 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 $y^{\prime }=f(y)$, its block form isWith $s$ 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.^[ref]