Rule: Q[2, 40, 44]

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)
0.99834630861965744153 + 8.2949910646854730493e-1722j 2.82e-480, 2.82e-480j
-0.93694626337399374781 - 2.2697401969991478772e-1738j 4.42e-481, 4.42e-481j
-0.92318502931499714444 + 1.2897169745250677997e-1738j 2.1e-481, 2.1e-481j
-0.99554382450969612291 - 2.3997024515993728415e-1736j 3.95e-480, 3.95e-480j
0.92318502931499714444 + 3.767525986092231343e-1736j 2.08e-481, 2.08e-481j
0.94938933590947429269 - 1.3387626549117077877e-1749j 8.53e-481, 8.53e-481j
0.98566309506018107986 + 1.0615889173953260629e-1772j 3.83e-480, 3.83e-480j
-0.90811820583256431509 + 3.2832161736632716229e-1802j 8.98e-482, 8.98e-482j
-0.99972534130981256278 - 5.0256806699814500169e-1800j 1.02e-480, 1.02e-480j
0.96050734061037693088 - 2.0670271206070231133e-1797j 1.46e-480, 1.46e-480j
0.87419643895694575533 - 6.4076851866738657359e-1816j 1.24e-482, 1.24e-482j
0.99554382450969612291 - 1.6803242744935220526e-1827j 3.75e-480, 3.75e-480j
-0.87419643895694575533 - 1.7366651798518981209e-1856j 1.32e-482, 1.32e-482j
-0.85539210975222010253 - 7.6756129267695442175e-1857j 4.42e-483, 4.42e-483j
0.99972534130981256278 + 2.0197257624220649037e-1851j 1.01e-480, 1.01e-480j
0.85539210975222010253 + 3.3120237202628095946e-1874j 4.11e-483, 4.11e-483j
0.90811820583256431509 - 2.2201423185658174031e-1871j 8.92e-482, 8.92e-482j
0.97866045836413075624 - 2.8731489954531548384e-1887j 3.04e-480, 3.04e-480j
-0.74407492042868702323 + 6.2766938750209460336e-1912j 4.1e-486, 4.1e-486j
-0.94938933590947429269 + 1.8364735520157773204e-1904j 8.5e-481, 8.5e-481j
0.8917779255885798882 - 9.0927820697055209845e-1906j 3.53e-482, 3.53e-482j
0.7185935550082764427 + 6.822704692839278422e-1922j 8.25e-487, 8.25e-487j
-0.97866045836413075624 + 4.6602528172031402483e-1912j 3.09e-480, 3.09e-480j
-0.8353859830515983726 + 1.8945770520847455493e-1918j 1.25e-483, 1.25e-483j
-0.76853342423121491934 - 1.3546439396035423623e-1920j 1.96e-485, 1.96e-485j
0.99129759995849076786 - 2.7541544944967573106e-1913j 4.22e-480, 4.22e-480j
0.74407492042868702323 + 2.0475917363923904269e-1934j 4.34e-486, 4.34e-486j
0.8353859830515983726 - 2.0355829145789327515e-1930j 1.23e-483, 1.23e-483j
-0.79192310657424410098 - 6.0517927561686300779e-1932j 8.68e-485, 8.68e-485j
-0.66474533231405273004 - 6.8058714641200896114e-1935j 2.25e-488, 2.25e-488j
-0.8917779255885798882 + 5.802308255985549626e-1928j 3.51e-482, 3.51e-482j
0.81421623172909643055 - 1.0310622954707895437e-1931j 3.62e-484, 3.62e-484j
0.69213916679901767953 - 4.1129977094193227457e-1934j 1.51e-487, 1.51e-487j
-0.96050734061037693088 + 2.5593487885547832826e-1925j 1.44e-480, 1.44e-480j
-0.69213916679901767953 - 9.9221529874406867337e-1935j 1.38e-487, 1.38e-487j
-0.7185935550082764427 + 6.6303725087078772494e-1934j 7.8e-487, 7.8e-487j
0.93694626337399374781 + 9.7491370259435719434e-1927j 4.57e-481, 4.57e-481j
0.60729173964450804875 - 1.3253555459403882956e-1944j 4.41e-490, 4.41e-490j
0.79192310657424410098 + 1.239433988222793792e-1938j 9.08e-485, 9.08e-485j
-0.98566309506018107986 - 1.2978684838289589015e-1931j 3.93e-480, 3.93e-480j
-0.54665858125247613584 + 4.0935475308658267644e-1949j 5.8e-492, 5.8e-492j
0.10538269695809222176 - 4.2759178943549617904e-1963j 4.56e-507, 4.56e-507j
0.76853342423121491934 - 1.5677594698765117598e-1941j 2e-485, 2e-485j
