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Nodes and weights of Gauss–Laguerre quadrature
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Numbers
$n$
$k$
$x_k$ or $w_k$
1
1
$w_k$:
1
equals: One
comment: $w_1=1=\int_0^\infty e^{-x}\,\mathrm{d}x$
1
1
$x_k$:
1
equals: One
comment: $x_1=1$, the one-point rule
2
1
$w_k$:
0.8535533905932737622004221810524245196424179688442370182941699344976831196155267597125968835819103932
comment: $w_1=(2+\sqrt{2})/4$
2
1
$x_k$:
0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273
equals: Algebraic_numbers_of_degree_2#1,-4,2,1
comment: $x_1=2-\sqrt{2}$
2
2
$w_k$:
0.1464466094067262377995778189475754803575820311557629817058300655023168803844732402874031164180896068
comment: $w_2=(2-\sqrt{2})/4$
2
2
$x_k$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
equals: Algebraic_numbers_of_degree_2#1,-4,2,2
comment: $x_2=2+\sqrt{2}$
3
1
$w_k$:
0.7110930099291730154495901911425944313093937962895534451317172443619021551221322358203721265797945735
comment: $w_1$ is the largest root of $1944w^3-1944w^2+405w-4$ [9]
3
1
$x_k$:
0.4157745567834790833115338731282744735466174126931184650939659543223250199369133149571962921305347523
comment: $x_1$ is the smallest root of $x^3-9x^2+18x-6=-6\,L_3(x)$
3
2
$w_k$:
0.2785177335692408488014448884567264810348900309863886718567349484344940965793657530357422874532912931
comment: $w_2$ is the second smallest root of $1944w^3-1944w^2+405w-4$ [9]
3
2
$x_k$:
2.294280360279041719822050361359593868959861721060280834035201248084030451337166446563187803002855713
comment: $x_2$ is the second smallest root of $x^3-9x^2+18x-6=-6\,L_3(x)$
3
3
$w_k$:
0.01038925650158613574896492040067908765571617272405788301154780720360374829850201114388558596691413345
comment: $w_3$ is the smallest root of $1944w^3-1944w^2+405w-4$ [9]
3
3
$x_k$:
6.289945082937479196866415765512131657493520866246600700870832797593644528725920238479615904866609535
comment: $x_3$ is the largest root of $x^3-9x^2+18x-6=-6\,L_3(x)$
4
1
$w_k$:
0.6031541043416336016359660238180782113018371867659489319846731614418089861258586735779491056652975941
comment: $w_1$ is the largest root of $1990656w^4-1990656w^3+504576w^2-16960w+9$ [9]
4
1
$x_k$:
0.3225476896193923118003614591043674797437572244742957671884518538069686787077040098685851872116594585
comment: $x_1$ is the smallest root of $x^4-16x^3+72x^2-96x+24=24\,L_4(x)$
4
2
$w_k$:
0.3574186924377996866414920174580912817635783649193409217482250466757641592070271151436284656291221577
comment: $w_2$ is the second largest root of $1990656w^4-1990656w^3+504576w^2-16960w+9$ [9]
4
2
$x_k$:
1.745761101158346575686816712517947023673874515531072501782782660998456057442197164140130458088740493
comment: $x_2$ is the second smallest root of $x^4-16x^3+72x^2-96x+24=24\,L_4(x)$
4
3
$w_k$:
0.03888790851500538427243816815620991372230719134827690218163529240452576291017698099984331190929904305
comment: $w_3$ is the second smallest root of $1990656w^4-1990656w^3+504576w^2-16960w+9$ [9]
4
3
$x_k$:
4.536620296921127983279285384957137880125784353386804649748057587555828450875143158976538801118043394
comment: $x_3$ is the second largest root of $x^4-16x^3+72x^2-96x+24=24\,L_4(x)$
4
4
$w_k$:
0.0005392947055613274501037905676205932122772569664332440854664994779010917569372302785791167962812052268
comment: $w_4$ is the smallest root of $1990656w^4-1990656w^3+504576w^2-16960w+9$ [9]
4
4
$x_k$:
9.395070912301133129233536443420547616456583906607827081280707897638746812974955667014745553581556655
comment: $x_4$ is the largest root of $x^4-16x^3+72x^2-96x+24=24\,L_4(x)$
5
1
$w_k$:
0.5217556105828086524758609287924500399119554812970853251314192904208501990560984890476349015517106502
5
1
$x_k$:
0.2635603197181409102030619433608333346890075699055169278626150283831131438046884853888014286367681425
5
2
$w_k$:
0.3986668110831759274541333481444192823834784622487164483031347604840264800165979934144477416138614673
5
2
$x_k$:
1.413403059106516792218407980187557749539096004380880280554408297900742377592354294632071875549162505
