Rule: Q[1, 2, 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 + 1.5633101277616895057e-1775j 5.92e-478, 5.92e-478j
-0.95567355593646219643 + 8.6335900684667942803e-1787j 1.52e-478, 1.52e-478j
-0.94475755206460393205 - 3.2135429279274664414e-1812j 8.4e-479, 8.4e-479j
-0.99975980509124899355 - 2.107701717847832344e-1830j 1.65e-478, 1.65e-478j
0.91944075690578682193 - 6.4094145073346635897e-1857j 1.91e-479, 1.91e-479j
0.93267685135591888185 + 7.4324717727444222312e-1856j 4.2e-479, 4.2e-479j
0.9654205626957687594 - 1.884992589494149452e-1854j 2.49e-478, 2.49e-478j
-0.91944075690578682193 + 9.9883370569650782054e-1858j 2.07e-479, 2.07e-479j
-0.90507391275050144198 + 4.9894506936895159546e-1865j 8.37e-480, 8.37e-480j
-0.99855341163032368546 - 1.6236996282019332285e-1876j 4.56e-478, 4.56e-478j
0.87303398795525178662 - 4.6394818533224053789e-1908j 1.11e-480, 1.11e-480j
-0.99238613019668857555 - 6.0999299375438538221e-1902j 6.78e-478, 6.78e-478j
-0.88960044761246962429 + 6.2115178346559969878e-1916j 3.14e-480, 3.14e-480j
-0.87303398795525178662 + 9.7127188956860959226e-1917j 1.14e-480, 1.14e-480j
0.99855341163032368546 + 1.5751610055231276488e-1913j 4.43e-478, 4.43e-478j
0.81698927374426241067 + 4.7496029055147765687e-1920j 2.94e-482, 2.94e-482j
0.90507391275050144198 + 1.5139720802629603215e-1916j 8.14e-480, 8.14e-480j
-0.97397883967229151808 + 2.991343636359778542e-1913j 3.85e-478, 3.85e-478j
-0.77459666924148337704 - 1.5035049216382622019e-1928j 1.78e-483, 1.78e-483j
-0.9654205626957687594 + 1.5254847246488295422e-1921j 2.62e-478, 2.62e-478j
0.09583383970537608263 - 2.4683704721619936185e-1960j 3.93e-507, 3.93e-507j
0.79628103698434134174 - 3.3064806592545178981e-1934j 7.6e-483, 7.6e-483j
0.99975980509124899355 + 3.0847610947982508031e-1927j 1.61e-478, 1.61e-478j
-0.85539075792856720776 - 1.4402666851698305704e-1936j 3.75e-481, 3.75e-481j
-0.79628103698434134174 - 1.0436368331735804431e-1938j 7.33e-483, 7.33e-483j
0.94475755206460393205 + 2.2545868719117163504e-1932j 9.01e-479, 9.01e-479j
0.72843646453669396918 - 1.7566145962049804021e-1939j 6.98e-485, 6.98e-485j
0.83669935424244029912 + 4.1272845683312575511e-1936j 1.05e-481, 1.05e-481j
0.99238613019668857555 + 2.0766584283128052827e-1931j 6.48e-478, 6.48e-478j
-0.65267393903817924216 - 3.1788170877912937651e-1946j 3.04e-487, 3.04e-487j
0.70402249979687315825 - 2.1782445837421484065e-1944j 1.2e-485, 1.2e-485j
0.77459666924148337704 + 7.739136713475185243e-1942j 1.77e-483, 1.77e-483j
0.67875502647205929737 + 6.8201274415272060241e-1945j 2.08e-486, 2.08e-486j
-0.99610148483727753794 + 3.3491168752455515422e-1933j 6.21e-478, 6.21e-478j
-0.67875502647205929737 - 5.6349556830434332162e-1969j 1.98e-486, 1.98e-486j
-0.7519699587372642935 + 1.6319093780525079897e-1966j 3.65e-484, 3.65e-484j
