  In many commercial and academic settings, numerical solvers fail to achieve
their theoretical performance levels due to issues in the system definition,
parameterization, and even implementation. We propose a pair of methods for
detecting and localizing these convergence rate issues in applications that can
be treated as homotopy problems including numerical continuation and the
evolution of differential algebraic equations. Both approaches are rooted in
dynamical systems theory, in particular, the numerical techniques used to
perform bifurcation studies on "black-box" systems, and can be applied across a
range of numerical solvers and systems without significant modification. In a
general problem, these methods will detect certain classes of convergence
anomalies, and indicate the states that are affected by their existence.
However in applications like circuit simulation, certain classes of anomalies
can be localized to an individual component through the use of a small number
of additional computational experiments. To demonstrate the efficacy of the
approaches for detection and localization, we apply them to a pair of nonlinear
circuits: a diode bridge and a simplified model of a power channel
