Rule: Q[18, 20, 39]

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)
0.99978494779442052958 - 6.6496105886555845297e-1759j 4.13e-483, 4.13e-483j
-0.96745178896399913487 - 9.7286932959516497778e-1762j 2.35e-483, 2.35e-483j
-0.9274951474090583421 - 2.7930073091704916566e-1788j 2.65e-484, 2.65e-484j
-0.99978494779442052958 - 3.3266210436531475492e-1813j 4.02e-483, 4.02e-483j
0.94249070272268796989 + 1.8457083129530920166e-1849j 6.11e-484, 6.11e-484j
0.9274951474090583421 + 2.9652466829856562277e-1850j 2.71e-484, 2.71e-484j
0.91086167316030996451 - 1.1400773866065983418e-1850j 1.12e-484, 1.12e-484j
-0.91086167316030996451 - 3.7638606491754659289e-1849j 1.07e-484, 1.07e-484j
0.95582394957139775518 - 1.1639862502933441033e-1856j 1.24e-483, 1.24e-483j
-0.99156516842093094673 - 8.1706851044239513371e-1854j 7.91e-483, 7.91e-483j
0.89260246649755573921 + 3.9684649368911302812e-1878j 3.71e-485, 3.71e-485j
0.99156516842093094673 - 4.3042362161098629795e-1875j 7.86e-483, 7.86e-483j
-0.94249070272268796989 + 3.1395075522078945397e-1875j 5.89e-484, 5.89e-484j
-0.8512614839549979707 - 2.6962893933233053419e-1885j 3.8e-486, 3.8e-486j
0.99593997734518047069 + 3.0933620031594572658e-1882j 1.01e-482, 1.01e-482j
0.82823638185176396428 - 1.9872226922253518702e-1887j 1.05e-486, 1.05e-486j
0.8512614839549979707 - 3.144828950408324597e-1886j 3.7e-486, 3.7e-486j
-0.99859651643055150742 - 3.7965998014596847773e-1880j 9.3e-483, 9.3e-483j
-0.77772775029255652518 - 9.4882934502575308632e-1906j 5.97e-488, 5.97e-488j
-0.95582394957139775518 + 4.3606244050535076031e-1900j 1.19e-483, 1.19e-483j
0.96745178896399913487 - 5.1799774569081633854e-1918j 2.19e-483, 2.19e-483j
0.75036505635902229785 - 2.023975347056066376e-1924j 1.17e-488, 1.17e-488j
0.99859651643055150742 - 1.2760378566636311626e-1916j 9.27e-483, 9.27e-483j
-0.82823638185176396428 + 3.6603041586889703552e-1922j 1.1e-486, 1.1e-486j
-0.80370495897252311568 + 1.2900523026342952426e-1926j 2.65e-487, 2.65e-487j
0.87272885201684050249 + 7.5030171694552187581e-1930j 1.28e-485, 1.28e-485j
0.72166992327418943232 + 4.4358715040457547577e-1933j 2.03e-489, 2.03e-489j
0.97731704962157666034 - 3.1848203340998771778e-1926j 3.9e-483, 3.9e-483j
-0.98536477117224722794 + 2.4261098080027052866e-1924j 5.88e-483, 5.88e-483j
-0.62802376237896618507 - 1.1496785197034838552e-1954j 5.84e-492, 5.84e-492j
-0.89260246649755573921 + 1.4396759925088932824e-1945j 4.01e-485, 4.01e-485j
0.77772775029255652518 + 2.754050119583331794e-1959j 5.31e-488, 5.31e-488j
0.69168704306035320787 - 3.0029781020324767902e-1961j 3.18e-490, 3.18e-490j
-0.99593997734518047069 + 2.2704506099898209256e-1951j 1.04e-482, 1.04e-482j
-0.69168704306035320787 - 3.3965517894533334667e-1983j 3.18e-490, 3.18e-490j
-0.75036505635902229785 - 1.8364484932170236756e-1981j 1.14e-488, 1.14e-488j
0.98536477117224722794 - 2.4666721448257433656e-1977j 5.72e-483, 5.72e-483j
0.59443989838629185438 + 2.1746797347098853701e-1989j 6.36e-493, 6.36e-493j
