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Nodes and weights of Gauss–Lobatto quadrature
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Numbers
$n$
$k$
$x_k$ or $w_k$
2
1
$x_k$:
-1
comment: the trapezoidal rule
equals: Integers#-1
2
1
$w_k$:
1
comment: $w_1=1$, the trapezoidal rule
equals: One
2
2
$x_k$:
1
equals: One
2
2
$w_k$:
1
comment: $w_2=1$
equals: One
3
1
$x_k$:
-1
equals: Integers#-1
3
1
$w_k$:
1/3
comment: $w_1=1/3$, Simpson's rule
3
2
$x_k$:
0
comment: $x_2=0$: Simpson's rule
equals: Zero
3
2
$w_k$:
4/3
comment: $w_2=4/3$
3
3
$x_k$:
1
equals: One
3
3
$w_k$:
1/3
comment: $w_3=1/3$
4
1
$x_k$:
-1
equals: Integers#-1
4
1
$w_k$:
1/6
comment: $w_1=1/6$
4
2
$x_k$:
-0.4472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
comment: $x_2=-1/\sqrt{5}$
equals: Algebraic_numbers_of_degree_2#5,0,-1,1
4
2
$w_k$:
5/6
comment: $w_2=5/6$
4
3
$x_k$:
0.4472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
comment: $x_3=1/\sqrt{5}$
equals: Algebraic_numbers_of_degree_2#5,0,-1,2
4
3
$w_k$:
5/6
comment: $w_3=5/6$
4
4
$x_k$:
1
equals: One
4
4
$w_k$:
1/6
comment: $w_4=1/6$
5
1
$x_k$:
-1
equals: Integers#-1
5
1
$w_k$:
1/10
comment: $w_1=1/10$
5
2
$x_k$:
-0.6546536707079771437982924562468583555692080823954245575153203034152669179353958409434802227847778618
comment: $x_2=-\sqrt{3/7}$
5
2
$w_k$:
49/90
comment: $w_2=49/90$
5
3
$x_k$:
0
equals: Zero
5
3
$w_k$:
32/45
comment: $w_3=32/45$
5
4
$x_k$:
0.6546536707079771437982924562468583555692080823954245575153203034152669179353958409434802227847778618
comment: $x_4=\sqrt{3/7}$
5
4
$w_k$:
49/90
comment: $w_4=49/90$
5
5
$x_k$:
1
equals: One
5
5
$w_k$:
1/10
comment: $w_5=1/10$
6
1
$x_k$:
-1
equals: Integers#-1
6
1
$w_k$:
1/15
comment: $w_1=1/15$
6
2
$x_k$:
-0.7650553239294646928510029739593381503657356885361042393275823558370553170388265141234972707297338731
comment: $x_2=-\sqrt{\tfrac13+\tfrac{2\sqrt{7}}{21}}$
6
2
$w_k$:
0.3784749562978469803166128082120246524763246938972516606543888513599643725589905457413202371175152130
comment: $w_2=(14-\sqrt{7})/30$
6
3
$x_k$:
-0.2852315164806450963141509940408790719190034727264253918197504165276979309141995981816757280050630173
comment: $x_3=-\sqrt{\tfrac13-\tfrac{2\sqrt{7}}{21}}$
6
3
$w_k$:
0.5548583770354863530167205251213086808570086394360816726789444819733689607743427875920130962158181204
comment: $w_3=(14+\sqrt{7})/30$
6
4
$x_k$:
0.2852315164806450963141509940408790719190034727264253918197504165276979309141995981816757280050630173
comment: $x_4=\sqrt{\tfrac13-\tfrac{2\sqrt{7}}{21}}$
6
4
$w_k$:
0.5548583770354863530167205251213086808570086394360816726789444819733689607743427875920130962158181204
comment: $w_4=(14+\sqrt{7})/30$
6
5
$x_k$:
0.7650553239294646928510029739593381503657356885361042393275823558370553170388265141234972707297338731
comment: $x_5=\sqrt{\tfrac13+\tfrac{2\sqrt{7}}{21}}$
6
5
$w_k$:
0.3784749562978469803166128082120246524763246938972516606543888513599643725589905457413202371175152130
comment: $w_5=(14-\sqrt{7})/30$
6
6
$x_k$:
1
equals: One
6
6
$w_k$:
1/15
comment: $w_6=1/15$
7
1
$x_k$:
-1
equals: Integers#-1
7
1
$w_k$:
1/21
comment: $w_1=1/21$
7
2
$x_k$:
-0.8302238962785669298720322139674651395871703648719960915908132120328845201726904752621497012460797537
comment: $x_2=-\sqrt{\tfrac{5}{11}+\tfrac{2}{11}\sqrt{5/3}}$
7
2
$w_k$:
0.2768260473615659480107004062900662934976272801798824691825342389634446238755461336153285381771122439
comment: $w_2=(124-7\sqrt{15})/350$
7
3
$x_k$:
-0.4688487934707142138037718819087663294055974716718447571026694593629555301171277087967223357575895753
comment: $x_3=-\sqrt{\tfrac{5}{11}-\tfrac{2}{11}\sqrt{5/3}}$
7
3
$w_k$:
0.4317453812098626234178710222813622779309441483915461022460371896079839475530252949561000332514591846
comment: $w_3=(124+7\sqrt{15})/350$
7
4
$x_k$:
0
equals: Zero
7
4
$w_k$:
256/525
comment: $w_4=256/525$
7
5
$x_k$:
0.4688487934707142138037718819087663294055974716718447571026694593629555301171277087967223357575895753
comment: $x_5=\sqrt{\tfrac{5}{11}-\tfrac{2}{11}\sqrt{5/3}}$
7
5
$w_k$:
0.4317453812098626234178710222813622779309441483915461022460371896079839475530252949561000332514591846
comment: $w_5=(124+7\sqrt{15})/350$
7
6
$x_k$:
0.8302238962785669298720322139674651395871703648719960915908132120328845201726904752621497012460797537
comment: $x_6=\sqrt{\tfrac{5}{11}+\tfrac{2}{11}\sqrt{5/3}}$
7
6
$w_k$:
0.2768260473615659480107004062900662934976272801798824691825342389634446238755461336153285381771122439
comment: $w_6=(124-7\sqrt{15})/350$
7
7
$x_k$:
1
equals: One
7
7
$w_k$:
1/21
comment: $w_7=1/21$
8
1
$x_k$:
-1
equals: Integers#-1
8
1
$w_k$:
1/28
8
2
$x_k$:
-0.8717401485096066153374457612206634381037806696769833549194152855738989940929685407261495266579023059
8
2
$w_k$:
0.2107042271435060393829920657757563244553461661610477680833405816426352499564855140066884340966389096
8
3
$x_k$:
-0.5917001814331423021445107313979531899457009895173324967422595914701481800491328878626145796220236189
8
3
$w_k$:
0.3411226924835043647642406771077481717751109756055816094750020738885508832361964289632553455870648109
8
4
$x_k$:
-0.2092992179024788687686572603453512552955454050866810988908466923221309236462829953885222153651896099
8
4
$w_k$:
0.4124587946587038815670529714022097894838285725190849081559430587545281525216037713157705060305819937
8
5
$x_k$:
0.2092992179024788687686572603453512552955454050866810988908466923221309236462829953885222153651896099
8
5
$w_k$:
0.4124587946587038815670529714022097894838285725190849081559430587545281525216037713157705060305819937
8
6
$x_k$:
0.5917001814331423021445107313979531899457009895173324967422595914701481800491328878626145796220236189
8
6
$w_k$:
0.3411226924835043647642406771077481717751109756055816094750020738885508832361964289632553455870648109
8
7
$x_k$:
0.8717401485096066153374457612206634381037806696769833549194152855738989940929685407261495266579023059
8
7
$w_k$:
0.2107042271435060393829920657757563244553461661610477680833405816426352499564855140066884340966389096
8
8
$x_k$:
1
equals: One
8
8
$w_k$:
1/28
9
1
$x_k$:
-1
equals: Integers#-1
9
1
$w_k$:
1/36
9
2
$x_k$:
-0.8997579954114601573123452444183379580514802955661044224293126684848132745663187182910131203124947968
9
2
$w_k$:
0.1654953615608055250463397200292083058391072292251593414462235945468554095218166317856751903722796987
9
3
$x_k$:
-0.6771862795107377534458854270913424507110296476139065795409633754970985279846502737808238783160437056
9
3
$w_k$:
0.2745387125001617352807056185793727259412512209865092865367936289129110522005503381286950946305814404
9
4
$x_k$:
-0.3631174638261781587107520687086592130206422776008781503202737307716104276337989580551330718474162880
9
4
$w_k$:
0.3464285109730463451151315321397182879475327062509164060305882187171042865769527579767861775821728745
9
5
$x_k$:
0
equals: Zero
9
5
$w_k$:
4096/11025
9
6
$x_k$:
0.3631174638261781587107520687086592130206422776008781503202737307716104276337989580551330718474162880
9
6
$w_k$:
0.3464285109730463451151315321397182879475327062509164060305882187171042865769527579767861775821728745
9
7
$x_k$:
0.6771862795107377534458854270913424507110296476139065795409633754970985279846502737808238783160437056
9
7
$w_k$:
0.2745387125001617352807056185793727259412512209865092865367936289129110522005503381286950946305814404
9
8
$x_k$:
0.8997579954114601573123452444183379580514802955661044224293126684848132745663187182910131203124947968
9
8
$w_k$:
0.1654953615608055250463397200292083058391072292251593414462235945468554095218166317856751903722796987
9
9
$x_k$:
1
equals: One
9
9
$w_k$:
1/36
10
1
$x_k$:
-1
equals: Integers#-1
10
1
$w_k$:
1/45
10
2
$x_k$:
-0.9195339081664588138289326608223381341535430754462758714896261445842674414779132813248797662240801089
10
2
$w_k$:
0.1333059908510701111262271707553928981096258749977974148878909035444654526063995820130973091708676738
10
3
$x_k$:
-0.7387738651055050750031061748598307250161851013769266854651089930551533157399098016742897862545735154
10
3
$w_k$:
0.2248893420631264521194578217310478427539129139617506411396587337100536859442260604103610979777172817
10
4
$x_k$:
-0.4779249498104444956611750927312579978867728933305664689142568396916225510884771393766715908981831667
10
4
$w_k$:
0.2920426836796837578755822573744438922074883896877075445913727105404627639886476614773619763162625393
10
5
$x_k$:
-0.1652789576663870246262197659581735332311503435494837311328995721047882964544051550467682742209501636
10
5
$w_k$:
0.3275397611838974566565105279168931447067505991305221771588554299827958752385044738769573943129302829
10
6
$x_k$:
0.1652789576663870246262197659581735332311503435494837311328995721047882964544051550467682742209501636
10
6
$w_k$:
0.3275397611838974566565105279168931447067505991305221771588554299827958752385044738769573943129302829
10
7
$x_k$:
0.4779249498104444956611750927312579978867728933305664689142568396916225510884771393766715908981831667
10
7
$w_k$:
0.2920426836796837578755822573744438922074883896877075445913727105404627639886476614773619763162625393
10
8
$x_k$:
0.7387738651055050750031061748598307250161851013769266854651089930551533157399098016742897862545735154
10
8
$w_k$:
0.2248893420631264521194578217310478427539129139617506411396587337100536859442260604103610979777172817
10
9
$x_k$:
0.9195339081664588138289326608223381341535430754462758714896261445842674414779132813248797662240801089
10
9
$w_k$:
0.1333059908510701111262271707553928981096258749977974148878909035444654526063995820130973091708676738
10
10
$x_k$:
1
equals: One
10
10
$w_k$:
1/45
11
1
$x_k$:
-1
equals: Integers#-1
11
1
$w_k$:
1/55
11
2
$x_k$:
-0.9340014304080591343322741360993836345399173301099622258916267532713560175096163419832464338313941214
11
2
$w_k$:
0.1096122732669948644614034495803507100494741436472767307977058813186330943543027718016420292839785457
11
3
$x_k$:
-0.7844834736631444186224178161084581035071974550940636216826353407685847121448766028125501919521124182
11
3
$w_k$:
0.1871698817803052041081415218994349195691942376251283186073324374328434076851509337025928732097993126
11
4
$x_k$:
-0.5652353269962050064709639694777516642830521455620169756911572480798775881488430902381620740464246016
11
4
$w_k$:
0.2480481042640283140400848664218727533436731696477138792559827219777901041393002717628900625284000859
11
5
$x_k$:
-0.2957581355869393914319115155590575089410064343486041094678993738016199617764552790264359949210823253
11
5
$w_k$:
0.2868791247790080886792224033315352311836535474559318283421169147896412936815268353946400776952457257
11
6
$x_k$:
0
equals: Zero
11
6
$w_k$:
65536/218295
11
7
$x_k$:
0.2957581355869393914319115155590575089410064343486041094678993738016199617764552790264359949210823253
11
7
$w_k$:
0.2868791247790080886792224033315352311836535474559318283421169147896412936815268353946400776952457257
11
8
$x_k$:
0.5652353269962050064709639694777516642830521455620169756911572480798775881488430902381620740464246016
11
8
$w_k$:
0.2480481042640283140400848664218727533436731696477138792559827219777901041393002717628900625284000859
11
9
$x_k$:
0.7844834736631444186224178161084581035071974550940636216826353407685847121448766028125501919521124182
11
9
$w_k$:
0.1871698817803052041081415218994349195691942376251283186073324374328434076851509337025928732097993126
11
10
$x_k$:
0.9340014304080591343322741360993836345399173301099622258916267532713560175096163419832464338313941214
11
10
$w_k$:
0.1096122732669948644614034495803507100494741436472767307977058813186330943543027718016420292839785457
11
11
$x_k$:
1
equals: One
11
11
$w_k$:
1/55
12
1
$x_k$:
-1
equals: Integers#-1
12
1
$w_k$:
1/66
12
2
$x_k$:
-0.9448992722228822234075801383032187136112565519500273237138918970765479228194259726073045436862317808
12
2
$w_k$:
0.09168451741319613066834259413407928631030614141778273862122550621980722715567697368617014897346317911
12
3
$x_k$:
-0.8192793216440066783486415817169026606904666579036442894233167261168767275911619423604924893591670874
12
3
$w_k$:
0.1579747055643701151646710627003402682511063164234571597262326661054164247768244962308749824370664735
12
4
$x_k$:
-0.6328761530318606776624048544436558582438437454015009929651535646470522420044593704451480454489772889
12
4
$w_k$:
0.2125084177610211453583020773668662701953456227653408334960748032543605105597134203445710317272840054
12
5
$x_k$:
-0.3995309409653489322643497915669669005277480327953070364879376720500962439073847761953217745808715195
12
5
$w_k$:
0.2512756031992012802932444121475961833334384773498036455703232861442177504950262627958231949368519054
12
6
$x_k$:
-0.1365529328549275548640618557396938968984141112820583854515815660670527439020029298850214139374604924
12
6
$w_k$:
0.2714052409106961770002883384996028403946519268921004710709922231246829354976073317910454904101829214
12
7
$x_k$:
0.1365529328549275548640618557396938968984141112820583854515815660670527439020029298850214139374604924
