Rule: Q[2, 3, 40, 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)
-0.99355051257660271205 + 1.4554578574761680674e-1715j 5.18e-477, 5.18e-477j
-0.9602422396877912789 + 5.2145382787629370624e-1722j 7.37e-478, 7.37e-478j
-0.94997131415289343196 - 3.4454369384148177537e-1723j 3.46e-478, 3.46e-478j
0.98379866320997716242 + 8.3454885655879057853e-1722j 3.3e-477, 3.3e-477j
0.92582009977255146157 - 2.3624470117207313803e-1737j 6.46e-479, 6.46e-479j
0.93849425326436405777 + 1.04656165241731222e-1736j 1.59e-478, 1.59e-478j
0.94997131415289343196 + 2.3028859826534431454e-1750j 3.73e-478, 3.73e-478j
-0.89693610670527040585 + 6.0535072502139769138e-1772j 9.13e-480, 9.13e-480j
-0.92582009977255146157 - 2.052130538700298301e-1770j 6.53e-479, 6.53e-479j
0.98925284094580390444 - 6.4411106451830087617e-1785j 4.39e-477, 4.39e-477j
0.88075950002989760094 - 1.4690223875615645541e-1813j 2.86e-480, 2.86e-480j
-0.99672704917551474314 + 1.3433792233682125018e-1809j 5.23e-477, 5.23e-477j
-0.93849425326436405777 - 2.5299979623058701158e-1823j 1.61e-478, 1.61e-478j
-0.91196216281098786624 + 4.0893372381384041786e-1824j 2.46e-479, 2.46e-479j
0.99980344139641647465 - 3.1993807137626760141e-1818j 1.57e-477, 1.57e-477j
0.86345229817332822548 + 2.2997308254147300781e-1849j 8.73e-481, 8.73e-481j
0.91196216281098786624 + 8.949550384663564591e-1847j 2.59e-479, 2.59e-479j
-0.99980344139641647465 + 4.1004614480852916478e-1844j 1.64e-477, 1.64e-477j
-0.80497834318825253117 - 2.0039551083540398509e-1857j 1.5e-482, 1.5e-482j
-0.96930322336369854382 - 7.5729604488362835538e-1852j 1.36e-477, 1.36e-477j
-0.15970914358069582725 + 2.3351932709970166231e-1878j 1.02e-504, 1.02e-504j
0.80497834318825253117 - 5.6388573460554096639e-1856j 1.47e-482, 1.47e-482j
0.99672704917551474314 + 4.9769522618271713573e-1848j 5.42e-477, 5.42e-477j
-0.84503677782738091062 + 4.1038689139186664479e-1876j 2.54e-481, 2.54e-481j
-0.86345229817332822548 - 5.5582886320305351639e-1876j 9.18e-481, 9.18e-481j
0.99355051257660271205 + 1.1388803377543800399e-1868j 5.21e-477, 5.21e-477j
0.73723144389431457554 - 3.4340018501932298024e-1890j 1.2e-484, 1.2e-484j
0.84503677782738091062 - 1.5979397308328629196e-1886j 2.47e-481, 2.47e-481j
-0.97715408227203967263 + 4.1340456366267039648e-1879j 2.16e-477, 2.16e-477j
-0.633923118011879036 - 4.4533789724212466131e-1892j 5.78e-488, 5.78e-488j
0.51792962223584562884 - 5.1246027872948492181e-1897j 7.45e-492, 7.45e-492j
0.89693610670527040585 + 6.5002497980411245421e-1884j 9.09e-480, 9.09e-480j
0.7127275162572423519 - 1.5997759249520468011e-1890j 1.88e-485, 1.88e-485j
-0.98925284094580390444 + 6.7077381927153597886e-1879j 4.39e-477, 4.39e-477j
-0.7127275162572423519 - 3.8222838632603757643e-1891j 2.03e-485, 2.03e-485j
-0.78338823319794908055 - 1.1757456042894803532e-1888j 3.16e-483, 3.16e-483j