5
3
$w_k$:
0.07594244968170759538765331140554090387328735861979070143775048603629220815672504729698376576306614711
5
3
$x_k$:
3.596425771040722081223186588782971665671150940710574538253074419444595035340244283770723676354534464
5
4
$w_k$:
0.003611758679922048454461262573038192553590828374073124348730999398594824744560758646873845788473851871
5
4
$x_k$:
7.085810005858837556922124181108086000385935670784940882578921655737736629317011861316381315331093202
5
5
$w_k$:
0.00002336997238577622789114908455158127768786946033440077896446366023628802601771159405974528288788349981
5
5
$x_k$:
12.64080084427578265943321930656055124971480981421808737075098059853381281394570107489202170412844169
6
1
$w_k$:
0.4589646739499635935682848777094125187193189956115607330572342769321511556451416916036996398845333020
6
1
$x_k$:
0.2228466041792606894643548267866737242822252688576002963291303937536401044309010515955776164183680980
6
2
$w_k$:
0.4170008307721209941133775661932916139825635081573587341952710263999265379132355587735399512288030434
6
2
$x_k$:
1.188932101672623030743150921935052745154296570089671016518761222862943356860203312806216636083315730
6
3
$w_k$:
0.1133733820740449757387061850982938081191985037433346413175327825830452631969599912725774402696417692
6
3
$x_k$:
2.992736326059314077691325284513675843160301186514696050581976320985026057506756326143056854935719884
6
4
$w_k$:
0.01039919745314907489891330284694784799373826290660856099244149931429194793398672467307177974045848454
6
4
$x_k$:
5.775143569104510501839830369433571311334116782700457560326826376525846664574720250069518682808674813
6
5
$w_k$:
0.0002610172028149320594792428600013863110565067874138197142532963132991486571137957257383171329530977911
6
5
$x_k$:
9.837467418382589917715547029942540818219135128979579386971656464085086484377693500300087127510565284
6
6
$w_k$:
8.985479064296212388252920528248741242227937235107232671184572859466535622379513728717436103030608550e-7
6
6
$x_k$:
15.98287398060170178254579156738848555784992506285799568927164922178745733224972555908554308224335619
7
1
$w_k$:
0.4093189517012739021304328800177787947669431832074338493649607110843932761253139365052675697532997358
7
1
$x_k$:
0.1930436765603624138382478850038237245912098489135675062022513620100641534578255987111800382336511561
7
2
$w_k$:
0.4218312778617197799292810054174989342590027059320029991901022434788137857776505436191417390405709368
7
2
$x_k$:
1.026664895339191950345199443174833535102196639113658933360710734930004545668668231595189210056106893
7
3
$w_k$:
0.1471263486575052783953741846365457173544765949455906715670008217886514033646663159537616602917905845
7
3
$x_k$:
2.567876744950746206907786226659590196836958161644470099286606330324050332197448782138334620001190572
7
4
$w_k$:
0.02063351446871693986570561496420129570134551659585448028302140110661333409366283420693098658683648616
7
4
$x_k$:
4.900353084526484568101714378102427218232682253669132382444273301292698837093801276835860894024831209
7
5
$w_k$:
0.001074010143280745522131959628430313790425752197955201860520306160868503752638765016731679063637537508
7
5
$x_k$:
8.182153444562860791081827551233961629640766851752429664304834245004144902520264945938674710602231569
7
6
$w_k$:
0.00001586546434856420126873262232340559538609480705009171784269678973607464897495100198286767200606375999
7
6
$x_k$:
12.73418029179781375801264245819510909085333614144626191746020093558286361258664675932815214335414068
7
7
$w_k$:
3.170315478995580562271322153853242015231411270601655181959092362223709265369618349759185865536038926e-8
7
7
$x_k$:
19.39572786226254031171258205763025460474285010346047949694112309085617361647534440545260838372784792
8
1
$w_k$:
0.3691885893416375299205828393757039441152848439042111370237387410544499872689961454628842475252966956
8
1
$x_k$:
0.1702796323051009997888618566082972446825182470854812212556888713838303736469716843276134760672408411
8
2
$w_k$:
0.4187867808143429560769785813333343195310764594887222161751793894957644292183829071651819356842177967