12
7
$w_k$:
0.2714052409106961770002883384996028403946519268921004710709922231246829354976073317910454904101829214
12
8
$x_k$:
0.3995309409653489322643497915669669005277480327953070364879376720500962439073847761953217745808715195
12
8
$w_k$:
0.2512756031992012802932444121475961833334384773498036455703232861442177504950262627958231949368519054
12
9
$x_k$:
0.6328761530318606776624048544436558582438437454015009929651535646470522420044593704451480454489772889
12
9
$w_k$:
0.2125084177610211453583020773668662701953456227653408334960748032543605105597134203445710317272840054
12
10
$x_k$:
0.8192793216440066783486415817169026606904666579036442894233167261168767275911619423604924893591670874
12
10
$w_k$:
0.1579747055643701151646710627003402682511063164234571597262326661054164247768244962308749824370664735
12
11
$x_k$:
0.9448992722228822234075801383032187136112565519500273237138918970765479228194259726073045436862317808
12
11
$w_k$:
0.09168451741319613066834259413407928631030614141778273862122550621980722715567697368617014897346317911
12
12
$x_k$:
1
equals: One
12
12
$w_k$:
1/66
13
1
$x_k$:
-1
equals: Integers#-1
13
1
$w_k$:
1/78
13
2
$x_k$:
-0.9533098466421639118969054647554491516265078886973631851893726028508995822987312174530645080392673580
13
2
$w_k$:
0.07780168674681892779358898833313358506783481749350292104662574588315140490751679338744019555597529621
13
3
$x_k$:
-0.8463475646518723168659256070987533595780366597144086256886470126661065700625862600658774829431382265
13
3
$w_k$:
0.1349819266896083491199147625893708892561726920926794185228045007573994402618228391567222045528187995
13
4
$x_k$:
-0.6861884690817574260727590395663555529291761981243844270626913458954686580001189492394932932293524224
13
4
$w_k$:
0.1836468652035500920074942587468107232464736869682379991450275214582184575635877139109345312997112396
13
5
$x_k$:
-0.4829098210913362017469372336369336207721932621185900104777240920488131791072973870944467798255766930
13
5
$w_k$:
0.2207677935661100860855340083793962974672628733339952047316294081877186991343448769715975904131790357
13
6
$x_k$:
-0.2492869301062399925686737003742269814888113124929758592985159086337259627994991444191134230101181864
13
6
$w_k$:
0.2440157903066763564585781483601562125312622004165560602268151979145874979095262546704848740590102110
13
7
$x_k$:
0
equals: Zero
13
7
$w_k$:
524288/2081079
13
8
$x_k$:
0.2492869301062399925686737003742269814888113124929758592985159086337259627994991444191134230101181864
13
8
$w_k$:
0.2440157903066763564585781483601562125312622004165560602268151979145874979095262546704848740590102110
13
9
$x_k$:
0.4829098210913362017469372336369336207721932621185900104777240920488131791072973870944467798255766930
13
9
$w_k$:
0.2207677935661100860855340083793962974672628733339952047316294081877186991343448769715975904131790357
13
10
$x_k$:
0.6861884690817574260727590395663555529291761981243844270626913458954686580001189492394932932293524224
13
10
$w_k$:
0.1836468652035500920074942587468107232464736869682379991450275214582184575635877139109345312997112396
13
11
$x_k$:
0.8463475646518723168659256070987533595780366597144086256886470126661065700625862600658774829431382265
13
11
$w_k$:
0.1349819266896083491199147625893708892561726920926794185228045007573994402618228391567222045528187995
13
12
$x_k$:
0.9533098466421639118969054647554491516265078886973631851893726028508995822987312174530645080392673580
13
12
$w_k$:
0.07780168674681892779358898833313358506783481749350292104662574588315140490751679338744019555597529621
13
13
$x_k$:
1
equals: One
13
13
$w_k$:
1/78
14
1
$x_k$:
-1
equals: Integers#-1
14
1
$w_k$:
1/91
14
2
$x_k$:
-0.9599350452672609013551001620154243890663915185726479363641582338520266975949515550398084572643659674
14
2
$w_k$:
0.06683728449768128463407066074605282505764205345068243384444966606018453118350095103439416597173414204
14
3
$x_k$:
-0.8678010538303472510002202029082642132498723530944361347588261841082505336490490534407108649721395669
14
3
$w_k$:
0.1165866558987116515409966706546502300047058549000732791952179064708648155750113438808879397500121359
14
4
$x_k$:
-0.7288685990913261405846724005208815956573395316943218026122439652276387638101153919729865082928397945
14
4
$w_k$:
0.1600218517629521424128209979875946405038025496328810701687340445386838606608999549224029807244747607
14
5
$x_k$:
-0.5506394029286470553166227058590806344621383195539141338237929139680389573232203183613676618968424806
14
5
$w_k$:
0.1948261493734161186403317783758845121486287560518647562764465495406211999280006869472617741128982296
14
6
$x_k$:
-0.3427240133427128450439034036416746448331135341403067465549148328709868936637677405603405447947626805
14
6
$w_k$:
0.2191262530097707548711625239541676129583321144588758261361451439123298155048813181293538909803852819
14
7
$x_k$:
-0.1163318688837038676587767097361601679415090442562807682033248605879716226627798595377186650776445107
14
7
$w_k$:
0.2316127944684570588896283572926391903158996605166116453680176784883047881367167340966882594495064389
14
8
$x_k$:
0.1163318688837038676587767097361601679415090442562807682033248605879716226627798595377186650776445107
14
8
$w_k$:
0.2316127944684570588896283572926391903158996605166116453680176784883047881367167340966882594495064389
14
9
$x_k$:
0.3427240133427128450439034036416746448331135341403067465549148328709868936637677405603405447947626805
14
9
$w_k$:
0.2191262530097707548711625239541676129583321144588758261361451439123298155048813181293538909803852819
14
10
$x_k$:
0.5506394029286470553166227058590806344621383195539141338237929139680389573232203183613676618968424806
14
10
$w_k$:
0.1948261493734161186403317783758845121486287560518647562764465495406211999280006869472617741128982296
14
11
$x_k$:
0.7288685990913261405846724005208815956573395316943218026122439652276387638101153919729865082928397945
14
11
$w_k$:
0.1600218517629521424128209979875946405038025496328810701687340445386838606608999549224029807244747607
14
12
$x_k$:
0.8678010538303472510002202029082642132498723530944361347588261841082505336490490534407108649721395669
14
12
$w_k$:
0.1165866558987116515409966706546502300047058549000732791952179064708648155750113438808879397500121359
14
13
$x_k$:
0.9599350452672609013551001620154243890663915185726479363641582338520266975949515550398084572643659674
14
13
$w_k$:
0.06683728449768128463407066074605282505764205345068243384444966606018453118350095103439416597173414204
14
14
$x_k$:
1
equals: One
14
14
$w_k$:
1/91
15
1
$x_k$:
-1
equals: Integers#-1
15
1
$w_k$:
1/105
15
2
$x_k$:
-0.9652459265038385727958513920696011777076501359970933400724984584271122372020720352005295295355551906
15
2
$w_k$:
0.05802989302860124909688058402528199543568672682601490377257574548226308682533611555027521323114848197
15
3
$x_k$:
-0.8850820442229762988254016314822296519887140852074779728173816306810879949568031777118343319627061712
15
3
$w_k$:
0.1016600703257180676036661707888007154809835937166059049922191940406373316436137750309801948850106158
15
4
$x_k$:
-0.7635196899518152007041184759762916181773685203152903367547056050470770910393527506063639020211546339
15
4
$w_k$:
0.1405116998024281094604468056436728906234843546024079581894227450724808656700394546742072539319780871
15
5
$x_k$:
-0.6062532054698457111235299386367335071797310337599197856858295039791967797105383640859511993841613812
15
5
$w_k$:
0.1727896472536009490520770994083505873566356630268052907284171476973541422586303642734439167856134124
15
6
$x_k$:
-0.4206380547136724809218969387385804129843382054924303683850933609375690682585863152230475106616618212
15
6
$w_k$:
0.1969872359646133560925003465074065948636039508334904284696850653209466689043360995088021224300942184
15
7
$x_k$:
-0.2153539553637942382256794462729177126521579012030410634745051476417103556667084443595313539535215865
15
7
$w_k$:
0.2119735859268209201274300769772209166236557453943602631827790878349537029841921274598781459733520719
15
8
$x_k$:
0
equals: Zero
15
8
$w_k$:
4194304/19324305
15
9
$x_k$:
0.2153539553637942382256794462729177126521579012030410634745051476417103556667084443595313539535215865
15
9
$w_k$:
0.2119735859268209201274300769772209166236557453943602631827790878349537029841921274598781459733520719
15
10
$x_k$:
0.4206380547136724809218969387385804129843382054924303683850933609375690682585863152230475106616618212
15
10
$w_k$:
0.1969872359646133560925003465074065948636039508334904284696850653209466689043360995088021224300942184
15
11
$x_k$:
0.6062532054698457111235299386367335071797310337599197856858295039791967797105383640859511993841613812
15
11
$w_k$:
0.1727896472536009490520770994083505873566356630268052907284171476973541422586303642734439167856134124
15
12
$x_k$:
0.7635196899518152007041184759762916181773685203152903367547056050470770910393527506063639020211546339
15
12
$w_k$:
0.1405116998024281094604468056436728906234843546024079581894227450724808656700394546742072539319780871
15
13
$x_k$:
0.8850820442229762988254016314822296519887140852074779728173816306810879949568031777118343319627061712
15
13
$w_k$:
0.1016600703257180676036661707888007154809835937166059049922191940406373316436137750309801948850106158
15
14
$x_k$:
0.9652459265038385727958513920696011777076501359970933400724984584271122372020720352005295295355551906
15
14
$w_k$:
0.05802989302860124909688058402528199543568672682601490377257574548226308682533611555027521323114848197
15
15
$x_k$:
1
equals: One
15
15
$w_k$:
1/105
16
1
$x_k$:
-1
equals: Integers#-1
16
1
$w_k$:
1/120
16
2
$x_k$:
-0.9695680462702179329522427383674592413889907465038347300686787322825410518539767563204383759083418474
16
2
$w_k$:
0.05085036100591990540324491956545531449387182753778698391280669444405898738303777494474467401745527689
16
3
$x_k$:
-0.8992005330934720929946282615198494767499976090451444254244274117884763971828569153067399167695146826
16
3
$w_k$:
0.08939369732593080099105208016608371493388721003779897847694466427930268399363454663689488039156834394
16
4
$x_k$:
-0.7920082918618150639310882709631457058080738279801990226844106498659962900279950004423665805791327572
16
4
$w_k$:
0.1242553821325140983495363326573132007743980312698995402106772728062540631179359845262497012517331687
16
5
$x_k$:
-0.6523887028824930894678832196405814803215580128295743373857812842222370430804283345794668654782173996
16
5
$w_k$:
0.1540269808071642808156449404849945154454162419836413228570130454501063688327650429031557807768682357
16
6
$x_k$:
-0.4860594218871376117818907858468746968889773042982525713766568676067844974335224194567516219224068002
16
6
$w_k$:
0.1774919133917041253010756695283577701583147451761712135149639501665627333512951466881384462967015472
16
7
$x_k$:
-0.2998304689007632080983534547223006478154609769077776646543100976440408769710927899653410977957777990
16
7
$w_k$:
0.1936900238252035843169135988535220303063579940184073251836659133469699576859756035822499491078971064
16
8
$x_k$:
-0.1013262735219494478430330050459177625332409144001911399175502002543079173795781589887611675908666249
16
8
$w_k$:
0.2019583081782298714891991254109401205544206166429613025105951261734118723020225673852332348244429878
16
9
$x_k$:
0.1013262735219494478430330050459177625332409144001911399175502002543079173795781589887611675908666249
16
9
$w_k$:
0.2019583081782298714891991254109401205544206166429613025105951261734118723020225673852332348244429878
16
10
$x_k$:
0.2998304689007632080983534547223006478154609769077776646543100976440408769710927899653410977957777990
16
10
$w_k$:
0.1936900238252035843169135988535220303063579940184073251836659133469699576859756035822499491078971064
16
11
$x_k$:
0.4860594218871376117818907858468746968889773042982525713766568676067844974335224194567516219224068002
16
11
$w_k$:
0.1774919133917041253010756695283577701583147451761712135149639501665627333512951466881384462967015472
16
12
$x_k$:
0.6523887028824930894678832196405814803215580128295743373857812842222370430804283345794668654782173996
16
12
$w_k$:
0.1540269808071642808156449404849945154454162419836413228570130454501063688327650429031557807768682357
16
13
$x_k$:
0.7920082918618150639310882709631457058080738279801990226844106498659962900279950004423665805791327572
16
13
$w_k$:
0.1242553821325140983495363326573132007743980312698995402106772728062540631179359845262497012517331687
16
14
$x_k$:
0.8992005330934720929946282615198494767499976090451444254244274117884763971828569153067399167695146826
16
14
$w_k$:
0.08939369732593080099105208016608371493388721003779897847694466427930268399363454663689488039156834394
16
15
$x_k$:
0.9695680462702179329522427383674592413889907465038347300686787322825410518539767563204383759083418474
16
15
$w_k$:
0.05085036100591990540324491956545531449387182753778698391280669444405898738303777494474467401745527689
16
16
$x_k$:
1
equals: One
16
16
$w_k$:
1/120
17
1
