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Title: Adams–Bashforth coefficients Definition: For $s\geq1$, the Adams–Bashforth coefficients $\beta_{s,j}$ are the rational- coefficients in the explicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s-1}\beta_{s,j}f_{n+j}$- CITE{Wiki}.+ numbers for which the explicit $s$-step formula $y_{n+s}=y_{n+s-1}+h\sum_{j=0}^{s-1}\beta_{s,j}f_{n+j}$+ is exact whenever $f(t,y(t))$ is a polynomial in $t$ of degree at most $s-1$ CITE{Wiki}. Parameters: s:
constraints: $0\leq j<s$ Comments:- comment-notation: Here $f_{n+j}=f(t_{n+j},y_{n+j})$ and $t_{n+j}=t_0+(n+j)h$. The- coefficients are chosen so the formula is exact when $f(t,y(t))$ is a polynomial- in $t$ of degree at most $s-1$; this is the moment condition in the Formulas section.- An absent row has coefficient $0$.- 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 $h=1$.+ comment-notation: Here $f_{n+j}=f(t_{n+j},y_{n+j})$ and $t_{n+j}=t_0+(n+j)h$.+ comment-step: The step size is factored out, so 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 $5/12,-4/3,23/12$. Formulas:- formula-integral: If $\ell_{s,j}$ is the Lagrange basis polynomial for the nodes- $0,1,\ldots,s-1$, 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: $\beta_{s,j}=\int_{s-1}^{s}\ell_{s-1,j}(x)\,\mathrm{d}x$, where+ $\ell_{s-1,j}$ is the HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[Lagrange+ basis polynomial] for the nodes $0,1,\ldots,s-1$, so $\ell_{s-1,j}(i)=1$ for $i=j$+ and $0$ for the other nodes. formula-moments: $\sum_{j=0}^{s-1}\beta_{s,j}j^m=(s^{m+1}-(s-1)^{m+1})/(m+1)$ for $0\leq m\leq s-1$.
- table: HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[Lagrange basis polynomials for equally spaced nodes]- relation: the coefficients are integrals of these basis polynomials over the next- unit interval+ relation: each $\beta_{s,j}$ is the integral of $\ell_{s-1,j}$ over $[s-1,s]$ - table: HREF{Newton_Cotes_weights}[Newton–Cotes weights] relation: integrals of the same basis over the interval spanned by the interpolation
title: 'Wikipedia: Linear multistep method, revision 1347229011' url: https://en.wikipedia.org/w/index.php?title=Linear_multistep_method&oldid=1347229011+ VODE:+ title: 'Netlib: VODE source'+ url: https://netlib.org/ode/vode.f Keywords: linear multistep method, explicit multistep method, Adams method, Adams predictor, Euler method
rigour: exact complete: 'no'- complete-note: it holds every nonzero coefficient with $1\leq s\leq12$+ complete-note: it holds every coefficient with $1\leq s\leq12$, matching VODE's+ default maximum Adams order CITE{VODE} rigour details: The generator builds the moment equations over $\mathbb{Q}$ for the nodes $0,1,\ldots,s-1$ and solves them exactly. Each completed method is checked
'0': number: '1'- comment: 'Euler method: $1$.'+ comment: This is the forward Euler method. '2': '0':
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