In this paper, initial value problem for dynamical astronomy will be established using parabolic cylindrical coordinates. Computation algorithm is developed for the initial value problem of gravity perturbed trajectories. Applications of the algorithm for the problem of final state predication are illustrated by numerical examples of seven test orbits of different eccentricities. The numerical results are extremely accurate and efficient in predicating final state for gravity perturbed trajectories which is of extreme importance for scientific researches as well as for military purposes. Moreover, an additional efficiency of the algorithm is that, for each of the test orbits, the step size used for solving the differential equations of motion is larger than 70% of the step size used for obtaining its reference final state solution.