The Newton algorithm is one of the most promising methods for determining the steady-state response of nonlinear circuits. However, the method has ever larger memory storage capacity requirements, as the circuit size increases. State elimination is efficient to reduce the required memory storage. In this paper, an automatic state elimination method is proposed. The Newton method has another problem, since it fails to converge to a steady-state solution in certain cases. A damped Newton method, which considers nonlinearity of the system, is discussed as a method to improve convergence. For autonomous systems, such as oscillatory circuits, another modification is discussed for improving reliability and convergence. The Newton algorithm with these modifications has been implemented in Toshiba's version of SPICE2, and experimental results have shown reduction in memory storage capacity requirements, less computational effort and convergence improvements.