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A steady-state periodic solution is essential in the study of vibration theory. A new computer-aided scheme to determine a periodic solution and evaluate its stability is discussed. It is a remarkably precise, speedy and powerful method following after some classical schemes with the same aim, e.g. the well-known harmonic balance method. Since the algorithm consists of a purely numerical procedure, the scheme is very effective for cases in which it is difficult to find steady solutions via conventional analytical methods. The successive improvement scheme for determining particular initial conditions that generate the periodic solution exclusively, is based on a certain strategy which involves the Newton-Raphson method and numerical integration, such as the RKG scheme, for relevant differential equations. Illustrative examples are provided which demonstrate the convergence of successive trajectories in a phase plane. A full listing of the program is given with its instruction manual.