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Nodes and weights of Gauss–Kronrod quadrature
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Numbers
$n$
$k$
$x_k$ or $w_k$
1
1
$x_k$:
-0.7745966692414833770358530799564799221665843410583181653175147532226966183873958067038574753717347036
comment: $x_1=-\sqrt{3/5}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,1,x
1
1
$w_k$:
5/9
comment: $w_1=5/9$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,1,w
1
2
$x_k$:
0
comment: $x_2=0$; the Kronrod extension of the midpoint rule is the three-point Gauss rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,2,x
1
2
$w_k$:
8/9
comment: $w_2=8/9$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,2,w
1
3
$x_k$:
0.7745966692414833770358530799564799221665843410583181653175147532226966183873958067038574753717347036
comment: $x_3=\sqrt{3/5}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,3,x
1
3
$w_k$:
5/9
comment: $w_3=5/9$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,3,w
2
1
$x_k$:
-0.9258200997725514615665667765839995225293149010083352213873363425442310539777206244857658438008014622
comment: $x_1=-\sqrt{6/7}$
2
1
$w_k$:
98/495
comment: $w_1=98/495$
2
2
$x_k$:
-0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252
comment: $x_2=-1/\sqrt{3}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#2,1,x
2
2
$w_k$:
27/55
comment: $w_2=27/55$
2
3
$x_k$:
0
comment: $x_3=0$
equals: Zero
2
3
$w_k$:
28/45
comment: $w_3=28/45$
2
4
$x_k$:
0.5773502691896257645091487805019574556476017512701268760186023264839776723029333456937153955857495252
comment: $x_4=1/\sqrt{3}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#2,2,x
2
4
$w_k$:
27/55
comment: $w_4=27/55$
2
5
$x_k$:
0.9258200997725514615665667765839995225293149010083352213873363425442310539777206244857658438008014622
comment: $x_5=\sqrt{6/7}$
2
5
$w_k$:
98/495
comment: $w_5=98/495$
3
1
$x_k$:
-0.9604912687080202834235070926290799626697822363652913171329795419177186755110607277765769797684345261
comment: $x_1=-\tfrac13\sqrt{5+2\sqrt{30/11}}$
3
1
$w_k$:
0.1046562260264672651938238571920730382422021606251036520630504326666374903364810422862726982741409519
comment: $w_1=(4057614-130977\sqrt{330})/16036300$
3
2
$x_k$:
-0.7745966692414833770358530799564799221665843410583181653175147532226966183873958067038574753717347036
comment: $x_2=-\sqrt{3/5}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,1,x
3
2
$w_k$:
12500/46557
comment: $w_2=12500/46557$
3
3
$x_k$:
-0.4342437493468025580020715028446278172828985569550271062941594695509705441342701652634607982329774245
comment: $x_3=-\tfrac13\sqrt{5-2\sqrt{30/11}}$
3
3
$w_k$:
0.4013974147759622229050518186184318787274230022865405551418534416684647464637783691739768605393010516
comment: $w_3=(4057614+130977\sqrt{330})/16036300$
3
4
$x_k$:
0
comment: $x_4=0$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,2,x
3
4
$w_k$:
22016/48825
comment: $w_4=22016/48825$
3
5
$x_k$:
0.4342437493468025580020715028446278172828985569550271062941594695509705441342701652634607982329774245
comment: $x_5=\tfrac13\sqrt{5-2\sqrt{30/11}}$
3
5
$w_k$:
0.4013974147759622229050518186184318787274230022865405551418534416684647464637783691739768605393010516
comment: $w_5=(4057614+130977\sqrt{330})/16036300$
3
6
$x_k$:
0.7745966692414833770358530799564799221665843410583181653175147532226966183873958067038574753717347036
comment: $x_6=\sqrt{3/5}$
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#3,3,x
3
6
$w_k$:
12500/46557
comment: $w_6=12500/46557$
3
7
$x_k$:
0.9604912687080202834235070926290799626697822363652913171329795419177186755110607277765769797684345261
comment: $x_7=\tfrac13\sqrt{5+2\sqrt{30/11}}$
3
7
$w_k$:
0.1046562260264672651938238571920730382422021606251036520630504326666374903364810422862726982741409519
comment: $w_7=(4057614-130977\sqrt{330})/16036300$
4
1
$x_k$:
-0.9765602507375731115345053593699196268337590533023609391747263402307933234640747912383270108094904064
4
1
$w_k$:
0.06297737366547301476549248855281867632941703873821368398447968870382653638722102093112154538177202385
4
2
$x_k$:
-0.8611363115940525752239464888928095050957253796297176376157219209065294714950488657041623398844793052
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#4,1,x
4
2
$w_k$:
0.1700536053357227268027388532962065872619470485872028155254609757993697849254229770754766808154270869
4
3
$x_k$:
-0.6402862174963099824046890231574920183560098001869829980711768186624953946251243069526088655746852887
4
3
$w_k$:
0.2667983404522844480327706284178556624766503773329987023238987103757181930562850623411383098787853047
4
4
$x_k$:
-0.3399810435848562648026657591032446872005758697709143525929539768210200304632370344778752804355548115
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#4,2,x
4
4
$w_k$:
0.3269491896014516295584594656173191857743738004940775929353781585662612072295567657803660147907940937
4
5
$x_k$:
0
comment: $x_{5}=0$, the central node of every rule
equals: Zero
4
5
$w_k$:
201344/581175
4
6
$x_k$:
0.3399810435848562648026657591032446872005758697709143525929539768210200304632370344778752804355548115
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#4,3,x
4
6
$w_k$:
0.3269491896014516295584594656173191857743738004940775929353781585662612072295567657803660147907940937
4
7
$x_k$:
0.6402862174963099824046890231574920183560098001869829980711768186624953946251243069526088655746852887
4
7
$w_k$:
0.2667983404522844480327706284178556624766503773329987023238987103757181930562850623411383098787853047
4
8
$x_k$:
0.8611363115940525752239464888928095050957253796297176376157219209065294714950488657041623398844793052
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#4,4,x
4
8
$w_k$:
0.1700536053357227268027388532962065872619470485872028155254609757993697849254229770754766808154270869
4
9
$x_k$:
0.9765602507375731115345053593699196268337590533023609391747263402307933234640747912383270108094904064
4
9
$w_k$:
0.06297737366547301476549248855281867632941703873821368398447968870382653638722102093112154538177202385
5
1
$x_k$:
-0.9840853600948424644961729346361394995805528241884719950608016533645331701671188096220348194284127022
5
1
$w_k$:
0.04258203675108183286450945084767009187528571052993374559900656835475405584035607690865649713310590328
5
2
$x_k$:
-0.9061798459386639927976268782993929651256519107625308628737622865437707949166868469411429895535422619
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#5,1,x
5
2
$w_k$:
0.1152333166224733940246268458805735391695962921801944271568656551359490215131267512750627477663138266
5
3
$x_k$:
-0.7541667265708492204408171669461158663862998043714840971052302472871081140430851012111371067697093960
5
3
$w_k$:
0.1868007965564926574678000268784859712873998237470783060412387748327111203590361796662431668289829322
5
4
$x_k$:
-0.5384693101056830910363144207002088049672866069055599562022316270594711853677552910358036672505709316
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#5,2,x
5
4
$w_k$:
0.2410403392286475866999426112232621112960798350994144822297850922744668657684649807520582437477270142
5
5
$x_k$:
-0.2796304131617831934134665227489774362421188153561726461128143950089739910670368919195145582562943315
5
5
$w_k$:
0.2728498019125589223409932644844555182626101274631601015242771158034740386076752304811695827557596808
5
6
$x_k$:
0
comment: $x_{6}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#5,3,x
5
6
$w_k$:
118298624/418034925
5
7
$x_k$:
0.2796304131617831934134665227489774362421188153561726461128143950089739910670368919195145582562943315
5
7
$w_k$:
0.2728498019125589223409932644844555182626101274631601015242771158034740386076752304811695827557596808
5
8
$x_k$:
0.5384693101056830910363144207002088049672866069055599562022316270594711853677552910358036672505709316
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#5,4,x
5
8
$w_k$:
0.2410403392286475866999426112232621112960798350994144822297850922744668657684649807520582437477270142
5
9
$x_k$:
0.7541667265708492204408171669461158663862998043714840971052302472871081140430851012111371067697093960
5
9
$w_k$:
0.1868007965564926574678000268784859712873998237470783060412387748327111203590361796662431668289829322
5
10
$x_k$:
0.9061798459386639927976268782993929651256519107625308628737622865437707949166868469411429895535422619
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#5,5,x
5
10
$w_k$:
0.1152333166224733940246268458805735391695962921801944271568656551359490215131267512750627477663138266
5
11
$x_k$:
0.9840853600948424644961729346361394995805528241884719950608016533645331701671188096220348194284127022
5
11
$w_k$:
0.04258203675108183286450945084767009187528571052993374559900656835475405584035607690865649713310590328
6
1
$x_k$:
-0.9887032026126788575046459517121850761363052662449377880764957968384380126252193458613107844483535111
6
1
$w_k$:
0.03039615411981976885196454467602788471477518727112560837216200538583129985073337895108841544661912559
6
2
$x_k$:
-0.9324695142031520278123015544939946091347657377122898248725496165266135008442001962762887399219259850
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,1,x
6
2
$w_k$:
0.08369444044690662613284560348241110839210934697369541772821800589921374007671640914173390135194135530
6
3
$x_k$:
-0.8213733408650279400456498342439502511175068505980280772179567281190827579951897037702249709424102068
6
3
$w_k$:
0.1373206046344469230871498725337818084362811130970035312786101050647635935233650361862762278369200099
6
4
$x_k$:
-0.6612093864662645136613995950199053470064485643951700708145267058521834966071431009442864037464614564
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,2,x
6
4
$w_k$:
0.1810719943231376151869920933155119377662277289718007677981447181545882289678422811529294950061314039
6
5
$x_k$:
-0.4631182124753046121567583640191766343144650077857935038771262347079228332038599312908409298835429967
6
5
$w_k$:
0.2132096522719622791628941635168893046782881023625370002425031822487221749313854987368148576633472526
6
6
$x_k$:
-0.2386191860831969086305017216807119354186106301400213501813951645742749342756398422492244272573491316
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,3,x
6
6
$w_k$:
0.2337708641169944066228357259889983727949444403735290659667389723632407501036772472785760615928967388
6
7
$x_k$:
0
comment: $x_{7}=0$, the central node of every rule
equals: Zero
6
7
$w_k$:
9302400/38587549
6
8
$x_k$:
0.2386191860831969086305017216807119354186106301400213501813951645742749342756398422492244272573491316
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,4,x
6
8
$w_k$:
0.2337708641169944066228357259889983727949444403735290659667389723632407501036772472785760615928967388
6
9
$x_k$:
0.4631182124753046121567583640191766343144650077857935038771262347079228332038599312908409298835429967
6
9
$w_k$:
0.2132096522719622791628941635168893046782881023625370002425031822487221749313854987368148576633472526
6
10
$x_k$:
0.6612093864662645136613995950199053470064485643951700708145267058521834966071431009442864037464614564
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,5,x
6
10
$w_k$:
0.1810719943231376151869920933155119377662277289718007677981447181545882289678422811529294950061314039
6
11
$x_k$:
0.8213733408650279400456498342439502511175068505980280772179567281190827579951897037702249709424102068
6
11
$w_k$:
0.1373206046344469230871498725337818084362811130970035312786101050647635935233650361862762278369200099
6
12
$x_k$:
0.9324695142031520278123015544939946091347657377122898248725496165266135008442001962762887399219259850
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#6,6,x
6
12
$w_k$:
0.08369444044690662613284560348241110839210934697369541772821800589921374007671640914173390135194135530
6
13
$x_k$:
0.9887032026126788575046459517121850761363052662449377880764957968384380126252193458613107844483535111
6
13
$w_k$:
0.03039615411981976885196454467602788471477518727112560837216200538583129985073337895108841544661912559
7
1
$x_k$:
-0.9914553711208126392068546975263285166420443383703347012910874135724417393465340723592450350962684176
7
1
$w_k$:
0.02293532201052922496373200805896959199356081127574699226750743025471181578797607594615636816815628948
7
2
$x_k$:
-0.9491079123427585245261896840478512624007709376706177835487691039130633303548401408057307700279257241
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,1,x
7
2
$w_k$:
0.06309209262997855329070066318920428666507115721155070711360554514698399747796487492819917026450444200
7
3
$x_k$:
-0.8648644233597690727897127886409262012109723070740881486014577127670677081325957210358584785960459054
7
3
$w_k$:
0.1047900103222501838398763225415180174437566542138306118933906513396374632157628952416757162750931133
7
4
$x_k$:
-0.7415311855993944398638647732807884070741476471413902601199553519674298746721805137928268323668632471
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,2,x
7
4
$w_k$:
0.1406532597155259187451895905102379203998897572479985755617454689331270809309095040809737912241555591
7
5
$x_k$:
-0.5860872354676911302941448382587295984367807506043609513049928931988037360744440746451167449893594210
7
5
$w_k$:
0.1690047266392679028265834265985502841062449003029442414973400675569568092161902911293670240385535991
7
6
$x_k$:
-0.4058451513773971669066064120769614633473820140993701263870432517946638132261256553283126897277465878
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,3,x
7
6
$w_k$:
0.1903505780647854099132564024210136828260780754553583558854408803674405807241021267960596460510637759
7
7
$x_k$:
-0.2077849550078984676006894037732449134797844071451706497138457346198669384494352022691034322718369853
7
7
$w_k$:
0.2044329400752988924141619992346490847165176041807183574244709531204546769854659887934837429200934755
7
8
$x_k$:
0
comment: $x_{8}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,4,x
7
8
$w_k$:
3496355037184/16690468309515
7
9
$x_k$:
0.2077849550078984676006894037732449134797844071451706497138457346198669384494352022691034322718369853
7
9
$w_k$:
0.2044329400752988924141619992346490847165176041807183574244709531204546769854659887934837429200934755
7
10
$x_k$:
0.4058451513773971669066064120769614633473820140993701263870432517946638132261256553283126897277465878
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,5,x
7
10
$w_k$:
0.1903505780647854099132564024210136828260780754553583558854408803674405807241021267960596460510637759
7
11
$x_k$:
0.5860872354676911302941448382587295984367807506043609513049928931988037360744440746451167449893594210
7
11
$w_k$:
0.1690047266392679028265834265985502841062449003029442414973400675569568092161902911293670240385535991
7
12
$x_k$:
0.7415311855993944398638647732807884070741476471413902601199553519674298746721805137928268323668632471
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,6,x
7
12
$w_k$:
0.1406532597155259187451895905102379203998897572479985755617454689331270809309095040809737912241555591
7
13
$x_k$:
0.8648644233597690727897127886409262012109723070740881486014577127670677081325957210358584785960459054
7
13
$w_k$:
0.1047900103222501838398763225415180174437566542138306118933906513396374632157628952416757162750931133
