Rule: Q[3, 40, 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.

Warning: This rule has negative weights!

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)
-0.99622490528349537804 - 6.3333704161522137803e-1712j 2.1e-479, 2.1e-479j
-0.97052802277417746491 - 1.0885801063628344615e-1715j 7.81e-480, 7.81e-480j
-0.92491477735576524157 - 3.0165380388673925351e-1717j 6.42e-481, 6.42e-481j
-0.98544547375341350032 + 9.7383918307487178855e-1715j 8.18e-480, 8.18e-480j
0.9111298583517110982 - 1.4446363298767133071e-1714j 2.88e-481, 2.88e-481j
0.92491477735576524157 - 1.3860524493725168057e-1728j 6.6e-481, 6.6e-481j
0.9924318645134292499 - 8.3324271743297281464e-1753j 1.85e-479, 1.85e-479j
-0.9111298583517110982 - 8.2998599741158684468e-1792j 2.72e-481, 2.72e-481j
-0.9775397615886434099 + 1.2953440360224467592e-1788j 8.48e-480, 8.48e-480j
0.9775397615886434099 + 8.8059203187928033425e-1788j 8.83e-480, 8.83e-480j
0.89446325430860287476 - 9.3406379414832294714e-1816j 9.99e-482, 9.99e-482j
0.99500543531732884972 - 2.4322232433794597854e-1826j 3.43e-479, 3.43e-479j
-0.93737166642545683026 + 4.3218109797146061983e-1860j 1.18e-480, 1.18e-480j
-0.85912690287453322245 - 6.1143682794601113112e-1863j 1.24e-482, 1.24e-482j
0.99933888183715517208 - 2.7105395111009738487e-1856j 1.15e-480, 1.15e-480j
0.84040142237434780595 + 1.0112863595992748084e-1884j 3.9e-483, 3.9e-483j
0.85912690287453322245 + 1.2521568258734015734e-1883j 1.33e-482, 1.33e-482j
0.99622490528349537804 - 8.4667945871865787541e-1882j 2.06e-479, 2.06e-479j
-0.79671554390300336293 + 2.0760122266050230359e-1901j 2.42e-484, 2.42e-484j
-0.95031557461792315437 + 1.2835592893981862237e-1896j 2.05e-480, 2.05e-480j
0.87663768802434542877 - 3.7012121732817948964e-1900j 3.45e-482, 3.45e-482j
0.77459666924148337704 + 9.6270997298286684729e-1903j 6.1e-485, 6.1e-485j
-0.9924318645134292499 + 5.2205785803719468103e-1896j 1.77e-479, 1.77e-479j
-0.84040142237434780595 - 5.3151442437884289354e-1905j 3.68e-483, 3.68e-483j
-0.77459666924148337704 - 1.6360557856973996166e-1905j 6.29e-485, 6.29e-485j
-0.9620826879393802647 + 3.341561796010700806e-1899j 4.27e-480, 4.27e-480j
0.75139187580841733814 - 6.1889738392243993647e-1909j 1.24e-485, 1.24e-485j
0.95031557461792315437 - 6.2090313016516292363e-1903j 2.27e-480, 2.27e-480j
-0.87663768802434542877 + 4.1271556855199973004e-1915j 3.46e-482, 3.46e-482j
-0.64602510655924630182 - 4.791642631063904939e-1923j 1.01e-488, 1.01e-488j
-0.89446325430860287476 - 1.2754266990848490029e-1914j 9.36e-482, 9.36e-482j
0.79671554390300336293 + 1.5746155527294908032e-1917j 2.51e-484, 2.51e-484j
0.72586431796511602165 - 2.4432887346377689264e-1920j 2.29e-486, 2.29e-486j
-0.99500543531732884972 + 3.5323075759262998307e-1911j 3.51e-479, 3.51e-479j
-0.75139187580841733814 - 2.6057539489483005106e-1920j 1.22e-485, 1.22e-485j
-0.72586431796511602165 + 7.7206878569140935285e-1921j 2.12e-486, 2.12e-486j
-0.36095839141911379512 + 9.4589074158542531782e-1932j 4.93e-498, 4.93e-498j
0.67316760748285544209 - 9.1120105279955588412e-1922j 6.45e-488, 6.45e-488j
0.81909853184898019962 - 9.2164760511048190366e-1918j 9.57e-484, 9.57e-484j
0.98544547375341350032 - 2.3228509836792829502e-1911j 7.88e-480, 7.88e-480j
-0.5867388869148402037 - 2.4653321061844267033e-1935j 1.33e-490, 1.33e-490j