$x_k$:
-1
equals: Integers#-1
17
1
$w_k$:
1/136
17
2
$x_k$:
-0.9731321766314183141569795018737214305889591491225094864464918430960781374664116321261968810329806435
17
2
$w_k$:
0.04492194054325420964740095462321244331819970844427089849666792409495706693606108109358522113048760331
17
3
$x_k$:
-0.9108799959155735956238025063977264675308794518687327643218125593109666600726691632381709537014958992
17
3
$w_k$:
0.07919827050368711919026442995283450409836421852684677352252635517763809122039863582189172810601324529
17
4
$x_k$:
-0.8156962512217703071067505532375266547164023970671235581498639615868404795860011773165331690580634415
17
4
$w_k$:
0.1105929090070281613757727052200768036394322245072764301600690330496478571575715630182088299060896579
17
5
$x_k$:
-0.6910289806276847053949193573724532968064130621904154284073980972380380447834283064689255536088149186
17
5
$w_k$:
0.1379877462019265590562015749540251706659032657800088053930789374866468747149965365312515201173873884
17
6
$x_k$:
-0.5413853993301015391237334075040632516751466479648311453209454114543599741873482218201321309867412673
17
6
$w_k$:
0.1603946619976215395163283658647512902658180413173828064999139015213634690245882310907782726512343926
17
7
$x_k$:
-0.3721744335654770419072346807352578125598173144002805734651033435799707238838426135335731144688919649
17
7
$w_k$:
0.1770042535156578704369457453632924918210764791784449377000611086187065631847131282967432592265146736
17
8
$x_k$:
-0.1895119735183173883042630147531139713449924229224985705778712711676797919215154129754950986657078208
17
8
$w_k$:
0.1872163396776192358920884828606217362671297525942505920880572736738940077594373775153560202915492590
17
9
$x_k$:
0
equals: Zero
17
9
$w_k$:
134217728/703956825
17
10
$x_k$:
0.1895119735183173883042630147531139713449924229224985705778712711676797919215154129754950986657078208
17
10
$w_k$:
0.1872163396776192358920884828606217362671297525942505920880572736738940077594373775153560202915492590
17
11
$x_k$:
0.3721744335654770419072346807352578125598173144002805734651033435799707238838426135335731144688919649
17
11
$w_k$:
0.1770042535156578704369457453632924918210764791784449377000611086187065631847131282967432592265146736
17
12
$x_k$:
0.5413853993301015391237334075040632516751466479648311453209454114543599741873482218201321309867412673
17
12
$w_k$:
0.1603946619976215395163283658647512902658180413173828064999139015213634690245882310907782726512343926
17
13
$x_k$:
0.6910289806276847053949193573724532968064130621904154284073980972380380447834283064689255536088149186
17
13
$w_k$:
0.1379877462019265590562015749540251706659032657800088053930789374866468747149965365312515201173873884
17
14
$x_k$:
0.8156962512217703071067505532375266547164023970671235581498639615868404795860011773165331690580634415
17
14
$w_k$:
0.1105929090070281613757727052200768036394322245072764301600690330496478571575715630182088299060896579
17
15
$x_k$:
0.9108799959155735956238025063977264675308794518687327643218125593109666600726691632381709537014958992
17
15
$w_k$:
0.07919827050368711919026442995283450409836421852684677352252635517763809122039863582189172810601324529
17
16
$x_k$:
0.9731321766314183141569795018737214305889591491225094864464918430960781374664116321261968810329806435
17
16
$w_k$:
0.04492194054325420964740095462321244331819970844427089849666792409495706693606108109358522113048760331
17
17
$x_k$:
1
equals: One
17
17
$w_k$:
1/136
18
1
$x_k$:
-1
equals: Integers#-1
18
1
$w_k$:
1/153
18
2
$x_k$:
-0.9761055574121985428645189243417000667618134427191940954825046942975192557171124374628083252940893780
18
2
$w_k$:
0.03997062881091406613759917641010087929430544990845948510148994164624314984747500296887783956622134618
18
3
$x_k$:
-0.9206491853475338738378546254312774235623534861890405934475841223054291520230014944589235694027162113
18
3
$w_k$:
0.07063716688563366499922296016778633310129662626355358376892146181051992236508790829383137852268796319
18
4
$x_k$:
-0.8355935352180902137136463623279372574336707591658185577284920511394500084472105361127638894325545808
18
4
$w_k$:
0.09901627171750280239442360531867210791806868286343937095094187295057360608766283976824358945526015914
18
5
$x_k$:
-0.7236793292832426813062103653020706791495252041547588903174835466501650007237435088644679504692246921
18
5
$w_k$:
0.1242105331329671002633963588967467799607895515050267557271609883105250171919116545300048348341250964
18
6
$x_k$:
-0.5885048343186617611735358931935594690008367893162179067642833939821063801398583147610889659324827265
18
6
$w_k$:
0.1454119615738022679830032104944267724854777170655443028789033067797931689226415372262860560813936724
18
7
$x_k$:
-0.4344150369121239753422871367406747958497584451636926419439426526195414335125734303778433004091653531
18
7
$w_k$:
0.1619395172376024892643267067002284719607940715542834454140771451345940156203622886084477330151528578
18
8
$x_k$:
-0.2663626528782809841676653320255959420651361893182584476224536316082908162510753387747628916372289639
18
8
$w_k$:
0.1732621094894562260106144038266834217245773588113460686184129644593026460652447306231091420282444931
18
9
$x_k$:
-0.08974909348465211102264501008856173496060390104112480446895797164167366419984590869078556863156238078
18
9
$w_k$:
0.1790158634397030822938188069435251681952134178453404515923799006077948791283722079158399493727314053
18
10
$x_k$:
0.08974909348465211102264501008856173496060390104112480446895797164167366419984590869078556863156238078
18
10
$w_k$:
0.1790158634397030822938188069435251681952134178453404515923799006077948791283722079158399493727314053
18
11
$x_k$:
0.2663626528782809841676653320255959420651361893182584476224536316082908162510753387747628916372289639
18
11
$w_k$:
0.1732621094894562260106144038266834217245773588113460686184129644593026460652447306231091420282444931
18
12
$x_k$:
0.4344150369121239753422871367406747958497584451636926419439426526195414335125734303778433004091653531
18
12
$w_k$:
0.1619395172376024892643267067002284719607940715542834454140771451345940156203622886084477330151528578
18
13
$x_k$:
0.5885048343186617611735358931935594690008367893162179067642833939821063801398583147610889659324827265
18
13
$w_k$:
0.1454119615738022679830032104944267724854777170655443028789033067797931689226415372262860560813936724
18
14
$x_k$:
0.7236793292832426813062103653020706791495252041547588903174835466501650007237435088644679504692246921
18
14
$w_k$:
0.1242105331329671002633963588967467799607895515050267557271609883105250171919116545300048348341250964
18
15
$x_k$:
0.8355935352180902137136463623279372574336707591658185577284920511394500084472105361127638894325545808
18
15
$w_k$:
0.09901627171750280239442360531867210791806868286343937095094187295057360608766283976824358945526015914
18
16
$x_k$:
0.9206491853475338738378546254312774235623534861890405934475841223054291520230014944589235694027162113
18
16
$w_k$:
0.07063716688563366499922296016778633310129662626355358376892146181051992236508790829383137852268796319
18
17
$x_k$:
0.9761055574121985428645189243417000667618134427191940954825046942975192557171124374628083252940893780
18
17
$w_k$:
0.03997062881091406613759917641010087929430544990845948510148994164624314984747500296887783956622134618
18
18
$x_k$:
1
equals: One
18
18
$w_k$:
1/153
19
1
$x_k$:
-1
equals: Integers#-1
19
1
$w_k$:
1/171
19
2
$x_k$:
-0.9786117662220800951526340631102225628142773378108094414443306485152804533162302589498854902546061857
19
2
$w_k$:
0.03579336518617647711542556903512235646156658500968525039796127294172259843582075878628685651251345034
19
3
$x_k$:
-0.9289015281525862437179402587965486124501681822519506066895230057440856850793788137292456869372531746
19
3
$w_k$:
0.06338189176262973685169569041831737955798443274103630816370259328297731814282293564853866256619607653
19
4
$x_k$:
-0.8524605777966460930859559700410626252370953808388652482693253050694431716255079857107952560344951489
19
4
$w_k$:
0.08913175709920708444800879055615317792087774316940079191366058323559026732267529615787347264719680884
19
5
$x_k$:
-0.7514942025526130141636374896339440404036593556657957034300796078416135779710499247782082573304419544
19
5
$w_k$:
0.1123153414773050440709100154637810241907728592260972924201908316228576855916929055282125499382275452
19
6
$x_k$:
-0.6289081372652204977668323062287325470686111571895613469126061034682481710781251690820010236774468912
19
6
$w_k$:
0.1322672804487507769260467339097253636246054196796275817420815554334471884901555646776269786835375322
19
7
$x_k$:
-0.4882292856807135027779096376249233697712155996514848064311705188483890220850554576473213248673913366
19
7
$w_k$:
0.1484139425959388850096806436684098502919324452218966812371990774556141519658960160378338916707633362
19
8
$x_k$:
-0.3335048478244986102985001038449270119229633754777301189473097117939650883999383699854134201855343162
19
8
$w_k$:
0.1602909240440612419799109681835940193731934344161586656537799158636978919904602147225321961195606333
19
9
$x_k$:
-0.1691860234092815713751541534448804237528955507658513527604511669169941239106128406482886220525586527
19
9
$w_k$:
0.1675565845271428672701372777402609494132705547366170166756278863371727979253377060331023853944441117
19
10
$x_k$:
0
equals: Zero
19
10
$w_k$:
4294967296/25264228275
19
11
$x_k$:
0.1691860234092815713751541534448804237528955507658513527604511669169941239106128406482886220525586527
19
11
$w_k$:
0.1675565845271428672701372777402609494132705547366170166756278863371727979253377060331023853944441117
19
12
$x_k$:
0.3335048478244986102985001038449270119229633754777301189473097117939650883999383699854134201855343162
19
12
$w_k$:
0.1602909240440612419799109681835940193731934344161586656537799158636978919904602147225321961195606333
19
13
$x_k$:
0.4882292856807135027779096376249233697712155996514848064311705188483890220850554576473213248673913366
19
13
$w_k$:
0.1484139425959388850096806436684098502919324452218966812371990774556141519658960160378338916707633362
19
14
$x_k$:
0.6289081372652204977668323062287325470686111571895613469126061034682481710781251690820010236774468912
19
14
$w_k$:
0.1322672804487507769260467339097253636246054196796275817420815554334471884901555646776269786835375322
19
15
$x_k$:
0.7514942025526130141636374896339440404036593556657957034300796078416135779710499247782082573304419544
19
15
$w_k$:
0.1123153414773050440709100154637810241907728592260972924201908316228576855916929055282125499382275452
19
16
$x_k$:
0.8524605777966460930859559700410626252370953808388652482693253050694431716255079857107952560344951489
19
16
$w_k$:
0.08913175709920708444800879055615317792087774316940079191366058323559026732267529615787347264719680884
19
17
$x_k$:
0.9289015281525862437179402587965486124501681822519506066895230057440856850793788137292456869372531746
19
17
$w_k$:
0.06338189176262973685169569041831737955798443274103630816370259328297731814282293564853866256619607653
19
18
$x_k$:
0.9786117662220800951526340631102225628142773378108094414443306485152804533162302589498854902546061857
19
18
$w_k$:
0.03579336518617647711542556903512235646156658500968525039796127294172259843582075878628685651251345034
19
19
$x_k$:
1
equals: One
19
19
$w_k$:
1/171
20
1
$x_k$:
-1
equals: Integers#-1
20
1
$w_k$:
1/190
20
2
$x_k$:
-0.9807437048939141719254464385842309152299106231262486805732511888660871853057070601746047987326822398
20
2
$w_k$:
0.03223712318848894149160502811729403010325214029274621910200615103781913033476216867736864816585742596
20
3
$x_k$:
-0.9359344988126654357161815849306269299155738331810513579118471447570406483015670106630777640195763982
20
3
$w_k$:
0.05718180212756682600475362717324278250520601408816342919406943044379967034483826130586578659612208976
20
4
$x_k$:
-0.8668779780899501413098472146162852139629112883169920640305565984640075517954652921730675138650575987
20
4
$w_k$:
0.08063176399611960314477684611372057166603645034176204013283038596658160861480052110568843771151228787
20
5
$x_k$:
-0.7753682609520558704143175275946913433727218594765318265521851154726658175635403801558182531648736052
20
5
$w_k$:
0.1019914996994508156837812057328865208675063489620363885846534344298637233594818658163092541177193776
20
6
$x_k$:
-0.6637764022903112898464033229711588524757457419914912939199393702444361536718001763291823151336837192
20
6
$w_k$:
0.1207092276286747250994297050023933598723181358251691366493842022023988888559672631523110261217489803
20
7
$x_k$:
-0.5349928640318862616481359618289839830068515691375233260866881776497631641017200824081116481766882507
20
7
$w_k$:
0.1363004823587241844897807929890319817440766626796339938277388454517972257701613882435036973258789551
20
8
$x_k$:
-0.3923531837139092993864747038158243666652033292989106133402097184674561726688079711978709630419496086
20
8
$w_k$:
0.1483615540709168258147130137339666049311512942338487174164589814639023010138999385837811946659067987
