Values of the Hastings–McLeod solution $q(s)$ of Painlevé II

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Numbers

$s$

$q(s)$

-6:

1.731024958831778696439750036134546689878380010507474620889719462214571389378920575487212313277153177e+0

-23/4:

1.694437378913070765647688349495193361417659305142158317424309912200265874228136855692625182077244033e+0

-17/3:

1.682061511007656438224946269029052474639665848647148857968812125781046706439342633050076157023357369e+0

-11/2:

1.657027002412415423359970267164941587368143524537505469736077600450508456169838922058446422045010025e+0

-16/3:

1.631600053581130319859483342345792099955933644428628811363868099344038947789112142519909168907123421e+0

-21/4:

1.618733246944635278116301028051974608275525953009414121993136091708085844789146575860172358955649555e+0

-5:

1.579487087847008038869945726240767633873487153648173253658319354754089450244023052716027257118598368e+0

-19/4:

1.539209143091647139785182503205704844118530999865080114718135378083838450258024063800655732210917307e+0

-14/3:

1.525538360157677268721141991689835886269983072472531856723037650837274962709309866014071017160097784e+0

-9/2:

1.497807160080573217971626453971009421863258346368308954896132839059060596033964292629072099333675129e+0

-13/3:

1.469528617736832350530908049842918472706860871257382715366785679621922982229583857998448180617229828e+0

-17/4:

1.455172716221326681283770466210038357389297466890690295182985907735272817828977937366263361415227407e+0

-4:

1.411176929362393977046581747037281425737524054746095215727191311928837694418826603188919066678521649e+0

-15/4:

1.365665013887688902794799882237614196677819218883817752839561489364018199611624536622080775586753497e+0

-11/3:

1.350126163187494628909006240961040937550910005866298289708648587070943871794930271576528457451447335e+0

-7/2:

1.318449678908555518383388065390104288988443325766083206556487111766488573897222190313645691004657144e+0

-10/3:

1.285916124062301911016039597627781876962218449965216387085526327272173615292149644336881141371559712e+0

-13/4:

1.269303750026317550222909993163623875294638754622836045295607195799857568985589688930028504410811915e+0

-3:

1.217953146253253723272725965169483368686425324234852730610637953607842569169189898801992544050709742e+0

-11/4:

1.164072608050911119316206252524480905912812658800744707349357581014338592759855185371163312358102848e+0

-8/3:

1.145486811406670276334703160374224591006603218240277539668252719745203433658455534345428095280259636e+0

-5/2:

1.107288423474072701501692114686374547328724311267919294068418067593721404785554393914914309784824885e+0

-7/3:

1.067619063467825076805316696201388757679746923038571862795511968894147904467247468536753902152580264e+0

-9/4:

1.047194657597528613945568600411477007653655877237910269099632537472343330272636334932381232874959507e+0

-2:

9.833913497278053435785458331373015966681726027277879509171354839744865462948147618529034250294676892e-1

-7/4:

9.155532797002253076429244928636530843077155710376321802340132166180500404224334514523516433058909329e-1

-5/3:

8.920110401978354951855386860779737287067876747067303057843473958020176961085816796066662893583945385e-1

-3/2:

8.435338539145317776736904072651076908117630781100437632634519190398230647867319838816690789211437286e-1

-4/3:

7.932622481679416611008713246663878397687968936485837455271829281742187730457473230853197487297500093e-1

-5/4:

7.674980026554697191532164239624037031881474799467879737035494298740368999403604980290508588644923218e-1

-1:

6.880603646051180816545045115304314911509294437277505239330283588990821766858868561549671494546692143e-1

-3/4:

6.063852802504198405601108513564803939021146476046639917664632700313600712022602504148154187115351396e-1

-2/3:

5.789462377045783947732720383673001949959503791479060560964320831884511502365311259279712267474024131e-1

-1/2:

5.241947440594490551134652537064873969656350208198864098396984952291881902125971334799348175080324198e-1

-1/3:

4.701679931522968884152116729061800956999954458410535363760313451655080309748995892259988185899363446e-1

-1/4:

4.436434572753099129479435109697027193329628329546698181542783468977777187587720571048166294026532049e-1

3.670615515480784277477921131756109615121920536131394537086149438391963071001863772228898876675201135e-1

1/4:

2.966207689346244566191435903380972284856228780716803406866021614514808129638995037788287054023810250e-1

1/3:

2.748169909042112047208272334339398828675575594787650942568891442141733189434108234601295603557188029e-1

1/2:

2.340181520218428535854538400248236674791500803328767494283150599625208329359974525859229106504175048e-1

2/3:

1.971690287090297025129596868025588654998137766319187084183957543588046291882047839041016724779282868e-1

