57e-478j
-0.95567355593646219643 - 9.8805227599944535758e-1787j 1.45e-478, 1.45e-478j
-0.94475755206460393205 + 2.490694144455243961e-1813j 7.62e-479, 7.62e-479j
-0.99975980509124899355 - 1.25670093709874307e-1830j 1.45e-478, 1.45e-478j
0.91944075690578682193 - 1.2581386077053138328e-1856j 1.83e-479, 1.83e-479j
0.93267685135591888185 - 5.2889115550723601962e-1856j 4.15e-479, 4.15e-479j
0.9654205626957687594 + 5.5897937750383613275e-1854j 2.38e-478, 2.38e-478j
-0.91944075690578682193 + 6.0025139285604499967e-1855j 1.91e-479, 1.91e-479j
-0.90507391275050144198 - 2.523580438231910206e-1863j 7.62e-480, 7.62e-480j
-0.99855341163032368546 + 2.5708228515876420354e-1879j 3.77e-478, 3.77e-478j
0.87303398795525178662 + 3.4689304450859934906e-1906j 1.07e-480, 1.07e-480j
-0.99238613019668857555 + 4.6544130094592172992e-1901j 6.18e-478, 6.18e-478j
-0.88960044761246962429 + 5.9588192149409654455e-1917j 3.03e-480, 3.03e-480j
-0.87303398795525178662 + 1.3480831015038796367e-1917j 1.08e-480, 1.08e-480j
0.99855341163032368546 - 7.7474840999701476189e-1914j 4.13e-478, 4.13e-478j
0.81698927374426241067 + 3.7551487257047751511e-1920j 2.96e-482, 2.96e-482j
0.90507391275050144198 - 5.5330329303464848062e-1918j 7.74e-480, 7.74e-480j
-0.97397883967229151808 + 9.1130827558857390993e-1914j 3.51e-478, 3.51e-478j
-0.77459666924148337704 + 1.4124224870715161073e-1929j 1.62e-483, 1.62e-483j
-0.9654205626957687594 - 1.2058827908126117024e-1923j 2.29e-478, 2.29e-478j
0.09583383970537608263 + 4.4537994754364271859e-1960j 3.99e-507, 3.99e-507j
0.79628103698434134174 + 1.0490280499030348805e-1935j 7.48e-483, 7.48e-483j
0.99975980509124899355 - 3.6152430792606489523e-1928j 1.43e-478, 1.43e-478j
-0.85539075792856720776 + 1.024162966222818883e-1938j 3.64e-481, 3.64e-481j
-0.79628103698434134174 - 4.1433651884508162186e-1940j 6.81e-483, 6.81e-483j
0.94475755206460393205 - 2.4029028359319237084e-1935j 7.66e-479, 7.66e-479j
0.72843646453669396918 - 6.9246987309867857278e-1942j 6.88e-485, 6.88e-485j
0.83669935424244029912 - 6.8107518152756199332e-1938j 1.05e-481, 1.05e-481j
0.99238613019668857555 - 1.6972920964910985081e-1933j 5.94e-478, 5.94e-478j
-0.65267393903817924216 + 4.7659796796064019143e-1947j 3.12e-487, 3.12e-487j
0.70402249979687315825 - 3.8638202783421006101e-1947j 1.19e-485, 1.19e-485j
0.77459666924148337704 - 7.3127094466444721712e-1945j 1.8e-483, 1.8e-483j
0.67875502647205929737 + 3.4136268194208534858e-1947j 1.93e-486, 1.93e-486j
-0.99610148483727753794 + 3.4204292950059991735e-1936j 5.37e-478, 5.37e-478j
-0.67875502647205929737 + 5.5227127607771791158e-1969j 2.15e-486, 2.15e-486j
-0.7519699587372642935 - 8.1254384743852447401e-1967j 3.43e-484, 3.43e-484j
-0.98132380834652692847 + 3.7609007657520854157e-1959j 4.89e-478, 4.89e-478j
0.59823082513832843119 - 5.2464893854685024064e-1988j 5.53e-489, 5.53e-489j