-0.98132380834652692847 - 2.3660327080269572493e-1958j 5.43e-478, 5.43e-478j
0.59823082513832843119 - 7.4753346741556274939e-1987j 5.44e-489, 5.44e-489j
0.7519699587372642935 + 1.9544191532981938371e-1983j 3.57e-484, 3.57e-484j
0.98132380834652692847 + 5.1976372535456318312e-1974j 5.1e-478, 5.1e-478j
-0.56993154558925436082 - 2.0108072372149839527e-1992j 6.78e-490, 6.78e-490j
-0.83669935424244029912 - 6.0479999522321273144e-1984j 1.08e-481, 1.08e-481j
0.62582200827490670876 - 1.2208350696889362209e-1990j 4.14e-488, 4.14e-488j
0.65267393903817924216 + 1.0971655644148753701e-1989j 3.09e-487, 3.09e-487j
-0.81698927374426241067 - 1.0480452606212730108e-1984j 2.93e-482, 2.93e-482j
-0.59823082513832843119 + 4.9117997066421910994e-1991j 5.59e-489, 5.59e-489j
-0.72843646453669396918 - 6.5785508432186214592e-1987j 6.79e-485, 6.79e-485j
0.98745485656843704766 + 7.6898387775184205384e-1978j 6.66e-478, 6.66e-478j
0.481267232117080114 - 8.6114148521003615646e-1996j 6.74e-493, 6.74e-493j
0.95567355593646219643 + 3.4476490034408038075e-1979j 1.66e-478, 1.66e-478j
-0.16054504865663444924 - 1.7408930580334369371e-2007j 1.39e-504, 1.39e-504j
-0.5114047255945475266 + 1.9461593165693613908e-1994j 6.86e-492, 6.86e-492j
0.88960044761246962429 - 2.5890612434222522365e-1982j 3.13e-480, 3.13e-480j
0.5114047255945475266 + 2.7218837220391832994e-1995j 7.09e-492, 7.09e-492j
0.56993154558925436082 + 3.7803125804047327478e-1993j 6.12e-490, 6.12e-490j
-0.70402249979687315825 + 3.9007214751354583488e-1988j 1.2e-485, 1.2e-485j
-0.481267232117080114 - 1.0491608769648696703e-1995j 6.69e-493, 6.69e-493j
-0.62582200827490670876 - 8.1843808548621370216e-1991j 4.3e-488, 4.3e-488j
-0.32374172538777687861 + 1.5578270368924787239e-2001j 1.73e-498, 1.73e-498j
0.45059415417275136725 + 9.2500860149579405358e-1998j 5.77e-494, 5.77e-494j
-0.93267685135591888185 - 2.160320337174064774e-1980j 4.35e-479, 4.35e-479j
-0.54097189243552464403 + 1.5112168356438079668e-2009j 6.72e-491, 6.72e-491j
-0.45059415417275136725 - 2.3088610584258004604e-2013j 5.64e-494, 5.64e-494j
0.2912651101355163025 - 2.4881526087599532839e-2019j 1.13e-499, 1.13e-499j
0.41944544253465535545 + 4.2698570375197099851e-2014j 4.84e-495, 4.84e-495j
0.54097189243552464403 - 6.3670499406663439362e-2010j 7.2e-491, 7.2e-491j
2.1573637858766415569e-2042 - 4.1660546306340608943e-2042j 4.27e-2040, 4.27e-2040j
-0.38789183847016823359 + 1.4742402002989485917e-2015j 3.49e-496, 3.49e-496j
-0.98745485656843704766 + 1.5794107878958027585e-1998j 6.18e-478, 6.18e-478j
0.85539075792856720776 + 2.7898253392281235474e-2008j 3.59e-481, 3.59e-481j
0.38789183847016823359 - 2.6270252474358580548e-2026j 3.83e-496, 3.83e-496j
0.1280820709211599132 - 3.179703802825424725e-2035j 7.79e-506, 7.79e-506j
-0.41944544253465535545 - 5.0714580204637409162e-2025j 4.71e-495, 4.71e-495j