0.96930322336369854382 - 8.4047701282172484277e-1880j 1.28e-477, 1.28e-477j
0.66103715218122828137 - 4.1172229187568566164e-1904j 4.13e-487, 4.13e-487j
0.97715408227203967263 + 3.8219006226337353896e-1892j 2.13e-477, 2.13e-477j
0.99879689537787841368 + 5.3290269367422788944e-1909j 4.26e-477, 4.26e-477j
-0.60601406092358698678 - 4.7873462369864403762e-1939j 7.13e-489, 7.13e-489j
-0.98379866320997716242 + 8.0176026365663289546e-1926j 3.32e-477, 3.32e-477j
0.78338823319794908055 - 2.1154182405702158874e-1936j 3.23e-483, 3.23e-483j
0.633923118011879036 - 4.0228990693021342207e-1942j 5.98e-488, 5.98e-488j
-0.73723144389431457554 + 4.5468562522773431238e-1937j 1.18e-484, 1.18e-484j
-0.57735026918962576451 + 3.1373374822361240522e-1943j 7.48e-490, 7.48e-490j
-0.68731761895496321609 + 5.2181609009476567425e-1939j 2.99e-486, 2.99e-486j
-0.99879689537787841368 - 6.6067939057754315692e-1933j 4.16e-477, 4.16e-477j
0.45602503276591046247 - 1.543902835929775846e-1950j 5.47e-494, 5.47e-494j
0.9602422396877912789 + 7.0849697160963119319e-1940j 7.04e-478, 7.04e-478j
-0.76079571384507294163 - 3.3652392755781811223e-1957j 6.45e-484, 6.45e-484j
-0.48726358419044074812 - 7.0037248374296471612e-1968j 6.75e-493, 6.75e-493j
0.82553695034210628946 - 1.2072409919863454465e-1956j 6.56e-482, 6.56e-482j
0.60601406092358698678 + 5.6070524125580782029e-1964j 6.59e-489, 6.59e-489j
0.57735026918962576451 + 3.6724645767048633295e-1965j 7.77e-490, 7.77e-490j
-0.45602503276591046247 - 2.1307895268050589321e-1969j 5.51e-494, 5.51e-494j
-0.51792962223584562884 - 1.0668263607478973471e-1966j 7.51e-492, 7.51e-492j
-0.82553695034210628946 + 1.9109949580748351139e-1954j 6.53e-482, 6.53e-482j
0.54797401518989522221 - 5.6405049177382945532e-1966j 8.37e-491, 8.37e-491j
0.39204406362290903475 + 5.1674857182612419222e-1971j 3.03e-496, 3.03e-496j
-0.88075950002989760094 + 1.1064023063743065356e-1964j 2.95e-480, 2.95e-480j
-0.66103715218122828137 - 5.1757662278439635987e-1970j 4.32e-487, 4.32e-487j
-0.42426640800363132138 + 6.6676894175170686965e-1978j 4.45e-495, 4.45e-495j
0.12678997759091780718 - 1.2039416572113965736e-1988j 5.65e-506, 5.65e-506j
0.42426640800363132138 - 2.7028515942149290784e-1976j 4.18e-495, 4.18e-495j
0.48726358419044074812 - 1.2912637710732530258e-1974j 6.66e-493, 6.66e-493j
0.031054829959228759423 + 1.4013447164203549032e-1992j 7.98e-510, 7.98e-510j
-0.39204406362290903475 - 1.4701771480564413621e-1979j 3.09e-496, 3.09e-496j
-0.54797401518989522221 - 5.9077656615382924094e-1974j 8.36e-491, 8.36e-491j
0.68731761895496321609 - 6.7302799081152826863e-1966j 2.97e-486, 2.97e-486j
0.35941905515279233024 + 1.612416615993780705e-1979j 2.59e-497, 2.59e-497j
0.76079571384507294163 - 5.7067421873965971196e-1983j 6.4e-484, 6.4e-484j
-0.32645864332483180038 + 1.2422115596804310407e-1998j 1.56e-498, 1.56e-498j
