  The Obreshkov method is a single-step multi-derivative method used in the
numerical solution of differential equations and has been used in recent years
in efficient circuit simulation. It has been shown that it can be made of
arbitrary high local order of convergence while maintaining unconditional
numerical stability. Nevertheless, the theoretical basis for the high order of
convergence has been known only for the special case where the underlying
system of differential equations is of the ordinary type, i.e., for ordinary
differential equations (ODE). On the other hand, theoretical analysis of the
order of convergence for the more general case of a system consisting of
differential and algebraic equations (DAE) is still lacking in the literature.
  This paper presents the theoretical characterization for the local order of
convergence of the Obreshkov method when used in the numerical solution of a
system of DAE. The contribution presented in this paper demonstrates that, in
DAE, the local order of convergence is a function of the differentiation index
of the system and, under certain conditions, becomes lower than the order
obtained in ODE.