0.7519699587372642935 - 9.888894518726371082e-1983j 3.67e-484, 3.67e-484j
0.98132380834652692847 + 8.9762414177136230426e-1976j 4.56e-478, 4.56e-478j
-0.56993154558925436082 + 1.8381641027999804642e-1991j 6.02e-490, 6.02e-490j
-0.83669935424244029912 - 1.3687639364101852203e-1982j 1.04e-481, 1.04e-481j
0.62582200827490670876 - 4.0854031568624928358e-1989j 4.27e-488, 4.27e-488j
0.65267393903817924216 - 1.1728192093832241545e-1988j 2.96e-487, 2.96e-487j
-0.81698927374426241067 + 5.5341890272353536071e-1983j 2.92e-482, 2.92e-482j
-0.59823082513832843119 - 1.8088347662888968485e-1990j 5.52e-489, 5.52e-489j
-0.72843646453669396918 + 3.8377207309579676935e-1986j 6.69e-485, 6.69e-485j
0.98745485656843704766 + 1.6508247087645116e-1977j 5.83e-478, 5.83e-478j
0.481267232117080114 + 6.6678285791780790988e-1996j 5.85e-493, 5.85e-493j
0.95567355593646219643 - 5.0695448247014732476e-1979j 1.39e-478, 1.39e-478j
-0.16054504865663444924 - 1.7592474159934604506e-2008j 1.42e-504, 1.42e-504j
-0.5114047255945475266 - 1.438510581670338422e-1994j 6.61e-492, 6.61e-492j
0.88960044761246962429 + 6.6559590789239444682e-1982j 3.31e-480, 3.31e-480j
0.5114047255945475266 - 3.7784482927477686718e-1995j 6.13e-492, 6.13e-492j
0.56993154558925436082 + 1.6660173020380241081e-1993j 6.28e-490, 6.28e-490j
-0.70402249979687315825 - 2.8560689146176549402e-1987j 1.18e-485, 1.18e-485j
-0.481267232117080114 + 4.749657602282848462e-1996j 5.94e-493, 5.94e-493j
-0.62582200827490670876 + 3.2909499053351928222e-1990j 4.12e-488, 4.12e-488j
-0.32374172538777687861 + 5.024964085537332858e-2003j 1.65e-498, 1.65e-498j
0.45059415417275136725 + 3.9025707802853067598e-1997j 5.37e-494, 5.37e-494j
-0.93267685135591888185 - 6.3564133815323586574e-1980j 4.11e-479, 4.11e-479j
-0.54097189243552464403 - 1.4418273285984222694e-2008j 6.53e-491, 6.53e-491j
-0.45059415417275136725 + 4.5217713602164733902e-2012j 5.35e-494, 5.35e-494j
0.2912651101355163025 - 1.9802197010074815927e-2018j 1.06e-499, 1.06e-499j
0.41944544253465535545 - 3.1308145754460829711e-2013j 4.31e-495, 4.31e-495j
0.54097189243552464403 - 3.8685432764678953087e-2009j 6.8e-491, 6.8e-491j
3.0436046314833006274e-2042 - 5.8774617669119034494e-2042j 6.02e-2040, 6.02e-2040j
-0.38789183847016823359 + 4.5026796193576488666e-2014j 3.55e-496, 3.55e-496j
-0.98745485656843704766 + 8.5584793772037404693e-1998j 5.79e-478, 5.79e-478j
0.85539075792856720776 - 9.5511040015755485954e-2009j 3.56e-481, 3.56e-481j
0.38789183847016823359 - 1.6159546816732457718e-2025j 3.46e-496, 3.46e-496j
0.1280820709211599132 - 2.2889670865765834662e-2035j 7.71e-506, 7.71e-506j
-0.41944544253465535545 + 1.1961049104957560394e-2025j 4.31e-495, 4.31e-495j
-0.35597824201184734416 + 5.1573715140544757506e-2029j 2.54e-497, 2.54e-497j