-0.35941905515279233024 + 1.8636404886746676307e-1997j 2.15e-497, 2.15e-497j
-6.005458769324213439e-2064 - 8.5464700688030255978e-2064j 9.51e-2062, 9.51e-2062j
0.32645864332483180038 + 1.4837852834126298421e-1998j 1.48e-498, 1.48e-498j
0.22637832569994118012 - 3.3904469200469224516e-2002j 2.93e-502, 2.93e-502j
-0.031054829959228759423 + 3.1427205823069105385e-2010j 8.2e-510, 8.2e-510j
-0.29323849689562363458 - 1.0868067483870747377e-1999j 8.55e-500, 8.55e-500j
-0.12678997759091780718 + 7.2836354063021481465e-2007j 5.6e-506, 5.6e-506j
0.19295466862645680272 + 4.8683821075887041494e-2003j 1.72e-503, 1.72e-503j
0.29323849689562363458 + 4.2195929185062266194e-2000j 8.37e-500, 8.37e-500j
0.062435425649932677909 - 1.1327920322414094179e-2008j 1.97e-508, 1.97e-508j
-0.22637832569994118012 + 5.8342800478664506922e-2002j 2.99e-502, 2.99e-502j
-0.19295466862645680272 - 5.1921732352518933937e-2003j 1.75e-503, 1.75e-503j
0.09433678237282067438 - 2.256131124508794487e-2007j 3.11e-507, 3.11e-507j
0.25984509307920656244 - 3.7930614486482278768e-2001j 5.09e-501, 5.09e-501j
0.15970914358069582725 - 2.2673098669651756675e-2004j 1.04e-504, 1.04e-504j
-0.09433678237282067438 + 1.4674722240825472555e-2008j 3.46e-507, 3.46e-507j
-0.25984509307920656244 - 1.3797350360626745745e-2001j 4.95e-501, 4.95e-501j
-0.062435425649932677909 - 1.1911809794263955319e-2008j 1.88e-508, 1.88e-508j

Weights

Weight (ball center) Error (ball radius)
0.0037326914731755294592 + 5.4848429370298823936e-1718j 3.39e-123, 1.1e-353j
0.0096662085543855975883 - 1.9044737271965004998e-1716j 1.69e-123, 5.45e-354j
0.01087504761990691905 + 1.4236051029515950758e-1716j 1.1e-123, 3.55e-354j
0.0060451716330680708647 + 3.2369772313613764592e-1718j 2.91e-124, 9.38e-355j
0.013268506024008534488 + 3.085666720432673804e-1718j 1.4e-124, 4.53e-355j
0.012077543505248958781 - 3.1356525362561850391e-1718j 1.1e-124, 3.56e-355j
0.01087504761990691905 + 3.191984465909203909e-1718j 5.45e-125, 1.76e-355j
0.015604444704268081554 + 5.9121480813009825177e-1717j 2.07e-126, 6.66e-357j
0.013268506024008534488 + 8.7445019885667169648e-1717j 3.58e-126, 1.16e-356j
0.0048691321130693249026 - 3.0675609095836046257e-1718j 1.93e-127, 6.22e-358j
0.016745441073996164433 - 3.0113329110760760635e-1718j 1.63e-127, 5.25e-358j
0.0026222276066948062247 + 1.3857121256789057765e-1715j 8.65e-128, 2.79e-358j
0.012077543505248958781 - 1.1004045658720668908e-1716j 2.13e-126, 6.87e-357j
0.014444752281688216579 - 7.115826309331241217e-1717j 6.03e-127, 1.95e-357j
0.00053303160221777263132 - 5.2538079304577475395e-1719j 2.34e-129, 7.54e-360j
0.017865237021652696338 + 3.0178891861399138574e-1718j 4.7e-129, 1.52e-359j
0.014444752281688216579 - 3.046722091696187715e-1718j 8.35e-128, 2.69e-358j
0.00053303160221777263132 + 1.6746837917069542406e-1716j 8.26e-129, 2.67e-359j