-0.14718617801380025857 - 2.1465373634578167002e-1949j 1.32e-505, 1.32e-505j
0.5867388869148402037 - 8.1768480533065895956e-1935j 1.34e-490, 1.34e-490j
0.61679433055058544069 + 3.3517225540181196961e-1933j 1.21e-489, 1.21e-489j
0.97052802277417746491 + 5.6366362452337590235e-1922j 7.67e-480, 7.67e-480j
-0.61679433055058544069 + 1.6500826307375502279e-1943j 1.18e-489, 1.18e-489j
-0.81909853184898019962 + 1.0196159280181251798e-1936j 9.38e-484, 9.38e-484j
0.9620826879393802647 + 7.8412433062128297941e-1932j 4.35e-480, 4.35e-480j
0.46128184088785847193 - 8.5952355405484898115e-1958j 1.21e-494, 1.21e-494j
-0.11109493391388428188 - 1.0046421441586093413e-1969j 5.71e-507, 5.71e-507j
-0.6993711330167845415 - 1.4062388289929711369e-1950j 3.78e-487, 3.78e-487j
-0.49426799723683859419 + 4.6216996580653094028e-1956j 1.44e-493, 1.44e-493j
0.93737166642545683026 + 6.6559689405607163502e-1947j 1.16e-480, 1.16e-480j
0.52659128058949802663 - 1.7570267334550541258e-1960j 1.77e-492, 1.77e-492j
0.5570523665246407664 + 2.5788034022779852515e-1959j 1.62e-491, 1.62e-491j
-0.99933888183715517208 - 4.7749068477239146602e-1962j 1.17e-480, 1.17e-480j
-0.46128184088785847193 + 3.1950168778747505371e-1976j 1.17e-494, 1.17e-494j
-0.67316760748285544209 - 1.2361386767995653841e-1969j 6.77e-488, 6.77e-488j
0.6993711330167845415 - 4.2774647021742816977e-1970j 3.73e-487, 3.73e-487j
0.39569138851584868174 + 1.1628421274764160152e-1982j 7.74e-497, 7.74e-497j
0.64602510655924630182 + 8.8266485155687658385e-1982j 9.06e-489, 9.06e-489j
-0.5570523665246407664 - 5.1762789847766413086e-1985j 1.55e-491, 1.55e-491j
-0.52659128058949802663 + 4.0925635615109906303e-1989j 1.76e-492, 1.76e-492j
0.073794608601719502957 - 3.5055874001236418942e-2004j 2.4e-508, 2.4e-508j
0.42878412349565830447 - 3.9770414169958991929e-1994j 9.87e-496, 9.87e-496j
0.49426799723683859419 - 1.3524790253201164729e-1990j 1.47e-493, 1.47e-493j
0.036394968383913911559 + 3.2482935645039107021e-2006j 9.73e-510, 9.73e-510j
-0.42878412349565830447 + 2.7590821555284025878e-1991j 9.74e-496, 9.74e-496j
-0.39569138851584868174 - 1.3307985660358824762e-1992j 7.57e-497, 7.57e-497j
0.21977387039115541258 + 3.6334132702352365599e-1999j 5.33e-503, 5.33e-503j
0.36095839141911379512 - 2.3519613079538377379e-1994j 4.5e-498, 4.5e-498j
0.18307356617022244873 - 7.1198616383928129533e-2001j 2.23e-504, 2.23e-504j
-0.32574039046727197686 - 4.8387476062658503972e-1996j 2.94e-499, 2.94e-499j
-0.29116123856529143669 - 2.938471199606668515e-1997j 1.81e-500, 1.81e-500j
-3.6841965829236379199e-2059 - 6.0219112443633603945e-2059j 6e-2057, 6e-2057j
0.32574039046727197686 - 6.3262319520266977154e-1996j 3.01e-499, 3.01e-499j
0.25617854793502086118 - 2.6632840402643100062e-1997j 1.07e-501, 1.07e-501j
-0.036394968383913911559 - 3.1329456779845892777e-2006j 1.09e-509, 1.09e-509j
-0.25617854793502086118 + 2.6407840950503462626e-1998j 1.08e-501, 1.08e-501j
-0.073794608601719502957 + 7.7299947140644815539e-2004j 2.89e-508, 2.89e-508j
0.14718617801380025857 + 2.5339315994160350818e-2001j 1.09e-505, 1.09e-505j
0.29116123856529143669 + 1.4804435232362904723e-1996j 1.88e-500, 1.88e-500j
0.11109493391388428188 - 9.9326260337385404217e-2003j 5.68e-507, 5.68e-507j
-0.18307356617022244873 + 1.3512617852773573156e-2000j 2.39e-504, 2.39e-504j
-0.21977387039115541258 - 5.6181670210723184393e-1999j 5.61e-503, 5.61e-503j

