 Solution

For a Runge-Kutta method with real stage matrix $A=(a_{ij})$ and weights $b_i$, defineThe method has algebraic stability of a Runge-Kutta method ifThe matrix condition means $z^{T}Mz\ge 0$ for every real vector $z$. Its significance is nonlinear contractivity: an algebraically stable method, on well-defined stage solutions, is B-stable for a dissipative vector field satisfying $⟨f(t,y)-f(t,z),y-z⟩\le 0$. This is distinct from scalar A-stability, which concerns the linear test equation. The contractivity proof in part (b) explains the matrix criterion directly.