7
14
$x_k$:
0.9491079123427585245261896840478512624007709376706177835487691039130633303548401408057307700279257241
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#7,7,x
7
14
$w_k$:
0.06309209262997855329070066318920428666507115721155070711360554514698399747796487492819917026450444200
7
15
$x_k$:
0.9914553711208126392068546975263285166420443383703347012910874135724417393465340723592450350962684176
7
15
$w_k$:
0.02293532201052922496373200805896959199356081127574699226750743025471181578797607594615636816815628948
8
1
$x_k$:
-0.9933798758817161559358880690196707954694123351161213263786244885745180623751643296018558602024737550
8
1
$w_k$:
0.01782238332071035515278696120274978980090967589535097962891925560361760977514759763492862475836126838
8
2
$x_k$:
-0.9602898564975362316835608685694729904282352343014520382716397773724248977434192844394389592633122683
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,1,x
8
2
$w_k$:
0.04943939500213930850036396944699689465826327628242587825569974121581095620535162426153093155119618299
8
3
$x_k$:
-0.8941209068474564219483610175382513245672899093811670433207884536114282821747867724923127979654842894
8
3
$w_k$:
0.08248229893135833068862519344560789547593191478910299786454104780022350316682058893541348789093424132
8
4
$x_k$:
-0.7966664774136267395915539364758304368371717316159648320701702950392173056764730921471519272957259390
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,2,x
8
4
$w_k$:
0.1116463708268396132221081589339414845397596388530349347822865976218512393952526374891923500903036891
8
5
$x_k$:
-0.6723540709451586771563107380928310497598795061737575836053834962263344584537084036175044201798814291
8
5
$w_k$:
0.1362631092551722152623387452545062032301192309852634465410113535910128040073110454353275403471292448
8
6
$x_k$:
-0.5255324099163289858177390491892463490419642431203928577508570992724548207685612725239614001936319821
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,3,x
8
6
$w_k$:
0.1566526061681884004902480884869687372763250363770561495681940911395305358527731786960836466208921517
8
7
$x_k$:
-0.3607010979281319571925486222968914391823042338828422188456650430092793527610436695842830402507642285
8
7
$w_k$:
0.1720706085552113118572948802038570866070057925848462833692048982598345545157098098621109704517313113
8
8
$x_k$:
-0.1834346424956498049394761423601839806667578129129737823171884736992044742215421141160682237111233537
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,4,x
8
8
$w_k$:
0.1814000250680346430617485251725504435592029631271780289270692335685122031664126243310478736646763060
8
9
$x_k$:
0
comment: $x_{9}=0$, the central node of every rule
equals: Zero
8
9
$w_k$:
687527723008/3727520307225
8
10
$x_k$:
0.1834346424956498049394761423601839806667578129129737823171884736992044742215421141160682237111233537
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,5,x
8
10
$w_k$:
0.1814000250680346430617485251725504435592029631271780289270692335685122031664126243310478736646763060
8
11
$x_k$:
0.3607010979281319571925486222968914391823042338828422188456650430092793527610436695842830402507642285
8
11
$w_k$:
0.1720706085552113118572948802038570866070057925848462833692048982598345545157098098621109704517313113
8
12
$x_k$:
0.5255324099163289858177390491892463490419642431203928577508570992724548207685612725239614001936319821
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,6,x
8
12
$w_k$:
0.1566526061681884004902480884869687372763250363770561495681940911395305358527731786960836466208921517
8
13
$x_k$:
0.6723540709451586771563107380928310497598795061737575836053834962263344584537084036175044201798814291
8
13
$w_k$:
0.1362631092551722152623387452545062032301192309852634465410113535910128040073110454353275403471292448
8
14
$x_k$:
0.7966664774136267395915539364758304368371717316159648320701702950392173056764730921471519272957259390
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,7,x
8
14
$w_k$:
0.1116463708268396132221081589339414845397596388530349347822865976218512393952526374891923500903036891
8
15
$x_k$:
0.8941209068474564219483610175382513245672899093811670433207884536114282821747867724923127979654842894
8
15
$w_k$:
0.08248229893135833068862519344560789547593191478910299786454104780022350316682058893541348789093424132
8
16
$x_k$:
0.9602898564975362316835608685694729904282352343014520382716397773724248977434192844394389592633122683
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#8,8,x
8
16
$w_k$:
0.04943939500213930850036396944699689465826327628242587825569974121581095620535162426153093155119618299
8
17
$x_k$:
0.9933798758817161559358880690196707954694123351161213263786244885745180623751643296018558602024737550
8
17
$w_k$:
0.01782238332071035515278696120274978980090967589535097962891925560361760977514759763492862475836126838
9
1
$x_k$:
-0.9946781606773402425263042362754331948771024276522767361747855043445180239944611239571714266452152764
9
1
$w_k$:
0.01430477564383893723193222141644869285568603149552731809061765987022293487345672170324872003626835934
9
2
$x_k$:
-0.9681602395076260898355762029036728700494048004919253295500233118490803743966007530618737492268941116
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,1,x
9
2
$w_k$:
0.03963189516026125507820628405408056822267924300210332295958283235449857586934142391832784751167040885
9
3
$x_k$:
-0.9149635072496778539613835957264172115308937407999780835282952693530749885748387431705368886592407317
9
3
$w_k$:
0.06651815594027414323762257382083183242312026595057734300674911339347224239151986797215307925975456271
9
4
$x_k$:
-0.8360311073266357942994297880697348765441067181246759961043719796394550068815901188939461970258575403
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,2,x
9
4
$w_k$:
0.09079068168872638685105183330601498905183631167653644208388264543293956126825296350972627002541666607
9
5
$x_k$:
-0.7344867651839337916097817318431993285577767708803940266028036409856275373827494735207165245096120890
9
5
$w_k$:
0.1117891346844182733403397109583932799842533464373497702636211268761178202798987474751254585526701223
9
6
$x_k$:
-0.6133714327005903973087020393414741847857206049405646928728129422812673464910011985832400139035685846
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,3,x
9
6
$w_k$:
0.1300014068553411967245726550322585733227473065959336057279782206824247168521555377944268359381204117
9
7
$x_k$:
-0.4754624791124598889553958477113145061269300890130407805487301507698059728103858430254315981950988036
9
7
$w_k$:
0.1452395883843661621706415284882886756292205389210294182243400547098365723714635189243206558281032156
9
8
$x_k$:
-0.3242534234038089290385380146433366085719562607369730888270474768421865795351242491930986016984975672
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,4,x
9
8
$w_k$:
0.1564135277884838655771294400482961487228560503385181464584555826986084200162863353533118777212262160
9
9
$x_k$:
-0.1642235636149867614393813701714323989848938043420049970012325806919051592620714589214267704134119523
9
9
$w_k$:
0.1628628274401150631741673119359344883038463303869095960638916227875248658799208676331946298186535814
9
10
$x_k$:
0
comment: $x_{10}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,5,x
9
10
$w_k$:
644961921847525376/3911325148406751075
9
11
$x_k$:
0.1642235636149867614393813701714323989848938043420049970012325806919051592620714589214267704134119523
9
11
$w_k$:
0.1628628274401150631741673119359344883038463303869095960638916227875248658799208676331946298186535814
9
12
$x_k$:
0.3242534234038089290385380146433366085719562607369730888270474768421865795351242491930986016984975672
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,6,x
9
12
$w_k$:
0.1564135277884838655771294400482961487228560503385181464584555826986084200162863353533118777212262160
9
13
$x_k$:
0.4754624791124598889553958477113145061269300890130407805487301507698059728103858430254315981950988036
9
13
$w_k$:
0.1452395883843661621706415284882886756292205389210294182243400547098365723714635189243206558281032156
9
14
$x_k$:
0.6133714327005903973087020393414741847857206049405646928728129422812673464910011985832400139035685846
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,7,x
9
14
$w_k$:
0.1300014068553411967245726550322585733227473065959336057279782206824247168521555377944268359381204117
9
15
$x_k$:
0.7344867651839337916097817318431993285577767708803940266028036409856275373827494735207165245096120890
9
15
$w_k$:
0.1117891346844182733403397109583932799842533464373497702636211268761178202798987474751254585526701223
9
16
$x_k$:
0.8360311073266357942994297880697348765441067181246759961043719796394550068815901188939461970258575403
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,8,x
9
16
$w_k$:
0.09079068168872638685105183330601498905183631167653644208388264543293956126825296350972627002541666607
9
17
$x_k$:
0.9149635072496778539613835957264172115308937407999780835282952693530749885748387431705368886592407317
9
17
$w_k$:
0.06651815594027414323762257382083183242312026595057734300674911339347224239151986797215307925975456271
9
18
$x_k$:
0.9681602395076260898355762029036728700494048004919253295500233118490803743966007530618737492268941116
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#9,9,x
9
18
$w_k$:
0.03963189516026125507820628405408056822267924300210332295958283235449857586934142391832784751167040885
9
19
$x_k$:
0.9946781606773402425263042362754331948771024276522767361747855043445180239944611239571714266452152764
9
19
$w_k$:
0.01430477564383893723193222141644869285568603149552731809061765987022293487345672170324872003626835934
10
1
$x_k$:
-0.9956571630258080807355272806890028479212605872194789243633791611175702304677486735715232599691207672
10
1
$w_k$:
0.01169463886737187427806439606219204839621733248193188892759814752562222205806499265180673670496996725
10
2
$x_k$:
-0.9739065285171717200779640120844520534282699466923821192312120666965952032346361596257235649562685563
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,1,x
10
2
$w_k$:
0.03255816230796472747881897245938976061738893984566260957153750423271412182016549869238160760538462649
10
3
$x_k$:
-0.9301574913557082260012071800595083462251679099819392423034940686682841598309167305501119457285100788
10
3
$w_k$:
0.05475589657435199603138130024458017637372111405833355752443261580478409892781897532511630156900329809
10
4
$x_k$:
-0.8650633666889845107320966884234930485275430149653304525219597318453747551380555613567907289460457707
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,2,x
10
4
$w_k$:
0.07503967481091995276704314091619000939521938200091008817369704804843040434285849517881380873064655409
10
5
$x_k$:
-0.7808177265864168970637175783450423771634075202981571797469485999950560798276142065452697723423899624
10
5
$w_k$:
0.09312545458369760553506546508336634439001882888076003197008503876017773567220077523741412306161582747
10
6
$x_k$:
-0.6794095682990244062343273651148735757692947118348094676648171889525585753950749246150785735704803795
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,3,x
10
6
$w_k$:
0.1093871588022976418992105903258049602718132998343452200781967582982655037289143216868389943267455384
10
7
$x_k$:
-0.5627571346686046833390000992726941408430138819419669588603462145877926635321632754971208785416999242
10
7
$w_k$:
0.1234919762620658510779581098310741595123003495286483276446799412097405423897545468968153862236373823
10
8
$x_k$:
-0.4333953941292471907992659431657841622000718376562464965027015131437669890777035012251027579501177212
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,4,x
10
8
$w_k$:
0.1347092173114733259280540017717068327609919130085597140663666849132029140012128203667695315948827177
10
9
$x_k$:
-0.2943928627014601981311266031038655661626866251569579186488822917272461116633273788844552317826823736
10
9
$w_k$:
0.1427759385770600807970942731387170608859790565319055556074100474397077044990934002781113170628375643
10
10
$x_k$:
-0.1488743389816312108848260011297199846175648594206916957079892535159036173556685213711776297994636912
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,5,x
10
10
$w_k$:
0.1477391049013384913748415159720680455237316254852066045181919543988599301673569640573270395918288225
10
11
$x_k$:
0
comment: $x_{11}=0$, the central node of every rule
equals: Zero
10
11
$w_k$:
237005715742720000/1585900077951426237
10
12
$x_k$:
0.1488743389816312108848260011297199846175648594206916957079892535159036173556685213711776297994636912
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,6,x
10
12
$w_k$:
0.1477391049013384913748415159720680455237316254852066045181919543988599301673569640573270395918288225
10
13
$x_k$:
0.2943928627014601981311266031038655661626866251569579186488822917272461116633273788844552317826823736
10
13
$w_k$:
0.1427759385770600807970942731387170608859790565319055556074100474397077044990934002781113170628375643
10
14
$x_k$:
0.4333953941292471907992659431657841622000718376562464965027015131437669890777035012251027579501177212
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,7,x
10
14
$w_k$:
0.1347092173114733259280540017717068327609919130085597140663666849132029140012128203667695315948827177
10
15
$x_k$:
0.5627571346686046833390000992726941408430138819419669588603462145877926635321632754971208785416999242
10
15
$w_k$:
0.1234919762620658510779581098310741595123003495286483276446799412097405423897545468968153862236373823
10
16
$x_k$:
0.6794095682990244062343273651148735757692947118348094676648171889525585753950749246150785735704803795
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,8,x
10
16
$w_k$:
0.1093871588022976418992105903258049602718132998343452200781967582982655037289143216868389943267455384
10
17
$x_k$:
0.7808177265864168970637175783450423771634075202981571797469485999950560798276142065452697723423899624
10
17
$w_k$:
0.09312545458369760553506546508336634439001882888076003197008503876017773567220077523741412306161582747
10
18
$x_k$:
0.8650633666889845107320966884234930485275430149653304525219597318453747551380555613567907289460457707
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,9,x
10
18
$w_k$:
0.07503967481091995276704314091619000939521938200091008817369704804843040434285849517881380873064655409
10
19
$x_k$:
0.9301574913557082260012071800595083462251679099819392423034940686682841598309167305501119457285100788
10
19
$w_k$:
0.05475589657435199603138130024458017637372111405833355752443261580478409892781897532511630156900329809
10
20
$x_k$:
0.9739065285171717200779640120844520534282699466923821192312120666965952032346361596257235649562685563
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#10,10,x
10
20
$w_k$:
0.03255816230796472747881897245938976061738893984566260957153750423271412182016549869238160760538462649
10
21
$x_k$:
0.9956571630258080807355272806890028479212605872194789243633791611175702304677486735715232599691207672
10
21
$w_k$:
0.01169463886737187427806439606219204839621733248193188892759814752562222205806499265180673670496996725
11
1
$x_k$:
-0.9963696138895426343601645733351607733670255420718933140205778155451740907666856531023195550771640705