8
2
$x_k$:
0.9037017767993799121860202235550898922166525972984565685800781122657835136746594348388051108904352877
8
3
$w_k$:
0.1757949866371718056996598667767736577732253097598447857556377858000586994394099571176486191073134202
8
3
$x_k$:
2.251086629866130689307118366968635153838187135894803716041951465658100718832541825933407162858830094
8
4
$w_k$:
0.03334349226121565152213253493440643668469857378962185270628216817303966492194707717330598118769111495
8
4
$x_k$:
4.266700170287658793649421826900639634482122279353236896629915337513518021208079105778811582275100478
8
5
$w_k$:
0.002794536235225672524938924147928639749927236532777305557343164201195164038087804816466560587753385432
8
5
$x_k$:
7.045905402393465697279325482119358498008738749252416568251406205168924995928762110456241988299804916
8
6
$w_k$:
0.00009076508773358213104238501493357342695902615520218861968107702434225890029408106229846278728848152549
8
6
$x_k$:
10.75851601018099522405995678803203414783260221914068729751765726386058646711827661814619538259317237
8
7
$w_k$:
8.485746716272531544868018308932031317521418994128998832630966242549832336027322440307273556457113486e-7
8
7
$x_k$:
15.74067864127800457802876115840283587414603204161376176729554143092699245598645270464710673839885594
8
8
$w_k$:
1.048001174871510381615088535515696798227721101262254411154525541229648424469970162393083459853169945e-9
8
8
$x_k$:
22.86313173688926410570053429741310955479314673036115596442776131322226345360425651587181855861656007
9
1
$w_k$:
0.3361264217979625196734677176059899425015216620719263298482171325661103823708090327985141364673083612
9
1
$x_k$:
0.1523222277318082474281070731274225752170461586109733678823087434243429180427990980129361690874573355
9
2
$w_k$:
0.4112139804239843873091469427928767253767970767387264151196453659480210467859426413857096648260543513
9
2
$x_k$:
0.8072200227422558477414192109515515746567930900973004271277443890030712436509751431776370074706225227
9
3
$w_k$:
0.1992875253708855808605756072117874219485121404588039509793149151821636800516401402266879242165941089
9
3
$x_k$:
2.005135155619347122983033247014097840533579582569755606888655934952185041338815124339357855528712739
9
4
$w_k$:
0.04746056276565159926211636004787839326286045922229387552350305297470879368653766188382261632713092005
9
4
$x_k$:
3.783473973331232991675406093640176587812058522725840549817817904487695006654730531891318377878476158
9
5
$w_k$:
0.005599626610794583177004199005555369978091996038996761216730357514237030819060189435923591829273475189
9
5
$x_k$:
6.204956777876612606973535210061584188613138223812335014276922687010558756588211692964618285600710339
9
6
$w_k$:
0.0003052497670932105663054128242912248629008961140182305701272903022882153457120065811693952046046880370
9
6
$x_k$:
9.372985251687576201809710732145639152388203840652377196545137664502976512854670936614307097255435763
9
7
$w_k$:
0.000006592123026075352392255722848751027975051388021489394257677144181539776220618406305007144691773265917
9
7
$x_k$:
13.46623691109209357109788183971171823812319879836926266469767660699224250639202741615909644762919249
9
8
$w_k$:
4.110769330349548442902410403300909049066314162650327323516583200880750429413253317973796340641370788e-8
9
8
$x_k$:
18.83359778899169661414989929962587340096424002299527198855217238487375263766148451346012413725331093
9
9
$w_k$:
3.290874030350707576466813803225022730407132084493097320245730235657341514933448424637891557048960223e-11
9
9
$x_k$:
26.37407189092737679614100729372193644169174176016688318421156368475317537681628554338060462229608172
10
1
$w_k$:
0.3084411157650201415474708346778606956287288865383374421155571802001855620293255700304068546057974227
10
1
$x_k$:
0.1377934705404924308307725056527111881079916807457832941633665113734459644764620865625437524173411653
10
2
$w_k$:
0.4011199291552735515157803099128195147954836169621130175749677551676609150002367690707827147111537014
10
2
$x_k$:
0.7294545495031704981603731216760787810760727333122492460078110488062678230730515720465249917862963194
