History of Adams–Moulton coefficients

 Title: Adams–Moulton coefficients Definition: For $s\geq0$, the Adams–Moulton coefficients $\beta^*_{s,j}$ are the rational-  coefficients in the implicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s}\beta^*_{s,j}f_{n+j}$-  CITE{Wiki}. The method has order $s+1$, and an absent row has coefficient $0$.+  numbers for which the implicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s}\beta^*_{s,j}f_{n+j}$+  is exact whenever $f(t,y(t))$ is a polynomial in $t$ of degree at most $s$ CITE{Wiki}. Parameters:   s:

   j:     type: Z-    title: history index, counted from the oldest value+    title: index of the node, counted from the oldest value     display: $j$     constraints: $0\leq j\leq s$

   comment-notation: Here $f_{n+j}=f(t_{n+j},y_{n+j})$ and $t_{n+j}=t_0+(n+j)h$. The     case $s=0$ is backward Euler, and the case $s=1$ is the trapezoidal rule. The-    coefficients are chosen so the formula is exact when $f(t,y(t))$ is a polynomial-    in $t$ of degree at most $s$; this is the moment condition in the Formulas section.-  comment-step: The step size appears as the single factor $h$ before the sum. Each-    stored number is the coefficient multiplying one $f_{n+j}$ when $h=1$.+    method has order $s+1$.+  comment-step: The step size appears as the single factor $h$ before the sum. The+    coefficient of $f_{n+j}$ in the formula is $h\beta^*_{s,j}$.   comment-ordering: Entries are listed from the oldest value to the newest value.     Wikipedia CITE{Wiki} writes the displayed formulas from newest to oldest; for

     as $1/24,-5/24,19/24,3/8$. Formulas:-  formula-integral: If $\ell_{s,j}$ is the Lagrange basis polynomial for the nodes-    $0,1,\ldots,s$, with $\ell_{s,j}(i)=1$ for $i=j$ and $0$ otherwise, then $\beta^*_{s,j}=\int_{s-1}^{s}\ell_{s,j}(x)\,\mathrm{d}x$.+  formula-integral: If $\ell_{s,j}$ is HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[the+    Lagrange basis polynomial] for the nodes $0,1,\ldots,s$, with $\ell_{s,j}(i)=1$+    for $i=j$ and $0$ otherwise, then $\beta^*_{s,j}=\int_{s-1}^{s}\ell_{s,j}(x)\,\mathrm{d}x$.   formula-moments: $\sum_{j=0}^{s}\beta^*_{s,j}j^m=(s^{m+1}-(s-1)^{m+1})/(m+1)$ for     $0\leq m\leq s$.-  formula-gregory: If $\gamma^*_m$ denotes the Gregory coefficient, then $\gamma^*_m=(-1)^m\psi_m(0)/m!$,-    where $\psi_m$ is HREF{Bernoulli_polynomials_of_the_second_kind}[the Bernoulli-    polynomial of the second kind], and $\beta^*_{s,0}=(-1)^s\gamma^*_s$.+  formula-gregory: $\beta^*_{s,0}=G_s=\psi_s(0)/s!$, where $G_s$ is the Gregory coefficient+    CITE{OEISGregory} and $\psi_s$ is HREF{Bernoulli_polynomials_of_the_second_kind}[the+    Bernoulli polynomial of the second kind]. Similar tables: - table: HREF{Adams_Bashforth_coefficients}[Adams–Bashforth coefficients]

 - table: HREF{Bernoulli_polynomials_of_the_second_kind}[Bernoulli polynomials of the     second kind]-  relation: their value at $0$ gives the oldest Adams–Moulton coefficient through-    the Gregory coefficients+  relation: 'their value at $0$ gives the oldest Adams–Moulton coefficient: $\beta^*_{s,0}=\psi_s(0)/s!$' - table: HREF{Butcher_tableaux_of_explicit_Runge_Kutta_methods}[Butcher tableaux of     explicit Runge–Kutta methods]

   rigour: exact   complete: 'no'-  complete-note: it holds every nonzero coefficient with $0\leq s\leq12$+  complete-note: it holds every coefficient with $0\leq s\leq12$, matching HREF{Adams_Bashforth_coefficients}[the+    Adams–Bashforth coefficient table] so each predictor there has its Adams–Moulton+    corrector   rigour details: The generator builds the moment equations over $\mathbb{Q}$ for     the nodes $0,1,\ldots,s$ and solves them exactly. Each completed method is checked