-0.35597824201184734416 + 1.8164906380382892231e-2027j 2.55e-497, 2.55e-497j
0.97397883967229151808 + 1.8509170666230950052e-2034j 3.76e-478, 3.76e-478j
0.35597824201184734416 + 3.9520068326143586281e-2058j 2.48e-497, 2.48e-497j
0.32374172538777687861 - 6.7286327971544857933e-2060j 1.78e-498, 1.78e-498j
-0.031723719975947408734 - 1.032475113633250379e-2070j 1.08e-509, 1.08e-509j
-0.2912651101355163025 + 4.2542560555138004421e-2062j 1.2e-499, 1.2e-499j
-0.1280820709211599132 + 2.3018498374066611058e-2067j 7.78e-506, 7.78e-506j
0.16054504865663444924 - 7.3412927673502570181e-2066j 1.56e-504, 1.56e-504j
0.25865636042642628176 + 2.7227020790096048229e-2062j 7e-501, 7e-501j
0.063692883543839287524 - 1.6154010270400125015e-2069j 1.94e-508, 1.94e-508j
-0.22596893272086916499 - 3.0181195461548773648e-2065j 4.18e-502, 4.18e-502j
-0.19322693907740572908 + 4.8647327881699629236e-2065j 2.52e-503, 2.52e-503j
0.031723719975947408734 + 1.3048358246243972471e-2070j 1.08e-509, 1.08e-509j
0.22596893272086916499 - 2.0594730645940051377e-2063j 4.35e-502, 4.35e-502j
0.19322693907740572908 - 9.306907480801041509e-2065j 2.54e-503, 2.54e-503j
-0.09583383970537608263 - 3.704709179752729222e-2068j 4.33e-507, 4.33e-507j
-0.25865636042642628176 + 1.0282619053550715383e-2063j 7.15e-501, 7.15e-501j
-0.063692883543839287524 + 2.4845777095304351365e-2070j 2.2e-508, 2.2e-508j

Weights

Weight (ball center) Error (ball radius)
0.0030912480181254993765 - 2.8525564498564057476e-1776j 1.93e-122, 1.08e-352j
0.010333145885033823531 - 8.959564003823188487e-1778j 9.35e-123, 5.22e-353j
0.011498632294562236256 + 1.0259561428966018965e-1777j 8.74e-123, 4.88e-353j
0.00064691450912965161761 - 5.0316305318990561268e-1779j 5.16e-124, 2.88e-354j
0.01380626243884675078 + 3.278507359886825959e-1776j 2.85e-124, 1.59e-354j
0.012661440408129241529 - 3.5391286013840375557e-1776j 2.71e-124, 1.51e-354j
0.0091573665583527762929 + 4.9454320415604220237e-1776j 6.16e-125, 3.44e-355j
0.01380626243884675078 + 1.3120710950296605626e-1777j 6.58e-125, 3.67e-355j
0.014923180352346962197 - 1.4734828900854206563e-1777j 2.9e-125, 1.62e-355j
0.0018116371862682237368 + 1.4820033628260927451e-1778j 8.08e-126, 4.51e-356j
0.017108647322502197037 - 2.7943114319768668187e-1776j 6.18e-129, 3.45e-359j
0.0043300385953446060712 + 3.3979043289214990456e-1778j 2.47e-126, 1.38e-356j
0.016021792486641351753 + 1.648965962103588535e-1777j 1.66e-126, 9.27e-357j
0.017108647322502197037 - 1.8398287251425487397e-1777j 3.15e-127, 1.76e-357j
0.0018116371862682237368 - 1.2056172606477087606e-1775j 2.34e-130, 1.31e-360j
0.020212102090768345322 + 2.571754096227373014e-1776j 1.07e-130, 5.97e-361j
0.014923180352346962197 - 3.0774735123800708784e-1776j 4.56e-129, 2.55e-359j
0.0079544220081816952949 - 6.5895157737582653972e-1778j 5.13e-128, 2.87e-358j