0.021079061326398417203 - 3.0031778698542748193e-1717j 1.29e-131, 4.16e-362j
0.0084557437162531390437 + 2.6610270303114229151e-1716j 4.62e-128, 1.49e-358j
0.033107421979817124581 - 5.8894873134293342924e-1717j 3.07e-134, 9.89e-365j
0.021079061326398417203 - 3.1487480399742469625e-1718j 5.29e-132, 1.71e-362j
0.0026222276066948062247 - 2.2119004796752926189e-1718j 3.89e-130, 1.26e-360j
0.018961798080897250742 - 3.7662404388474588755e-1717j 8.26e-132, 2.67e-362j
0.017865237021652696338 + 4.307741628020942186e-1717j 2.44e-131, 7.87e-362j
0.0037326914731755294592 + 2.734457336509393233e-1718j 5.37e-130, 1.73e-360j
0.024039572760047951363 + 3.4807808902284074867e-1718j 4.2e-135, 1.35e-365j
0.018961798080897250742 - 3.0421876847346565904e-1718j 2.15e-132, 6.93e-363j
0.0072464614740703807272 - 3.9596279163085259011e-1716j 1.37e-130, 4.42e-361j
0.02751810492624298107 + 1.9866816511841653852e-1717j 3.59e-137, 1.16e-367j
0.030363162597014043968 + 6.2956167285110697347e-1718j 2.57e-138, 8.3e-369j
0.015604444704268081554 + 3.0213869327697097613e-1718j 8.85e-132, 2.86e-362j
0.024962670403330540202 - 3.6484791758528952393e-1718j 4.26e-136, 1.37e-366j
0.0048691321130693249026 - 1.4148669866003374253e-1715j 2.17e-131, 6.99e-362j
0.024962670403330540202 - 2.2168231898245910775e-1717j 1.86e-137, 6e-368j
0.02209630399146618355 + 2.7345380880252311243e-1717j 1.04e-135, 3.35e-366j
0.0084557437162531390437 + 3.286827057290520683e-1718j 4.43e-133, 1.43e-363j
0.026703543484510778473 - 4.0963403265322062302e-1718j 3.06e-139, 9.86e-370j
0.0072464614740703807272 - 3.2936685351449148232e-1718j 7.11e-133, 2.29e-363j
0.0015210535640796514323 + 1.4762964371256846164e-1718j 1.74e-132, 5.61e-363j
0.028293299306403450514 - 1.9574226874530455408e-1717j 9.22e-142, 2.98e-372j
0.0060451716330680708647 + 6.5633988526381666065e-1716j 1.14e-134, 3.67e-365j
0.02209630399146618355 + 3.2341738394821365983e-1718j 1.39e-137, 4.5e-368j
0.02751810492624298107 + 4.3900109292524582744e-1718j 6.3e-141, 2.03e-371j
0.024039572760047951363 + 2.3503927077050029359e-1717j 4.61e-140, 1.49e-370j
0.029027239121093419838 + 1.9497906408101604906e-1717j 2.28e-143, 7.36e-374j
0.02585122134549715641 + 2.1141408637576656467e-1717j 3.52e-141, 1.13e-371j
0.0015210535640796514323 - 5.6057473892590566766e-1716j 1.02e-135, 3.29e-366j
0.031507635152031610648 + 7.9785633622436445782e-1718j 7.51e-147, 2.42e-377j
0.0096662085543855975883 - 3.2465317676957309853e-1718j 2.3e-136, 7.44e-367j
0.023083622207614317293 - 2.5204163204755400025e-1717j 4.36e-140, 1.41e-370j
0.03096073707511874696 - 2.0622129077585790094e-1717j 1.72e-146, 5.54e-377j
0.020033608355427060022 + 3.0853495048868162991e-1718j 1.14e-139, 3.67e-370j
0.028293299306403450514 - 4.7423549992940503466e-1718j 6.84e-145, 2.21e-375j
0.029027239121093419838 + 5.1658328361058103147e-1718j 8.44e-146, 2.72e-376j