Weights

Weight (ball center) Error (ball radius)
0.006135302381468397343 - 4.5179864309526987904e-1711j 1.67e-129, 9.47e-360j
0.0067528979026245891915 + 2.6798895678146934649e-1712j 7.85e-130, 4.46e-360j
0.012501542826540675337 + 1.3758791905812533922e-1712j 1.94e-130, 1.1e-360j
0.0075881460235888154626 + 2.7376871036196025348e-1712j 2.88e-130, 1.64e-360j
0.015372394574014444213 - 5.3512358501316629241e-1714j 2.19e-131, 1.25e-361j
0.012501542826540675337 + 6.7247421203164706802e-1714j 1.92e-131, 1.09e-361j
0.0068584913419369783957 - 2.3113816406210119088e-1714j 8.31e-133, 4.73e-363j
0.015372394574014444213 - 1.1470800112038360352e-1712j 2.59e-131, 1.47e-361j
0.0077910221644058569791 - 2.657125129849493943e-1712j 3.29e-130, 1.87e-360j
0.0077910221644058569791 - 2.6754641988726597288e-1714j 3.51e-133, 1.99e-363j
0.017608650368993399455 + 3.8150989811947851282e-1714j 5.32e-133, 3.02e-363j
-0.0038340088414281268223 + 3.6017497079256341156e-1714j 2.87e-133, 1.63e-363j
0.012813564929980027029 - 1.3827988085445025618e-1712j 9.35e-133, 5.32e-363j
0.017629308636108387913 + 9.3018520116828828355e-1713j 1.04e-134, 5.9e-365j
0.0017051383043378171418 + 9.734618884764992206e-1716j 2.66e-134, 1.51e-364j
0.02010247052903410573 - 6.5381785003488816652e-1714j 3.81e-136, 2.17e-366j
0.017629308636108387913 + 6.3154492594745831652e-1714j 7.85e-136, 4.46e-366j
0.006135302381468397343 - 2.0590715154937427714e-1714j 9.37e-134, 5.33e-364j
0.022290566860453298341 - 7.0920283707978309498e-1713j 9.62e-139, 5.47e-369j
0.012713887138843940361 + 1.3993202326380280114e-1712j 9.04e-136, 5.14e-366j
0.017727367303569019492 - 5.2667395178393666286e-1714j 9.14e-136, 5.2e-366j
0.022241672786748671461 + 8.4957764708084600615e-1714j 6.35e-139, 3.61e-369j
0.0068584913419369783957 - 1.1436439181210310702e-1711j 1.63e-136, 9.24e-367j
0.02010247052903410573 - 8.1897865436827382567e-1713j 1.6e-138, 9.1e-369j
0.022241672786748671461 + 6.9931996319836700067e-1713j 3.14e-140, 1.78e-370j
0.010429617463937363602 - 1.9408646326630072935e-1712j 3.48e-136, 1.98e-366j
0.024428010969977580104 - 8.6531254445924659217e-1714j 7.52e-142, 4.27e-372j
0.012713887138843940361 + 3.6712558156556930738e-1714j 2.11e-139, 1.2e-369j
0.017727367303569019492 - 9.4115169526023729594e-1713j 3.48e-138, 1.98e-368j
0.028234360350637892946 - 5.1161586212601962535e-1713j 2.19e-144, 1.25e-374j
0.017608650368993399455 + 9.5557828132558594502e-1713j 6.03e-138, 3.43e-368j
0.022290566860453298341 - 7.6114209047688482629e-1714j 3.29e-142, 1.87e-372j
0.026340037898502226051 + 8.8124763368066197868e-1714j 4.93e-144, 2.8e-374j
-0.0038340088414281268223 + 5.5645898088619932785e-1711j 2.23e-137, 1.27e-367j
0.024428010969977580104 - 6.3327199995246343846e-1713j 2.58e-142, 1.47e-372j
0.026340037898502226051 + 5.7327768140237984527e-1713j 2.45e-143, 1.4e-373j
0.035273834943073891508 + 3.354161778272061685e-1713j 8.55e-150, 4.86e-380j
0.026294917356826307861 + 1.0586740493722468478e-1713j 2.21e-147, 1.26e-377j
0.022189186644821545055 + 6.7187621443950296518e-1714j 1.46e-143, 8.29e-374j
0.0075881460235888154626 + 1.581966481842358755e-1714j 1.2e-143, 6.83e-374j
0.029898341398616813225 - 4.6327358178969589005e-1713j 1.33e-148, 7.54e-379j
0.035675772579771702052 + 2.7900295086070015577e-1713j 7.11e-153, 4.04e-383j