10
3
$w_k$:
0.2180682876118094215886485234746467267427785384121889405665039813207866459828519234900012632360320998
10
3
$x_k$:
1.808342901740316048232920075750608833283060282371448151693209768373296766404342823915338360612401224
10
4
$w_k$:
0.06208745609867774739290212931351795369590906568380209236867676030392386646299027694371840017667495622
10
4
$x_k$:
3.401433697854899514482532221408390679273156614203473229426766102182326922866694641039509587084778059
10
5
$w_k$:
0.009501516975181100553839072194171991225862450401579753364631897084868032245134075078424087790157087697
10
5
$x_k$:
5.552496140063803632417558486868762857974064287317813828490478833625263608429043546057187425993506486
10
6
$w_k$:
0.0007530083885875387754559643536756639017920391401436288650616401436776153273255358771563784949667883715
10
6
$x_k$:
8.330152746764496700238767197274522182709438972030989344744629428357395432053247695166582449185828958
10
7
$w_k$:
0.00002825923349599565567422563826850021282803316474437467711708679525932250866427518788195598127822373840
10
7
$x_k$:
11.84378583790006556491853891914161398580281690946533587333959149527622015069710823271721252760324171
10
8
$w_k$:
4.249313984962686372586576659747123546481080198644157008125460482902179134442969550391268304080928492e-7
10
8
$x_k$:
16.27925783137810209953265393583362233525599560303305907778839216500282735093259794022584150759651917
10
9
$w_k$:
1.839564823979630780921535224355938247982612776590658409513276706558695584589034651680586700279282934e-9
10
9
$x_k$:
21.99658581198076195127709019559449397680673234000188778824150351169310637579899882306768653334698036
10
10
$w_k$:
9.911827219609008558377547283244736064581094611244769243130436668625454356386541965226117258699716166e-13
10
10
$x_k$:
29.92069701227389155990879334079919517971067057751796016610425113530984960526845263920157286437310656
11
1
$w_k$:
0.2849332128942006050560510247235555629642220481678877322460422807016237418298299188319702619492680685
11
1
$x_k$:
0.1257964421879675226757945775164576086114765222531486009632809943790173003197742994873648093442608809
11
2
$w_k$:
0.3897208895278493779375535080480987881352993126582550735118947944260672726279906003763638061812869469
11
2
$x_k$:
0.6654182558392278416781278394198778634300465539768208100973814029841421607606141688981869622690935855
11
3
$w_k$:
0.2327818318489913339402237955434576368051934156730032596917934805546786296094628062960040167946273271
11
3
$x_k$:
1.647150545872169309587003213653052975718073741977940779100446692095604091034014027423029341141129298
11
4
$w_k$:
0.07656445354619668640085417901324513154290766222758013755037568775026608368789998900093513058295565406
11
4
$x_k$:
3.091138143035254953301959342585854751123044478824415912270276825642138285346089086647698468389852683
11
5
$w_k$:
0.01439328276735069509186391874087044403571884012925644693573304819437786937931180055074539556108343039
11
5
$x_k$:
5.029284401579833212369995083660750980296427909783737334197642383093817438711313507259932575511625987
11
6
$w_k$:
0.001518880846484873069847776400424957604253589032316169790046860230727489805233406238041678422939296104
11
6
$x_k$:
7.509887863806616819410997144500091909927580197825845447929873029016889411063677074151456302899189204
11
7
$w_k$:
0.00008513122435471922597204241706003829312641764564222416560996643627744884227624023856842381667645647617
11
7
$x_k$:
10.60595099954696778055592164572198436779768375103757839861875645755844864751191615806812653132285180
11
8
$w_k$:
0.000002292403879574504078576832707092393716963898139746434679647256893763611045631214374770413430356022314
11
8
$x_k$:
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Definition
For $n\geq 1$ the Gauss–Laguerre quadrature rule with $n$ points is the unique rule $\int_{0}^{\infty} f(x)\,e^{-x}\,\mathrm{d}x \approx \sum_{k=1}^{n} w_k f(x_k)$ that is exact for every polynomial $f$ of degree at most $2n-1$ [4]. Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$.
