\section{(Backward Euler Method)}
For the backward euler method, the characterisitc polynomials are:
\[\rho(r) = r - 1\]
\[\sigma(r) = r\]
\[z = \rho(e^{i \theta}) / \sigma(e^{i \theta}) = 1 - e^{-i \theta} = 1 - \cos{\theta} + i \sin{\theta}\]

Hence, the boundary locus is the unit circle shifted one unit to the
right.  This divides the complex plane into two regions: the region
inside this circle and everything outside of it.  All linear multistep
methods are necessarily absolutely unstable for small positive values
of $Re(z)$, so the region of absolute stability for the backward euler
method is everything outside of the circle.  This is "good" because it
means that the entire left half plane is included in the region of
absolute stable.