       \ rhs))))\n\nadams_moulton_coefficients(3)" Numbers:-- params:-    s: '0'-    j: '0'-  number: '1'-  comment: 'Backward Euler method: $1$.'-- params:-    s: '1'-    j: '0'-  number: 1/2-  comment: 'Trapezoidal rule, from newest to oldest: $\tfrac{1}{2}(1,1)$.'-- params:-    s: '1'-    j: '1'-  number: 1/2-- params:-    s: '2'-    j: '0'-  number: -1/12-  comment: '2-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{12}(5,8,-1)$.'-- params:-    s: '2'-    j: '1'-  number: 2/3-- params:-    s: '2'-    j: '2'-  number: 5/12-- params:-    s: '3'-    j: '0'-  number: 1/24-  comment: '3-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{24}(9,19,-5,1)$.'-- params:-    s: '3'-    j: '1'-  number: -5/24-- params:-    s: '3'-    j: '2'-  number: 19/24-- params:-    s: '3'-    j: '3'-  number: 3/8-- params:-    s: '4'-    j: '0'-  number: -19/720-  comment: '4-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{720}(251,646,-264,106,-19)$.'-- params:-    s: '4'-    j: '1'-  number: 53/360-- params:-    s: '4'-    j: '2'-  number: -11/30-- params:-    s: '4'-    j: '3'-  number: 323/360-- params:-    s: '4'-    j: '4'-  number: 251/720-- params:-    s: '5'-    j: '0'-  number: 3/160-  comment: '5-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{1440}(475,1427,-798,482,-173,27)$.'-- params:-    s: '5'-    j: '1'-  number: -173/1440-- params:-    s: '5'-    j: '2'-  number: 241/720-- params:-    s: '5'-    j: '3'-  number: -133/240-- params:-    s: '5'-    j: '4'-  number: 1427/1440-- params:-    s: '5'-    j: '5'-  number: 95/288-- params:-    s: '6'-    j: '0'-  number: -863/60480-  comment: '6-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{60480}(19087,65112,-46461,37504,-20211,6312,-863)$.'-- params:-    s: '6'-    j: '1'-  number: 263/2520-- params:-    s: '6'-    j: '2'-  number: -6737/20160-- params:-    s: '6'-    j: '3'-  number: 586/945-- params:-    s: '6'-    j: '4'-  number: -15487/20160-- params:-    s: '6'-    j: '5'-  number: 2713/2520-- params:-    s: '6'-    j: '6'-  number: 19087/60480-- params:-    s: '7'-    j: '0'-  number: 275/24192-  comment: '7-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{120960}(36799,139849,-121797,123133,-88547,41499,-11351,1375)$.'-- params:-    s: '7'-    j: '1'-  number: -11351/120960-- params:-    s: '7'-    j: '2'-  number: 1537/4480-- params:-    s: '7'-    j: '3'-  number: -88547/120960-- params:-    s: '7'-    j: '4'-  number: 123133/120960-- params:-    s: '7'-    j: '5'-  number: -4511/4480-- params:-    s: '7'-    j: '6'-  number: 139849/120960-- params:-    s: '7'-    j: '7'-  number: 5257/17280-- params:-    s: '8'-    j: '0'-  number: -33953/3628800-  comment: '8-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{3628800}(1070017,4467094,-4604594,5595358,-5033120,3146338,-1291214,312874,-33953)$.'-- params:-    s: '8'-    j: '1'-  number: 156437/1814400-- params:-    s: '8'-    j: '2'-  number: -645607/1814400-- params:-    s: '8'-    j: '3'-  number: 1573169/1814400-- params:-    s: '8'-    j: '4'-  number: -31457/22680-- params:-    s: '8'-    j: '5'-  number: 2797679/1814400-- params:-    s: '8'-    j: '6'-  number: -2302297/1814400-- params:-    s: '8'-    j: '7'-  number: 2233547/1814400-- params:-    s: '8'-    j: '8'-  number: 1070017/3628800-- params:-    s: '9'-    j: '0'-  number: 8183/1036800-  comment: '9-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{7257600}(2082753,9449717,-11271304,16002320,-17283646,13510082,-7394032,2687864,-583435,57281)$.'