To compute the condition
\kappa_{\mathrm{abs}}^\infty := \max\{\| W(t,t_0, y_0) \|: t_0 \leq t \leq T \}
of an initial value problem on $[t_0, T]$, I need to compute the Wronski matrix $W$.
The approximation of the solution is denoted by the discrete evolution operator $\Psi^{t_0, t}$ and the exact solution is denoted $\Phi^{t_0, t}$ and the Wronski matrix is given as $ W(t,t_0, y) := \frac{d}{dy} \Phi^{t_0, t}y \Big |_{y=z}$
My first question: is the condition really a function of the exact solution? According to my understanding numerical analysis deals with finding good approximation functions so the condition should be a measure of how good (stable?) a given approximation is, therefore the Wronski matrix should be a function of $\Psi$, not of $\Phi$.
My second question: how do I explicitly calculate the Wronski matrix for a given problem? Say, I'm using the explicit Euler method $y_{k+1} = y(t_k) + h*f(t_k, y(t_k))$ to solve the ODE $\dot{y} = (\alpha - \beta y)y$ or the system $\dot{u} = (\alpha - \beta v)u, \dot{v} = (\delta u - \gamma)v$.
Many thanks for your help.