20
9
$x_k$:
-0.2395517059229864951824013569270880719415178099273774376727425552936633390663940111807541925660964244
20
9
$w_k$:
0.1565801026474754871581698967936433723030979365909109506698204019536450222730992517894454345196432786
20
10
$x_k$:
-0.08054593723882183797594451815955446302239287009290846501883204471692448342994898660817383393791043247
20
10
$w_k$:
0.1607432863878457490077267264490839339020918590909922823177750091554555873277261834309896786703476482
20
11
$x_k$:
0.08054593723882183797594451815955446302239287009290846501883204471692448342994898660817383393791043247
20
11
$w_k$:
0.1607432863878457490077267264490839339020918590909922823177750091554555873277261834309896786703476482
20
12
$x_k$:
0.2395517059229864951824013569270880719415178099273774376727425552936633390663940111807541925660964244
20
12
$w_k$:
0.1565801026474754871581698967936433723030979365909109506698204019536450222730992517894454345196432786
20
13
$x_k$:
0.3923531837139092993864747038158243666652033292989106133402097184674561726688079711978709630419496086
20
13
$w_k$:
0.1483615540709168258147130137339666049311512942338487174164589814639023010138999385837811946659067987
20
14
$x_k$:
0.5349928640318862616481359618289839830068515691375233260866881776497631641017200824081116481766882507
20
14
$w_k$:
0.1363004823587241844897807929890319817440766626796339938277388454517972257701613882435036973258789551
20
15
$x_k$:
0.6637764022903112898464033229711588524757457419914912939199393702444361536718001763291823151336837192
20
15
$w_k$:
0.1207092276286747250994297050023933598723181358251691366493842022023988888559672631523110261217489803
20
16
$x_k$:
0.7753682609520558704143175275946913433727218594765318265521851154726658175635403801558182531648736052
20
16
$w_k$:
0.1019914996994508156837812057328865208675063489620363885846534344298637233594818658163092541177193776
20
17
$x_k$:
0.8668779780899501413098472146162852139629112883169920640305565984640075517954652921730675138650575987
20
17
$w_k$:
0.08063176399611960314477684611372057166603645034176204013283038596658160861480052110568843771151228787
20
18
$x_k$:
0.9359344988126654357161815849306269299155738331810513579118471447570406483015670106630777640195763982
20
18
$w_k$:
0.05718180212756682600475362717324278250520601408816342919406943044379967034483826130586578659612208976
20
19
$x_k$:
0.9807437048939141719254464385842309152299106231262486805732511888660871853057070601746047987326822398
20
19
$w_k$:
0.03223712318848894149160502811729403010325214029274621910200615103781913033476216867736864816585742596
20
20
$x_k$:
1
equals: One
20
20
$w_k$:
1/190
21
1
$x_k$:
-1
equals: Integers#-1
21
1
$w_k$:
1/210
21
2
$x_k$:
-0.9825722966045480282344812765554058768591715882364144152940340004574392390388403520795260686648380683
21
2
$w_k$:
0.02918484009850545860945854361317081426572115115480448763057170182031255903311588407551832664250924683
21
3
$x_k$:
-0.9419762969597455342961026506614351766496508740440096902983784068981074619292938712341108848281715293
21
3
$w_k$:
0.05184316900084962507272297185282975603041435902245542870077170086753439900197215124271106276752539783
21
4
$x_k$:
-0.8792947553235904644511535963049440477105815515091964416298779433900487269464192685967102121768159768
21
4
$w_k$:
0.07327391818507414425254786104189370365728029405193696595672487452115924823895701333337836081220488103
21
5
$x_k$:
-0.7960019260777124047443125896603586390904196605497765163466262433300305019307110009278813624957687386
21
5
$w_k$:
0.09298546795788606530113766414921440605982096796649198517667950263491732009128591297625330767476751951
21
6
$x_k$:
-0.6940510260622232326273163931946666287577160061058454970586647036851350336176323851976002758544129348
21
6
$w_k$:
0.1105170832191233352670004867843877747623086939384557480950041987424634926055614053247377815107413485
21
7
$x_k$:
-0.5758319602618306869270218703380852873357730085584824889052869979690922507719639655431768961098524170
21
7
$w_k$:
0.1254581211908689480151575357079992317038953214827590012751481086520041142794705227396320979067189693
21
8
$x_k$:
-0.4441157832790021011945163496073512847350574865670648529345013322233760447449114204588301671948079183
21
8
$w_k$:
0.1374584628600413435808996174151452635177249848248065823121896554927932062606700692656116387689859839
21
9
$x_k$:
-0.3019898565087648872753518678587522320210710340603862323419132483737947680413248048258907613826076984
21
9
$w_k$:
0.1462368624479774592672705306343935032195060608189875277951060784085758368766128222133503852561570479
21
10
$x_k$:
-0.1527855158021854660063583284856694355177489933132764808650513361389059360067866977907779957661145903
21
10
$w_k$:
0.1515875751116813844532506815052909110500412340828656529966979156527298005383075178037876264142329161
21
11
$x_k$:
0
equals: Zero
21
11
$w_k$:
34359738368/224009490705
21
12
$x_k$:
0.1527855158021854660063583284856694355177489933132764808650513361389059360067866977907779957661145903
21
12
$w_k$:
0.1515875751116813844532506815052909110500412340828656529966979156527298005383075178037876264142329161
21
13
$x_k$:
0.3019898565087648872753518678587522320210710340603862323419132483737947680413248048258907613826076984
21
13
$w_k$:
0.1462368624479774592672705306343935032195060608189875277951060784085758368766128222133503852561570479
21
14
$x_k$:
0.4441157832790021011945163496073512847350574865670648529345013322233760447449114204588301671948079183
21
14
$w_k$:
0.1374584628600413435808996174151452635177249848248065823121896554927932062606700692656116387689859839
21
15
$x_k$:
0.5758319602618306869270218703380852873357730085584824889052869979690922507719639655431768961098524170
21
15
$w_k$:
0.1254581211908689480151575357079992317038953214827590012751481086520041142794705227396320979067189693
21
16
$x_k$:
0.6940510260622232326273163931946666287577160061058454970586647036851350336176323851976002758544129348
21
16
$w_k$:
0.1105170832191233352670004867843877747623086939384557480950041987424634926055614053247377815107413485
21
17
$x_k$:
0.7960019260777124047443125896603586390904196605497765163466262433300305019307110009278813624957687386
21
17
$w_k$:
0.09298546795788606530113766414921440605982096796649198517667950263491732009128591297625330767476751951
21
18
$x_k$:
0.8792947553235904644511535963049440477105815515091964416298779433900487269464192685967102121768159768
21
18
$w_k$:
0.07327391818507414425254786104189370365728029405193696595672487452115924823895701333337836081220488103
21
19
$x_k$:
0.9419762969597455342961026506614351766496508740440096902983784068981074619292938712341108848281715293
21
19
$w_k$:
0.05184316900084962507272297185282975603041435902245542870077170086753439900197215124271106276752539783
21
20
$x_k$:
0.9825722966045480282344812765554058768591715882364144152940340004574392390388403520795260686648380683
21
20
$w_k$:
0.02918484009850545860945854361317081426572115115480448763057170182031255903311588407551832664250924683
21
21
$x_k$:
1
equals: One
21
21
$w_k$:
1/210
22
1
$x_k$:
-1
equals: Integers#-1
22
1
$w_k$:
1/231
22
2
$x_k$:
-0.9841524384576461765522896222120702966055135361195164558938104807722919277828458226634488927559461582
22
2
$w_k$:
0.02654574768250175791162790452054291344755730156773687382128199134198499459753345607997482107409498781
22
3
$x_k$:
-0.9472042839992286805242137666157295099120620453413636141672028958261752687283685780871623567490940603
22
3
$w_k$:
0.04721446529374075212377573486479198440114997893371302187104439640992650197638998023237608149355579109
22
4
$x_k$:
-0.8900622901909044705296578257790867901995340828471472872660922171939265252802339805748762112038543271
22
4
$w_k$:
0.06686560586455307601240419415709709256915337926709912646127750943224201411026645179412424679971267012
22
5
$x_k$:
-0.8139489276119211360454418480561350424386685149070968984936908383092126988616969222672951085268896497
22
5
$w_k$:
0.08509006039183844781571123609574813671437818108439013946165289134207714319079169643132210223298601889
22
6
$x_k$:
-0.7204872399612021581198818963984657585933454261195007005238777282001468242780708203574459384876895090
22
6
$w_k$:
0.1015005748016476743724373037496001908822035328750081331017544121394485138340918166680366064845059448
22
7
$x_k$:
-0.6116694382842589712262116058699265993454403046076955965561556692969522731350329222178689510170369211
22
7
$w_k$:
0.1157476446539390665900363677214561681422099637534815493210067470621503037950766907775941672371796302
22
8
$x_k$:
-0.4898148751899023498087512356832700416712716357951518174370716874656714480664242191368328573717364345
22
8
$w_k$:
0.1275276966534302755308444593088298433725501722076401441547587899238188830618669493643980451299744307
22
9
$x_k$:
-0.3575207101389195380609572802401791292833071039429362794169419157382978628115421901343711145204183253
22
9
$w_k$:
0.1365896886137414266861773622061702562810841629804189191848786180728406390280984489808301787443868798
22
10
$x_k$:
-0.2176065851592850417879550934653927632750066940141930651237351584646823159941197049805300299555281100
22
10
$w_k$:
0.1427404922713614003362359935667889380885729508888788355233936383357721829223303111291756477756989187
22
11
$x_k$:
-0.07305454001089833476108879046410735619277923633351649896897140269702017767130949132930652739833055587
22
11
$w_k$:
0.1458490194442417936164204394799701470968113721126289280946220016107344944792251942131637740235757236
22
12
$x_k$:
0.07305454001089833476108879046410735619277923633351649896897140269702017767130949132930652739833055587
22
12
$w_k$:
0.1458490194442417936164204394799701470968113721126289280946220016107344944792251942131637740235757236
22
13
$x_k$:
0.2176065851592850417879550934653927632750066940141930651237351584646823159941197049805300299555281100
22
13
$w_k$:
0.1427404922713614003362359935667889380885729508888788355233936383357721829223303111291756477756989187
22
14
$x_k$:
0.3575207101389195380609572802401791292833071039429362794169419157382978628115421901343711145204183253
22
14
$w_k$:
0.1365896886137414266861773622061702562810841629804189191848786180728406390280984489808301787443868798
22
15
$x_k$:
0.4898148751899023498087512356832700416712716357951518174370716874656714480664242191368328573717364345
22
15
$w_k$:
0.1275276966534302755308444593088298433725501722076401441547587899238188830618669493643980451299744307
22
16
$x_k$:
0.6116694382842589712262116058699265993454403046076955965561556692969522731350329222178689510170369211
22
16
$w_k$:
0.1157476446539390665900363677214561681422099637534815493210067470621503037950766907775941672371796302
22
17
$x_k$:
0.7204872399612021581198818963984657585933454261195007005238777282001468242780708203574459384876895090
22
17
$w_k$:
0.1015005748016476743724373037496001908822035328750081331017544121394485138340918166680366064845059448
22
18
$x_k$:
0.8139489276119211360454418480561350424386685149070968984936908383092126988616969222672951085268896497
22
18
$w_k$:
0.08509006039183844781571123609574813671437818108439013946165289134207714319079169643132210223298601889
22
19
$x_k$:
0.8900622901909044705296578257790867901995340828471472872660922171939265252802339805748762112038543271
22
19
$w_k$:
0.06686560586455307601240419415709709256915337926709912646127750943224201411026645179412424679971267012
22
20
$x_k$:
0.9472042839992286805242137666157295099120620453413636141672028958261752687283685780871623567490940603
22
20
$w_k$:
0.04721446529374075212377573486479198440114997893371302187104439640992650197638998023237608149355579109
22
21
$x_k$:
0.9841524384576461765522896222120702966055135361195164558938104807722919277828458226634488927559461582
22
21
$w_k$:
0.02654574768250175791162790452054291344755730156773687382128199134198499459753345607997482107409498781
22
22
$x_k$:
1
equals: One
22
22
$w_k$:
1/231
23
1
$x_k$:
-1
equals: Integers#-1
23
1
$w_k$:
1/253
23
2
$x_k$:
-0.9855271558787325780814627667380990990206107921396450417608657493151684105024065291915283115715619012
23
2
$w_k$:
0.02424860077153173651739965893709704115977772714704124669336334803060701775766945560908057755192783985
23
3
$x_k$:
-0.9517579557107102041356396798514291558483519254487993944251280487109981038986297815165248098405882937
23
3
$w_k$:
0.04317587117024183474887646561204227115245703635276407270205231357499734406959019290204765211586354885
23
4
$x_k$:
-0.8994585580403450109501603203473671579117983481392947345553115789633287533816137109567164060244209818
23
4
$w_k$:
0.06125247712955420638138284744035549333790842765512580275582147479913058537059498306741857460703907541
23
5
$x_k$:
-0.8296510966512858862232006192900048845985118830133272845011106438139352435165537428961289367383134012
23
5
$w_k$:
0.07813544947556998974193425534796471313534000907562796909515759996387152445702047330112034583795603057
23
6
$x_k$:
-0.7436950411720606839451635430669967912872192289538602142745877930619208741996662646311889089347390430
23
6
$w_k$:
0.09349724616351234183350070690669710412683215672378285465712182209969345385337519201505389837403724236
23
7
$x_k$:
-0.6432636444601362084761455336027687438913118818022987213744832501390915217839011217474001307059022720
23
7
$w_k$:
0.1070391017243365115351836279154666291744070682097397496410631923703132464520298042687418775825726584
23
8
$x_k$:
-0.5303117711368441681301153201522998111303465149273427103964248515692387523482981487436412782848590782
23
8
$w_k$:
0.1184975106627491313021260047242605185684919960834971465983435220325383492868857726981226822293730620
23
9
$x_k$:
-0.4070379379144748291959504882150956395519537239941739528811159688203533174757652073042545458729810407
23
9
$w_k$:
0.1276494747017588766361485530556707677955193775622753944713138848019181176216658111316328301044899751
23
10
$x_k$:
-0.2758415489457930671068776326791352041731911066094172959368076651834509592592350918586315962648104370
23
10
$w_k$:
0.1343168726386038199015648977007108491281493386838845140820293308318256292571852176062691756012078734
23
11
$x_k$:
-0.1392762040406683985918626129827669339085444571744417385234077065216125900569160500359975336614873213
23
11
$w_k$:
0.1383699363858073945235027338629444393870379739292738234658440526438475393162307138293708637571832200
23
12
$x_k$:
0
equals: Zero
23
12
$w_k$:
274877906944/1967210618373
23
13
$x_k$:
0.1392762040406683985918626129827669339085444571744417385234077065216125900569160500359975336614873213
23
13
$w_k$:
0.1383699363858073945235027338629444393870379739292738234658440526438475393162307138293708637571832200
23
14
$x_k$:
0.2758415489457930671068776326791352041731911066094172959368076651834509592592350918586315962648104370
23
14
$w_k$:
0.1343168726386038199015648977007108491281493386838845140820293308318256292571852176062691756012078734
23
15
$x_k$:
0.4070379379144748291959504882150956395519537239941739528811159688203533174757652073042545458729810407
23
15
$w_k$:
0.1276494747017588766361485530556707677955193775622753944713138848019181176216658111316328301044899751
23
16
$x_k$:
0.5303117711368441681301153201522998111303465149273427103964248515692387523482981487436412782848590782
23
16
$w_k$:
0.1184975106627491313021260047242605185684919960834971465983435220325383492868857726981226822293730620
23
17
$x_k$:
0.6432636444601362084761455336027687438913118818022987213744832501390915217839011217474001307059022720
23
17
$w_k$:
0.1070391017243365115351836279154666291744070682097397496410631923703132464520298042687418775825726584
23
18
$x_k$:
0.7436950411720606839451635430669967912872192289538602142745877930619208741996662646311889089347390430
23
18
$w_k$:
0.09349724616351234183350070690669710412683215672378285465712182209969345385337519201505389837403724236
23
19
$x_k$:
0.8296510966512858862232006192900048845985118830133272845011106438139352435165537428961289367383134012
23
19
$w_k$:
0.07813544947556998974193425534796471313534000907562796909515759996387152445702047330112034583795603057
23
20
$x_k$:
0.8994585580403450109501603203473671579117983481392947345553115789633287533816137109567164060244209818
23
20
$w_k$:
0.06125247712955420638138284744035549333790842765512580275582147479913058537059498306741857460703907541
23
21
$x_k$:
0.9517579557107102041356396798514291558483519254487993944251280487109981038986297815165248098405882937
23
21
$w_k$:
0.04317587117024183474887646561204227115245703635276407270205231357499734406959019290204765211586354885
23
22
$x_k$:
0.9855271558787325780814627667380990990206107921396450417608657493151684105024065291915283115715619012
23
22
$w_k$:
0.02424860077153173651739965893709704115977772714704124669336334803060701775766945560908057755192783985
23
23
$x_k$:
1
equals: One
23
23
$w_k$:
1/253
24
1
$x_k$:
-1
equals: Integers#-1
24
1
$w_k$:
1/276
24
2
$x_k$:
-0.9867305535051608835530867381544749753719197924133019006401869628866431442334840817363529094639429809
24
2
$w_k$:
0.02223685346471120899296043481729852401003523844783393857391447809405990532712363806466148761514415415
24
3
$x_k$:
-0.9557482209298863580269771305506448310707330429557406963650629658436128763207325938434625782810012186
24
3
$w_k$:
0.03963168133346780946926216336347172695861848804210116293464319447991749791467555803324723571088139417
24
4
$x_k$:
-0.9077056751135065219951529964662077492084201138782763032042750386726226564087719840190389393517230765
24
4
$w_k$:
0.05630984872464619902094833967811799432844844562904498669452999473617091601097458042822869227033463195
24
5
$x_k$:
-0.8434640701548720406233050374233422858410761008103305711861685318945247151543537735979276937675670058
24
5
$w_k$:
0.07198186205529398221563906066868012598006509010265436848264937705982829001380008283669996077352049666
24
6
$x_k$:
-0.7641704824204933077873752809522936513210604492368998647771278612376657227876813380282104127023753921
24
6
$w_k$:
0.08636902996792906821656281930658319302172822325475831645942996484036600125131769455342037433527969880
24
7
$x_k$:
-0.6712401052641286998356648581870067565740232889464325476651151518161518245183633089684629349210713228
24
7
$w_k$:
0.09921482768408358741414977099571242888261038314786619486476006102272453354078392843292925635782002395
24
8
$x_k$:
-0.5663313579792953121894095445422837704388949971264806735762986229512311689011078358422746868960932193
24
8
$w_k$:
0.1102900868929686041104241230275966044063286128769524651186326775036943074381664591295358767927888556
24
9
$x_k$:
-0.4513163732143226182482184915696224488230882183124894098058567252768718091731067326234981303762319419
24
9
$w_k$:
0.1193971937024913190322938883507179929071315534019036887585330133870538189786129341424609573414144871
24
10
$x_k$:
-0.3282476133755109120333891793596093437011778687726996946231212845307817982436295434431059326406940876
24
10
$w_k$:
0.1263736420280208001272480906913731160124240173743181084944500688132230562141853091021466380514824817
24
11
$x_k$:
-0.1993212533908326672365725391249907308118755914214762137574416696222416721047861035246938296454596029
24
11
$w_k$:
0.1310949418736039423544502258165301143074297432579824830557677730873699990072682442350566329818684499
24
12
$x_k$:
-0.06683799373722857811364180839167730979622320891762773171395628935857359697103285334877084566128250734
24
12
$w_k$:
0.1334768438669863775967857209650775994750352769283524025047183824828380511146857739401636124071464855
24
13
$x_k$:
0.06683799373722857811364180839167730979622320891762773171395628935857359697103285334877084566128250734
24
13
$w_k$:
0.1334768438669863775967857209650775994750352769283524025047183824828380511146857739401636124071464855
24
14
$x_k$:
0.1993212533908326672365725391249907308118755914214762137574416696222416721047861035246938296454596029
24
14
$w_k$:
0.1310949418736039423544502258165301143074297432579824830557677730873699990072682442350566329818684499
24
15
$x_k$:
0.3282476133755109120333891793596093437011778687726996946231212845307817982436295434431059326406940876
24
15
$w_k$:
0.1263736420280208001272480906913731160124240173743181084944500688132230562141853091021466380514824817
24
16
$x_k$:
0.4513163732143226182482184915696224488230882183124894098058567252768718091731067326234981303762319419
24
16
$w_k$:
0.1193971937024913190322938883507179929071315534019036887585330133870538189786129341424609573414144871
24
17
$x_k$:
0.5663313579792953121894095445422837704388949971264806735762986229512311689011078358422746868960932193
24
17
$w_k$:
0.1102900868929686041104241230275966044063286128769524651186326775036943074381664591295358767927888556
24
18
$x_k$:
0.6712401052641286998356648581870067565740232889464325476651151518161518245183633089684629349210713228
24
18
$w_k$:
0.09921482768408358741414977099571242888261038314786619486476006102272453354078392843292925635782002395
24
19
$x_k$:
0.7641704824204933077873752809522936513210604492368998647771278612376657227876813380282104127023753921
24
19
$w_k$:
0.08636902996792906821656281930658319302172822325475831645942996484036600125131769455342037433527969880
24
20
$x_k$:
0.8434640701548720406233050374233422858410761008103305711861685318945247151543537735979276937675670058
24
20
$w_k$:
0.07198186205529398221563906066868012598006509010265436848264937705982829001380008283669996077352049666
24
21
$x_k$:
0.9077056751135065219951529964662077492084201138782763032042750386726226564087719840190389393517230765
24
21
$w_k$:
0.05630984872464619902094833967811799432844844562904498669452999473617091601097458042822869227033463195
24
22
$x_k$:
0.9557482209298863580269771305506448310707330429557406963650629658436128763207325938434625782810012186
24
22
$w_k$:
0.03963168133346780946926216336347172695861848804210116293464319447991749791467555803324723571088139417
24
23
$x_k$:
0.9867305535051608835530867381544749753719197924133019006401869628866431442334840817363529094639429809
24
23
$w_k$:
0.02223685346471120899296043481729852401003523844783393857391447809405990532712363806466148761514415415
24
24
$x_k$:
1
equals: One
24
24
$w_k$:
1/276
25
1
$x_k$:
-1
equals: Integers#-1
25
1
$w_k$:
1/300
25
2
$x_k$:
-0.9877899449314937092718040708731605181529839936234190775218182382939438119743149048753074077787831126
25
2
$w_k$:
0.02046516893297438530854247181916285110283367836479675150229617899321606180026585406055614451686567963
25
3
$x_k$:
-0.9592641382525344788598451788395232025265029302852769919194837170136535229969317577387308808842448518
25
3
$w_k$:
0.03650473879427137203238298875510974047758133661619788701674841337897025511638017963039137737259262875
25
4
$x_k$:
-0.9149827707346225783231499337368465516468972691268483097690408060694197233249408836998609459804560511
25
4
$w_k$:
0.05193622836849147464333388971348937801051203813906883408134343681160698477946772633422415893969641393
25
5
$x_k$:
-0.8556764658353165775238132601717480961140676160118140901874996655034987091714861840442919233886194847
25
5
$w_k$:
0.06651372867531278469386999331359944109273595941288986985810513837749928603324170266130894510671659226
25
6
$x_k$:
-0.7823196592407167803991879522287473574794630260720353430612565322893751644986620894861857283343660569
25
6
$w_k$:
0.07999877483629298180162643613883603449558866474311303159853659497977028203299776712586483806260453324
25
7
$x_k$:
-0.6961170488151343667603654337893775512606287340051854677043294426773398871150623667890559794361275988
25
7
$w_k$:
0.09217013991062042191268962271404854571213798326952547726664803401127991106892719074996180073521754373
25
8
$x_k$:
-0.5984841472799932680976209971810661579661803781166909973991479507345999095998460645111878632139783873
25
8
$w_k$:
0.1028280303479578308275036449071341655836616218748194977812669286374618253883283717440390709036417050
25
9
$x_k$:
-0.4910241148188783826189599225688031610645484675613731067551551859865583617526353667825242144122849732
25
9
$w_k$:
0.1117974662683208881562442324310391869069086308402578590493706046448770170172898522445921636316131881
25
10
$x_k$:
-0.3755014578592272332287146122607180326863163064877193818637228633931827888802318492311404625136208054
25
10
$w_k$:
0.1189311794068118254094442446341871571147586810411110841209700395217823285376186463390353298497549715
25
11
$x_k$:
-0.2538130641688765801798868816872755298639191983595341837752587409800243526429273589309426304350481463
25
11
$w_k$:
0.1241120389379502906952153124392969703595937889068961189806862747226550992634233155577072528083041021
25
12
$x_k$:
-0.1279570594831069727089846250946545968955413236815924910882052235133028565694945171158630297218285160
25
12
$w_k$:
0.1272549775383314470171144167387579132904409357922862630193657531516559448218190699665891653776543168
25
13
$x_k$:
0
equals: Zero
25
13
$w_k$:
4398046511104/34277154714075
25
14
$x_k$:
0.1279570594831069727089846250946545968955413236815924910882052235133028565694945171158630297218285160
25
14
$w_k$:
0.1272549775383314470171144167387579132904409357922862630193657531516559448218190699665891653776543168
25
15
$x_k$:
0.2538130641688765801798868816872755298639191983595341837752587409800243526429273589309426304350481463
25
15
$w_k$:
0.1241120389379502906952153124392969703595937889068961189806862747226550992634233155577072528083041021
25
16
$x_k$:
0.3755014578592272332287146122607180326863163064877193818637228633931827888802318492311404625136208054
25
16
$w_k$:
0.1189311794068118254094442446341871571147586810411110841209700395217823285376186463390353298497549715
25
17
$x_k$:
0.4910241148188783826189599225688031610645484675613731067551551859865583617526353667825242144122849732
25
17
$w_k$:
0.1117974662683208881562442324310391869069086308402578590493706046448770170172898522445921636316131881
25
18
$x_k$:
0.5984841472799932680976209971810661579661803781166909973991479507345999095998460645111878632139783873
25
18
$w_k$:
0.1028280303479578308275036449071341655836616218748194977812669286374618253883283717440390709036417050
25
19
$x_k$:
0.6961170488151343667603654337893775512606287340051854677043294426773398871150623667890559794361275988
25
19
$w_k$:
0.09217013991062042191268962271404854571213798326952547726664803401127991106892719074996180073521754373
25
20
$x_k$:
0.7823196592407167803991879522287473574794630260720353430612565322893751644986620894861857283343660569
25
20
$w_k$:
0.07999877483629298180162643613883603449558866474311303159853659497977028203299776712586483806260453324
25
21
$x_k$:
0.8556764658353165775238132601717480961140676160118140901874996655034987091714861840442919233886194847
25
21
$w_k$:
0.06651372867531278469386999331359944109273595941288986985810513837749928603324170266130894510671659226
25
22
$x_k$:
0.9149827707346225783231499337368465516468972691268483097690408060694197233249408836998609459804560511
25
22
$w_k$:
0.05193622836849147464333388971348937801051203813906883408134343681160698477946772633422415893969641393
25
23
$x_k$:
0.9592641382525344788598451788395232025265029302852769919194837170136535229969317577387308808842448518
25
23
$w_k$:
0.03650473879427137203238298875510974047758133661619788701674841337897025511638017963039137737259262875
25
24
$x_k$:
0.9877899449314937092718040708731605181529839936234190775218182382939438119743149048753074077787831126
25
24
$w_k$:
0.02046516893297438530854247181916285110283367836479675150229617899321606180026585406055614451686567963
25
25
$x_k$:
1
equals: One
25
25
$w_k$:
1/300
26
1
$x_k$:
-1
equals: Integers#-1
26
1
$w_k$:
1/325
26
2
$x_k$:
-0.9887274123114756544292466903713013409946335426896035580107805341391344751442735008068942985549455404
26
2
$w_k$:
0.01889685802426346558135204767626720193674486556590662206905252923667949775628021590517100311939337818
26
3
$x_k$:
-0.9623778747677173297506626712821301620310446378206946957693116661756169174137331947949550738054382101
26
3
$w_k$:
0.03373230368595599937752274858275590999183915751436549623677223233779880483499923428558612336809489167
26
4
$x_k$:
-0.9214355468175573411251305217486883551267646485542595429653663388461446588154742070757931380413043511
26
4
$w_k$:
0.04804839908118062731597931278066204195101628706221631905302994966088980301585283008534823137858979632
26
5
$x_k$:
-0.8665243239591235709744514978097700547408603872075592676464236705145788759809390184835628543885449425
26
5
$w_k$:
0.06163502514254740278218005410427926873187544724090825859278029014634280012892134316643514891948063454
26
6
$x_k$:
-0.7984771831074374390367876385703433977563385736152783161992831479102718647669612095858586338133653204
26
6
$w_k$:
0.07428705012229113731608408298624873774879158492433384825417458343936268141742080465951353616300801884
26
7
$x_k$:
-0.7183258163626650805254053879039260159976785084039183331427039956235298271012411164093421031212918508
26
7
$w_k$:
0.08581286398000436218748125235906840728978945903974740924638928453705558934584008065632748570475595202
26
8
$x_k$:
-0.6272852994923168878005189071612210361810055064045715853095245704212365481246179272256652734232055585
26
8
$w_k$:
0.09603780235390131080369739637086240321145648854212394417417683313493867231160213106662003532525116640
26
9
$x_k$:
-0.5267357420298785452982433943809943124256275014322974459475137973567866242985910096748694828855130448
26
9
$w_k$:
0.1048068862307370529899058368913641761605785419951773111771268106731089726467357292828827461990380474
26
10
$x_k$:
-0.4182013870662467855636838823338498242423604705872464730877040054753085078166366416253860346829577308
26
10
$w_k$:
0.1119871941198603353002108798798862410435081861852638258175233414152167230506899124476079203887525091
26
11
$x_k$:
-0.3033275128592527207740410379021101814115551888264973654310110196668034967186154749141568252429990419
26
11
$w_k$:
0.1174698840938090070766956202964820942229803388904521254816055470066440350693889233184563039680148228
26
12
$x_k$:
-0.1838554952700549013212056718857597523762721598761359844950275065496258375400410212965475600733807214
26
12
$w_k$:
0.1211718462884433460442866957042263912328607028862209709493124081842967314737033683907349558226482769
26
13
$x_k$:
-0.06159641178191972820548727478175133466309396121606723409906730343647865340226168128681451372531076392
26
13
$w_k$:
0.1230369638000828763015271492909740495554820170763607920249792671507426120254885036583934327198955828
26
14
$x_k$:
0.06159641178191972820548727478175133466309396121606723409906730343647865340226168128681451372531076392
26
14
$w_k$:
0.1230369638000828763015271492909740495554820170763607920249792671507426120254885036583934327198955828
26
15
$x_k$:
0.1838554952700549013212056718857597523762721598761359844950275065496258375400410212965475600733807214
26
15
$w_k$:
0.1211718462884433460442866957042263912328607028862209709493124081842967314737033683907349558226482769
26
16
$x_k$:
0.3033275128592527207740410379021101814115551888264973654310110196668034967186154749141568252429990419
26
16
$w_k$:
0.1174698840938090070766956202964820942229803388904521254816055470066440350693889233184563039680148228
26
17
$x_k$:
0.4182013870662467855636838823338498242423604705872464730877040054753085078166366416253860346829577308
26
17
$w_k$:
0.1119871941198603353002108798798862410435081861852638258175233414152167230506899124476079203887525091
26
18
$x_k$:
0.5267357420298785452982433943809943124256275014322974459475137973567866242985910096748694828855130448
26
18
$w_k$:
0.1048068862307370529899058368913641761605785419951773111771268106731089726467357292828827461990380474
26
19
$x_k$:
0.6272852994923168878005189071612210361810055064045715853095245704212365481246179272256652734232055585
26
19
$w_k$:
0.09603780235390131080369739637086240321145648854212394417417683313493867231160213106662003532525116640
26
20
$x_k$:
0.7183258163626650805254053879039260159976785084039183331427039956235298271012411164093421031212918508
26
20
$w_k$:
0.08581286398000436218748125235906840728978945903974740924638928453705558934584008065632748570475595202
26
21
$x_k$:
0.7984771831074374390367876385703433977563385736152783161992831479102718647669612095858586338133653204
26
21
$w_k$:
0.07428705012229113731608408298624873774879158492433384825417458343936268141742080465951353616300801884
26
22
$x_k$:
0.8665243239591235709744514978097700547408603872075592676464236705145788759809390184835628543885449425
26
22
$w_k$:
0.06163502514254740278218005410427926873187544724090825859278029014634280012892134316643514891948063454
26
23
$x_k$:
0.9214355468175573411251305217486883551267646485542595429653663388461446588154742070757931380413043511
26
23
$w_k$:
0.04804839908118062731597931278066204195101628706221631905302994966088980301585283008534823137858979632
26
24
$x_k$:
0.9623778747677173297506626712821301620310446378206946957693116661756169174137331947949550738054382101
26
24
$w_k$:
0.03373230368595599937752274858275590999183915751436549623677223233779880483499923428558612336809489167
26
25
$x_k$:
0.9887274123114756544292466903713013409946335426896035580107805341391344751442735008068942985549455404
26
25
$w_k$:
0.01889685802426346558135204767626720193674486556590662206905252923667949775628021590517100311939337818
26
26
$x_k$:
1
equals: One
26
26
$w_k$:
1/325
27
1
$x_k$:
-1
equals: Integers#-1
27
1
$w_k$:
1/351
27
2
$x_k$:
-0.9895609637285506211488746400545194376668790296678082016575066062044525437516659384640188789065580712
27
2
$w_k$:
0.01750197487606557901936973436271924879870807353525209813248439650870780515867878539491868679919628424
27
3
$x_k$:
-0.9651484024508189192642668031211300546689807306819202452552449087136669338494250293774679556101493946
27
3
$w_k$:
0.03126295173520238432478499006479536351611086136412874373812810725258120200361656298372027314592162341
27
4
$x_k$:
-0.9271834587251157788597467640727187315604306861922375572983688179080387528164747846145702538261750946
27
4
$w_k$:
0.04457765793306169874407474224138623556378454515179361670149739139848960288553075916597929485429454735
27
5
$x_k$:
-0.8762020862145224786665430491941408419267125131335132914288410826375531276641640330403062153321959251
27
5
$w_k$:
0.05726556968016273173908722825394062390162126505887369402835897455923579540565861281432426965237712222
27
6
$x_k$:
-0.8129204868958122882580918647585379120733178333219033746991290967924651991528653118372543103308226464
27
6
$w_k$:
0.06914934236004327628063473503012721935333560860087654229637488760869720818027676890789868087966034607
27
7
$x_k$:
-0.7382271498464599108394272257484052740029931410339381272368499515276645153304583233322073243403075709
27
7
$w_k$:
0.08006232197053845816823807048395035378269442147010534332672136780893049125896667907827492259650124662
27
8
$x_k$:
-0.6531706636968095151787048751824609481943777912148402651240288666049914501473200134044391658215268050
27
8
$w_k$:
0.08985136525929055997213630194606260203281856986003769129686779235562811822192006282510754195969131167
27
9
$x_k$:
-0.5589450609425611800626680748681247749079325637076692633594110338868707281013598396898335846247975724
27
9
$w_k$:
0.09837907458595276317930893126998863277389880881964088682719367920647690957118845116337548411326138808
27
10
$x_k$:
-0.4568730756140824036786126508392420222427712408047451311508149393815958957640028089876798874483153858
27
10
$w_k$:
0.1055257478212530112183819894799018876933140715205704752410296946363782987954798979565288550557441834
27
11
$x_k$:
-0.3483875819890286704370865785871665957634751517230021214873988145674395089778958363573519561629682393
27
11
$w_k$:
0.1111910652574370329282619593421546638897390226247015784304229781097036196462463444187003174367284229
27
12
$x_k$:
-0.2350114831029181333368015805133316637814493669196037113352210408673351030083798356333384073312140520
27
12
$w_k$:
0.1152955002546519828137533333747300879826996749202348510234773429908912355574332688973270929813487992
27
13
$x_k$:
-0.1183363338985210484387082682753163282194296918936552485723397830866647397827217312947819987954089301
27
13
$w_k$:
0.1177814365859561590664990794483642372011284017160419392106099934145414530773252303337374860677722186
27
14
$x_k$:
0
equals: Zero
27
14
$w_k$:
70368744177664/593258446974375
27
15
$x_k$:
0.1183363338985210484387082682753163282194296918936552485723397830866647397827217312947819987954089301
27
15
$w_k$:
0.1177814365859561590664990794483642372011284017160419392106099934145414530773252303337374860677722186
27
16
$x_k$:
0.2350114831029181333368015805133316637814493669196037113352210408673351030083798356333384073312140520
27
16
$w_k$:
0.1152955002546519828137533333747300879826996749202348510234773429908912355574332688973270929813487992
27
17
$x_k$:
0.3483875819890286704370865785871665957634751517230021214873988145674395089778958363573519561629682393
27
17
$w_k$:
0.1111910652574370329282619593421546638897390226247015784304229781097036196462463444187003174367284229
27
18
$x_k$:
0.4568730756140824036786126508392420222427712408047451311508149393815958957640028089876798874483153858
27
18
$w_k$:
0.1055257478212530112183819894799018876933140715205704752410296946363782987954798979565288550557441834
27
19
$x_k$:
0.5589450609425611800626680748681247749079325637076692633594110338868707281013598396898335846247975724
27
19
$w_k$:
0.09837907458595276317930893126998863277389880881964088682719367920647690957118845116337548411326138808
27
20
$x_k$:
0.6531706636968095151787048751824609481943777912148402651240288666049914501473200134044391658215268050
27
20
$w_k$:
0.08985136525929055997213630194606260203281856986003769129686779235562811822192006282510754195969131167
27
21
$x_k$:
0.7382271498464599108394272257484052740029931410339381272368499515276645153304583233322073243403075709
27
21
$w_k$:
0.08006232197053845816823807048395035378269442147010534332672136780893049125896667907827492259650124662
27
22
$x_k$:
0.8129204868958122882580918647585379120733178333219033746991290967924651991528653118372543103308226464
27
22
$w_k$:
0.06914934236004327628063473503012721935333560860087654229637488760869720818027676890789868087966034607
27
23
$x_k$:
0.8762020862145224786665430491941408419267125131335132914288410826375531276641640330403062153321959251
27
23
$w_k$:
0.05726556968016273173908722825394062390162126505887369402835897455923579540565861281432426965237712222
27
24
$x_k$:
0.9271834587251157788597467640727187315604306861922375572983688179080387528164747846145702538261750946
27
24
$w_k$:
0.04457765793306169874407474224138623556378454515179361670149739139848960288553075916597929485429454735
27
25
$x_k$:
0.9651484024508189192642668031211300546689807306819202452552449087136669338494250293774679556101493946
27
25
$w_k$:
0.03126295173520238432478499006479536351611086136412874373812810725258120200361656298372027314592162341
27
26
$x_k$:
0.9895609637285506211488746400545194376668790296678082016575066062044525437516659384640188789065580712
27
26
$w_k$:
0.01750197487606557901936973436271924879870807353525209813248439650870780515867878539491868679919628424
27
27
$x_k$:
1
equals: One
27
27
$w_k$:
1/351
28
1
$x_k$:
-1
equals: Integers#-1
28
1
$w_k$:
1/378
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2
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-0.9903054026184541255170929118645980985914687336596068181653055732716206152496790536842494139664523827
28
2
$w_k$:
0.01625588395750421819899015289710968311934590382944507628708408761277022931445126030440636537053001291
28
3
$x_k$:
-0.9676242858571313043437708791434725615400058731765246410357953450112913610826309657292200794879387727
28
3
$w_k$:
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28
4
$x_k$:
-0.9323251671215585249261424186089904324584322196012392920013406584243828507392594406131579475063888146
28
4
$w_k$:
0.04146691524300672103798584877862711073545160695358481104137176912829356304386527350060207084008928746
28
5
$x_k$:
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28
5
$w_k$:
0.05333807704732748840032551347150020650376199079603349601223895910636511637896279762081323550540500266
28
6
$x_k$:
-0.8258809700563381960445144261966045009106000157532008514296088746201820967991429967982828667798490616
28
6
$w_k$:
0.06451365808035453839024569346247843130985429004866611956893233227196733889182499913350642298633981527
28
7
$x_k$:
-0.7561241940055697651520957252661909240276667654448534346163110336007906220947421763883429707352406806
28
7
$w_k$:
0.07484812350970770278056793914316175042342879687461068162142015514980286455889778971421445889942430791
28
8
$x_k$:
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28
8
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28
9
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28
9
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28
10
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28
10
$w_k$:
0.09952311041249567287594535475091513510824188149959485825469243620016504334494344179682044473562169311
28
11
$x_k$:
-0.3894631375763628079358841552041097446482275672495814037793745389329312723894579020156746745124081816
28
11
$w_k$:
0.1052810937610558944823775517949101678535287110162783838817485816333911359921345681166323798448770955
28
12
$x_k$:
-0.2818722666216023719541525089093325053541076149240158669365482997688889881533496839240041208023092969
28
12
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0.1096665737959766250885858637912060418834322354381598398327846867598586563071043070450221585668524803
28
13
$x_k$:
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28
13
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0.1126223800772390125406364241136602114109825091849781029867009942903870863397213074615366062784629020
28
14
$x_k$:
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28
14
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28
15
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0.05711712169351289762654365157374676322606056346143366069206444080768594409746226405305609848520260209
28
15
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28
16
$x_k$:
0.1706067553080043608827026777487650829849007999261806385319573571417830008672190775288229829177211540
28
16
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0.1126223800772390125406364241136602114109825091849781029867009942903870863397213074615366062784629020
28
17
$x_k$:
0.2818722666216023719541525089093325053541076149240158669365482997688889881533496839240041208023092969
28
17
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0.1096665737959766250885858637912060418834322354381598398327846867598586563071043070450221585668524803
28
18
$x_k$:
0.3894631375763628079358841552041097446482275672495814037793745389329312723894579020156746745124081816
28
18
$w_k$:
0.1052810937610558944823775517949101678535287110162783838817485816333911359921345681166323798448770955
28
19
$x_k$:
0.4919767539315793804296166829444798429627332597771520100905889426941832443313598690829536617765189323
28
19
$w_k$:
0.09952311041249567287594535475091513510824188149959485825469243620016504334494344179682044473562169311
28
20
$x_k$:
0.5880766898371756064783120797617120774052941535661503251108472237119520461948499895359486418711876618
28
20
$w_k$:
0.09246768599771217572203539469938608889463043455388095470359391068371602531749486030559011982830480504
28
21
$x_k$:
0.6765101289295733177193812450733884889580471950362445682317679252815355518364157663184556014730929073
28
21
$w_k$:
0.08420679512151032573949118779791983198950732681831128548688324460197419352755849484199525705603424195
28
22
$x_k$:
0.7561241940055697651520957252661909240276667654448534346163110336007906220947421763883429707352406806
28
22
$w_k$:
0.07484812350970770278056793914316175042342879687461068162142015514980286455889778971421445889942430791
28
23
$x_k$:
0.8258809700563381960445144261966045009106000157532008514296088746201820967991429967982828667798490616
28
23
$w_k$:
0.06451365808035453839024569346247843130985429004866611956893233227196733889182499913350642298633981527
28
24
$x_k$:
0.8848710172113028384123063382572496734016611896957733143941857552711898995345724849449169865779039524
28
24
$w_k$:
0.05333807704732748840032551347150020650376199079603349601223895910636511637896279762081323550540500266
28
25
$x_k$:
0.9323251671215585249261424186089904324584322196012392920013406584243828507392594406131579475063888146
28
25
$w_k$:
0.04146691524300672103798584877862711073545160695358481104137176912829356304386527350060207084008928746
28
26
$x_k$:
0.9676242858571313043437708791434725615400058731765246410357953450112913610826309657292200794879387727
28
26
$w_k$:
0.02905422067797914470685277982358242820183286883845615962429793026550766828657106748878364137276927517
28
27
$x_k$:
0.9903054026184541255170929118645980985914687336596068181653055732716206152496790536842494139664523827
28
27
$w_k$:
0.01625588395750421819899015289710968311934590382944507628708408761277022931445126030440636537053001291
28
28
$x_k$:
1
equals: One
28
28
$w_k$:
1/378
29
1
$x_k$:
-1
equals: Integers#-1
29
1
$w_k$:
1/406
29
2
$x_k$:
-0.9909729882685698162787451132995713026874663409369348571590479260907087937053532974031199312715934986
29
2
$w_k$:
0.01513816985996759679085792660685573856344843865455112920410554457866280996984454188369930045037015570
29
3
$x_k$:
-0.9698458072879362937124013878878207457352757632496935636571187669571934697470968347104825463109378798
29
3
$w_k$:
0.02707080629682482754085142544502585388025522126984244741650692569291720746341212953579871759829112341
29
4
$x_k$:
-0.9369427185209825115420368482833629224097154426156435491146680828478050163198652173271675839491113846
29
4
$w_k$:
0.03866843997971297974783257426712441099428510620310339272950233296195979051234407668876786914250023370
29
5
$x_k$:
-0.8926657199760882787642299140530868780160165015095563983921184107383081653012059632209019295873401150
29
5
$w_k$:
0.04979580909323756092266834117547757161059273184359804724891209854178940237470194596563463524840835691
29
6
$x_k$:
-0.8375527362817865652420733161868245695155236980556465194930916411990991509452427067268270433061883736
29
6
$w_k$:
0.06031850382852273502806446351763368108900617211203733815726170756948436092750780760010175098081759313
29
7
$x_k$:
-0.7722728972064687413541706341364529832846259590580789402300909261406101653026256468248822782205613810
29
7
$w_k$:
0.07010893800059799918526685842401360869493482506754792689290802299739213678678994358259955603501108506
29
8
$x_k$:
-0.6976186613563679883939744285770931107884282636932362943255674291299162112455210289165843019972237655
29
8
$w_k$:
0.07904831302788560178417023217510940180440587243029549934743097211278426593208268196041426352300002281
29
9
$x_k$:
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29
9
$w_k$:
0.08702813344113560563762615521563306036882552495411760702850619595783601907276215459742911992563586048
29
10
$x_k$:
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29
10
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0.09395154211479631249783650698757002608906866902402522301169254112833295853881389466853675759476228741
29
11
$x_k$:
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29
11
$w_k$:
0.09973450163714986065356917753406848351607864659110458558314543431803389579525323994690411471569109700
29
12
$x_k$:
-0.3248493828419110062600140230788373752796047303415255636608226938599542996166901923151811384945868917
29
12
$w_k$:
0.1043068164636765374227071121912422879805778546025568676734166209883282698831522874688462869917649146
29
13
$x_k$:
-0.2187820582842610198090518523249946684617763511843255265013278454028163701910051620336961009683012849
29
13
$w_k$:
0.1076129858335685393735868540466068836272250294497734137179269478189308416972156022183634877962224272
29
14
$x_k$:
-0.1100590133955921097077876022556808549513887048685024608664607887480910514906314685204450936458990662
29
14
$w_k$:
0.1096128778458904201321324107565953741915880188894849339515847724529518623099964490217862354478232821
29
15
$x_k$:
0
equals: Zero
29
15
$w_k$:
562949953421312/5104630373416875
29
16
$x_k$:
0.1100590133955921097077876022556808549513887048685024608664607887480910514906314685204450936458990662
29
16
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0.1096128778458904201321324107565953741915880188894849339515847724529518623099964490217862354478232821
29
17
$x_k$:
0.2187820582842610198090518523249946684617763511843255265013278454028163701910051620336961009683012849
29
17
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0.1076129858335685393735868540466068836272250294497734137179269478189308416972156022183634877962224272
29
18
$x_k$:
0.3248493828419110062600140230788373752796047303415255636608226938599542996166901923151811384945868917
29
18
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0.1043068164636765374227071121912422879805778546025568676734166209883282698831522874688462869917649146
29
19
$x_k$:
0.4269734717134943876375764109683589091992527000641955153556542042881256340395164614825733803321967824
29
19
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0.09973450163714986065356917753406848351607864659110458558314543431803389579525323994690411471569109700
29
20
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29
20
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29
21
$x_k$:
0.6144962522034329499709036174243587188960534866587591323798345357125214564572787045091753739954024282
29
21
$w_k$:
0.08702813344113560563762615521563306036882552495411760702850619595783601907276215459742911992563586048
29
22
$x_k$:
0.6976186613563679883939744285770931107884282636932362943255674291299162112455210289165843019972237655
29
22
$w_k$:
0.07904831302788560178417023217510940180440587243029549934743097211278426593208268196041426352300002281
29
23
$x_k$:
0.7722728972064687413541706341364529832846259590580789402300909261406101653026256468248822782205613810
29
23
$w_k$:
0.07010893800059799918526685842401360869493482506754792689290802299739213678678994358259955603501108506
29
24
$x_k$:
0.8375527362817865652420733161868245695155236980556465194930916411990991509452427067268270433061883736
29
24
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0.06031850382852273502806446351763368108900617211203733815726170756948436092750780760010175098081759313
29
25
$x_k$:
0.8926657199760882787642299140530868780160165015095563983921184107383081653012059632209019295873401150
29
25
$w_k$:
0.04979580909323756092266834117547757161059273184359804724891209854178940237470194596563463524840835691
29
26
$x_k$:
0.9369427185209825115420368482833629224097154426156435491146680828478050163198652173271675839491113846
29
26
$w_k$:
0.03866843997971297974783257426712441099428510620310339272950233296195979051234407668876786914250023370
29
27
$x_k$:
0.9698458072879362937124013878878207457352757632496935636571187669571934697470968347104825463109378798
29
27
$w_k$:
0.02707080629682482754085142544502585388025522126984244741650692569291720746341212953579871759829112341
29
28
$x_k$:
0.9909729882685698162787451132995713026874663409369348571590479260907087937053532974031199312715934986
29
28
$w_k$:
0.01513816985996759679085792660685573856344843865455112920410554457866280996984454188369930045037015570
29
29
$x_k$:
1
equals: One
29
29
$w_k$:
1/406
30
1
$x_k$:
-1
equals: Integers#-1
30
1
$w_k$:
1/435
30
2
$x_k$:
-0.9915739428405002933388182259086506253976046001460166901938800917727302597012775939282581775572933087
30
2
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30
3
$x_k$:
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30
3
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30
4
$x_k$:
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30