0.57735026918962576451 - 1.4475671140842403376e-1946j 5.25e-491, 5.25e-491j
-0.99129759995849076786 - 8.113115553519314436e-1933j 4.41e-480, 4.41e-480j
-0.57735026918962576451 - 9.728522631342512288e-1948j 5.25e-491, 5.25e-491j
-0.63644433637359227139 - 8.244226364576144013e-1946j 3.1e-489, 3.1e-489j
-0.99834630861965744153 - 3.0541508823735468693e-1934j 2.76e-480, 2.76e-480j
0.48319124771609154445 - 6.4331020267636697285e-1953j 5.49e-494, 5.49e-494j
0.14069036445817512175 + 1.5203644695534448792e-1964j 8.74e-506, 8.74e-506j
-0.60729173964450804875 + 3.8830934068947414956e-1949j 4.36e-490, 4.36e-490j
-0.51524935790704568498 - 7.1400491433096677283e-1951j 5.88e-493, 5.88e-493j
-0.9702745993611777786 + 5.8386087744713269257e-1941j 2.22e-480, 2.22e-480j
0.51524935790704568498 + 3.2422905780360658765e-1956j 6.09e-493, 6.09e-493j
0.54665858125247613584 - 1.3025127009566588772e-1954j 5.81e-492, 5.81e-492j
0.0051321412927769956822 + 4.7863700440095964408e-1973j 5.08e-511, 5.08e-511j
-0.48319124771609154445 + 7.2867816277104772018e-1956j 5.61e-494, 5.61e-494j
-0.81421623172909643055 - 1.6499556354973336886e-1950j 3.44e-484, 3.44e-484j
0.66474533231405273004 + 3.7869284684129431147e-1956j 2.16e-488, 2.16e-488j
0.45056824768285576066 + 9.8847738274760457036e-1963j 4.86e-495, 4.86e-495j
0.9702745993611777786 - 1.1527303116450593558e-1970j 2.39e-480, 2.39e-480j
-0.41742067199362992495 - 5.620764618357859932e-1990j 4.14e-496, 4.14e-496j
-0.38376918304185822055 + 1.7244263107159198985e-1992j 2.96e-497, 2.96e-497j
-0.069885022268966105213 - 9.7575756542603297602e-2002j 2.09e-508, 2.09e-508j
0.41742067199362992495 + 6.3868699864219987733e-1990j 4.11e-496, 4.11e-496j
0.38376918304185822055 - 4.3035982066969167818e-1990j 3e-497, 3e-497j
-0.035111246300125718722 + 6.2347054656174758952e-2003j 1.13e-509, 1.13e-509j
-0.34970233849925494868 - 4.9833764215084339178e-1992j 2.07e-498, 2.07e-498j
-0.45056824768285576066 + 5.2182671525277369438e-1989j 4.87e-495, 4.87e-495j
0.24582280961046555355 - 6.9575425680913755641e-1995j 5.67e-502, 5.67e-502j
0.34970233849925494868 + 1.2688883579459893089e-1991j 2.14e-498, 2.14e-498j
0.63644433637359227139 - 1.0646872060376313602e-1993j 3.15e-489, 3.15e-489j
-0.24582280961046555355 + 3.0501844249612558582e-2005j 5.53e-502, 5.53e-502j
-0.28073526299799082661 - 1.9540489802662816489e-2004j 9.38e-501, 9.38e-501j
-0.0051321412927769956822 + 3.0471502895704814885e-2014j 5.08e-511, 5.08e-511j
0.31534817041554394372 - 2.9477487572919993515e-2003j 1.45e-499, 1.45e-499j
0.21072820550076014014 + 7.9056255476648993523e-2007j 3.15e-503, 3.15e-503j
-0.10538269695809222176 + 1.294737572167793487e-2010j 4.14e-507, 4.14e-507j
-0.31534817041554394372 + 5.576548934632448711e-2003j 1.6e-499, 1.6e-499j
-0.14069036445817512175 + 1.6458123605560052303e-2010j 9.34e-506, 9.34e-506j
0.17569618181294266073 - 4.6267075426897127822e-2010j 1.83e-504, 1.83e-504j
0.28073526299799082661 + 1.0871855176858899793e-2004j 9.64e-501, 9.64e-501j
0.069885022268966105213 + 6.7219172002224363138e-2012j 2.11e-508, 2.11e-508j
-0.17569618181294266073 + 1.1365808884612917562e-2008j 1.78e-504, 1.78e-504j
-0.21072820550076014014 - 1.0245915798003684263e-2006j 3.11e-503, 3.11e-503j
0.035111246300125718722 - 3.0143769929603131738e-2013j 1.13e-509, 1.13e-509j

Weights

Weight (ball center) Error (ball radius)