11
1
$w_k$:
0.009765441045960758022479172609643522163694469405185463334885137933841331943681209440531842718097720796
11
2
$x_k$:
-0.9782286581460569928039380011228573907714224089197844154258010659836637993808899882003193981673447699
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,1,x
11
2
$w_k$:
0.02715655468210426205172140161785167941281344912488029029841290913477031686215645035776794876251809925
11
3
$x_k$:
-0.9416771085780679464553967303521349621887508937383184623824558387012555494696002242409994853810213235
11
3
$w_k$:
0.04582937856442641598526161637154832107049886748562641958315178306865641956835426354672218172246312165
11
4
$x_k$:
-0.8870625997680952990751577693039272666316757512253143849674110555376113138573726674422905951265973742
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,2,x
11
4
$w_k$:
0.06309742475037490658454053049537146778131754091445865261757547710038577981517821385824072570472632977
11
5
$x_k$:
-0.8160574566562209423922613551926258792778640464249351554136242461884651368612930405915007300046006812
11
5
$w_k$:
0.07866457193222732928421712412387330212537013927185540580724871403138289435247716931955321947807810388
11
6
$x_k$:
-0.7301520055740493240934162520311534580496430620261303119783783396870132450585192295954234109712837001
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,3,x
11
6
$w_k$:
0.09295309859690082776929366791242916193983947355308010256071312793630263264034913019120880758817617144
11
7
$x_k$:
-0.6305995201619650921682633124053903188224253083530672238832869311367449686803443989104621380013137871
11
7
$w_k$:
0.1058720744813893964818918982319811205549624780907203319296467711108833716727290448670096159586632252
11
8
$x_k$:
-0.5190961292068118159257256694586095544802271151199284890209226114866959264510728928255987801045490553
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,4,x
11
8
$w_k$:
0.1167395024610472708108110608932828323249092145991439080871545801818431893882677856483659368557917453
11
9
$x_k$:
-0.3979441409523775736750739432982322772591121672889972654041942696693169452294420645881565700703156878
11
9
$w_k$:
0.1251587991003195050600671896097701470444367151101039242536676577243138239173417778670802946348633792
11
10
$x_k$:
-0.2695431559523449723315319854008615246796218624390522816239256318800570662236947357038215922442673013
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,5,x
11
10
$w_k$:
0.1312806842298056442559775373399571616706094357653728102145210770767705951932583565796520310145949960
11
11
$x_k$:
-0.1361130007993618157983643559949529343446588487825888943925526812025798286337705000141798233533389557
11
11
$w_k$:
0.1351935727998845331842618531415332171561349328360709052954346192514658780408931811116755405731881397
11
12
$x_k$:
0
comment: $x_{12}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,6,x
11
12
$w_k$:
3510224158170177077248/25701280106292582482691
11
13
$x_k$:
0.1361130007993618157983643559949529343446588487825888943925526812025798286337705000141798233533389557
11
13
$w_k$:
0.1351935727998845331842618531415332171561349328360709052954346192514658780408931811116755405731881397
11
14
$x_k$:
0.2695431559523449723315319854008615246796218624390522816239256318800570662236947357038215922442673013
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,7,x
11
14
$w_k$:
0.1312806842298056442559775373399571616706094357653728102145210770767705951932583565796520310145949960
11
15
$x_k$:
0.3979441409523775736750739432982322772591121672889972654041942696693169452294420645881565700703156878
11
15
$w_k$:
0.1251587991003195050600671896097701470444367151101039242536676577243138239173417778670802946348633792
11
16
$x_k$:
0.5190961292068118159257256694586095544802271151199284890209226114866959264510728928255987801045490553
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,8,x
11
16
$w_k$:
0.1167395024610472708108110608932828323249092145991439080871545801818431893882677856483659368557917453
11
17
$x_k$:
0.6305995201619650921682633124053903188224253083530672238832869311367449686803443989104621380013137871
11
17
$w_k$:
0.1058720744813893964818918982319811205549624780907203319296467711108833716727290448670096159586632252
11
18
$x_k$:
0.7301520055740493240934162520311534580496430620261303119783783396870132450585192295954234109712837001
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,9,x
11
18
$w_k$:
0.09295309859690082776929366791242916193983947355308010256071312793630263264034913019120880758817617144
11
19
$x_k$:
0.8160574566562209423922613551926258792778640464249351554136242461884651368612930405915007300046006812
11
19
$w_k$:
0.07866457193222732928421712412387330212537013927185540580724871403138289435247716931955321947807810388
11
20
$x_k$:
0.8870625997680952990751577693039272666316757512253143849674110555376113138573726674422905951265973742
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,10,x
11
20
$w_k$:
0.06309742475037490658454053049537146778131754091445865261757547710038577981517821385824072570472632977
11
21
$x_k$:
0.9416771085780679464553967303521349621887508937383184623824558387012555494696002242409994853810213235
11
21
$w_k$:
0.04582937856442641598526161637154832107049886748562641958315178306865641956835426354672218172246312165
11
22
$x_k$:
0.9782286581460569928039380011228573907714224089197844154258010659836637993808899882003193981673447699
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#11,11,x
11
22
$w_k$:
0.02715655468210426205172140161785167941281344912488029029841290913477031686215645035776794876251809925
11
23
$x_k$:
0.9963696138895426343601645733351607733670255420718933140205778155451740907666856531023195550771640705
11
23
$w_k$:
0.009765441045960758022479172609643522163694469405185463334885137933841331943681209440531842718097720796
12
1
$x_k$:
-0.9969339225295954269123502372583852660991365510837240238787803531658916745551531964435798598486574885
12
1
$w_k$:
0.008257711433168395757693922439211577331315791742950042460194043732860531024100518320331271731941018324
12
2
$x_k$:
-0.9815606342467192506905490901492808229601551998137315104626821218077932443182539822252572678904522358
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,1,x
12
2
$w_k$:
0.02303608403898223259108458036796898034480967610739777657562120180147713342782470525872446487476429921
12
3
$x_k$:
-0.9505377959431212965490601951316186347247906585873649733510868795826948234822232103609854139780488460
12
3
$w_k$:
0.03891523046929947711508963228586289496673895692377658876950107537664366183729980227395113359836218984
12
4
$x_k$:
-0.9041172563704748566784658661190961925375967092132975465540757606812347957292357904869694278237332678
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,2,x
12
4
$w_k$:
0.05369701760775625122888916332045818473957033402184624215423603304618583157677125793399586856924885817
12
5
$x_k$:
-0.8435581241611532447921418850598392608308772241692436877891385384102030272700034887770181888747909286
12
5
$w_k$:
0.06725090705083993030494094004731607383829331185252386242293224222661194484389855432368767747880144443
12
6
$x_k$:
-0.7699026741943046870368938332128180759849257500189316376644190642491165431084712240164249992234219106
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,3,x
12
6
$w_k$:
0.07992027533360170149339260952978335472176635880533618057937917238703748724530826323487852766594462978
12
7
$x_k$:
-0.6840598954700558939449291003411544833088924958324662931959967019704951839843424312236994765784675397
12
7
$w_k$:
0.09154946829504921052817193973961417194826372089348634808716607015222671885645194975284598442107495439
12
8
$x_k$:
-0.5873179542866174472967024189405342803690985140480524815102708796673406993758952624357107649887482019
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,4,x
12
8
$w_k$:
0.1016497322790602777156887704912275146117757204528510417841015228945478634410795276993067213817610496
12
9
$x_k$:
-0.4813394504781570929359436150188322625614475139251727657515164337206366365964848967341909860542340262
12
9
$w_k$:
0.1100226049776440726359073987422496062680322556942200166406642296380487549808531731409438002807274943
12
10
$x_k$:
-0.3678314989981801937526915366437175612563601413354096213117998795040899295167878738787344285005465772
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,5,x
12
10
$w_k$:
0.1167120535017568262935807453057300497193680342866879394062702316642637422296928448365004244309650951
12
11
$x_k$:
-0.2485057483204692762677909603627180986780297376704472980771153491885443852558553240122334147120484977
12
11
$w_k$:
0.1216263035239483832460997580913096286010507357848945833726692110755175584116491420104412513445092658
12
12
$x_k$:
-0.1252334085114689154724413694638531299833969163054442732129217547484620562413896887428682984694913596
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,6,x
12
12
$w_k$:
0.1245841645361560734373124732092289926973821693872056024791044425753511455594869449034639161633610799
12
13
$x_k$:
0
comment: $x_{13}=0$, the central node of every rule
equals: Zero
12
13
$w_k$:
234860696567318732341248/1870551980555794105986325
12
14
$x_k$:
0.1252334085114689154724413694638531299833969163054442732129217547484620562413896887428682984694913596
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,7,x
12
14
$w_k$:
0.1245841645361560734373124732092289926973821693872056024791044425753511455594869449034639161633610799
12
15
$x_k$:
0.2485057483204692762677909603627180986780297376704472980771153491885443852558553240122334147120484977
12
15
$w_k$:
0.1216263035239483832460997580913096286010507357848945833726692110755175584116491420104412513445092658
12
16
$x_k$:
0.3678314989981801937526915366437175612563601413354096213117998795040899295167878738787344285005465772
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,8,x
12
16
$w_k$:
0.1167120535017568262935807453057300497193680342866879394062702316642637422296928448365004244309650951
12
17
$x_k$:
0.4813394504781570929359436150188322625614475139251727657515164337206366365964848967341909860542340262
12
17
$w_k$:
0.1100226049776440726359073987422496062680322556942200166406642296380487549808531731409438002807274943
12
18
$x_k$:
0.5873179542866174472967024189405342803690985140480524815102708796673406993758952624357107649887482019
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,9,x
12
18
$w_k$:
0.1016497322790602777156887704912275146117757204528510417841015228945478634410795276993067213817610496
12
19
$x_k$:
0.6840598954700558939449291003411544833088924958324662931959967019704951839843424312236994765784675397
12
19
$w_k$:
0.09154946829504921052817193973961417194826372089348634808716607015222671885645194975284598442107495439
12
20
$x_k$:
0.7699026741943046870368938332128180759849257500189316376644190642491165431084712240164249992234219106
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,10,x
12
20
$w_k$:
0.07992027533360170149339260952978335472176635880533618057937917238703748724530826323487852766594462978
12
21
$x_k$:
0.8435581241611532447921418850598392608308772241692436877891385384102030272700034887770181888747909286
12
21
$w_k$:
0.06725090705083993030494094004731607383829331185252386242293224222661194484389855432368767747880144443
12
22
$x_k$:
0.9041172563704748566784658661190961925375967092132975465540757606812347957292357904869694278237332678
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,11,x
12
22
$w_k$:
0.05369701760775625122888916332045818473957033402184624215423603304618583157677125793399586856924885817
12
23
$x_k$:
0.9505377959431212965490601951316186347247906585873649733510868795826948234822232103609854139780488460
12
23
$w_k$:
0.03891523046929947711508963228586289496673895692377658876950107537664366183729980227395113359836218984
12
24
$x_k$:
0.9815606342467192506905490901492808229601551998137315104626821218077932443182539822252572678904522358
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#12,12,x
12
24
$w_k$:
0.02303608403898223259108458036796898034480967610739777657562120180147713342782470525872446487476429921
12
25
$x_k$:
0.9969339225295954269123502372583852660991365510837240238787803531658916745551531964435798598486574885
12
25
$w_k$:
0.008257711433168395757693922439211577331315791742950042460194043732860531024100518320331271731941018324
13
1
$x_k$:
-0.9973661769948249447836087361473290562933529894187991366905204221515036279972401839353866307342726424
13
1
$w_k$:
0.007087846351248645045178045006627245190242800695162386898032614075416549606334805043020804372799208990
13
2
$x_k$:
-0.9841830547185881494728294488071096110649905619258749086940073204285952378756268418605692872614185884
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,1,x
13
2
$w_k$:
0.01975374638270591561563638859871553771949659908639615866215071124491315864818051930869570857034281409
13
3
$x_k$:
-0.9575524683860811822644890950135119114293779240073167388835495363654570408778270149276678559620277429
13
3
$w_k$:
0.03344358998955241519719568699503724826232703851040721229388012734638042311474733246590758411513644131
13
4
$x_k$:
-0.9175983992229779652065478365007195123904747901116832958952853456596592085896099142834853903405829245
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,2,x
13
4
$w_k$:
0.04627901797383071573696863287899217370376381432204749903261056393140601990627718807205897509496852208
13
5
$x_k$:
-0.8653331602663444620846168909584260321640877229879270500825820278847596946059082011982702912131657332
13
5
$w_k$:
0.05811521042311458463304700969225037287351994639623725092518192311308419596352169575686468065110761626
13
6
$x_k$:
-0.8015780907333099127942064895828598903056157247905000298973847138592221162266401220300298741853170603
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,3,x
13
6
$w_k$:
0.06930363324778149775245320873890761840542825465871165050817056952161099767379467408955430493831710005
13
7
$x_k$:
-0.7269488493206320380657037153090094916179807600574958963785236703217725600373888178678925259937536643
13
7
$w_k$:
0.07980596216947628980439178482293905295319653166504194983098060439687011255971613095090874475907133975
13
8
$x_k$:
-0.6423493394403402206439846069955156500716973982615768573891424079186435531449935749143883001141915551
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,4,x
13
8
$w_k$:
0.08916844187753961022682206914031715174510461374356474884928112694802882093925495259999766742427515128
13
9
$x_k$:
-0.5490799579565368579439904908194850636145706005218892945120240416208573882565025480965337877241659534
13
9
$w_k$:
0.09714173487607860843750247795288621600899951619128694430445507614644003930303892977173788435151339669
13
10
$x_k$:
-0.4484927510364468528779128521276398678019216674417578789582829474582440920662171436450382085547330101
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,5,x
13
10
$w_k$:
0.1038306011690399633519673575622429637629065209837617773183690603672817130346781714889745536790836127
13
11
$x_k$:
-0.3418324630218063763201496321542144086679330988729366378087112819818243024284863820965309511615993973
13
11
$w_k$:
0.1092663510952850190520479720739638184085283062210924424013792433913533944445055130811371712083701498