Parameters
$n$
—   number of nodes ($n\geq 1$)
$k$
—   index of the node, in increasing order ($1\leq k\leq n$)
Formulas
(1)
$\dfrac{j_{0,k}^2}{4n+2}<x_k<\dfrac{(4k+2)\bigl(2k+1+\sqrt{(2k+1)^2+\frac14}\bigr)}{4n+2}$ for $1\leq k\leq n$ [6], where $j_{0,k}$ is the $k$-th positive zero of the Bessel function $J_0$, in the table of zeros of $J_\alpha$; the ratio of $x_k$ to the lower bound tends to $1$ as $n\to\infty$.
(2)
$w_k=\dfrac{x_k}{(n+1)^2\,L_{n+1}(x_k)^2}=\dfrac{1}{x_k\,L_n'(x_k)^2}$ [1], the two equal because $x L_n'(x)=n L_n(x)-n L_{n-1}(x)$ and $(n+1)L_{n+1}(x_k)=-nL_{n-1}(x_k)$ at a root of $L_n$. Also $\dfrac{1}{w_k}=\sum_{j=0}^{n-1}L_j(x_k)^2$, the Christoffel function at the node, the $L_j$ being orthonormal for $e^{-x}$.
(3)
$\sum_{k=1}^{n} w_k x_k^m=\int_{0}^{\infty}x^m e^{-x}\,\mathrm{d}x=m!$ for $0\leq m\leq 2n-1$, and not for $m=2n$; the moments are in the table of factorials. In particular $w_k>0$ and $\sum_{k=1}^{n}w_k=1$.
(4)
$w_k=\dfrac{q_n(x_k)}{L_n'(x_k)}$ with $q_n(x)=\int_{0}^{\infty}\dfrac{L_n(t)-L_n(x)}{t-x}\,e^{-t}\,\mathrm{d}t$ the secondary polynomial of $L_n$ for the density $e^{-x}$, a polynomial of degree $n-1$ with rational coefficients; the same construction for the Legendre polynomials gives their secondary polynomials.
(5)
The nodes of the rules with $n$ and $n-1$ points interlace: $x_k^{(n)}<x_k^{(n-1)}<x_{k+1}^{(n)}$ for $1\leq k\leq n-1$, as the roots of consecutive orthogonal polynomials do.
(6)
For the weight $x^{-1/2}e^{-x}$ the $n$-point Gauss rule has nodes $t_k^2$ and weights $2u_k$, where $t_1<\cdots<t_n$ are the positive nodes of the Gauss–Hermite rule with $2n$ points and $u_k$ their weights; for $x^{1/2}e^{-x}$ it has nodes $t_k^2$ and weights $2u_k t_k^2$, taken from the Gauss–Hermite rule with $2n+1$ points. Both follow from $\int_{0}^{\infty}g(x)\,x^{\mp 1/2}e^{-x}\,\mathrm{d}x=\int_{-\infty}^{\infty} g(t^2)\,t^{1\mp 1}\,e^{-t^2}\,\mathrm{d}t$ and the uniqueness of the Gauss rule.
(7)
$\int_{0}^{\infty}f(x)\,\mathrm{d}x\approx\sum_{k=1}^{n}w_k\,e^{x_k}f(x_k)$ [4], and on $[a,\infty)$ with the weight $e^{-x}$, $\int_{a}^{\infty}f(x)\,e^{-x}\,\mathrm{d}x\approx e^{-a}\sum_{k=1}^{n}w_k\,f(x_k+a)$.
Comments
(8)
The nodes are unbounded above; the largest at $n=30$ is $104.15\ldots$, and the weight there, $8.7\cdot 10^{-45}$, is small because $e^{-x}$ is.
(9)
This is the generalised Gauss–Laguerre rule for the weight $x^{\alpha}e^{-x}$ at $\alpha=0$ only, the case of Abramowitz–Stegun Table 25.9, DLMF Tables 3.5.6–3.5.9 and numpy's laggauss. The rules for $\alpha=\pm\frac12$ are the Gauss–Hermite rules folded onto the half-line, by the formula (6); other $\alpha$ are not listed.
(10)
The nodes are the roots of $L_n$, the Laguerre polynomial of degree $n$, orthogonal for the weight $e^{-x}$ on $[0,\infty)$ and normalised by $L_n(0)=1$ (the roots do not depend on the normalisation). Its $n$ roots are real, simple and positive, and $x_1<\cdots<x_n$ is their increasing order. Placing the nodes at these roots is what lets a rule with $n$ points reach degree $2n-1$; no other $n$ points do, and $n$ points in general position reach only $n-1$.