0.0020707099781681290016 - 2.503194692887047425e-1722j 4.52e-132, 1.54e-363j
0.013102708798487112758 + 6.4020354718935631918e-1724j 2.19e-132, 7.45e-364j
0.014417968395364576818 - 7.3847022916854089231e-1724j 2.01e-132, 6.86e-364j
0.0035331446443693992253 - 1.412443989383112871e-1724j 1.58e-133, 5.39e-365j
0.014417968395364576818 + 1.8879317922826374928e-1722j 6.4e-134, 2.18e-365j
0.011782865777918646796 + 2.1910718652737443799e-1722j 4.23e-134, 1.44e-365j
0.0063174171319568077661 + 4.082971919956765351e-1722j 1.12e-134, 3.8e-366j
0.015709861623071644553 + 8.4819455880236881206e-1724j 1.93e-134, 6.57e-366j
0.00073964366346800885348 - 2.9161528785208743775e-1725j 1.88e-135, 6.4e-367j
0.010448579499322199539 - 2.4289296622967075613e-1722j 1.23e-134, 4.18e-366j
0.018195301003874741262 - 1.6786033925317087627e-1722j 3.23e-136, 1.1e-367j
0.0035331446443693992253 + 1.0049149338680243115e-1721j 4.82e-135, 1.64e-366j
0.018195301003874741262 + 1.1129166085415766257e-1723j 4.71e-137, 1.6e-368j
0.019410186497610372652 - 1.2723900323152787139e-1723j 1.98e-137, 6.75e-369j
0.00073964366346800885348 - 4.2251952655203245856e-1722j 3.64e-135, 1.24e-366j
0.019410186497610372652 + 1.6499538346847641406e-1722j 8.24e-138, 2.81e-369j
0.015709861623071644553 - 1.7921831200681723879e-1722j 6.36e-137, 2.16e-368j
0.007691558203709417738 - 3.2789875980970675637e-1722j 3.81e-136, 1.3e-367j
0.024978903313099832692 + 2.5422856889323007608e-1723j 5.25e-142, 1.79e-373j
0.011782865777918646796 - 5.5073308236448072475e-1724j 2.64e-140, 9e-372j
0.01696500435639955501 + 1.7245851728470188711e-1722j 2.32e-137, 7.9e-369j
0.025975037498127117608 + 1.8081646195973471375e-1722j 3.28e-142, 1.11e-373j
0.007691558203709417738 + 3.2650196207659121705e-1724j 1.68e-141, 5.7e-373j
0.020595351527149154875 + 1.455487920103323373e-1723j 5.59e-141, 1.9e-372j
0.0239301173946149638 - 2.2030014409194535739e-1723j 2.68e-142, 9.11e-374j
0.0049483333727581500311 - 5.6117968459653650363e-1722j 5.85e-137, 1.99e-368j
0.024978903313099832692 - 1.7421278053409582076e-1722j 5.57e-143, 1.9e-374j
0.020595351527149154875 - 1.6378067421791711629e-1722j 5.27e-141, 1.79e-372j
0.022845037720093301629 + 1.9150188202377763948e-1723j 9.65e-143, 3.28e-374j
0.027854404317256892961 - 4.0057037082865894396e-1723j 7.47e-146, 2.54e-377j
0.01696500435639955501 - 9.7235012357098643732e-1724j 7.44e-142, 2.53e-373j
0.021737075934579191162 + 1.6419372586368319053e-1722j 1.26e-142, 4.3e-374j
0.026928475662831030452 - 1.892674759345460218e-1722j 1.43e-145, 4.88e-377j
0.010448579499322199539 + 4.6919376454988828351e-1724j 2.08e-143, 7.07e-375j
0.026928475662831030452 + 3.4283082397482043299e-1723j 1.44e-145, 4.92e-377j
0.025975037498127117608 - 2.9461662695989428991e-1723j 6.89e-145, 2.34e-376j
0.013102708798487112758 - 2.0178831538026447227e-1722j 1.78e-141, 6.06e-373j
0.029556006521237097583 + 2.283567433465479008e-1722j 2.33e-149, 7.94e-381j
0.022845037720093301629 - 1.6608596269318855397e-1722j 1.33e-144, 4.54e-376j
0.0063174171319568077661 - 2.6101290005047091664e-1724j 1.01e-146, 3.44e-378j
0.031058075357855082814 - 7.9151021799364258623e-1723j 6.44e-152, 2.19e-383j
0.035492445637309036549 - 2.762775247553622298e-1721j 3.6e-155, 1.23e-386j
0.0239301173946149638 + 1.6937425452711084825e-1722j 1.5e-145, 5.09e-377j
