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Two new numerical methods which may be used to calculate solutions to optimal control problems are developed. These methods involve guessing initial values for unknown Lagrange multipliers and a control sequence. They are similar to the successive sweep method in that Riccati equations are used to calculate corrections to these guessed variables. They integrate Riccati equations, however, which differ from the one integrated by the standard sweep method. The effectiveness of these methods along with a standard method based on the integration of linear equations are compared for two example problems, the Brachistochrone and an Earth-to-Mars low thrust transfer problem.