In the computer-aided analysis of nonlinear autonomous oscillators, the steady-state periodic response is usually found by integrating the system equations from some initial state until the transient response appears to be negligible. In lightly damped systems, convergence to the steady-state response is very slow, and this integration could extend over many periods making the computation costly. Also, one is never sure if a stable orbit exists or if the response might eventually decay to a singular point. If a stable orbit does exist, sometimes it is difficult to determine the period of the orbit. In this paper, a Newton algorithm is defined which in the neighborhood of an orbit converges to it rapidly and gives a precise value for the period of this oscillation. This algorithm represents a substantial step forward in the analysis of nonlinear systems. In addition, the algorithm meshes easily with most computer-aided circuit-analysis programs, and the initial iterates give information on the transient behavior of the circuit.