0.031507635152031610648 + 2.151628522802525499e-1717j 2.51e-148, 8.09e-379j
0.030363162597014043968 + 2.0006945308833088101e-1717j 5.23e-147, 1.69e-377j
0.020033608355427060022 + 3.3405457493622196404e-1717j 4.22e-142, 1.36e-372j
0.029717882642455730878 - 5.6765787625309080816e-1718j 4.82e-147, 1.56e-377j
0.032434304477807096685 + 1.0569117773110747468e-1717j 3.25e-151, 1.05e-381j
0.016745441073996164433 - 5.0041665125678005013e-1717j 5.64e-142, 1.82e-372j
0.026703543484510778473 - 2.0383481536163466005e-1717j 5.58e-145, 1.8e-375j
0.032000233459788170214 - 2.2735988788395804619e-1717j 2.51e-150, 8.1e-381j
0.032706884014278559247 + 5.3853365640036951516e-1717j 1.43e-155, 4.62e-386j
0.032000233459788170214 - 9.1289753537303521562e-1718j 2.36e-151, 7.63e-382j
0.03096073707511874696 - 7.0506440567184874731e-1718j 4.39e-150, 1.42e-380j
0.031171288317772426081 - 9.3551156413471243453e-1717j 2.88e-156, 9.3e-387j
0.032434304477807096685 + 2.4346428592400710851e-1717j 1.41e-152, 4.57e-383j
0.029717882642455730878 - 1.9638844030733407636e-1717j 5.71e-149, 1.84e-379j
0.02585122134549715641 + 3.8516806898900873449e-1718j 6.9e-149, 2.22e-379j
0.03280454085895522966 - 1.2391027191573309748e-1717j 2.96e-154, 9.55e-385j
0.023083622207614317293 - 3.3438980554852431178e-1718j 2.24e-148, 7.23e-379j
0.0331037113836281489 + 2.9130519492802572901e-1717j 1.27e-155, 4.1e-386j
0.03280454085895522966 - 2.6437270157541910042e-1717j 1.27e-154, 4.11e-385j
0.03099514719012105775 + 1.0027519148999367565e-1716j 9.21e-159, 2.97e-389j
0.0331037113836281489 + 1.4721648033636217987e-1717j 9.83e-157, 3.17e-387j
0.033466166058462997898 - 2.6855333533998892287e-1717j 8.58e-159, 2.77e-389j
0.031171288317772426081 - 9.958800418623828446e-1717j 1.65e-159, 5.33e-390j
0.03332221005745588547 - 3.2589424356284450721e-1717j 1.95e-158, 6.3e-389j
0.032706884014278559247 + 6.9608765263461226059e-1717j 5.66e-160, 1.83e-390j
0.033358579530749376302 + 3.3680369301994170295e-1717j 6.65e-160, 2.14e-390j
0.03332221005745588547 - 1.7736196146847457611e-1717j 4.76e-159, 1.54e-389j
0.031622857328326880026 + 8.1409093955429190834e-1717j 2.31e-160, 7.45e-391j
0.033466166058462997898 - 4.2704390389835857787e-1717j 4.42e-161, 1.43e-391j
0.033358579530749376302 + 4.9915270361520294434e-1717j 4.66e-161, 1.5e-391j
0.032184067928733912654 - 6.7248123630598298057e-1717j 4.01e-161, 1.29e-391j
0.033448004265830180175 + 2.1674270199027309014e-1717j 3.61e-161, 1.16e-391j
0.033107421979817124581 - 4.258274010119617914e-1717j 2.21e-161, 7.14e-392j
0.032184067928733912654 - 8.1358189763683137513e-1717j 8.64e-163, 2.79e-393j
0.033448004265830180175 + 3.7026384512750856298e-1717j 3.32e-163, 1.12e-393j
0.031622857328326880026 + 9.2326798999718002391e-1717j 1e-162, 3.21e-393j

Download

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