0.029898341398616813225 - 1.1857883635061787083e-1713j 5.84e-150, 3.32e-380j
0.029959178124191275524 + 1.0941834880596063966e-1713j 2.97e-149, 1.69e-379j
0.0067528979026245891915 + 3.7582418136326434745e-1714j 1.26e-144, 7.15e-375j
0.029959178124191275524 + 4.70785670443875343e-1713j 5.13e-150, 2.92e-380j
0.022189186644821545055 + 7.2087278932634483771e-1713j 8.2e-146, 4.66e-376j
0.010429617463937363602 - 3.6768722522165155732e-1714j 5.67e-145, 3.22e-375j
0.032767660488310223657 - 1.4167790885509821726e-1713j 2e-154, 1.13e-384j
0.036724055487141997864 - 2.6288921010169677109e-1713j 5.27e-156, 3e-386j
0.026379562916778022664 - 5.6396569011243601399e-1713j 6.32e-149, 3.59e-379j
0.032960439789155110533 + 3.9470888447060657061e-1713j 9.08e-154, 5.16e-384j
0.012813564929980027029 - 4.9125039674111308108e-1714j 1e-145, 5.69e-376j
0.031431578045524178361 - 1.2985783040028296574e-1713j 9.33e-154, 5.31e-384j
0.02972330787490541756 + 1.2785047690244937902e-1713j 5.79e-153, 3.29e-383j
0.0017051383043378171418 - 5.921949981791028282e-1713j 1.44e-148, 8.21e-379j
0.032767660488310223657 - 3.8854927510532478449e-1713j 3.31e-155, 1.88e-385j
0.026294917356826307861 + 5.5571889775720671929e-1713j 4.29e-151, 2.44e-381j
0.026379562916778022664 - 9.6970401148698454672e-1714j 4.38e-151, 2.49e-381j
0.033946997058123631984 - 1.538837256357735453e-1713j 4.41e-158, 2.51e-388j
0.028234360350637892946 - 1.0761020702146428517e-1713j 5.08e-152, 2.89e-382j
0.02972330787490541756 + 4.5645174512885034807e-1713j 3.59e-154, 2.04e-384j
0.031431578045524178361 - 4.2460046225387012305e-1713j 7.83e-155, 4.45e-385j
0.037641274008861946682 + 2.1308723783116094317e-1713j 2.5e-162, 1.42e-392j
0.032463150809657226021 + 1.5157411597479689513e-1713j 2.43e-158, 1.38e-388j
0.032960439789155110533 + 1.3194853588797862584e-1713j 6.79e-157, 3.86e-387j
0.036923970834279764574 - 2.2640382072163135989e-1713j 9.25e-163, 5.26e-393j
0.032463150809657226021 + 3.8286362143861095796e-1713j 3.81e-159, 2.16e-389j
0.033946997058123631984 - 3.585099022433769728e-1713j 6.4e-160, 3.64e-390j
0.036846092423823773469 + 1.8307841977837489265e-1713j 4.03e-164, 2.29e-394j
0.035273834943073891508 + 1.5629377851405615435e-1713j 2.63e-162, 1.5e-392j
0.036319526061966122282 - 1.948029277606170621e-1713j 2.22e-164, 1.26e-394j
0.034923886025430438987 - 3.302838473060204374e-1713j 3.11e-163, 1.77e-393j
0.034460706509963887248 + 3.2549153000992985176e-1713j 4.08e-164, 2.32e-394j
0.036083580584408554961 + 2.401525648071576935e-1713j 6.76e-166, 3.84e-396j
0.034923886025430438987 - 1.6685791840139225308e-1713j 9.01e-165, 5.12e-395j
0.035724328412297085681 - 1.8030517504660547138e-1713j 1.65e-165, 9.4e-396j
0.036923970834279764574 - 2.4366697297516634938e-1713j 1.05e-166, 5.99e-397j
0.035724328412297085681 - 3.0603158014382777353e-1713j 7.04e-167, 4.01e-397j
0.037641274008861946682 + 2.4737473123326244231e-1713j 6.11e-167, 3.47e-397j
0.035675772579771702052 + 2.0684384012448226476e-1713j 1.73e-167, 9.83e-398j
0.034460706509963887248 + 1.7765414699005451562e-1713j 3.91e-167, 2.22e-397j
0.036724055487141997864 - 2.0988959337648018114e-1713j 1.23e-167, 6.99e-398j
0.036319526061966122282 - 2.8308740329569325414e-1713j 7.05e-169, 4.01e-399j
0.036846092423823773469 + 2.8742449530533082243e-1713j 6.71e-169, 3.81e-399j

Download

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