0.022161926559098145849 + 3.1493100754534108009e-1777j 1.2e-131, 6.68e-362j
0.0091573665583527762929 + 7.7353408170891704196e-1778j 7.14e-128, 3.99e-358j
0.032185325811924593395 - 1.1126051045646880433e-1775j 1.99e-135, 1.11e-365j
0.021200943011649996879 - 2.5363278171825053311e-1776j 5.04e-133, 2.81e-363j
0.00064691450912965161761 + 2.7450949901443133269e-1776j 3.68e-132, 2.06e-362j
0.018172773393816053654 + 2.0496140448792762152e-1777j 6.43e-130, 3.59e-360j
0.021200943011649996879 - 2.8275781209214845045e-1777j 1.82e-131, 1.02e-361j
0.011498632294562236256 + 3.8782308749663049604e-1776j 2.96e-132, 1.65e-362j
0.023977527119731576717 + 2.5286502645997570241e-1776j 1.85e-135, 1.04e-365j
0.01920488709784730393 - 2.6242123489278079231e-1776j 1.6e-133, 8.91e-364j
0.0043300385953446060712 + 1.8185856605582433581e-1775j 2.77e-133, 1.55e-363j
0.026472843354401938109 - 5.541862883473677088e-1777j 9.91e-138, 5.53e-368j
0.02484620313971285517 - 2.5562659100707022454e-1776j 2.71e-136, 1.51e-366j
0.022161926559098145849 + 2.5175423489110971051e-1776j 5.15e-135, 2.88e-365j
0.025681864094918798963 + 2.5996422471637007315e-1776j 3.19e-137, 1.78e-367j
0.0030912480181254993765 - 2.4257464771087982606e-1778j 2.4e-133, 1.34e-363j
0.025681864094918798963 + 4.9257190365783453123e-1777j 8.33e-139, 4.65e-369j
0.023085305252914850361 - 3.5130039284736861164e-1777j 1.9e-137, 1.06e-367j
0.0067355490137734319443 + 5.4984845808023831133e-1778j 5.31e-134, 2.97e-364j
0.027951589757696146189 - 2.8369668383540908583e-1776j 1.33e-140, 7.43e-371j
0.023085305252914850361 - 2.5154399134336416228e-1776j 8.83e-137, 4.93e-367j
0.0067355490137734319443 + 7.3576130095881395267e-1776j 2.39e-137, 1.34e-367j
0.028638764165911251486 + 8.0452992441629082184e-1777j 1.63e-142, 9.13e-373j
0.01920488709784730393 - 2.2823267569526457071e-1777j 3.99e-137, 2.23e-367j
0.027225993612436188154 + 2.7395126308599297009e-1776j 1.69e-140, 9.47e-371j
0.026472843354401938109 - 2.6606157359165060151e-1776j 9.06e-140, 5.06e-370j
0.020212102090768345322 + 2.5405929645391366996e-1777j 1.14e-137, 6.37e-368j
0.027951589757696146189 - 7.0797402801994523951e-1777j 4.85e-143, 2.71e-373j
0.023977527119731576717 + 3.9247105268237907218e-1777j 4.97e-140, 2.78e-370j
0.0055299303205379025981 - 1.0166579254296713824e-1775j 9.45e-139, 5.28e-369j
0.030412675657503570807 - 3.4830328697122276494e-1776j 3.51e-146, 1.96e-376j
0.010333145885033823531 - 4.3254883135647772554e-1776j 3.55e-139, 1.98e-369j
0.032591162043710204139 - 6.6477996746925764574e-1776j 2.34e-150, 1.31e-380j
0.029857186566145197158 + 1.0535519966828779837e-1776j 7.2e-147, 4.02e-377j
0.016021792486641351753 + 2.9196507214956068317e-1776j 1.11e-139, 6.22e-370j
0.029857186566145197158 + 3.2767625278091377324e-1776j 3.43e-146, 1.92e-376j
0.028638764165911251486 + 2.956380353456889483e-1776j 1.17e-144, 6.55e-375j