0.97397883967229151808 + 5.0050368135968361575e-2035j 3.55e-478, 3.55e-478j
0.35597824201184734416 + 1.251253296074469835e-2058j 2.4e-497, 2.4e-497j
0.32374172538777687861 - 1.0676341664497843264e-2059j 1.69e-498, 1.69e-498j
-0.031723719975947408734 - 8.5431913308924786616e-2071j 1.11e-509, 1.11e-509j
-0.2912651101355163025 - 3.6108681678689696725e-2061j 1.06e-499, 1.06e-499j
-0.1280820709211599132 + 4.4085403354021276349e-2067j 8.41e-506, 8.41e-506j
0.16054504865663444924 + 1.7595440812100008789e-2065j 1.2e-504, 1.2e-504j
0.25865636042642628176 - 8.045728202968943764e-2062j 6.52e-501, 6.52e-501j
0.063692883543839287524 - 1.409751623674727441e-2069j 2.6e-508, 2.6e-508j
-0.22596893272086916499 - 4.8889891753294293411e-2063j 3.68e-502, 3.68e-502j
-0.19322693907740572908 + 3.8856604581729184012e-2064j 2.12e-503, 2.12e-503j
0.031723719975947408734 + 9.5201458683129406151e-2071j 9.29e-510, 9.29e-510j
0.22596893272086916499 + 2.8275481794365308136e-2063j 3.86e-502, 3.86e-502j
0.19322693907740572908 - 5.9947533724861591896e-2064j 2.21e-503, 2.21e-503j
-0.09583383970537608263 - 2.0234779253820793335e-2068j 3.79e-507, 3.79e-507j
-0.25865636042642628176 + 3.5649189458804481046e-2062j 6.43e-501, 6.43e-501j
-0.063692883543839287524 - 2.8094109716759675692e-2069j 2.34e-508, 2.34e-508j

Weights

Weight (ball center) Error (ball radius)
0.0030912480181254993765 - 3.6875357682188257843e-1778j 1.93e-122, 1.02e-352j
0.010333145885033823531 - 1.158214168177404871e-1779j 9.35e-123, 4.91e-353j
0.011498632294562236256 + 1.3262665778141363353e-1779j 8.74e-123, 4.59e-353j
0.00064691450912965161761 - 6.5044522347812516024e-1781j 5.16e-124, 2.71e-354j
0.01380626243884675078 + 4.2381678920807121973e-1778j 2.85e-124, 1.5e-354j
0.012661440408129241529 - 4.5750762642174279102e-1778j 2.71e-124, 1.42e-354j
0.0091573665583527762929 + 6.3930224916419216066e-1778j 6.16e-125, 3.23e-355j
0.01380626243884675078 + 1.6961309250944304127e-1779j 6.58e-125, 3.45e-355j
0.014923180352346962197 - 1.9047899839268587705e-1779j 2.9e-125, 1.52e-355j
0.0018116371862682237368 + 1.9158044333126601183e-1780j 8.08e-126, 4.24e-356j
0.017108647322502197037 - 3.6122416977066428236e-1778j 6.18e-129, 3.25e-359j
0.0043300385953446060712 + 4.3925137539308345708e-1780j 2.47e-126, 1.3e-356j
0.016021792486641351753 + 2.1316391636620989519e-1779j 1.66e-126, 8.71e-357j
0.017108647322502197037 - 2.3783698660061097603e-1779j 3.15e-127, 1.66e-357j
0.0018116371862682237368 - 1.5585166672596597427e-1777j 2.34e-130, 1.23e-360j
0.020212102090768345322 + 3.3245390175465819776e-1778j 1.07e-130, 5.61e-361j
0.014923180352346962197 - 3.9782888971145045298e-1778j 4.56e-129, 2.39e-359j
0.0079544220081816952949 - 8.5183498884480115821e-1780j 5.13e-128, 2.7e-358j
0.022161926559098145849 + 4.0711529480349513977e-1779j 1.2e-131, 6.28e-362j
