Accurate and efficient computer simulation of a proposed design for an integrated circuit ("chip") is essential because of the difficulty and expense of building prototypes for such devices. The transistors and diodes in such circuits are modeled with nonlinear equations, hence simulation of circuits requires the solution of systems of nonlinear equations involving hundreds or even thousands of variables. This paper discusses the application of probability-one homotopy methods to various systems of nonlinear equations which arise in circuit simulation. The so-called "coercivity conditions" which are required for such methods are proved using concepts from circuit theory. The theoretical claims of global convergence for such methods are substantiated by experiments with a collection of examples which have proved difficult for commercial simulation packages which do not use homotopy methods. Moreover, by careful design of the homotopy equations, the performance of the homotopy methods can be made quite reasonable