0.02484620313971285517 - 4.3916300177509749648e-1777j 1.68e-143, 9.38e-374j
0.030412675657503570807 - 1.2137691857654028095e-1776j 8.24e-149, 4.6e-379j
0.027225993612436188154 + 6.2542117878825328005e-1777j 1.83e-145, 1.02e-375j
0.032375089547853038741 + 2.6853770222214382713e-1776j 1.77e-152, 9.86e-383j
0.03092319503884305769 + 3.7284598080032939071e-1776j 1.77e-150, 9.87e-381j
0.012661440408129241529 - 1.1637829504721524165e-1777j 1.83e-143, 1.02e-373j
0.029271227882453244518 - 9.1855135404438907062e-1777j 1.23e-147, 6.84e-378j
0.03092319503884305769 + 1.4058949944163436033e-1776j 3.7e-150, 2.06e-380j
0.032558246020238127481 - 5.8555725999009686349e-1776j 5.89e-155, 3.29e-385j
0.031361552591500827665 - 4.0229370333499688938e-1776j 1.47e-152, 8.19e-383j
0.029271227882453244518 - 3.1021512456120915278e-1776j 4.81e-150, 2.69e-380j
0.031661548665296909227 + 1.2098651558496470509e-1775j 4.01e-157, 2.24e-387j
0.031738508639053386442 + 1.9216540570166690984e-1776j 1.24e-153, 6.94e-384j
0.0055299303205379025981 - 4.4317688270605484895e-1778j 3.34e-147, 1.87e-377j
0.018172773393816053654 + 2.6969120183536814244e-1776j 7.39e-149, 4.13e-379j
0.031738508639053386442 + 4.37276258364272072e-1776j 5.88e-154, 3.28e-384j
0.032336313198372286172 + 1.0224522888097974183e-1775j 2.37e-157, 1.32e-387j
0.031361552591500827665 - 1.6388371894988602877e-1776j 2.45e-153, 1.37e-383j
0.032083790685366557809 - 2.264831982745267575e-1776j 7.96e-155, 4.45e-385j
0.0079544220081816952949 - 5.8681388549976796437e-1776j 3.43e-150, 1.92e-380j
0.032083790685366557809 - 4.7838184957956474287e-1776j 2.41e-155, 1.35e-385j
0.032375089547853038741 + 5.2713990983093703796e-1776j 4.66e-156, 2.6e-386j
0.03183473459433664845 - 1.1438173857562525906e-1775j 1.3e-159, 7.26e-390j
0.032558246020238127481 - 3.2059403896335301393e-1776j 3.57e-158, 1.99e-388j
0.032336313198372286172 + 7.8946932815485486446e-1776j 7.07e-160, 3.95e-390j
0.032591162043710204139 - 9.202435785511041662e-1776j 1.02e-159, 5.67e-390j
0.032650695719921473112 + 6.542034864398599045e-1776j 4.13e-159, 2.3e-389j
0.032082349744490467679 + 1.180737863638234858e-1775j 3.76e-160, 2.1e-390j
0.032723073515840040673 - 4.6155718400848020778e-1776j 6.06e-161, 3.38e-391j
0.032741168361119022658 + 5.5420791367423008259e-1776j 6.01e-161, 3.36e-391j
0.03183473459433664845 - 1.2190703494223659238e-1775j 5.78e-161, 3.23e-391j
0.032723073515840040673 - 7.3241337356934690882e-1776j 2.89e-161, 1.62e-391j
0.032741168361119022658 + 8.2096913892105155667e-1776j 2.37e-161, 1.32e-391j
0.032185325811924593395 - 9.1730925351792098348e-1776j 7.43e-163, 4.16e-393j
0.032650695719921473112 + 3.8448787012232486427e-1776j 3.71e-163, 2.2e-393j
0.032082349744490467679 + 1.0388148614238374346e-1775j 8.06e-163, 4.45e-393j

Download

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