0.80370495897252311568 - 2.8860778297643626031e-1983j 2.54e-487, 2.54e-487j
-0.72166992327418943232 - 7.8051995036251850315e-1983j 1.97e-489, 1.97e-489j
-0.55977083107394753461 - 2.210652516386015112e-1988j 7.07e-494, 7.07e-494j
-0.97731704962157666034 - 9.2158816788940720237e-1986j 3.61e-483, 3.61e-483j
0.66045711694402080612 + 8.1092233950286530093e-2008j 4.57e-491, 4.57e-491j
0.62802376237896618507 - 1.8939929627190642542e-2008j 6.03e-492, 6.03e-492j
-0.87272885201684050249 + 1.9993326029740141806e-2010j 1.25e-485, 1.25e-485j
-0.59443989838629185438 - 1.1738706983516428015e-2016j 6.62e-493, 6.62e-493j
-0.66045711694402080612 + 4.0280334437538761961e-2023j 4.59e-491, 4.59e-491j
0.21055260102903231553 + 1.0961427225676660862e-2037j 2.62e-504, 2.62e-504j
0.48748173749781995875 + 7.4954874258906143602e-2028j 5.8e-496, 5.8e-496j
0.55977083107394753461 + 6.2979817378188859291e-2027j 7.38e-494, 7.38e-494j
-0.16885305857357954056 + 1.5495325597566356417e-2039j 1.25e-505, 1.25e-505j
-0.45001348478961689136 + 3.2662984573934853731e-2030j 5.28e-497, 5.28e-497j
-0.12689236510498180303 - 9.586855894188943703e-2041j 5.51e-507, 5.51e-507j
0.52409197316681862632 + 5.8478227481398002403e-2031j 6.73e-495, 6.73e-495j
0.45001348478961689136 - 3.6084385200350473627e-2033j 5.09e-497, 5.09e-497j
-0.0037762088409933203275 - 4.292786466020456787e-2046j 4.39e-511, 4.39e-511j
-0.48748173749781995875 + 2.351246145857577657e-2029j 5.64e-496, 5.64e-496j
-0.52409197316681862632 + 4.7872054612978661262e-2031j 6.3e-495, 6.3e-495j
0.042651889527379573632 - 5.3207778726392424485e-2045j 1.6e-509, 1.6e-509j
0.41175116146284264604 - 1.1778322808586951997e-2034j 3.94e-498, 3.94e-498j
0.084775013041735301242 + 7.0552320661443729463e-2044j 2.82e-508, 2.82e-508j
-0.41175116146284264604 + 4.6600775164517478144e-2034j 3.56e-498, 3.56e-498j
-0.37275055614765271772 - 1.2143111489037032139e-2035j 2.62e-499, 2.62e-499j
-0.042651889527379573632 + 5.4643226820286428e-2045j 1.6e-509, 1.6e-509j
0.37275055614765271772 - 3.9794273892921744076e-2037j 2.58e-499, 2.58e-499j
0.25188622569150550959 + 8.1259049171077754957e-2039j 5.44e-503, 5.44e-503j
-0.084775013041735301242 - 1.0072666397016282745e-2043j 2.82e-508, 2.82e-508j
-0.33306496018034047177 + 1.0178775688146317665e-2036j 1.83e-500, 1.83e-500j
-0.21055260102903231553 + 2.0636234926369267431e-2040j 2.6e-504, 2.6e-504j
0.12689236510498180303 - 1.4862746371425818931e-2042j 6.12e-507, 6.12e-507j
0.33306496018034047177 - 1.5226242783404048039e-2036j 1.71e-500, 1.71e-500j
0.0037762088409933203275 - 5.2862582722852941645e-2047j 4.39e-511, 4.39e-511j
-0.25188622569150550959 - 4.9494438017148649025e-2039j 5.47e-503, 5.47e-503j
-0.29275301062654972345 + 5.3152423678491313291e-2038j 9.65e-502, 9.65e-502j
7.5832567018799857494e-2082 + 5.167019214524905226e-2082j 7.07e-2080, 7.07e-2080j
0.29275301062654972345 + 4.3860432217256400288e-2038j 9.57e-502, 9.57e-502j
0.16885305857357954056 + 1.5566445763881048561e-2041j 1.28e-505, 1.28e-505j

Weights

Weight (ball center) Error (ball radius)