(11)
Gauss–Legendre quadrature is the same construction for the weight $1$ on $[-1,1]$, and Gauss–Hermite quadrature for $e^{-x^2}$ on the real line.
(12)
Every value here is an algebraic number: the one-point rule is $x_1=1$, $w_1=1$, written exactly; the two-point rule is $x=2\mp\sqrt{2}$ with $w=(2\pm\sqrt{2})/4$, and its nodes are also in the table of quadratic algebraic numbers. For $n\geq 2$ no node is rational: $L_n$ is irreducible over $\mathbb{Q}$ [2], so the $n$ nodes of a rule are conjugate algebraic numbers of degree exactly $n$. The nodes of the rules with $n=3,4$ are the roots of $x^3-9x^2+18x-6$ and $x^4-16x^3+72x^2-96x+24$, and the polynomials whose roots are the weights are OEIS A387347 [9]; the comments on those entries say which root each is.
(13)
The weights are not monotone in $k$ for every rule. For $n\leq 6$ they decrease from $w_1$ to $w_n$; for $7\leq n\leq 23$ the largest weight is $w_2$ ($w_1=0.4093\ldots$, $w_2=0.4218\ldots$ at $n=7$), and for $24\leq n\leq 30$ it is $w_3$. Beyond the largest weight they decrease, the last one being of the size of $e^{-x_n}$.
Programs
(P1)
Sage
R.<x> = QQ[]
p = R(laguerre(8, x))
nodes = p.roots(RealIntervalField(400), multiplicities=False)   # x_1 = 0.17027963230510099978...
weights = [r/(81*R(laguerre(9, x))(r)^2) for r in nodes]        # w_1 = 0.36918858934163752992...
(P2)
Python
import numpy.polynomial.laguerre as L
nodes, weights = L.laggauss(8)      # double precision: 0.17027963, 0.36918859
References
[1]
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972, §25.4.45 and Table 25.9.
[2]
I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen I, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. 1929, 125–136.
Links
Similar tables
Laguerre polynomials —   the nodes are the roots of $L_n$
Nodes and weights of Gauss–Legendre quadrature —   the same construction for the weight $1$ on $[-1,1]$
Nodes and weights of Gauss–Hermite quadrature —   the same construction for $e^{-x^2}$ on the real line; folded onto the half-line it gives the rules for $x^{\pm 1/2}e^{-x}$
Factorial of natural numbers —   the moments $\int_0^\infty x^m e^{-x}\,\mathrm{d}x=m!$ that the rule reproduces for $m\leq 2n-1$
Algebraic numbers of degree 2 —   holds the nodes $2\pm\sqrt{2}$ of $n=2$
Zeros of Bessel functions of the first kind —   $j_{0,k}^2/(4n+2)$ is a lower bound for $x_k$, sharp as $n\to\infty$
Secondary polynomials of the Legendre polynomials —   the Legendre analogue of the polynomial $q_n$ with $w_k=q_n(x_k)/L_n'(x_k)$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every rule with $n\leq 30$, where $n$ is the number of nodes; each rule is listed in full, every node with its weight. The nodes lie in $(0,\infty)$ and are not symmetric, so there are no halves to list)
How they were obtained:

$L_n$ is built exactly in $\mathbb{Q}[x]$; its roots 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 461 bits; $w_k$ is $x_k/((n+1)^2L_{n+1}(x_k)^2)$ in ball arithmetic, and the digits written are those the ball supports (every entry supports more than 120; the evaluation of $L_{31}$ near $x=40$ loses about fifteen digits to cancellation, which is why the working precision is what it is).

more

Before a rule is written it must also satisfy the Christoffel formula for the weights, $\sum w_k=1$, exactness on $x^m$ for $m\leq 2n-1$ with failure at $m=2n$, and the polynomials named in the entry comments. Outside the generator the values were compared with the closed forms for $n\leq 2$, with the stored polynomials $L_n$, with fourteen OEIS expansions to 100 digits, with the OEIS A387347 weight polynomials for $n\leq 6$, with all hundred values of DLMF Tables 3.5.6–3.5.9 ($n=5,10,15,20$; one printed weight at $n=10$ differs from the ball by one unit in its eighteenth digit, the other ninety-nine agree to the last digit), with MathWorld's rows for $n\leq 5$, and with the folded Gauss–Hermite rules of the stored table, with the controls that must fail failing.



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