13
12
$x_k$:
-0.2304583159551347940655281210979888352115423758835311634692614978371620836885819508128029447646801293
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,6,x
13
12
$w_k$:
0.1132102591715298668829737874987394055846196949793066464427568016000783158923129425176802832779499221
13
13
$x_k$:
-0.1159710897449335503122815172191185966438141609731023762733290048578012465789051627060177124313903828
13
13
$w_k$:
0.1154887990912864256525380886836510685512887873915378603966715338674260611926737473575752238121670521
13
14
$x_k$:
0
comment: $x_{14}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,7,x
13
14
$w_k$:
367330563065503421462485139456/3160930972886244765355667644125
13
15
$x_k$:
0.1159710897449335503122815172191185966438141609731023762733290048578012465789051627060177124313903828
13
15
$w_k$:
0.1154887990912864256525380886836510685512887873915378603966715338674260611926737473575752238121670521
13
16
$x_k$:
0.2304583159551347940655281210979888352115423758835311634692614978371620836885819508128029447646801293
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,8,x
13
16
$w_k$:
0.1132102591715298668829737874987394055846196949793066464427568016000783158923129425176802832779499221
13
17
$x_k$:
0.3418324630218063763201496321542144086679330988729366378087112819818243024284863820965309511615993973
13
17
$w_k$:
0.1092663510952850190520479720739638184085283062210924424013792433913533944445055130811371712083701498
13
18
$x_k$:
0.4484927510364468528779128521276398678019216674417578789582829474582440920662171436450382085547330101
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,9,x
13
18
$w_k$:
0.1038306011690399633519673575622429637629065209837617773183690603672817130346781714889745536790836127
13
19
$x_k$:
0.5490799579565368579439904908194850636145706005218892945120240416208573882565025480965337877241659534
13
19
$w_k$:
0.09714173487607860843750247795288621600899951619128694430445507614644003930303892977173788435151339669
13
20
$x_k$:
0.6423493394403402206439846069955156500716973982615768573891424079186435531449935749143883001141915551
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,10,x
13
20
$w_k$:
0.08916844187753961022682206914031715174510461374356474884928112694802882093925495259999766742427515128
13
21
$x_k$:
0.7269488493206320380657037153090094916179807600574958963785236703217725600373888178678925259937536643
13
21
$w_k$:
0.07980596216947628980439178482293905295319653166504194983098060439687011255971613095090874475907133975
13
22
$x_k$:
0.8015780907333099127942064895828598903056157247905000298973847138592221162266401220300298741853170603
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,11,x
13
22
$w_k$:
0.06930363324778149775245320873890761840542825465871165050817056952161099767379467408955430493831710005
13
23
$x_k$:
0.8653331602663444620846168909584260321640877229879270500825820278847596946059082011982702912131657332
13
23
$w_k$:
0.05811521042311458463304700969225037287351994639623725092518192311308419596352169575686468065110761626
13
24
$x_k$:
0.9175983992229779652065478365007195123904747901116832958952853456596592085896099142834853903405829245
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,12,x
13
24
$w_k$:
0.04627901797383071573696863287899217370376381432204749903261056393140601990627718807205897509496852208
13
25
$x_k$:
0.9575524683860811822644890950135119114293779240073167388835495363654570408778270149276678559620277429
13
25
$w_k$:
0.03344358998955241519719568699503724826232703851040721229388012734638042311474733246590758411513644131
13
26
$x_k$:
0.9841830547185881494728294488071096110649905619258749086940073204285952378756268418605692872614185884
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#13,13,x
13
26
$w_k$:
0.01975374638270591561563638859871553771949659908639615866215071124491315864818051930869570857034281409
13
27
$x_k$:
0.9973661769948249447836087361473290562933529894187991366905204221515036279972401839353866307342726424
13
27
$w_k$:
0.007087846351248645045178045006627245190242800695162386898032614075416549606334805043020804372799208990
14
1
$x_k$:
-0.9977205937565431220161858742705826785441030166985389559875712717808293321182295133969544806355475937
14
1
$w_k$:
0.006139558686378131376066151763667321002996025013661071821872412059995319239359861487396429496470427470
14
2
$x_k$:
-0.9862838086968123388415972667040528016760914072392258816440708117777495541324916379106462396651517528
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,1,x
14
2
$w_k$:
0.01714845890993550645296678149021035251760784275790161375078491458376003606314818735814554833839341814
14
3
$x_k$:
-0.9631583382788532040076698240230578032090304427838928709413034904129736653268462940906153268103982332
14
3
$w_k$:
0.02904870126150850553510391500429234364133708192761640041596289741754471144818288116772784550800118081
14
4
$x_k$:
-0.9284348836635735173363911393778742644770392104098376187179624474821310935443598531114139056836575176
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,2,x
14
4
$w_k$:
0.04025059487268861216681661622513682419109763521659967149471864112229635913924082887625391122901272943
14
5
$x_k$:
-0.8829146632520570399351806244578696865379255472310041230619498428081933597445351141873858893187174446
14
5
$w_k$:
0.05069154326046537607003655156203169477519059626072928971263801432651397473392807668691027244355651430
14
6
$x_k$:
-0.8272013150697649931897947426503949610397011014750811815607090542414798308100288735704263901378895454
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,3,x
14
6
$w_k$:
0.06066712586674214967354827414230320947616370246546390271460317306225610926237579555584073364821811252
14
7
$x_k$:
-0.7617567525622055250576531387931963639746425008877642287723659942058399541698564341798454598760743513
14
7
$w_k$:
0.07010297900274698266320741349274302503752878280927575090410370130043114919448085791272341921003028859
14
8
$x_k$:
-0.6872929048116854701480198030193341375384012127471706756192664886281848961831332569473730705052118384
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,4,x
14
8
$w_k$:
0.07865579724962168225002901306436711780064955624222741771112190601609643683176104148920886712683904502
14
9
$x_k$:
-0.6047893659409215924120259116228219499824260323144694771363631790932779544349207580787617299959748681
14
9
$w_k$:
0.08618376286945813800926821516534004129094724577313583496159166444445719711503170691587964422771011566
14
10
$x_k$:
-0.5152486363581540919652907185511886623088852825693060369515047690927849518320556604520720203507728924
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,5,x
14
10
$w_k$:
0.09273683001785357143548802186791187943445500463107906363218594261636405918479039620614174663855575786
14
11
$x_k$:
-0.4196558976429790001573570757058000559807797057054098434609389711638140026674651463550475721084430113
14
11
$w_k$:
0.09826492647210373747161166832695497929386278948174508741284566676503124179688607159669200983509633193
14
12
$x_k$:
-0.3191123689278897604356718241684754668342612035338439565966501872573334405127927831649337054213464132
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,6,x
14
12
$w_k$:
0.1026166273213997903286759816166735395394129963186054981910530262630322957049380688306857836471893408
14
13
$x_k$:
-0.2148359185334849021050969526379601327244052594548632127257637959038847596659004939369514401804538062
14
13
$w_k$:
0.1057316398417643357016849237903892098862576975931755990835556053570617500999398264933204824074318277
14
14
$x_k$:
-0.1080549487073436620662446502198347476119516054742375570408210613080135290117300071301006881766893672
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,7,x
14
14
$w_k$:
0.1076264211141186523632943351668680678017301649366209071060049604795749491318378085746307718697661475
14
15
$x_k$:
0
comment: $x_{15}=0$, the central node of every rule
equals: Zero
14
15
$w_k$:
11157509282664595051249664/103052576235390068155503375
14
16
$x_k$:
0.1080549487073436620662446502198347476119516054742375570408210613080135290117300071301006881766893672
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,8,x
14
16
$w_k$:
0.1076264211141186523632943351668680678017301649366209071060049604795749491318378085746307718697661475
14
17
$x_k$:
0.2148359185334849021050969526379601327244052594548632127257637959038847596659004939369514401804538062
14
17
$w_k$:
0.1057316398417643357016849237903892098862576975931755990835556053570617500999398264933204824074318277
14
18
$x_k$:
0.3191123689278897604356718241684754668342612035338439565966501872573334405127927831649337054213464132
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,9,x
14
18
$w_k$:
0.1026166273213997903286759816166735395394129963186054981910530262630322957049380688306857836471893408
14
19
$x_k$:
0.4196558976429790001573570757058000559807797057054098434609389711638140026674651463550475721084430113
14
19
$w_k$:
0.09826492647210373747161166832695497929386278948174508741284566676503124179688607159669200983509633193
14
20
$x_k$:
0.5152486363581540919652907185511886623088852825693060369515047690927849518320556604520720203507728924
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,10,x
14
20
$w_k$:
0.09273683001785357143548802186791187943445500463107906363218594261636405918479039620614174663855575786
14
21
$x_k$:
0.6047893659409215924120259116228219499824260323144694771363631790932779544349207580787617299959748681
14
21
$w_k$:
0.08618376286945813800926821516534004129094724577313583496159166444445719711503170691587964422771011566
14
22
$x_k$:
0.6872929048116854701480198030193341375384012127471706756192664886281848961831332569473730705052118384
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,11,x
14
22
$w_k$:
0.07865579724962168225002901306436711780064955624222741771112190601609643683176104148920886712683904502
14
23
$x_k$:
0.7617567525622055250576531387931963639746425008877642287723659942058399541698564341798454598760743513
14
23
$w_k$:
0.07010297900274698266320741349274302503752878280927575090410370130043114919448085791272341921003028859
14
24
$x_k$:
0.8272013150697649931897947426503949610397011014750811815607090542414798308100288735704263901378895454
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,12,x
14
24
$w_k$:
0.06066712586674214967354827414230320947616370246546390271460317306225610926237579555584073364821811252
14
25
$x_k$:
0.8829146632520570399351806244578696865379255472310041230619498428081933597445351141873858893187174446
14
25
$w_k$:
0.05069154326046537607003655156203169477519059626072928971263801432651397473392807668691027244355651430
14
26
$x_k$:
0.9284348836635735173363911393778742644770392104098376187179624474821310935443598531114139056836575176
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,13,x
14
26
$w_k$:
0.04025059487268861216681661622513682419109763521659967149471864112229635913924082887625391122901272943
14
27
$x_k$:
0.9631583382788532040076698240230578032090304427838928709413034904129736653268462940906153268103982332
14
27
$w_k$:
0.02904870126150850553510391500429234364133708192761640041596289741754471144818288116772784550800118081
14
28
$x_k$:
0.9862838086968123388415972667040528016760914072392258816440708117777495541324916379106462396651517528
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#14,14,x
14
28
$w_k$:
0.01714845890993550645296678149021035251760784275790161375078491458376003606314818735814554833839341814
14
29
$x_k$:
0.9977205937565431220161858742705826785441030166985389559875712717808293321182295133969544806355475937
14
29
$w_k$:
0.006139558686378131376066151763667321002996025013661071821872412059995319239359861487396429496470427470
15
1
$x_k$:
-0.9980022986933970602851728401522712090734064423155572303483942797068334868283713456664897990776012528
15
1
$w_k$:
0.005377479872923348987792051430127649818308040243128419787648616953684863555435459921379317259649003899
15
2
$x_k$:
-0.9879925180204854284895657185866125811469728171237614899999975155873884373690194247127220503683191450
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,1,x
15
2
$w_k$:
0.01500794732931612253837476307580726809463943643738763497929175970089649474615433439896171022749040253
15
3
$x_k$:
-0.9677390756791391342573479787843372252833573373001316379746806222633580424945217480431938504820311851
15
3
$w_k$:
0.02546084732671532018687400101965335939727174504686464050837798498240090344700918526760520577881971285
15
4
$x_k$:
-0.9372733924007059043077589477102094712439962735153044579013630763502029737970455279505475861742680866
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,2,x
15
4
$w_k$:
0.03534636079137584622203794847836004812263067899242082086814802334090250183724768097843466272429681008
15
5
$x_k$:
-0.8972645323440819008825096564544958828317787114944278676397268760107853772147377122119539966191971612
15
5
$w_k$:
0.04458975132476487660822729937327969022325664966792109657098082321180545070005990636645503641889714959
15
6
$x_k$:
-0.8482065834104272162006483207742168513662561747369926340957275587606750751741454851976077197508214809
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,3,x
15
6
$w_k$:
0.05348152469092808726534314723943029677155476094711673981322288875272741361625962543971481247519898751
15
7
$x_k$:
-0.7904185014424659329676492948179473468621405199569761733236528064330830297463180705999473866422544553
15
7
$w_k$:
0.06200956780067064028513923096080293219040000421032972356914782939561837620627231733303058426830380864
15
8
$x_k$:
-0.7244177313601700474161860546139380096308992945841025635514234207041237816779252189961010976031343263
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,4,x
15
8
$w_k$:
0.06985412131872825870952007709914747578604543514067154969879809317799267562498799884974862877857066752
15
9
$x_k$:
-0.6509967412974169705337358953132746925469482260925996670896616057609330584104384079446039474722806037
15
9
$w_k$:
0.07684968075772037889443277748265900672210910116794700058408909711247082109203408441822473152769029191
15
10
$x_k$:
-0.5709721726085388475372267372539106412383863962827496048532654170541953798697585794834146285698261448
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,5,x
15
10
$w_k$:
0.08308050282313302103828924728610378960155418825336871760728160487523363064388505605763078922833708886
15
11
$x_k$:
-0.4850818636402396806936557402323506128663389308940731212936794360408023995516715597437184869084859528
15
11
$w_k$:
0.08856444305621177064727544369377430321226673269065596781799605257487714454474981426071883757632510992
15
12
$x_k$:
-0.3941513470775633698972073709810454683627527761586982550311653439516089577869614179754971141616597620
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,6,x
15
12
$w_k$:
0.09312659817082532122548687274734571856192788132131733056028587918905200287453185506011490899045871674
15
13
$x_k$:
-0.2991800071531688121667800242663889626616033827438208018412554573891808110251388446760232202015724356
15
13
$w_k$:
0.09664272698362367850517990762758933513665656863049519897340766888293439235996284182651140250466459219
15
14
$x_k$:
-0.2011940939974345223006283033945962078128364544626376796159497246099482390030201876018362580675210591
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,7,x
15
14
$w_k$:
0.09917359872179195933239317348460313105956726081671328173486009569365156306430874571705668012822379074
15
15
$x_k$:
-0.1011420669187174990270742314473923387874510574016418049580018950415109786245408305093132145154038100
15
15
$w_k$:
0.1007698455238755950449466626175697219163483801353637306927892902948812276082276107747506018596540833
15
16
$x_k$:
0
comment: $x_{16}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,8,x
15
16
$w_k$:
48573630978151702037927516933170659328/479360777810478327470528249974671314475
15
17
$x_k$:
0.1011420669187174990270742314473923387874510574016418049580018950415109786245408305093132145154038100
15
17
$w_k$:
0.1007698455238755950449466626175697219163483801353637306927892902948812276082276107747506018596540833
15
18
$x_k$:
0.2011940939974345223006283033945962078128364544626376796159497246099482390030201876018362580675210591
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,9,x
15
18
$w_k$:
0.09917359872179195933239317348460313105956726081671328173486009569365156306430874571705668012822379074
15
19
$x_k$:
0.2991800071531688121667800242663889626616033827438208018412554573891808110251388446760232202015724356
15
19
$w_k$:
0.09664272698362367850517990762758933513665656863049519897340766888293439235996284182651140250466459219
15
20
$x_k$:
0.3941513470775633698972073709810454683627527761586982550311653439516089577869614179754971141616597620
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,10,x
15
20
$w_k$:
0.09312659817082532122548687274734571856192788132131733056028587918905200287453185506011490899045871674
15
21
$x_k$:
0.4850818636402396806936557402323506128663389308940731212936794360408023995516715597437184869084859528
15
21
$w_k$:
0.08856444305621177064727544369377430321226673269065596781799605257487714454474981426071883757632510992
15
22
$x_k$:
0.5709721726085388475372267372539106412383863962827496048532654170541953798697585794834146285698261448
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,11,x
15
22
$w_k$:
0.08308050282313302103828924728610378960155418825336871760728160487523363064388505605763078922833708886
15
23
$x_k$:
0.6509967412974169705337358953132746925469482260925996670896616057609330584104384079446039474722806037
15
23
$w_k$:
0.07684968075772037889443277748265900672210910116794700058408909711247082109203408441822473152769029191
15
24
$x_k$:
0.7244177313601700474161860546139380096308992945841025635514234207041237816779252189961010976031343263
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,12,x
15
24
$w_k$:
0.06985412131872825870952007709914747578604543514067154969879809317799267562498799884974862877857066752
15
25
$x_k$:
0.7904185014424659329676492948179473468621405199569761733236528064330830297463180705999473866422544553
15
25
$w_k$:
0.06200956780067064028513923096080293219040000421032972356914782939561837620627231733303058426830380864
15
26
$x_k$:
0.8482065834104272162006483207742168513662561747369926340957275587606750751741454851976077197508214809
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,13,x
15
26
$w_k$:
0.05348152469092808726534314723943029677155476094711673981322288875272741361625962543971481247519898751
15
27
$x_k$:
0.8972645323440819008825096564544958828317787114944278676397268760107853772147377122119539966191971612
15
27
$w_k$:
0.04458975132476487660822729937327969022325664966792109657098082321180545070005990636645503641889714959
15
28
$x_k$:
0.9372733924007059043077589477102094712439962735153044579013630763502029737970455279505475861742680866
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,14,x
15
28
$w_k$:
0.03534636079137584622203794847836004812263067899242082086814802334090250183724768097843466272429681008
15
29
$x_k$:
0.9677390756791391342573479787843372252833573373001316379746806222633580424945217480431938504820311851
15
29
$w_k$:
0.02546084732671532018687400101965335939727174504686464050837798498240090344700918526760520577881971285
15
30
$x_k$:
0.9879925180204854284895657185866125811469728171237614899999975155873884373690194247127220503683191450
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#15,15,x
15
30
$w_k$:
0.01500794732931612253837476307580726809463943643738763497929175970089649474615433439896171022749040253
15
31
$x_k$:
0.9980022986933970602851728401522712090734064423155572303483942797068334868283713456664897990776012528
15
31
$w_k$:
0.005377479872923348987792051430127649818308040243128419787648616953684863555435459921379317259649003899
20
1
$x_k$:
-0.9988590315882776638383155765458630099995702043262966686666686033932441179331198296783912977285417988
20
1
$w_k$:
0.003073583718520531501218293246030987488033504688254344919846162821211433366559037815670626524141446931
20
2
$x_k$:
-0.9931285991850949247861223884713202782226471309016558961481841312179847176277537808394494024965722093
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,1,x
20
2
$w_k$:
0.008600269855642942198661787950102347252128922766707797662245060203142653536269643783844882800955453203
20
3
$x_k$:
-0.9815078774502502591933429947202169445672509398102375986907753331879309885746572346089806049188751136
20
3
$w_k$:
0.01462616925697125298378796030886835616388105016224977034210347463107696002974875195938048248430838229
20
4
$x_k$:
-0.9639719272779137912676661311972772219120603278061888560635375938920415807843830569800181252559647156
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,2,x
20
4
$w_k$:
0.02038837346126652359801023143275470512283862794018592936537186821443300653203035367125364030067915750
20
5
$x_k$:
-0.9408226338317547535199827222124433802742955737796529105953683997318679600655757122088821867677661845
20
5
$w_k$:
0.02588213360495115883450506709615314299947911804867494452699779775537430642162944039339242719886934579
20
6
$x_k$:
-0.9122344282513259058677524412032981130491847974236917747958822191580708912087190789364447261929213874
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,3,x
20
6
$w_k$:
0.03128730677703279895854311932380073788776928036281333735955459800532242326604799677192603106970504948
20
7
$x_k$:
-0.8782768112522819760774429951130784667112452682825116485389808699824814590474322074084026162424568388
20
7
$w_k$:
0.03660016975820079803055724070721100848745349674749800165107000944197328006148926607404498690143632430
20
8
$x_k$:
-0.8391169718222188233945290617015206853296293650656373732524927255328610939993248099192293405659576492
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,4,x
20
8
$w_k$:
0.04166887332797368626378830593689473804396084315301032486096635323527188959637972646220870208106871546
20
9
$x_k$:
-0.7950414288375511983506388332727879429593895991157802970385516389432269787171038286670177789025182462
20
9
$w_k$:
0.04643482186749767472023188092610751684212707100707792928999412793324322258593880439295393118514644607
20
10
$x_k$:
-0.7463319064601507926143050703556415903107306795691764441395459060685353550381550646811041136206475206
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,5,x
20
10
$w_k$:
0.05094457392372869193270767005034494866483636580926257974751714008611911347686673564105482257417319890
20
11
$x_k$:
-0.6932376563347513848054907118459315333864258514102141790468737845430119171073921901154667241632502275
20
11
$w_k$:
0.05519510534828599474483237241977732919475345622815311690981213121317782770788469291784545399953551882
20
12
$x_k$:
-0.6360536807265150254528366962262859367433891167993684639394466225465412625854301325587031954957613066
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,6,x
20
12
$w_k$:
0.05911140088063957237496722064859421713641936597704219174838804720401526284040769661150873283985195270
20
13
$x_k$:
-0.5751404468197103153429460365864251328138126401477168253741588549571746807472006201235778848904947021
20
13
$w_k$:
0.06265323755478116802587012217425498058581974469889788618655332415710042408891928450345159674258838634
20
14
$x_k$:
-0.5108670019508270980043640509552509984254913292024268334723486198947349703907657281440316830508677792
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,7,x
20
14
$w_k$:
0.06583459713361842211156355696939794314722350634338144370975174963994442031438429634750352381009684240
20
15
$x_k$:
-0.4435931752387251031999922134926401078401010108230030961331502834629954305931525860199347915698784743
20
15
$w_k$:
0.06864867292852161934562341188536780171548970495823986040043426417392380602958997094171122425796765104
20
16
$x_k$:
-0.3737060887154195606725481770249272373957463217056827118279486135156457643730595278958956836345333789
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,8,x
20
16
$w_k$:
0.07105442355344406830579036172321016741291215932221014392162827058640738187978952590108614647327809516
20
17
$x_k$:
-0.3016278681149130043205553568585922606153965050137309245692637442795695743597838411606649823476222022
20
17
$w_k$:
0.07303069033278666749518941765891311276062684523455274238017425077184974383166004096680480231246452772
20
18
$x_k$:
-0.2277858511416450780804961953685746247430889376829274723146357392071713418635558277949521251909687080
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,9,x
20
18
$w_k$:
0.07458287540049918898658141836248752861611649357209227308004704072696989956788736422766420264294235710
20
19
$x_k$:
-0.1526054652409226755052202410226775279116762248184173066017415670380913368575169635698799588639704972
20
19
$w_k$:
0.07570449768455667465954277537661655826336315590041432619485522327234883859609941484188674046837970728
20
20
$x_k$:
-0.07652652113349733375464040939883821100479626681349750080479524438425634204833697824154511418155621561
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,10,x
20
20
$w_k$:
0.07637786767208073670550283503806100180080103676494599671494643111693674554206194105000834504748250125
20
21
$x_k$:
0
comment: $x_{21}=0$, the central node of every rule
equals: Zero
20
21
$w_k$:
624992238612793816992362272777211740160000/8159091775567858242431146758900398828210807
20
22
$x_k$:
0.07652652113349733375464040939883821100479626681349750080479524438425634204833697824154511418155621561
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,11,x
20
22
$w_k$:
0.07637786767208073670550283503806100180080103676494599671494643111693674554206194105000834504748250125
20
23
$x_k$:
0.1526054652409226755052202410226775279116762248184173066017415670380913368575169635698799588639704972
20
23
$w_k$:
0.07570449768455667465954277537661655826336315590041432619485522327234883859609941484188674046837970728
20
24
$x_k$:
0.2277858511416450780804961953685746247430889376829274723146357392071713418635558277949521251909687080
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,12,x
20
24
$w_k$:
0.07458287540049918898658141836248752861611649357209227308004704072696989956788736422766420264294235710
20
25
$x_k$:
0.3016278681149130043205553568585922606153965050137309245692637442795695743597838411606649823476222022
20
25
$w_k$:
0.07303069033278666749518941765891311276062684523455274238017425077184974383166004096680480231246452772
20
26
$x_k$:
0.3737060887154195606725481770249272373957463217056827118279486135156457643730595278958956836345333789
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,13,x
20
26
$w_k$:
0.07105442355344406830579036172321016741291215932221014392162827058640738187978952590108614647327809516
20
27
$x_k$:
0.4435931752387251031999922134926401078401010108230030961331502834629954305931525860199347915698784743
20
27
$w_k$:
0.06864867292852161934562341188536780171548970495823986040043426417392380602958997094171122425796765104
20
28
$x_k$:
0.5108670019508270980043640509552509984254913292024268334723486198947349703907657281440316830508677792
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,14,x
20
28
$w_k$:
0.06583459713361842211156355696939794314722350634338144370975174963994442031438429634750352381009684240
20
29
$x_k$:
0.5751404468197103153429460365864251328138126401477168253741588549571746807472006201235778848904947021
20
29
$w_k$:
0.06265323755478116802587012217425498058581974469889788618655332415710042408891928450345159674258838634
20
30
$x_k$:
0.6360536807265150254528366962262859367433891167993684639394466225465412625854301325587031954957613066
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,15,x
20
30
$w_k$:
0.05911140088063957237496722064859421713641936597704219174838804720401526284040769661150873283985195270
20
31
$x_k$:
0.6932376563347513848054907118459315333864258514102141790468737845430119171073921901154667241632502275
20
31
$w_k$:
0.05519510534828599474483237241977732919475345622815311690981213121317782770788469291784545399953551882
20
32
$x_k$:
0.7463319064601507926143050703556415903107306795691764441395459060685353550381550646811041136206475206
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,16,x
20
32
$w_k$:
0.05094457392372869193270767005034494866483636580926257974751714008611911347686673564105482257417319890
20
33
$x_k$:
0.7950414288375511983506388332727879429593895991157802970385516389432269787171038286670177789025182462
20
33
$w_k$:
0.04643482186749767472023188092610751684212707100707792928999412793324322258593880439295393118514644607
20
34
$x_k$:
0.8391169718222188233945290617015206853296293650656373732524927255328610939993248099192293405659576492
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,17,x
20
34
$w_k$:
0.04166887332797368626378830593689473804396084315301032486096635323527188959637972646220870208106871546
20
35
$x_k$:
0.8782768112522819760774429951130784667112452682825116485389808699824814590474322074084026162424568388
20
35
$w_k$:
0.03660016975820079803055724070721100848745349674749800165107000944197328006148926607404498690143632430
20
36
$x_k$:
0.9122344282513259058677524412032981130491847974236917747958822191580708912087190789364447261929213874
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,18,x
20
36
$w_k$:
0.03128730677703279895854311932380073788776928036281333735955459800532242326604799677192603106970504948
20
37
$x_k$:
0.9408226338317547535199827222124433802742955737796529105953683997318679600655757122088821867677661845
20
37
$w_k$:
0.02588213360495115883450506709615314299947911804867494452699779775537430642162944039339242719886934579
20
38
$x_k$:
0.9639719272779137912676661311972772219120603278061888560635375938920415807843830569800181252559647156
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,19,x
20
38
$w_k$:
0.02038837346126652359801023143275470512283862794018592936537186821443300653203035367125364030067915750
20
39
$x_k$:
0.9815078774502502591933429947202169445672509398102375986907753331879309885746572346089806049188751136
20
39
$w_k$:
0.01462616925697125298378796030886835616388105016224977034210347463107696002974875195938048248430838229
20
40
$x_k$:
0.9931285991850949247861223884713202782226471309016558961481841312179847176277537808394494024965722093
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#20,20,x
20
40
$w_k$:
0.008600269855642942198661787950102347252128922766707797662245060203142653536269643783844882800955453203
