We have to look for numerical methods for the numerical solution of $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ that have 'great' regions of absolute stability.
Methods of which the region of absolute stability contains the whole left complex semiplane are calles $A-$ stable.
Since at applications we often see systems with matrices with real ( negative ) eigenvalues, or eigenvalues at a region $S_{\theta}, \theta \in \left( 0, \frac{\pi}{2} \right), S_{\theta}:=\{ z \in \mathbb{C} z=\rho e^{i \phi}, \pi-\theta<\phi< \pi+\theta, \rho>0 \} $, it isn't necessary to resort to a $A-$ stable method , but it suffices to work with a method, of which the region of absolute stability contains the negative real semiaxis or the region $S_{\theta}$, respectively. Such methods are called $A_0-$ stable and $A(\theta)-$stable, respectively.
Could you explain me why the graph for the $A(\theta)-$stable method looks like that?
Also, for the $A_0-$stable method, is the region of absolute stability this one: $\{ z \in \mathbb{C}: z=\rho e^{i \phi}, \pi<\phi< \pi, \rho>0 \}$? If so, could you explain me why it is like that?