0.00059577862883845374987 + 6.0983337998079270445e-1759j 2.09e-143, 2.25e-376j
0.010756590956444491917 + 6.8959859852915500546e-1762j 9.62e-144, 1.03e-376j
0.015817210986372928137 - 4.1469196337429311348e-1762j 7.44e-144, 8e-377j
0.00059577862883845374987 + 3.0247323359138364576e-1762j 2.22e-144, 2.39e-377j
0.014169961837318022996 + 2.2724737823488464353e-1760j 8.1e-145, 8.71e-378j
0.015817210986372928137 - 1.8757153773187898679e-1760j 5.21e-145, 5.61e-378j
0.017447910738201467284 + 1.5903526596656447374e-1760j 1.95e-145, 2.1e-378j
0.017447910738201467284 + 5.2755864095031316697e-1762j 1.65e-146, 1.77e-379j
0.012489272383547695662 - 2.8600952586618861763e-1760j 6.68e-146, 7.18e-379j
0.0052782076645404567212 - 1.0713543809775063459e-1761j 9.17e-148, 9.86e-381j
0.019068790588430750655 - 1.3761476541838217711e-1760j 3.19e-146, 3.43e-379j
0.0052782076645404567212 - 1.5237637928690620424e-1759j 2.23e-147, 2.39e-380j
0.014169961837318022996 + 2.2619359403652434509e-1762j 2.12e-148, 2.28e-381j
0.02225363399288419867 - 7.5372845715422959972e-1762j 2.59e-150, 2.79e-383j
0.0034884414422285052115 + 3.1621417799540814254e-1759j 2.52e-147, 2.71e-380j
0.023787888533297678234 + 9.9053852567790394865e-1761j 1.3e-149, 1.4e-382j
0.02225363399288419867 - 1.0860662651537618108e-1760j 4.81e-149, 5.17e-382j
0.0018643303349882083045 - 7.6044951551596875121e-1762j 1.21e-149, 1.3e-382j
0.026679543490899008336 + 9.9904241099656172885e-1762j 7.87e-153, 8.46e-386j
0.012489272383547695662 + 2.7797537126237539362e-1762j 9.72e-150, 1.05e-382j
0.010756590956444491917 + 3.7999432818155457225e-1760j 9.33e-149, 1e-381j
0.028036987079838109886 - 8.320552273523465665e-1761j 5.18e-153, 5.57e-386j
0.0018643303349882083045 - 8.268784396746733386e-1759j 2.36e-148, 2.54e-381j
0.023787888533297678234 + 8.2628445531107325612e-1762j 1.78e-153, 1.91e-386j
0.025264697172602544564 - 9.0683233031816094253e-1762j 6.51e-154, 7e-387j
0.020675101355583384733 + 1.2120242616215953658e-1760j 2.07e-150, 2.23e-383j
0.029345911297597195412 + 8.0821856895271729805e-1761j 1.72e-154, 1.85e-387j
0.0089643771344325544969 - 5.4362192373652694009e-1760j 2.49e-149, 2.68e-382j
0.0071256728248354104291 + 1.2093397619606575178e-1761j 1.29e-152, 1.39e-385j
0.033018438939476624374 - 1.7485122733425771537e-1761j 1.11e-158, 1.2e-391j
0.019068790588430750655 - 6.1191353193850758049e-1762j 9.45e-154, 1.02e-386j
0.026679543490899008336 + 8.6810566825379166744e-1761j 8.08e-155, 8.69e-388j
0.030613187924530234661 - 7.9463717187483785974e-1761j 4.9e-157, 5.27e-390j
0.0034884414422285052115 + 9.7269134976964886386e-1762j 3.36e-152, 3.61e-385j
0.030613187924530234661 - 1.3739801618745864502e-1761j 2.88e-158, 3.09e-391j
0.028036987079838109886 - 1.1057450760624531954e-1761j 3.81e-157, 4.1e-390j
0.0071256728248354104291 + 8.5644247037749608081e-1760j 4.01e-152, 4.31e-385j
0.034138365927625484136 + 8.1404113382536322435e-1761j 1.23e-160, 1.33e-393j
0.025264697172602544564 - 9.1954221880057771734e-1761j 1.91e-155, 2.06e-388j