0.0091573665583527762929 + 9.9995719004408473654e-1780j 7.14e-128, 3.75e-358j
0.032185325811924593395 - 1.4382786783342296965e-1777j 1.99e-135, 1.05e-365j
0.021200943011649996879 - 3.2787430189890977338e-1778j 5.04e-133, 2.65e-363j
0.00064691450912965161761 + 3.5486189813268168292e-1778j 3.68e-132, 1.93e-362j
0.018172773393816053654 + 2.6495619974921701375e-1779j 6.43e-130, 3.38e-360j
0.021200943011649996879 - 3.6552459855326304797e-1779j 1.82e-131, 9.56e-362j
0.011498632294562236256 + 5.0134380582331753044e-1778j 2.96e-132, 1.55e-362j
0.023977527119731576717 + 3.2688181498167753804e-1778j 1.85e-135, 9.73e-366j
0.01920488709784730393 - 3.3923524636697951228e-1778j 1.6e-133, 8.38e-364j
0.0043300385953446060712 + 2.3509086634184627972e-1777j 2.77e-133, 1.46e-363j
0.026472843354401938109 - 7.1640361953614672873e-1779j 9.91e-138, 5.2e-368j
0.02484620313971285517 - 3.3045172437022174812e-1778j 2.71e-136, 1.42e-366j
0.022161926559098145849 + 3.2544588067971958324e-1778j 5.15e-135, 2.7e-365j
0.025681864094918798963 + 3.3605903828068351815e-1778j 3.19e-137, 1.67e-367j
0.0030912480181254993765 - 3.1357930525986926324e-1780j 2.4e-133, 1.26e-363j
0.025681864094918798963 + 6.3675392585476145674e-1779j 8.33e-139, 4.37e-369j
0.023085305252914850361 - 4.5413045871591484395e-1779j 1.9e-137, 9.97e-368j
0.0067355490137734319443 + 7.107960247276457721e-1780j 5.31e-134, 2.79e-364j
0.027951589757696146189 - 3.6673828808505996776e-1778j 1.33e-140, 6.98e-371j
0.023085305252914850361 - 3.2517409619107330067e-1778j 8.83e-137, 4.64e-367j
0.0067355490137734319443 + 9.5112793097998712949e-1778j 2.39e-137, 1.26e-367j
0.028638764165911251486 + 1.0400260014227599508e-1778j 1.63e-142, 8.58e-373j
0.01920488709784730393 - 2.9503926594102434675e-1779j 3.99e-137, 2.09e-367j
0.027225993612436188154 + 3.5414025953218279439e-1778j 1.69e-140, 8.9e-371j
0.026472843354401938109 - 3.439411582228217486e-1778j 9.06e-140, 4.76e-370j
0.020212102090768345322 + 3.2842566483125434413e-1779j 1.14e-137, 5.99e-368j
0.027951589757696146189 - 9.1520697398755473655e-1779j 4.85e-143, 2.55e-373j
0.023977527119731576717 + 5.0735229090400039027e-1779j 4.97e-140, 2.61e-370j
0.0055299303205379025981 - 1.3142465468988478844e-1777j 9.45e-139, 4.96e-369j
0.030412675657503570807 - 4.5025606037150072815e-1778j 3.51e-146, 1.84e-376j
0.010333145885033823531 - 5.5916134007361804763e-1778j 3.55e-139, 1.87e-369j
0.032591162043710204139 - 8.5936946481134442391e-1778j 2.34e-150, 1.23e-380j
0.029857186566145197158 + 1.3619399811168546536e-1778j 7.2e-147, 3.78e-377j
0.016021792486641351753 + 3.7742693810683553257e-1778j 1.11e-139, 5.85e-370j
0.029857186566145197158 + 4.2359123261658956179e-1778j 3.43e-146, 1.8e-376j
0.028638764165911251486 + 3.8217502408158123843e-1778j 1.17e-144, 6.16e-375j
