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An efficient and stable algorithm based on the simple-shooting method to solve the fiber Raman-amplifier (FRA) equations is proposed and numerically demonstrated. In the algorithm, the conventional Newton-Raphson method is substituted by a more efficient and stable modified Newton method to serve as the correction mechanism in the shooting iterations. By introducing the Broyden's rank-one method into the modified Newton method, the time-consuming calculation of the Jacobian matrix is dramatically relieved, and the efficiency of the proposed algorithm is further remarkably improved. An effective initial-guess-providing technique for the shooting method has also been proposed. Simulation results show that the efficiency of the proposed algorithm is improved more than three times compared with those of the previously reported methods. This algorithm can be used in the design of various kinds of FRAs to predict their gain and noise performance accurately.