20
41
$x_k$:
0.9988590315882776638383155765458630099995702043262966686666686033932441179331198296783912977285417988
20
41
$w_k$:
0.003073583718520531501218293246030987488033504688254344919846162821211433366559037815670626524141446931
25
1
$x_k$:
-0.9992621049926098341934574865403405937045249604227961858622869776290452442816771907381874610223807598
25
1
$w_k$:
0.001987383892330315926507851882843409889429980428250597383765334629898562933682011875352309367530347688
25
2
$x_k$:
-0.9955569697904980979087849468939016172575626494048081712108049311329334813437279344872880263529470076
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,1,x
25
2
$w_k$:
0.005561932135356713758040236901065522070176929549629098405296121079381003885758172417102161010070879976
25
3
$x_k$:
-0.9880357945340772476373310145774062270724841520916074813144997219940518682134729368624540474203236050
25
3
$w_k$:
0.009473973386174151607207710523655323871645326848372633497139402960352930614035902318790470575471964303
25
4
$x_k$:
-0.9766639214595175114983153864795940677453705553144067446709874273161638675358805538964467094830061787
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,2,x
25
4
$w_k$:
0.01323622919557167481365640584697623807757808499786365473221386048856061458763439554400215625819258227
25
5
$x_k$:
-0.9616149864258425124181300336601672416921264296370967666662452014129289328118566691763640779082321089
25
5
$w_k$:
0.01684781770912829823151666753633631584040265462470613941117576927684218227007896007854459737264653264
25
6
$x_k$:
-0.9429745712289743394140111696584705319052015706089901419274524971372953225440492613089052181512734833
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,3,x
25
6
$w_k$:
0.02043537114588283545656829223593897367875800609766893722007453155016362256684188585595762310335444325
25
7
$x_k$:
-0.9207471152817015617463460845463306315745703599627719970064283650113138504263121240780895228170282018
25
7
$w_k$:
0.02400994560695321622009248916488108139293152820965933029073497234253601228219191306977865824197204777
25
8
$x_k$:
-0.8949919978782753688510420067828049541745548497535839030617016829591715109011994513711860069303917816
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,4,x
25
8
$w_k$:
0.02747531758785173780294845551781107861479601328871060319961362106972781035283546992610782204743356679
25
9
$x_k$:
-0.8658470652932755954489969695883400882028440940282369029396521324669143294818028012075670873806477906
25
9
$w_k$:
0.03079230016738748889110902021522858560087716239329248764454483055996538804799649270924861824908485148
25
10
$x_k$:
-0.8334426287608340014210211086935695694609641138235207860208647154617181324770901252532297394775916811
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,5,x
25
10
$w_k$:
0.03400213027432933783674879522955120322567052825005044308326419312152433906334485501025766054770802243
25
11
$x_k$:
-0.7978737979985000594104109049943065694086323000933826766170693449948865081764382407711895031444398403
25
11
$w_k$:
0.03711627148341554356033062536761987599599780268804776480562870276277300966939576058229452574858387571
25
12
$x_k$:
-0.7592592630373576305772828652043609763875220188983341209183897354450186288202624076076367972418523033
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,6,x
25
12
$w_k$:
0.04008382550403238207483928446707564640141054926659130871311587838683577731505845195561411615894961407
25
13
$x_k$:
-0.7177664068130843881866540797732977805977116755551558242349348682399161282097496508952290595376586033
25
13
$w_k$:
0.04287284502017004947689579243949516110199950419988332887791924251573895765525393204895136696080259234
25
14
$x_k$:
-0.6735663684734683644851206332476221758834167280727493170596569617782877368492842115819636856803093219
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,7,x
25
14
$w_k$:
0.04550291304992178890987058475266039304370776893569532731672425439279429956795703545820897059964169720
25
15
$x_k$:
-0.6268100990103174127881226816245178810195462899506851080652522200843726018418118305304523642384519875
25
15
$w_k$:
0.04798253713883671390639225575691475498359220742327116965123586519675791388033411781023551747732811003
25
16
$x_k$:
-0.5776629302412229677236898416126540673957350392915182566454835077610230127526320222767165964657964908
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,8,x
25
16
$w_k$:
0.05027767908071567196332525943344008444058763060477597514205096827974301464114140231030258454263355704
25
17
$x_k$:
-0.5263252843347191825996237781580101780368325232019111431300242518047145502250269530237100852060463834
25
17
$w_k$:
0.05236288580640747586436671213787271488735155072370759635090579365604665924854127659750456649799092631
25
18
$x_k$:
-0.4730027314457149605221821150091920413318177384616272909072308276956032758412860301031568477827936354
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,9,x
25
18
$w_k$:
0.05425112988854549014454337045987560682607683844126338307216329331293692347665093413024231502842204780
25
19
$x_k$:
-0.4178853821930377488518143945945724870933699814006952803495578506879693207696659954871722420510979730
25
19
$w_k$:
0.05595081122041231730824068638274734682027103511277180242893279106611515826833860701936583165546031473
25
20
$x_k$:
-0.3611723058093878377358217301276406674220783470433750697945787778467453823956965486032953150609376140
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,10,x
25
20
$w_k$:
0.05743711636156783285358269393950647199483285682389668297650941231336749572722438119997859824773708959
25
21
$x_k$:
-0.3030895389311078301674789099803393292004193787665519468573157845257312037233720971734961788211166242
25
21
$w_k$:
0.05868968002239420796197417585678776413979564625482831529324370030501256948605415761704968503150659186
25
22
$x_k$:
-0.2438668837209884320451903627974515864056331563259844764211356532503874727858559506797763677632503406
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,11,x
25
22
$w_k$:
0.05972034032417405997909929193256185383536304547618997548337220781614998846070829902077961237501063978
25
23
$x_k$:
-0.1837189394210488920159698887595284157852844783499055521503451265323675285110981561765186716064559124
25
23
$w_k$:
0.06053945537604586294536026751756542716231236571045707992348704314455474781068951440801358251548993091
25
24
$x_k$:
-0.1228646926107103963873598188080368055322053460497837384238935378927088349688584158264388499463310554
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,12,x
25
24
$w_k$:
0.06112850971705304830585903041629271192267855232196093835732202807039013376995203283120489556934775781
25
25
$x_k$:
-0.06154448300568507888654639236679663128172434803982354527430543175168727936155865854514104878102269107
25
25
$w_k$:
0.06147118987142531666154413196526417758653796287688502271111168350015170079619872655848336756653742288
25
26
$x_k$:
0
comment: $x_{26}=0$, the central node of every rule
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,13,x
25
26
$w_k$:
220315971287016536776250504422901142903111497756959045870876950775463936/3577672044634622119495981070234448702995951652702534015285612444926645625
25
27
$x_k$:
0.06154448300568507888654639236679663128172434803982354527430543175168727936155865854514104878102269107
25
27
$w_k$:
0.06147118987142531666154413196526417758653796287688502271111168350015170079619872655848336756653742288
25
28
$x_k$:
0.1228646926107103963873598188080368055322053460497837384238935378927088349688584158264388499463310554
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,14,x
25
28
$w_k$:
0.06112850971705304830585903041629271192267855232196093835732202807039013376995203283120489556934775781
25
29
$x_k$:
0.1837189394210488920159698887595284157852844783499055521503451265323675285110981561765186716064559124
25
29
$w_k$:
0.06053945537604586294536026751756542716231236571045707992348704314455474781068951440801358251548993091
25
30
$x_k$:
0.2438668837209884320451903627974515864056331563259844764211356532503874727858559506797763677632503406
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,15,x
25
30
$w_k$:
0.05972034032417405997909929193256185383536304547618997548337220781614998846070829902077961237501063978
25
31
$x_k$:
0.3030895389311078301674789099803393292004193787665519468573157845257312037233720971734961788211166242
25
31
$w_k$:
0.05868968002239420796197417585678776413979564625482831529324370030501256948605415761704968503150659186
25
32
$x_k$:
0.3611723058093878377358217301276406674220783470433750697945787778467453823956965486032953150609376140
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,16,x
25
32
$w_k$:
0.05743711636156783285358269393950647199483285682389668297650941231336749572722438119997859824773708959
25
33
$x_k$:
0.4178853821930377488518143945945724870933699814006952803495578506879693207696659954871722420510979730
25
33
$w_k$:
0.05595081122041231730824068638274734682027103511277180242893279106611515826833860701936583165546031473
25
34
$x_k$:
0.4730027314457149605221821150091920413318177384616272909072308276956032758412860301031568477827936354
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,17,x
25
34
$w_k$:
0.05425112988854549014454337045987560682607683844126338307216329331293692347665093413024231502842204780
25
35
$x_k$:
0.5263252843347191825996237781580101780368325232019111431300242518047145502250269530237100852060463834
25
35
$w_k$:
0.05236288580640747586436671213787271488735155072370759635090579365604665924854127659750456649799092631
25
36
$x_k$:
0.5776629302412229677236898416126540673957350392915182566454835077610230127526320222767165964657964908
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,18,x
25
36
$w_k$:
0.05027767908071567196332525943344008444058763060477597514205096827974301464114140231030258454263355704
25
37
$x_k$:
0.6268100990103174127881226816245178810195462899506851080652522200843726018418118305304523642384519875
25
37
$w_k$:
0.04798253713883671390639225575691475498359220742327116965123586519675791388033411781023551747732811003
25
38
$x_k$:
0.6735663684734683644851206332476221758834167280727493170596569617782877368492842115819636856803093219
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,19,x
25
38
$w_k$:
0.04550291304992178890987058475266039304370776893569532731672425439279429956795703545820897059964169720
25
39
$x_k$:
0.7177664068130843881866540797732977805977116755551558242349348682399161282097496508952290595376586033
25
39
$w_k$:
0.04287284502017004947689579243949516110199950419988332887791924251573895765525393204895136696080259234
25
40
$x_k$:
0.7592592630373576305772828652043609763875220188983341209183897354450186288202624076076367972418523033
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,20,x
25
40
$w_k$:
0.04008382550403238207483928446707564640141054926659130871311587838683577731505845195561411615894961407
25
41
$x_k$:
0.7978737979985000594104109049943065694086323000933826766170693449948865081764382407711895031444398403
25
41
$w_k$:
0.03711627148341554356033062536761987599599780268804776480562870276277300966939576058229452574858387571
25
42
$x_k$:
0.8334426287608340014210211086935695694609641138235207860208647154617181324770901252532297394775916811
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,21,x
25
42
$w_k$:
0.03400213027432933783674879522955120322567052825005044308326419312152433906334485501025766054770802243
25
43
$x_k$:
0.8658470652932755954489969695883400882028440940282369029396521324669143294818028012075670873806477906
25
43
$w_k$:
0.03079230016738748889110902021522858560087716239329248764454483055996538804799649270924861824908485148
25
44
$x_k$:
0.8949919978782753688510420067828049541745548497535839030617016829591715109011994513711860069303917816
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,22,x
25
44
$w_k$:
0.02747531758785173780294845551781107861479601328871060319961362106972781035283546992610782204743356679
25
45
$x_k$:
0.9207471152817015617463460845463306315745703599627719970064283650113138504263121240780895228170282018
25
45
$w_k$:
0.02400994560695321622009248916488108139293152820965933029073497234253601228219191306977865824197204777
25
46
$x_k$:
0.9429745712289743394140111696584705319052015706089901419274524971372953225440492613089052181512734833
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,23,x
25
46
$w_k$:
0.02043537114588283545656829223593897367875800609766893722007453155016362256684188585595762310335444325
25
47
$x_k$:
0.9616149864258425124181300336601672416921264296370967666662452014129289328118566691763640779082321089
25
47
$w_k$:
0.01684781770912829823151666753633631584040265462470613941117576927684218227007896007854459737264653264
25
48
$x_k$:
0.9766639214595175114983153864795940677453705553144067446709874273161638675358805538964467094830061787
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,24,x
25
48
$w_k$:
0.01323622919557167481365640584697623807757808499786365473221386048856061458763439554400215625819258227
25
49
$x_k$:
0.9880357945340772476373310145774062270724841520916074813144997219940518682134729368624540474203236050
25
49
$w_k$:
0.009473973386174151607207710523655323871645326848372633497139402960352930614035902318790470575471964303
25
50
$x_k$:
0.9955569697904980979087849468939016172575626494048081712108049311329334813437279344872880263529470076
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#25,25,x
25
50
$w_k$:
0.005561932135356713758040236901065522070176929549629098405296121079381003885758172417102161010070879976
25
51
$x_k$:
0.9992621049926098341934574865403405937045249604227961858622869776290452442816771907381874610223807598
25
51
$w_k$:
0.001987383892330315926507851882843409889429980428250597383765334629898562933682011875352309367530347688
30
1
$x_k$:
-0.9994844100504906375713258957058108194688739470185080192363264283074801667484358783065646882314543572
30
1
$w_k$:
0.001389013698677007624551591226759699681048841291963272453441105533230136713098986536695625155642382048
30
2
$x_k$:
-0.9968934840746495402716300509186952833408820381177507901080942978023876952101637408158820195580617174
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,1,x
30
2
$w_k$:
0.003890461127099884051267201844515503278515142984886464921420010128114473367645545106122627365594103835
30
3
$x_k$:
-0.9916309968704045948586283661094857248505003337461632551001992334980748960326079660555619149584357523
30
3
$w_k$:
0.006630703915931292173319826369750168133628388217781258597395559735783756827773192132773181584451259816
30
4
$x_k$:
-0.9836681232797472099700325816056628019403178547097113635171800101511442953647910437020759716603547137
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,2,x
30
4
$w_k$:
0.009273279659517763428441146892024360421270024938193107696495646914362666555743438549232578459634311215
30
5
$x_k$:
-0.9731163225011262683746938684237068848876379642834393385375585018562411895816683828830856170826148637
30
5
$w_k$:
0.01182301525349634174223289885325059289626440625060781832630243154826536515585518273940170003251914145
30
6
$x_k$:
-0.9600218649683075122168710255817976629303592174039233994856616724249399577070684292271894437038000238