-- params:-    s: '9'-    j: '1'-  number: -116687/1451520-- params:-    s: '9'-    j: '2'-  number: 335983/907200-- params:-    s: '9'-    j: '3'-  number: -462127/453600-- params:-    s: '9'-    j: '4'-  number: 6755041/3628800-- params:-    s: '9'-    j: '5'-  number: -8641823/3628800-- params:-    s: '9'-    j: '6'-  number: 200029/90720-- params:-    s: '9'-    j: '7'-  number: -1408913/907200-- params:-    s: '9'-    j: '8'-  number: 9449717/7257600-- params:-    s: '9'-    j: '9'-  number: 25713/89600-- params:-    s: '10'-    j: '0'-  number: -3250433/479001600-  comment: '10-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{479001600}(134211265,656185652,-890175549,1446205080,-1823311566,1710774528,-1170597042,567450984,-184776195,36284876,-3250433)$.'-- params:-    s: '10'-    j: '1'-  number: 9071219/119750400-- params:-    s: '10'-    j: '2'-  number: -12318413/31933440-- params:-    s: '10'-    j: '3'-  number: 23643791/19958400-- params:-    s: '10'-    j: '4'-  number: -21677723/8870400-- params:-    s: '10'-    j: '5'-  number: 2227571/623700-- params:-    s: '10'-    j: '6'-  number: -33765029/8870400-- params:-    s: '10'-    j: '7'-  number: 12051709/3991680-- params:-    s: '10'-    j: '8'-  number: -296725183/159667200-- params:-    s: '10'-    j: '9'-  number: 164046413/119750400-- params:-    s: '10'-    j: '10'-  number: 26842253/95800320-- params:-    s: '11'-    j: '0'-  number: 4671/788480-  comment: '11-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{958003200}(262747265,1374799219,-2092490673,3828828885,-5519460582,6043521486,-4963166514,3007739418,-1305971115,384709327,-68928781,5675265)$.'-- params:-    s: '11'-    j: '1'-  number: -68928781/958003200-- params:-    s: '11'-    j: '2'-  number: 384709327/958003200-- params:-    s: '11'-    j: '3'-  number: -87064741/63866880-- params:-    s: '11'-    j: '4'-  number: 501289903/159667200-- params:-    s: '11'-    j: '5'-  number: -91910491/17740800-- params:-    s: '11'-    j: '6'-  number: 1007253581/159667200-- params:-    s: '11'-    j: '7'-  number: -102212233/17740800-- params:-    s: '11'-    j: '8'-  number: 36465037/9123840-- params:-    s: '11'-    j: '9'-  number: -99642413/45619200-- params:-    s: '11'-    j: '10'-  number: 1374799219/958003200-- params:-    s: '11'-    j: '11'-  number: 4777223/17418240-- params:-    s: '12'-    j: '0'-  number: -13695779093/2615348736000-  comment: '12-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{2615348736000}(703604254357,3917551216986,-6616420957428,13465774256510,-21847538039895,27345870698436,-26204344465152,19058185652796,-10344711794985,4063327863170,-1092096992268,179842822566,-13695779093)$.'-- params:-    s: '12'-    j: '1'-  number: 2724891251/39626496000-- params:-    s: '12'-    j: '2'-  number: -30336027563/72648576000-- params:-    s: '12'-    j: '3'-  number: 406332786317/261534873600-- params:-    s: '12'-    j: '4'-  number: -229882484333/58118860800-- params:-    s: '12'-    j: '5'-  number: 529394045911/72648576000-- params:-    s: '12'-    j: '6'-  number: -4874320027/486486000-- params:-    s: '12'-    j: '7'-  number: 84400835489/8072064000-- params:-    s: '12'-    j: '8'-  number: -485500845331/58118860800-- params:-    s: '12'-    j: '9'-  number: 1346577425651/261534873600-- params:-    s: '12'-    j: '10'-  number: -551368413119/217945728000-- params:-    s: '12'-    j: '11'-  number: 6595204069/4402944000-- params:-    s: '12'-    j: '12'-  number: 703604254357/2615348736000+  '0':+    '0':+      number: '1'+      comment: This is the backward Euler method.+  '1':+    '0':+      number: 1/2+      comment: 'Trapezoidal rule, from newest to oldest: $\tfrac{1}{2}(1,1)$.'+    '1': 1/2+  '2':+    '0':+      number: -1/12+      comment: '2-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{12}(5,8,-1)$.'+    '1': 2/3+    '2': 5/12+  '3':+    '0':+      number: 1/24+      comment: '3-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{24}(9,19,-5,1)$.'