In this paper, we propose an efficient computational algorithm that can reduce significantly the computational complexity of the ML optimum multiuser detector known as the most excellent detector in synchronous code division multiple access channels. The proposed detector uses the sequential detection algorithm based on the alternating maximization appraoch to obtain the ML estimates. As initial estimates for this sequential algorithm, we can use the estimated values obtained by the conventional single-user detector, the linear decorrelating multiuser detector, or the decorrelating decision-feedback muliuser detector, the linear decorrelating multiuser detector, or the decorrelating decision-feedback multiuser detector. We have performed computer simulations in order to see the convergence behaviors and the detection performance of the propsoed algorithm in terms of initial algorithms and the number of users, and then to compare the computational complexity with that of the ML optimum multiuser detector. From the results, we have seen that the proposed alternating maximization detector has nearly equal detction performance with that of the ML optimum multiuser detctor in only a few iteration.