Theorems on the Unique Initial Solution for Globally Convergent Homotopy Methods

Finding DC operating points of nonlinear circuits is an important and difficult task. The Newton-Raphson method adopted in the SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. For the global convergence of homotopy methods, it is a necessary condition that a given initial solution is the unique solution to the homotopy equation. According to the conventional criterion, such an initial solution, however, is restricted in some very narrow region. In this paper, considering the circuit interpretation of homotopy equations, we prove theorems on the uniqueness of an initial solution for globally convergent homotopy methods. These theorems give new criteria extending the region wherein any desired initial solution satisfies the uniqueness condition.