0.030320863797747869394 - 2.4758916730729608822e-1722j 1.1e-150, 3.75e-382j
0.0049483333727581500311 + 1.9880904750030002188e-1724j 2.56e-147, 8.71e-379j
0.030320863797747869394 + 6.615109806678047384e-1723j 3.17e-152, 1.08e-383j
0.02873797955860862488 + 4.7049471089401440004e-1723j 4.12e-151, 1.4e-382j
0.0020707099781681290016 + 8.6024862405569192179e-1725j 6.76e-148, 2.3e-379j
0.032352277418506079383 + 3.337076686178081491e-1722j 6.66e-156, 2.27e-387j
0.035110814170624365496 + 2.0219530341683599761e-1721j 4.44e-158, 1.51e-389j
0.029556006521237097583 - 5.5616487127030413945e-1723j 4.44e-152, 1.51e-383j
0.031748530408392702834 + 9.5407469650586658916e-1723j 9.23e-155, 3.14e-386j
0.0090800453400589542752 - 3.9512672718848928912e-1724j 1.34e-148, 4.58e-380j
0.031748530408392702834 - 2.9892205364655946966e-1722j 3.98e-156, 1.35e-387j
0.031058075357855082814 + 2.7073729992885711759e-1722j 1.48e-155, 5.05e-387j
0.018342400164486675856 + 2.9691783454931028698e-1720j 2.88e-161, 9.79e-393j
0.032352277418506079383 - 1.1603566417578985845e-1722j 4.42e-157, 1.5e-388j
0.021737075934579191162 - 1.6679702186944418196e-1723j 1.7e-151, 5.79e-383j
0.027854404317256892961 + 1.996952295249193184e-1722j 3.87e-155, 1.32e-386j
0.032887416353163396554 - 3.7642940556408799801e-1722j 1.47e-158, 5.02e-390j
0.0090800453400589542752 + 2.7709560870795666133e-1722j 1.23e-152, 4.18e-384j
0.033406385587210028572 - 1.7587842933748864791e-1722j 9.43e-160, 3.21e-391j
0.033881421464415393288 + 2.1950958916180797423e-1722j 1.19e-160, 4.05e-392j
0.035343308482981151983 - 3.7344085477171762838e-1721j 2.17e-163, 7.38e-395j
0.033406385587210028572 + 4.2863123463500788058e-1722j 1.39e-160, 4.71e-392j
0.033881421464415393288 - 4.9365261270268676937e-1722j 1.34e-161, 4.56e-393j
0.033927168203673409825 + 8.1623508929615217143e-1721j 9.15e-164, 3.11e-395j
0.034227667472703735405 - 2.7743638096218049167e-1722j 8.9e-163, 3.03e-394j
0.032887416353163396554 + 1.4231327096726251911e-1722j 1.53e-160, 5.22e-392j
0.035043565637461766684 - 9.8870461808835332437e-1722j 6.68e-165, 2.27e-396j
0.034227667472703735405 + 5.7658400485586963276e-1722j 1.71e-163, 5.82e-395j
0.02873797955860862488 - 2.1253278812345313218e-1722j 3.18e-160, 1.08e-391j
0.035043565637461766684 + 5.9800829950577618801e-1722j 4.87e-166, 1.66e-397j
0.034761881983808872626 - 4.572075100106161748e-1722j 1.42e-165, 4.82e-397j
0.018342400164486675856 - 2.938807503361208441e-1720j 1.84e-165, 6.28e-397j
0.034476112941431022977 - 6.8197887611273171285e-1722j 1.56e-165, 5.31e-397j
0.035095666028148548277 + 1.2279830817576857132e-1721j 1.68e-166, 5.73e-398j
0.035492445637309036549 + 2.2352024371899815762e-1721j 2e-167, 6.8e-399j
0.034476112941431022977 + 3.5456516725381656214e-1722j 7.21e-167, 2.46e-398j
0.035110814170624365496 - 1.5224619887645572941e-1721j 5.06e-168, 1.72e-399j
0.034972281155945937564 - 1.5593854923364944281e-1721j 8.85e-168, 3.01e-399j
0.034761881983808872626 + 8.1493408445678400026e-1722j 1.17e-167, 3.98e-399j
0.035343308482981151983 + 4.2965843289916049526e-1721j 2.53e-168, 8.62e-400j
0.034972281155945937564 + 1.0926594935575856628e-1721j 2.16e-169, 7.37e-401j
0.035095666028148548277 - 7.9993556577420553397e-1722j 1.64e-169, 5.61e-401j
0.033927168203673409825 - 8.7574087844396275586e-1721j 1.27e-168, 4.32e-400j

Download

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