4
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0.03614209419940853531473268312335969981698347347151031698062549532406357648160714347465744608639969252
30
5
$x_k$:
-0.8996992181992768595533428231494541300392204118777323786167603777606412718282944819507620796394842906
30
5
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0.04659069453314292740188049144602166411533046522278927750084395701694948314759791705900368910825722220
30
6
$x_k$:
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30
6
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30
7
$x_k$:
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30
7
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30
8
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30
8
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30
9
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30
9
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30
10
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30
10
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30
11
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30
11
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30
12
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30
12
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30
13
$x_k$:
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30
13
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30
14
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30
14
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30
15
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30
15
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30
16
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0.05324511048548666936300783445135376415596309776683403766123721377267123419702559585705729456975046379
30
16
$w_k$:
0.1063895587236679249475823068099311408477991213883440765551039432360330235906852592802586029637218149
30
17
$x_k$:
0.1591320426258504678249485770988477186631453593591757306611377253837894712066223416670999715183786853
30
17
$w_k$:
0.1051841215964546498562129808591394061897722360464786987217600075052895017882103626859605897529651253
30
18
$x_k$:
0.2632159437195737912670994690000676107963263514143343059076758686728959809464761112126581165727397443
30
18
$w_k$:
0.1027869053072349894702808083353761610234467029604508057656452070349533239585771141413712135195149056
30
19
$x_k$:
0.3643175004224489977559851055517856540273346361128288735119735194129622725202788311662260111411605096
30
19
$w_k$:
0.09922507100429983065767596495430906143526109539616727418012225259977712650971889453092484924304478195
30
20
$x_k$:
0.4612911901682406852265572211071521348408482476095720762338188162675560219386153664222767097995534547
30
20
$w_k$:
0.09453897519386089178113271536403588934611422671411685178791363286312412069632081462000016908502468413
30
21
$x_k$:
0.5530382600950528523845502300406577703199504845325382179566082981031998013261192816041963009422115382
30
21
$w_k$:
0.08878171231976521016725643445592055957133267275995476230829628195806137573909208694164417212340714084
30
22
$x_k$:
0.6385191758075584073710934827097407277767510451297820714996243326055590431261439184515142417069042763
30
22
$w_k$:
0.08201851283340691479972154552715988550435910386975047736648941898783436863329026189209242825067152390
30
23
$x_k$:
0.7167653986370851316337985961298973077895498673626822140226089094798582652837226403402194681139933838
30
23
$w_k$:
0.07432600332471825383406764200083067078354778338067158478503059626769984933564370929375454707479255325
30
24
$x_k$:
0.7868903572375470804499595205924037600336790917026762326444248572283896767634087460421082589454267999
30
24
$w_k$:
0.06579133639779005494410137459313238256077320876118507272653987770946740041319317446708642928659466077
30
25
$x_k$:
0.8480994871801981095514238321830652098263746776098134595025560339791453001998283916273697299567666053
30
25
$w_k$:
0.05651119792308038330219371047230332357343614553902614061484065625036490940357389420900081725279369825
30
26
$x_k$:
0.8996992181992768595533428231494541300392204118777323786167603777606412718282944819507620796394842906
30
26
$w_k$:
0.04659069453314292740188049144602166411533046522278927750084395701694948314759791705900368910825722220
30
27
$x_k$:
0.9411047809510570823071895494155210722803079074402374824855328433947469276930974405004346463781775337
30
27
$w_k$:
0.03614209419940853531473268312335969981698347347151031698062549532406357648160714347465744608639969252
30
28
$x_k$:
0.9718466031662692416765929522091917287305673471440824631332574783539278475086769342168458578682726647
30
28
$w_k$:
0.02528316674055140220426825385004247990443170457178888935774385930191207878749792342201022260268326384
30
29
$x_k$:
0.9915739428405002933388182259086506253976046001460166901938800917727302597012775939282581775572933087
30
29
$w_k$:
0.01413179932790538764073216866820779026994079554994967939502182543872273507820983478683252479955422004
30
30
$x_k$:
1
equals: One
30
30
$w_k$:
1/435
Definition
For $n\geq 2$ the Gauss–Lobatto quadrature rule with $n$ points is the unique rule $\int_{-1}^{1} f(x)\,\mathrm{d}x \approx \sum_{k=1}^{n} w_k f(x_k)$ with $x_1=-1$ and $x_n=1$ that is exact for every polynomial $f$ of degree at most $2n-3$ [5]. Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$.
Parameters
$n$
—   number of nodes, the two endpoints included ($n\geq 2$)
$k$
—   index of the node, in increasing order ($1\leq k\leq n$)
Formulas
(1)
$w_k=\dfrac{2}{n(n-1)\,P_{n-1}(x_k)^2}$ for $1\leq k\leq n$ [1]; at the endpoints $P_{n-1}(\pm1)^2=1$, so $w_1=w_n=\dfrac{2}{n(n-1)}$.
(2)
$w_k=\dfrac{q(x_k)}{\omega_n'(x_k)}$, where $\omega_n=(x^2-1)P_{n-1}'=\dfrac{n(n-1)}{2n-1}\,(P_n-P_{n-2})$ is the polynomial whose roots are the nodes and $q(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,\mathrm{d}t =\dfrac{n(n-1)}{2n-1}\,(q_n-q_{n-2})$ with $q_n$ the secondary polynomial of $P_n$.
(3)
$\sum_{k=1}^{n} w_k x_k^m=\int_{-1}^{1}x^m\,\mathrm{d}x=\dfrac{1+(-1)^m}{m+1}$ for $0\leq m\leq 2n-3$, and not for $m=2n-2$.
(4)
$\int_{-1}^{1}f(x)\,\mathrm{d}x-\sum_{k=1}^{n}w_kf(x_k) =-\dfrac{n(n-1)^3\,2^{2n-1}\,((n-2)!)^4}{(2n-1)\,((2n-2)!)^3}\,f^{(2n-2)}(\xi)$ for some $\xi\in(-1,1)$ [1].
(5)
$x_{n+1-k}=-x_k$, $w_{n+1-k}=w_k$, $w_k>0$ and $\sum_{k=1}^{n}w_k=2$.
(6)
For odd $n$, $x_{(n+1)/2}=0$ and $w_{(n+1)/2}=\dfrac{2}{n(n-1)P_{n-1}(0)^2}=\dfrac{2\,((n-1)!!)^2}{n(n-1)\,((n-2)!!)^2}$: $\tfrac43$, $\tfrac{32}{45}$, $\tfrac{256}{525}$, $\tfrac{4096}{11025}$ for $n=3,5,7,9$.
(7)
$P_{n-1}'(x)=\dfrac{n}{2}\,P^{(1,1)}_{n-2}(x)$, so the interior nodes are the roots of the Jacobi polynomial $P^{(1,1)}_{n-2}$, orthogonal on $[-1,1]$ for the weight $1-x^2$.
(8)
$w_k=\int_{-1}^{1}\ell_k(x)\,\mathrm{d}x$ with $\ell_k(x)=\prod_{j\neq k}\frac{x-x_j}{x_k-x_j}$ the Lagrange basis polynomial of the nodes, as for every interpolatory rule.
(9)
On $[a,b]$ the rule is $\int_a^b f \approx \frac{b-a}{2}\sum_k w_k\, f\bigl(\frac{b-a}{2}x_k+\frac{a+b}{2}\bigr)$.
Comments
(10)
The interior nodes $x_2<\cdots<x_{n-1}$ are the roots of $P_{n-1}'$, the derivative of the Legendre polynomial of degree $n-1$: the points at which $P_{n-1}$ has its local extrema in $(-1,1)$. They are real, simple, symmetric about $0$, and interlace with the roots of $P_{n-1}$. Since $P_{n-1}'=\tfrac{n}{2}P^{(1,1)}_{n-2}$, they are also the roots of a Jacobi polynomial.
(11)
Fixing both endpoints costs two degrees of freedom: a rule with $n$ points reaches degree $2n-3$ here against $2n-1$ for the Gauss–Legendre rule with $n$ free points. The rule is used where the values of $f$ at the endpoints are wanted anyway: the nodes are the Legendre–Gauss–Lobatto collocation points of spectral and spectral-element methods [4], and an adaptive rule built on the four-point Lobatto rule [3] is the routine quadl of MATLAB and Octave.
(12)
Every value here is an algebraic number, and the ones that are rational are written exactly rather than to a hundred digits: the endpoints $x_1=-1$ and $x_n=1$, their weights $w_1=w_n=2/(n(n-1))$, the node $x=0$ of every rule of odd order with its weight $2/(n(n-1)P_{n-1}(0)^2)$, and all weights of the rules with $n\leq 5$. The rules with $n\leq 7$ have closed forms in square roots, given in the comments on their entries; the nodes $\pm 1/\sqrt{5}$ of $n=4$ are also in the table of quadratic algebraic numbers.
(13)
The rule with $n=2$ is the trapezoidal rule and the rule with $n=3$ is Simpson's rule, which are also the closed Newton–Cotes rules with two and three points. From $n=4$ on the interior nodes are not equally spaced ($\pm 1/\sqrt{5}$ here, $\pm 1/3$ for Simpson's $3/8$ rule) and the two families part; the Newton–Cotes weights are the integrals of the Lagrange basis polynomials for equally spaced nodes.
(14)
The same rule on $[0,1]$ has nodes $c_k=(1+x_k)/2$ and weights $b_k=w_k/2$, which is how it appears in the Butcher tableaux of the Lobatto IIIA, IIIB and IIIC Runge–Kutta methods [2]: $c=0,\tfrac12,1$ with $b=\tfrac16,\tfrac23,\tfrac16$ for three stages, and $c=0,(5-\sqrt5)/10,(5+\sqrt5)/10,1$ with $b=\tfrac1{12},\tfrac5{12},\tfrac5{12},\tfrac1{12}$ for four. Those values are an affine image of the entries here rather than a multiple of them, and they are not listed separately.
(15)
The Gauss–Radau rule fixes one endpoint only and reaches degree $2n-2$; the Gauss–Kronrod rule extends the Gauss–Legendre rule by $n+1$ points. The Radau rule is not listed here, nor are the Lobatto rules for other weight functions.
Programs
(P1)
Sage
R.<x> = QQ[]
n = 8
P = R(legendre_P(n - 1, x))
nodes = [-1] + P.derivative().roots(RealIntervalField(400), multiplicities=False) + [1]   # x_2 = -0.87174014850960661534...
weights = [2/(n*(n - 1)*P(r)^2) for r in nodes]                                          # w_1 = 1/28, w_2 = 0.21070422714350603938...
(P2)
Python
from sympy.integrals.quadrature import gauss_lobatto
nodes, weights = gauss_lobatto(8, 20)      # nodes[1] = -0.87174014850960661534, weights[1] = 0.21070422714350603938
References
[1]
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972, §25.4.32 and Table 25.6.
[2]
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed., Springer, 1996, §IV.5.
[3]
W. Gander and W. Gautschi, Adaptive quadrature — revisited, BIT 40 (2000), 84–101.
[4]
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, 2006, §2.3.
Links
Similar tables
Nodes and weights of Gauss–Legendre quadrature —   the rule with $n$ free points; the interior nodes here interlace with the nodes of its $(n-1)$-point rule
Legendre polynomials —   the interior nodes are the roots of $P_{n-1}'$
Jacobi polynomials —   $P_{n-1}'=\frac{n}{2}P^{(1,1)}_{n-2}$
Secondary polynomials of the Legendre polynomials —   $q=\frac{n(n-1)}{2n-1}(q_n-q_{n-2})$ in the weight formula
Algebraic numbers of degree 2 —   holds the nodes $\pm 1/\sqrt{5}$ of $n=4$
Lagrange basis polynomials for equally spaced nodes —   the weights are the integrals of the Lagrange basis of the nodes; on equally spaced nodes that gives Newton–Cotes, which agrees with this table for $n\leq 3$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every rule with $2\leq n\leq 30$, where $n$ is the number of nodes, the two endpoints included; each rule is listed in full, every node with its weight, and both halves of the symmetric set)
How they were obtained:

$P_{n-1}$ is built exactly in $\mathbb{Q}[x]$; the roots of $P_{n-1}'$ are isolated by Sage's real root isolation over the interval field, so each interior node is an interval provably containing one root, carried on as an arb ball at 397 bits; $w_k$ is $2/(n(n-1)P_{n-1}(x_k)^2)$ in ball arithmetic, and a hundred digits are written, which every ball supports (the widest, the weight $w_2$ at $n=30$, supports 107).

more

The endpoints, their weights, the central node and weight of every odd rule and all weights of $n\leq 5$ are exact. Before a rule is written it must also satisfy the secondary-polynomial weight formula (exactly in $\mathbb{Q}$ at the exact nodes), the interlacing with the roots of $P_{n-1}$, the symmetry, $\sum w_k=2$, exactness on $x^m$ for $m\leq 2n-3$ with failure at $m=2n-2$, and the closed forms of the entry comments. Outside the generator the values were compared with the rows for $n\leq 6$ on MathWorld and $n\leq 7$ on Wikipedia (the closed forms verified exactly in $\mathbb{Q}(\sqrt7)$ and $\mathbb{Q}(\sqrt{15})$), with the weights as integrals of the Lagrange basis polynomials, with the stored Legendre, secondary and Gauss–Legendre tables, with the explicit sum for $P^{(1,1)}_{n-2}$, with the Lobatto IIIA tableaux, and with sympy's gauss_lobatto on another machine to 50 digits, with the controls that must fail failing.



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