3/4:

1.802673872879670715459638870631975038460130173297999657035477280163767886922215235011513334606972809e-1

1.356435435044715939402150523352239368718141280394064191079266967860904862367049721525644738527873791e-1

5/4:

9.976941501192832128022676562343657733414963361987095776771295413166777802516307008318683378196957977e-2

4/3:

8.961800235756333570020792042870537130041888153699818743102217945248323155343531225978297788774240715e-2

3/2:

7.179144602049947017793734428710865308782699901313247810322582687926142485144554794176060524099787505e-2

5/3:

5.697539698010684075785083837141179292299996589100311560650099511763266132708529192179387877323164489e-2

7/4:

5.058321658842106543352725279882413067001714803328860839402653336461420669741132241720851589404750589e-2

3.492814926459571958921428955778459735345322608330103507014801854907177190796936867949892157068458802e-2

Definition

Let $q$ be the Hastings–McLeod solution of $q''(s)=s q(s)+2q(s)^3$, the real solution with $q(s)\sim\operatorname{Ai}(s)$ as $s\to+\infty$ [1] [2]. This table gives its values at rational $s$.

Parameters

$s$

— argument ($s\in\mathbb{Q}$)

Formulas

(1)

For the Tracy–Widom GUE distribution, $F_2(s)=\exp\left(-\int_s^\infty (x-s)q(x)^2\,dx\right)$ [2].

(2)

As $s\to-\infty$, $q(s)\sim\sqrt{-s/2}\left(1+\frac{1}{8s^3}-\frac{73}{128s^6}+\frac{10657}{1024s^9}\right)$ [3].

Comments

(3)

The sign convention is the Tracy–Widom convention: $q(s)\sim\operatorname{Ai}(s)$ as $s\to+\infty$. Changing the sign gives another solution of Painlevé II and leaves formulas involving $q(s)^2$ unchanged, so the asymptotic condition is part of the definition.

(4)

The selected arguments are the reduced rationals of denominator at most $4$ in $[-6,2]$, shared with the Tracy-Widom tables on this interval. The positive-tail Airy asymptotic describes limiting behaviour, not an equality with the Airy function at omitted finite arguments. The present selection makes no claim to cover the far tails.

Programs

(P1)

Python

from generate import HastingsMcLeodPainleveIIValues

generator = HastingsMcLeodPainleveIIValues()
print(generator.value({'s': '0'}, 100)['number'])

References

[1]

S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Archive for Rational Mechanics and Analysis 73 (1980), 31-51. (doi)

[2]

Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Communications in Mathematical Physics 159 (1994), 151-174. (arXiv)

[3]

Folkmar Bornemann, On the numerical evaluation of distributions in random matrix theory: a review, Markov Processes and Related Fields 16 (2010), 803-866. (arXiv)

Links

[4]

Wikipedia: Tracy-Widom distribution

[5]

DLMF 32.11: Painlevé transcendents, asymptotic approximations

Similar tables

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — gives the Airy function to which $q(s)$ is asymptotic as $s\to+\infty$

Random-matrix factors $g_G(k)$ — is another table in the corpus whose quantities come from random matrix theory

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds every rational $s=a/b$ in lowest terms with $-6\leq s\leq2$ and $1\leq b\leq4$ (49 entries). These integers, halves, thirds and quarters match the common argument range of the Tracy-Widom distribution and density tables. Other arguments and the far tails are not included.)

How they were obtained:

Values are computed with multiprecision Taylor integration of the Hastings-McLeod Painleve-II equation, starting from Airy data at a large positive boundary. The stored values come from a run using 280 decimal digits, 240 Taylor terms, maximum step 1/12, and right boundary 48. A second run using 230 decimal digits, 180 Taylor terms, maximum step 1/8, and right boundary 40 agreed on all 49 entries; the worst relative difference was 6.6318618168586463930e-146 at $s=-6$.

The Hamiltonian identity $\int_s^\infty q(x)^2\,dx=q'(s)^2-sq(s)^2-q(s)^4$ was checked during both integrations; the displayed maximum residuals were 0e-229 and 0e-279. Independent midpoint Nyström checks of the Airy-kernel resolvent formula for $q(s)$ used 260 Gauss-Legendre nodes on a shifted interval of length 52 at 210 decimal digits for $s=-6,0,2$; the worst relative difference from the rounded stored value was 1.4054574233560079477e-100 at $s=-6$.

These are heuristic numerical agreement checks, not rigorous enclosures. Arb is used for multiprecision arithmetic, but Taylor step results are advanced by midpoints. The Airy boundary error, Taylor remainders, Fredholm interval truncation, and Fredholm matrix conditioning are not rigorously bounded.