= General linear method
A <general linear method> advances an <ordinary differential equation> using internal stages $Y$ and a retained external vector $y^{[n]}$. For an autonomous equation $y'=f(y)$, its block form is
$$
Y=hAF(Y)+Uy^{[n-1]},\qquad y^{[n]}=hBF(Y)+Vy^{[n-1]}.
$$
With $s$ stages and $r$ external components, $A,U,B,V$ have sizes $s\times s$, $s\times r$, $r\times s$, $r\times r$ respectively.
The retained components can encode previous values or derivative information. Runge-Kutta and linear multistep methods fit this framework. With one scalar stage and external component, $A=0$, $U=B=V=1$ gives explicit Euler. The word linear describes these combinations, not a requirement that the differential equation itself be linear.<General linear methods by Zdzislaw Jackiewicz>[Definition section, equations (1)–(2)]{ref}
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