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,3,x
30
6
$w_k$:
0.01436972950704580481245143244358001019584189989500150587356589940300019866249582190614427468289422259
30
7
$x_k$:
-0.9443744447485599794158313240374391215856437149649809318174894013952091700065734275344887137684984852
30
7
$w_k$:
0.01692088918905327262757228942032209236856670378383519113988341084054667997855186104362008945168114602
30
8
$x_k$:
-0.9262000474292743258793242770804740040864745368253290609110371336794229956511023268167728801505588624
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,4,x
30
8
$w_k$:
0.01941414119394238117340895105012845585142101419143152577027606653649717907902554048607272611462876361
30
9
$x_k$:
-0.9055733076999077985465225589259583195689753636622284135640476639780376023944963191358507442684257416
30
9
$w_k$:
0.02182803582160919229716748573833899340150729605683491277363042235872043940338255907935605860239387980
30
10
$x_k$:
-0.8825605357920526815431164625302255900566891471464842320683260531216162626951916557292158382857321049
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,5,x
30
10
$w_k$:
0.02419116207808060136568637072523202676039137782818246243222894356294488526750107068800647096287174366
30
11
$x_k$:
-0.8572052335460610989586585106589438568208001706235961285050455173911988722571293268803112070465719564
30
11
$w_k$:
0.02650995488233310161060170933507541436651757952274856577086743833847213890365807761765252275993447490
30
12
$x_k$:
-0.8295657623827683974428981197325019164390686961703416788069529834536565065895816350829524435081401600
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,6,x
30
12
$w_k$:
0.02875404876504129284397878535433421114467916054207493003510228075913217481546983422785466051536600314
30
13
$x_k$:
-0.7997278358218390830136689423226832407356984293777845092364734954868666256732600722919520252418535647
30
13
$w_k$:
0.03090725756238776247288425294309227263527045852380715342684048696402208618987405694771744632818713127
30
14
$x_k$:
-0.7677774321048261949179773409745031316948836172329084532064943873651585701729950450526096025862396842
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,7,x
30
14
$w_k$:
0.03298144705748372603181419101685392751059929121385838571451934764145231658238100880499451534196920599
30
15
$x_k$:
-0.7337900624532268047261711313695276456693817277546854920870139951830001646361332538202466453159731880
30
15
$w_k$:
0.03497933802806002413749967073146787509722691279481871997220845723217778670200874421949847060384678447
30
16
$x_k$:
-0.6978504947933157969322923880266400683823538006539546563797228467399767212431599606953816364400890469
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,8,x
30
16
$w_k$:
0.03688236465182122922391106561713596773695516478103033767000519858419613497015416986258419336075124323
30
17
$x_k$:
-0.6600610641266269613700536681492707530383503748088339095506719733990493749973452207678802051702968819
30
17
$w_k$:
0.03867894562472759295034865153228105025092362982155384679037613067933740205662070055413910948753375956
30
18
$x_k$:
-0.6205261829892428611404775564311892992073646928295281325950511701243353149748891177411525844553278211
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,9,x
30
18
$w_k$:
0.04037453895153595911199527975246811421612606212603025563399828961381084676105974096183682880295957390
30
19
$x_k$:
-0.5793452358263616917560249321725404959070515888121528920812601661231283356781224190380997075178380821
30
19
$w_k$:
0.04196981021516424614714754128596975779008865671899237482038872032385265551120036579037994846200615695
30
20
$x_k$:
-0.5366241481420198992641697933110727941641780069302971054527434829120149086189783786311411600971899026
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,10,x
30
20
$w_k$:
0.04345253970135606931683172811707325807460330863170316806488880549573864083957386333394208411719654146
30
21
$x_k$:
-0.4924804678617785749936930612077087956442656409631869702607334098298842254639635277683704745226202598
30
21
$w_k$:
0.04481480013316266319235555161672324375743139279637300988968020119406350394790789918906106479211191904
30
22
$x_k$:
-0.4470337695380891767806099003228540001624075938614244097544773817276153517285842070040068887212418983
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,11,x
30
22
$w_k$:
0.04605923827100698811627173555937358059469287557182492400473237949229360400644605267225297343897863917
30
23
$x_k$:
-0.4004012548303943925354762115426606336110459329707839598318661065642917068931175906117552701571024738
30
23
$w_k$:
0.04718554656929915394526147818109948648288480730062845719414186155172553328949089702902027652560351550
30
24
$x_k$:
-0.3527047255308781134710372070893738606536310080214256265941844689002694162331910786643603967521135295
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,12,x
30
24
$w_k$:
0.04818586175708712914077949229830459260579923610842980005737335087243379358396936842894267206327029894
30
25
$x_k$:
-0.3040732022736250773726771071992565535311577898094627284442153699831215044238776730400142369990977859
30
25
$w_k$:
0.04905543455502977888752816536723817360588740529529656957949071790132821564459055524752287306524629747
30
26
$x_k$:
-0.2546369261678898464398051298178051078827893033025184261642859750889635315690788029063662813842362026
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,13,x
30
26
$w_k$:
0.04979568342707420635781156937994232853920960281369610895104739284294848264622037765509834192408925020
30
27
$x_k$:
-0.2045251166823098914389576710020247095241042645955637744760446502835032189466324549559256523531714782
30
27
$w_k$:
0.05040592140278234684089308565358502890219701825162223366424395921106671330863528371344774790797370079
30
28
$x_k$:
-0.1538699136085835469637946727432559204185519712443384617189629829157871485108161013969231065107407856
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,14,x
30
28
$w_k$:
0.05088179589874960649229747304980469185338491426091923992077194208097254264678057557113205625407092986
30
29
$x_k$:
-0.1028069379667370301470967513180005924719013329651584055210194691463278825391787273823479714078649021
30
29
$w_k$:
0.05122154784925877217065628260494420825114695242524632755350905680551101540127955397119041272296930862
30
30
$x_k$:
-0.05147184255531769583302521316672257374914145366656956425516084398796475521042710905587009070728548584
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,15,x
30
30
$w_k$:
0.05142612853745902593386287921578125982955203486239598726385582417276158925940689207206611068118422461
30
31
$x_k$:
0
comment: $x_{31}=0$, the central node of every rule
equals: Zero
30
31
$w_k$:
6807797268792384629824040124156301153658087332819473443421845152170246144000000/132203768118039210856178131041734703445769907460992923212020844025968606678266611
30
32
$x_k$:
0.05147184255531769583302521316672257374914145366656956425516084398796475521042710905587009070728548584
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,16,x
30
32
$w_k$:
0.05142612853745902593386287921578125982955203486239598726385582417276158925940689207206611068118422461
30
33
$x_k$:
0.1028069379667370301470967513180005924719013329651584055210194691463278825391787273823479714078649021
30
33
$w_k$:
0.05122154784925877217065628260494420825114695242524632755350905680551101540127955397119041272296930862
30
34
$x_k$:
0.1538699136085835469637946727432559204185519712443384617189629829157871485108161013969231065107407856
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,17,x
30
34
$w_k$:
0.05088179589874960649229747304980469185338491426091923992077194208097254264678057557113205625407092986
30
35
$x_k$:
0.2045251166823098914389576710020247095241042645955637744760446502835032189466324549559256523531714782
30
35
$w_k$:
0.05040592140278234684089308565358502890219701825162223366424395921106671330863528371344774790797370079
30
36
$x_k$:
0.2546369261678898464398051298178051078827893033025184261642859750889635315690788029063662813842362026
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,18,x
30
36
$w_k$:
0.04979568342707420635781156937994232853920960281369610895104739284294848264622037765509834192408925020
30
37
$x_k$:
0.3040732022736250773726771071992565535311577898094627284442153699831215044238776730400142369990977859
30
37
$w_k$:
0.04905543455502977888752816536723817360588740529529656957949071790132821564459055524752287306524629747
30
38
$x_k$:
0.3527047255308781134710372070893738606536310080214256265941844689002694162331910786643603967521135295
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,19,x
30
38
$w_k$:
0.04818586175708712914077949229830459260579923610842980005737335087243379358396936842894267206327029894
30
39
$x_k$:
0.4004012548303943925354762115426606336110459329707839598318661065642917068931175906117552701571024738
30
39
$w_k$:
0.04718554656929915394526147818109948648288480730062845719414186155172553328949089702902027652560351550
30
40
$x_k$:
0.4470337695380891767806099003228540001624075938614244097544773817276153517285842070040068887212418983
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,20,x
30
40
$w_k$:
0.04605923827100698811627173555937358059469287557182492400473237949229360400644605267225297343897863917
30
41
$x_k$:
0.4924804678617785749936930612077087956442656409631869702607334098298842254639635277683704745226202598
30
41
$w_k$:
0.04481480013316266319235555161672324375743139279637300988968020119406350394790789918906106479211191904
30
42
$x_k$:
0.5366241481420198992641697933110727941641780069302971054527434829120149086189783786311411600971899026
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,21,x
30
42
$w_k$:
0.04345253970135606931683172811707325807460330863170316806488880549573864083957386333394208411719654146
30
43
$x_k$:
0.5793452358263616917560249321725404959070515888121528920812601661231283356781224190380997075178380821
30
43
$w_k$:
0.04196981021516424614714754128596975779008865671899237482038872032385265551120036579037994846200615695
30
44
$x_k$:
0.6205261829892428611404775564311892992073646928295281325950511701243353149748891177411525844553278211
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,22,x
30
44
$w_k$:
0.04037453895153595911199527975246811421612606212603025563399828961381084676105974096183682880295957390
30
45
$x_k$:
0.6600610641266269613700536681492707530383503748088339095506719733990493749973452207678802051702968819
30
45
$w_k$:
0.03867894562472759295034865153228105025092362982155384679037613067933740205662070055413910948753375956
30
46
$x_k$:
0.6978504947933157969322923880266400683823538006539546563797228467399767212431599606953816364400890469
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,23,x
30
46
$w_k$:
0.03688236465182122922391106561713596773695516478103033767000519858419613497015416986258419336075124323
30
47
$x_k$:
0.7337900624532268047261711313695276456693817277546854920870139951830001646361332538202466453159731880
30
47
$w_k$:
0.03497933802806002413749967073146787509722691279481871997220845723217778670200874421949847060384678447
30
48
$x_k$:
0.7677774321048261949179773409745031316948836172329084532064943873651585701729950450526096025862396842
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,24,x
30
48
$w_k$:
0.03298144705748372603181419101685392751059929121385838571451934764145231658238100880499451534196920599
30
49
$x_k$:
0.7997278358218390830136689423226832407356984293777845092364734954868666256732600722919520252418535647
30
49
$w_k$:
0.03090725756238776247288425294309227263527045852380715342684048696402208618987405694771744632818713127
30
50
$x_k$:
0.8295657623827683974428981197325019164390686961703416788069529834536565065895816350829524435081401600
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,25,x
30
50
$w_k$:
0.02875404876504129284397878535433421114467916054207493003510228075913217481546983422785466051536600314
30
51
$x_k$:
0.8572052335460610989586585106589438568208001706235961285050455173911988722571293268803112070465719564
30
51
$w_k$:
0.02650995488233310161060170933507541436651757952274856577086743833847213890365807761765252275993447490
30
52
$x_k$:
0.8825605357920526815431164625302255900566891471464842320683260531216162626951916557292158382857321049
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,26,x
30
52
$w_k$:
0.02419116207808060136568637072523202676039137782818246243222894356294488526750107068800647096287174366
30
53
$x_k$:
0.9055733076999077985465225589259583195689753636622284135640476639780376023944963191358507442684257416
30
53
$w_k$:
0.02182803582160919229716748573833899340150729605683491277363042235872043940338255907935605860239387980
30
54
$x_k$:
0.9262000474292743258793242770804740040864745368253290609110371336794229956511023268167728801505588624
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,27,x
30
54
$w_k$:
0.01941414119394238117340895105012845585142101419143152577027606653649717907902554048607272611462876361
30
55
$x_k$:
0.9443744447485599794158313240374391215856437149649809318174894013952091700065734275344887137684984852
30
55
$w_k$:
0.01692088918905327262757228942032209236856670378383519113988341084054667997855186104362008945168114602
30
56
$x_k$:
0.9600218649683075122168710255817976629303592174039233994856616724249399577070684292271894437038000238
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,28,x
30
56
$w_k$:
0.01436972950704580481245143244358001019584189989500150587356589940300019866249582190614427468289422259
30
57
$x_k$:
0.9731163225011262683746938684237068848876379642834393385375585018562411895816683828830856170826148637
30
57
$w_k$:
0.01182301525349634174223289885325059289626440625060781832630243154826536515585518273940170003251914145
30
58
$x_k$:
0.9836681232797472099700325816056628019403178547097113635171800101511442953647910437020759716603547137
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,29,x
30
58
$w_k$:
0.009273279659517763428441146892024360421270024938193107696495646914362666555743438549232578459634311215
30
59
$x_k$:
0.9916309968704045948586283661094857248505003337461632551001992334980748960326079660555619149584357523
30
59
$w_k$:
0.006630703915931292173319826369750168133628388217781258597395559735783756827773192132773181584451259816
30
60
$x_k$:
0.9968934840746495402716300509186952833408820381177507901080942978023876952101637408158820195580617174
equals: Nodes_and_weights_of_Gauss_Legendre_quadrature#30,30,x
30
60
$w_k$:
0.003890461127099884051267201844515503278515142984886464921420010128114473367645545106122627365594103835
30
61
$x_k$:
0.9994844100504906375713258957058108194688739470185080192363264283074801667484358783065646882314543572
30
61
$w_k$:
0.001389013698677007624551591226759699681048841291963272453441105533230136713098986536695625155642382048
Definition
For $n\geq 1$ the Gauss–Kronrod rule with $2n+1$ points, $\int_{-1}^{1} f(x)\,\mathrm{d}x \approx \sum_{k=1}^{2n+1} w_k f(x_k)$, extends the Gauss–Legendre rule with $n$ points by $n+1$ nodes so as to be exact for every polynomial of degree at most $3n+1$ [8]. Listed are its nodes $x_1<\cdots<x_{2n+1}$ and its weights $w_k$.