+    '1': -5/24+    '2': 19/24+    '3': 3/8+  '4':+    '0':+      number: -19/720+      comment: '4-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{720}(251,646,-264,106,-19)$.'+    '1': 53/360+    '2': -11/30+    '3': 323/360+    '4': 251/720+  '5':+    '0':+      number: 3/160+      comment: '5-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{1440}(475,1427,-798,482,-173,27)$.'+    '1': -173/1440+    '2': 241/720+    '3': -133/240+    '4': 1427/1440+    '5': 95/288+  '6':+    '0':+      number: -863/60480+      comment: '6-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{60480}(19087,65112,-46461,37504,-20211,6312,-863)$.'+    '1': 263/2520+    '2': -6737/20160+    '3': 586/945+    '4': -15487/20160+    '5': 2713/2520+    '6': 19087/60480+  '7':+    '0':+      number: 275/24192+      comment: '7-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{120960}(36799,139849,-121797,123133,-88547,41499,-11351,1375)$.'+    '1': -11351/120960+    '2': 1537/4480+    '3': -88547/120960+    '4': 123133/120960+    '5': -4511/4480+    '6': 139849/120960+    '7': 5257/17280+  '8':+    '0':+      number: -33953/3628800+      comment: '8-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{3628800}(1070017,4467094,-4604594,5595358,-5033120,3146338,-1291214,312874,-33953)$.'+    '1': 156437/1814400+    '2': -645607/1814400+    '3': 1573169/1814400+    '4': -31457/22680+    '5': 2797679/1814400+    '6': -2302297/1814400+    '7': 2233547/1814400+    '8': 1070017/3628800+  '9':+    '0':+      number: 8183/1036800+      comment: '9-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{7257600}(2082753,9449717,-11271304,16002320,-17283646,13510082,-7394032,2687864,-583435,57281)$.'+    '1': -116687/1451520+    '2': 335983/907200+    '3': -462127/453600+    '4': 6755041/3628800+    '5': -8641823/3628800+    '6': 200029/90720+    '7': -1408913/907200+    '8': 9449717/7257600+    '9': 25713/89600+  '10':+    '0':+      number: -3250433/479001600+      comment: '10-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{479001600}(134211265,656185652,-890175549,1446205080,-1823311566,1710774528,-1170597042,567450984,-184776195,36284876,-3250433)$.'+    '1': 9071219/119750400+    '2': -12318413/31933440+    '3': 23643791/19958400+    '4': -21677723/8870400+    '5': 2227571/623700+    '6': -33765029/8870400+    '7': 12051709/3991680+    '8': -296725183/159667200+    '9': 164046413/119750400+    '10': 26842253/95800320+  '11':+    '0':+      number: 4671/788480+      comment: '11-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{958003200}(262747265,1374799219,-2092490673,3828828885,-5519460582,6043521486,-4963166514,3007739418,-1305971115,384709327,-68928781,5675265)$.'+    '1': -68928781/958003200+    '2': 384709327/958003200+    '3': -87064741/63866880+    '4': 501289903/159667200+    '5': -91910491/17740800+    '6': 1007253581/159667200+    '7': -102212233/17740800+    '8': 36465037/9123840+    '9': -99642413/45619200+    '10': 1374799219/958003200+    '11': 4777223/17418240+  '12':+    '0':+      number: -13695779093/2615348736000+      comment: '12-step Adams-Moulton formula, from newest to oldest: $\tfrac{1}{2615348736000}(703604254357,3917551216986,-6616420957428,13465774256510,-21847538039895,27345870698436,-26204344465152,19058185652796,-10344711794985,4063327863170,-1092096992268,179842822566,-13695779093)$.'+    '1': 2724891251/39626496000+    '2': -30336027563/72648576000+    '3': 406332786317/261534873600+    '4': -229882484333/58118860800+    '5': 529394045911/72648576000+    '6': -4874320027/486486000+    '7': 84400835489/8072064000+    '8': -485500845331/58118860800+    '9': 1346577425651/261534873600+    '10': -551368413119/217945728000+    '11': 6595204069/4402944000+    '12': 703604254357/2615348736000