Abstract: A parallel algorithm for solving the cyclic tridiagonal equations is developed in this paper. This algorithm is mainly based on the matrix decomposition. Firstly, a lower order cyclic tridiagonal equation can be formed by extracting the first and the last equation in each process. Secondly, the lower order cyclic tridiagonal equation is solved to obtain the values of the key variables. Finally, the values of the key variables are used in the corresponding processes and the related equations are solved in parallel. The cyclic tridiagonal equations can be solved efficiently and stably by using this algorithm if the coefficient matrix is diagonally dominant, which condition is also required by the traditional chasing method, so, the parallel algorithm developed in this paper to solve the tridiagonal equations does not need additional conditions to restrict the coefficient matrix. The computational cost of the parallel algorithm in one process is only O(17n) and equal to that of the traditional algorithm for solving the cyclic tridiagonal equations. Moreover, the distributed parallel computation can also be performed and the data communication cost is small. In fact, the data length for the communication from child processes to the master process and from the master process to child processes is 8 real numbers and 2 real numbers, respectively. Therefore, the computational cost and the data communication cost are smaller than the traditional parallel algorithm for solving the cyclic tridiagonal equations. The numerical experiments indicate that，as far as large scale cyclical tridiagonal equations are considered, the parallel computational efficiency is greater than 0.75 if the rank number is less than 16. Indeed, the traditional chasing algorithm is a particular case of this algorithm. So, this algorithm can be used to solve the tridiagonal equations naturally.