Parameters
$n$
—   number of points of the embedded Gauss rule; the Kronrod rule has $2n+1$ points ($n\geq 1$)
$k$
—   index of the node, in increasing order ($1\leq k\leq 2n+1$)
Formulas
(1)
The nodes are the roots of $\omega_n=P_n E_{n+1}$, where $P_n$ is the Legendre polynomial of degree $n$ and $E_{n+1}$ is the Stieltjes polynomial [9]: the monic polynomial of degree $n+1$ with $\int_{-1}^{1}P_n(x)E_{n+1}(x)\,x^j\,\mathrm{d}x=0$ for $0\leq j\leq n$.
(2)
$E_2=x^2-\tfrac35$, $E_3=x^3-\tfrac67x$, $E_4=x^4-\tfrac{10}{9}x^2+\tfrac{155}{891}$, $E_5=x^5-\tfrac{15}{11}x^3+\tfrac{615}{1573}x$, $E_6=x^6-\tfrac{21}{13}x^4+\tfrac{567}{845}x^2-\tfrac{8043}{186745}$. $E_{n+1}$ has the parity of $n+1$, so $x=0$ is a root of $E_{n+1}$ for even $n$ and of $P_n$ for odd $n$.
(3)
$w_k=\dfrac{q_{\omega_n}(x_k)}{\omega_n'(x_k)}$ with $q_{\omega_n}(x)=\int_{-1}^{1}\dfrac{\omega_n(t)-\omega_n(x)}{t-x}\,\mathrm{d}t$, and $q_{\omega_n}=c_n+E_{n+1}\,q_n$, where $c_n=\int_{-1}^{1}P_n(x)\,x^n\,\mathrm{d}x=\dfrac{2^{n+1}(n!)^2}{(2n+1)!}$ and $q_n$ is the secondary polynomial of $P_n$. Hence at a Gauss node $w_{2j}=\lambda_j+\dfrac{c_n}{P_n'(x_{2j})\,E_{n+1}(x_{2j})}$ with $\lambda_j=\dfrac{2}{(1-x_{2j}^2)P_n'(x_{2j})^2}$ the Gauss weight, and at a new node $w_k=\dfrac{c_n}{P_n(x_k)\,E_{n+1}'(x_k)}$ [3]. The correction to $\lambda_j$ is negative because $P_n'$ and $E_{n+1}$ take opposite signs at every root of $P_n$, by the interlacing.
(4)
$\sum_{k=1}^{2n+1} w_k x_k^m=\int_{-1}^{1}x^m\,\mathrm{d}x=\dfrac{1+(-1)^m}{m+1}$ for $0\leq m\leq 3n+1$ [5], and by the symmetry also for $m=3n+2$ when $n$ is odd; it fails at the next even $m$. So the degree of exactness is $3n+1$ for even $n$ and $3n+2$ for odd $n$.
(5)
$x_{2n+2-k}=-x_k$, $w_{2n+2-k}=w_k$, $w_k>0$ and $\sum_{k=1}^{2n+1}w_k=2$.
(6)
$x_{n+1}=0$ and $w_{n+1}=\dfrac{q_{\omega_n}(0)}{\omega_n'(0)}$ is rational: $\tfrac89$, $\tfrac{28}{45}$, $\tfrac{22016}{48825}$, $\tfrac{201344}{581175}$, $\tfrac{118298624}{418034925}$ for $n=1,\ldots,5$.
(7)
$w_k=\int_{-1}^{1}\ell_k(x)\,\mathrm{d}x$ with $\ell_k(x)=\prod_{j\neq k}\frac{x-x_j}{x_k-x_j}$ the Lagrange basis polynomial of the $2n+1$ nodes, as for every interpolatory rule; on equally spaced nodes the same integrals are the Newton–Cotes weights.
(8)
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
(9)
The roots of $E_{n+1}$ are real, simple, lie in $(-1,1)$ and interlace with the roots of $P_n$ [2], so in increasing order the Gauss nodes are $x_2,x_4,\ldots,x_{2n}$ and the new nodes are $x_1,x_3,\ldots,x_{2n+1}$; the entries $x_{2j}$ link to the corresponding entries of the table of Gauss–Legendre nodes and weights. $E_{n+1}$ is not the Stieltjes–Wigert polynomial and not a Heine–Stieltjes polynomial, which share the name.
(10)
At a Gauss node the weight $w_{2j}$ of the Kronrod rule is smaller than the weight $\lambda_j$ of the Gauss rule, so the two rules are different rules on a common set of points, and the Gauss weights are in the Gauss–Legendre table, not here. Evaluating $f$ once at the $2n+1$ nodes gives both $K=\sum_{k}w_kf(x_k)$ and $G=\sum_{j}\lambda_jf(x_{2j})$, and $|K-G|$ is used as an estimate of the error of $K$ [8]; that is how the rules are used in adaptive integration.
(11)
The rules with $n=7,10,15,20,25,30$ ($15$, $21$, $31$, $41$, $51$ and $61$ points) are the constants of QUADPACK's routines qk15 to qk61 [6], and so of every library built on it: scipy's quad, the GNU Scientific Library, Octave, R's integrate. The pair $G_7$, $K_{15}$ is the textbook example [7]; Boost and Julia's QuadGK ship the same six rules, and Kronrod's tables [1] list them to sixteen places.
(12)
Every value here is an algebraic number, and those that are rational are written exactly rather than to a hundred digits: the central node $x_{n+1}=0$ of every rule, its weight $w_{n+1}=q_{\omega_n}(0)/\omega_n'(0)$, all weights of the rules with $n\leq 2$, and the weights $12500/46557$ at the Gauss nodes $\pm\sqrt{3/5}$ of $n=3$. The rules with $n\leq 3$ have closed forms in square roots, given in the comments on their entries; $\pm 1/\sqrt{3}$ and $\pm\sqrt{3/5}$ are also in the table of quadratic algebraic numbers.
(13)
For $n=1$, $\omega_1=x(x^2-3/5)=\tfrac25P_3$, so the Kronrod extension of the midpoint rule is the three-point Gauss rule, entry by entry. For $n\geq 2$ the $(2n+1)$-point Gauss rule has degree $4n+1$ and the Kronrod rule at most $3n+2$, and they share no node but $0$.
(14)
The same extension of the Gauss–Lobatto and Gauss–Radau rules, and Patterson's repeated extensions of the Kronrod rules (the $2^m(n+1)-1$-point rules) [4], are not listed.
Programs
(P1)
Sage
R.<x> = QQ[]
n = 7
P = R(legendre_P(n, x))
I = lambda p: sum(c*(1 + (-1)^m)/(m + 1) for m, c in enumerate(p.list()))      # int_{-1}^1 p(x) dx
M = matrix(QQ, [[I(P*x^(i + j)) for i in range(n + 1)] for j in range(n + 1)])
c = M.solve_right(vector(QQ, [-I(P*x^(n + 1 + j)) for j in range(n + 1)]))
E = x^(n + 1) + sum(c[i]*x^i for i in range(n + 1))                            # the Stieltjes polynomial E_8
omega = P*E
nodes = omega.roots(RealIntervalField(400), multiplicities=False)              # x_1 = -0.99145537112081263920...
S = PolynomialRing(RealIntervalField(400), 'x')
weights = [I(S(omega).quo_rem(S([-r, 1]))[0])/omega.derivative()(r) for r in nodes]   # w_1 = 0.02293532201052922496...
References
[1]
A. S. Kronrod, Nodes and weights of quadrature formulas. Sixteen-place tables, Consultants Bureau, New York, 1965.
[2]
G. Szegő, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1935), 501–513.
[3]
G. Monegato, A note on extended Gaussian quadrature rules, Math. Comp. 30 (1976), 812–817.
[4]
T. N. L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 22 (1968), 847–856.
[5]
D. P. Laurie, Calculation of Gauss–Kronrod quadrature rules, Math. Comp. 66 (1997), 1133–1145.
[6]
R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber and D. K. Kahaner, QUADPACK, A subroutine package for automatic integration, Springer, 1983; the routines dqk15 to dqk61 at https://www.netlib.org/quadpack/.
[7]
D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, 1989, §5.5.
Links
Similar tables
Nodes and weights of Gauss–Legendre quadrature —   the embedded rule; its nodes are $x_2,x_4,\ldots,x_{2n}$ here, with different weights
Legendre polynomials —   $P_n$ is a factor of $\omega_n$, the polynomial whose roots are the nodes
Secondary polynomials of the Legendre polynomials —   $q_{\omega_n}=c_n+E_{n+1}q_n$ in the weight formula
Algebraic numbers of degree 2 —   holds the nodes $\pm\sqrt{3/5}$ of $n=1,3$ and $\pm 1/\sqrt{3}$ of $n=2$
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
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every rule with $n\leq 15$, from 3 to 31 points, together with the three larger rules that QUADPACK uses, $n=20,25,30$, of 41, 51 and 61 points; $n$ is the number of points of the embedded Gauss rule, and the Kronrod rule built on it has $2n+1$ points. Each rule is listed in full, every node with its weight, and both halves of the symmetric set)
How they were obtained:

$P_n$ is built exactly in $\mathbb{Q}[x]$ and $E_{n+1}$ by solving its defining linear system exactly over $\mathbb{Q}$; the roots of each are isolated by Sage's real root isolation over the interval field, so each node is an interval provably containing one root, carried on as an arb ball at 493 bits; $w_k$ is $q_{\omega_n}(x_k)/\omega_n'(x_k)$ in ball arithmetic, and a hundred digits are written, which every ball supports (the widest, a weight of the 61-point rule, supports 124; the working precision is what it is because $q_{\omega_n}$ and $\omega_n'$ for a polynomial of degree 61 cancel about twenty digits near the ends of the interval).

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Before a rule is written it must also satisfy the second weight formula through $c_n$, $P_n$ and $E_{n+1}$, the interlacing, the symmetry, $\sum w_k=2$, and exactness on $x^m$ for $m\leq 3n+1$ with failure at the first even $m$ beyond. Outside the generator the values were compared with the 33-digit constants of QUADPACK's six routines (281 values; the central weight of the 51-point rule is printed there one unit too large in its 33rd digit, the rational value being $0.0615808180678329350787598242400645531\ldots$), with Wikipedia's fifteen-point row, with closed forms for $n\leq 3$ computed in $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{330})$, with the Stieltjes polynomials from a second linear system in the Legendre basis, with the weights as integrals of the Lagrange basis polynomials, and with the stored Gauss–Legendre nodes and weights, with the controls that must fail failing.

Repeats values from: Nodes_and_weights_of_Gauss_Legendre_quadrature


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