0.029345911297597195412 + 1.2295833702762750544e-1761j 6.78e-158, 7.29e-391j
0.03518707096620657044 - 2.3119363168340129242e-1761j 4.07e-162, 4.38e-395j
0.0089643771344325544969 - 1.6723093297498294986e-1761j 9.25e-155, 9.94e-388j
0.031839507748624676671 + 7.9062032088947476925e-1761j 2.71e-159, 2.92e-392j
0.033018438939476624374 - 7.9662662738651766579e-1761j 2.95e-160, 3.18e-393j
0.020675101355583384733 + 6.8434965940391965615e-1762j 3.15e-156, 3.39e-389j
0.034138365927625484136 + 1.9988516839371535452e-1761j 9.62e-162, 1.03e-394j
0.031839507748624676671 + 1.5442254592291149227e-1761j 3.7e-160, 3.98e-393j
0.041534058548745681668 + 3.3180573742286033651e-1760j 1.79e-168, 1.92e-401j
0.03705073808753202216 - 9.5860846369598484969e-1761j 1.09e-165, 1.18e-398j
0.03518707096620657044 - 8.4494267993270255885e-1761j 8.48e-164, 9.11e-397j
0.041847439271721812397 - 3.471322143265852996e-1760j 3.29e-170, 3.54e-403j
0.037875134722857757431 + 3.898198027879471845e-1761j 2.57e-168, 2.76e-401j
0.04205687707027648853 + 6.411643298732910638e-1760j 2.09e-170, 2.25e-403j
0.036157429030320321806 + 8.919911868826785452e-1761j 6.84e-166, 7.36e-399j
0.037875134722857757431 + 1.0496411954977225282e-1760j 7.32e-168, 7.87e-401j
0.020185126623438251892 - 4.0727252284130546194e-1757j 7.07e-172, 7.6e-405j
0.03705073808753202216 - 3.2245519541920455317e-1761j 1.11e-168, 1.2e-401j
0.036157429030320321806 + 2.7102953950215532243e-1761j 5.58e-168, 6e-401j
0.042002580859928921563 + 6.7211419259931704866e-1759j 1.55e-172, 1.67e-405j
0.038640313785407878888 - 1.1727115555245801313e-1760j 7.81e-171, 8.4e-404j
0.042157030134640915482 - 1.7834153226373927942e-1759j 7.45e-173, 8.02e-406j
0.038640313785407878888 - 4.7971529383566551184e-1761j 1.13e-171, 1.21e-404j
0.039352150776063699473 + 6.0281798180625542675e-1761j 1.36e-172, 1.46e-405j
0.042002580859928921563 + 6.161625497177261947e-1759j 8.4e-174, 9.03e-407j
0.039352150776063699473 + 1.3406194066283761419e-1760j 1.19e-173, 1.28e-406j
0.041116196161189976316 - 2.445745348430628599e-1760j 9.36e-175, 1.01e-407j
0.042157030134640915482 - 1.49997679661828886e-1759j 2.47e-174, 2.66e-407j
0.040009403126751390546 - 7.7748694278156914871e-1761j 5.84e-175, 6.27e-408j
0.041534058548745681668 + 2.1464087840981913039e-1760j 2e-175, 2.16e-408j
0.04205687707027648853 + 8.3144573501329333116e-1760j 6.03e-176, 6.49e-409j
0.040009403126751390546 - 1.575692962346434182e-1760j 3.9e-176, 4.19e-409j
0.020185126623438251892 - 4.1041671246886950765e-1757j 1.89e-174, 2.03e-407j
0.041116196161189976316 - 1.4471719544674662593e-1760j 4.68e-177, 5.03e-410j
0.040602646075946892608 + 1.0371907330260971272e-1760j 4.09e-177, 4.4e-410j
0.0030119915516666671195 + 8.0713268024360521429e-1757j 2.8e-174, 3.01e-407j
0.040602646075946892608 + 1.9183447518787190419e-1760j 1.21e-178, 1.31e-411j
0.041847439271721812397 - 4.9129544343538245317e-1760j 3.64e-178, 3.89e-411j

Download

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