0.02484620313971285517 - 5.6771156349813666467e-1779j 1.68e-143, 8.82e-374j
0.030412675657503570807 - 1.569054766101320502e-1778j 8.24e-149, 4.33e-379j
0.027225993612436188154 + 8.0848986258265783684e-1779j 1.83e-145, 9.62e-376j
0.032375089547853038741 + 3.471420812899291498e-1778j 1.77e-152, 9.27e-383j
0.03092319503884305769 + 4.8198271082249732146e-1778j 1.77e-150, 9.28e-381j
0.012661440408129241529 - 1.5044369835017565829e-1779j 1.83e-143, 9.61e-374j
0.029271227882453244518 - 1.1874229443823160749e-1778j 1.23e-147, 6.43e-378j
0.03092319503884305769 + 1.8174182268304689074e-1778j 3.7e-150, 1.94e-380j
0.032558246020238127481 - 7.569572694034840354e-1778j 5.89e-155, 3.09e-385j
0.031361552591500827665 - 5.2005015386459620546e-1778j 1.47e-152, 7.7e-383j
0.029271227882453244518 - 4.0101901151644262572e-1778j 4.81e-150, 2.53e-380j
0.031661548665296909227 + 1.5640079758099224562e-1777j 4.01e-157, 2.11e-387j
0.031738508639053386442 + 2.4841464846768092611e-1778j 1.24e-153, 6.53e-384j
0.0055299303205379025981 - 5.7290034001457931934e-1780j 3.34e-147, 1.75e-377j
0.018172773393816053654 + 3.4863322449701322493e-1778j 7.39e-149, 3.88e-379j
0.031738508639053386442 + 5.6527254481478243616e-1778j 5.88e-154, 3.09e-384j
0.032336313198372286172 + 1.3217369941399230734e-1777j 2.37e-157, 1.25e-387j
0.031361552591500827665 - 2.1185455460946303659e-1778j 2.45e-153, 1.28e-383j
0.032083790685366557809 - 2.9277769259804776203e-1778j 7.96e-155, 4.18e-385j
0.0079544220081816952949 - 7.5858172488735187832e-1778j 3.43e-150, 1.8e-380j
0.032083790685366557809 - 6.1841026201824779016e-1778j 2.41e-155, 1.26e-385j
0.032375089547853038741 + 6.8144042264753415175e-1778j 4.66e-156, 2.45e-386j
0.03183473459433664845 - 1.4786271888240458653e-1777j 1.3e-159, 6.82e-390j
0.032558246020238127481 - 4.1443596562520845205e-1778j 3.57e-158, 1.87e-388j
0.032336313198372286172 + 1.0205569770315722002e-1777j 7.07e-160, 3.72e-390j
0.032591162043710204139 - 1.1896105027819757587e-1777j 1.02e-159, 5.33e-390j
0.032650695719921473112 + 8.4569711380170999887e-1778j 4.13e-159, 2.17e-389j
0.032082349744490467679 + 1.5263547570030415298e-1777j 3.76e-160, 1.97e-390j
0.032723073515840040673 - 5.9666080455321902454e-1778j 6.06e-161, 3.18e-391j
0.032741168361119022658 + 7.1643157359934742332e-1778j 6.01e-161, 3.16e-391j
0.03183473459433664845 - 1.575907645818481773e-1777j 5.78e-161, 3.04e-391j
0.032723073515840040673 - 9.468000232170720974e-1778j 2.89e-161, 1.52e-391j
0.032741168361119022658 + 1.0612771801525096258e-1777j 2.37e-161, 1.24e-391j
0.032185325811924593395 - 1.1858172640139446436e-1777j 7.43e-163, 3.91e-393j
0.032650695719921473112 + 4.9703232858156147394e-1778j 3.71e-163, 2.07e-393j
0.032082349744490467679 + 1.3428890987640673415e-1777j 8.06e-163, 4.19e-393j

Download

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