AbstractThe alternating direction implicit (ADI) iterative method is an efficient iterative method to solve systems of linear equations due to its extremely fast convergence. The ADI method has also been used successfully as a preconditioner in some other iterative methods, such as the preconditioned conjugate gradient. In this paper a parallel algorithm for the ADI preconditioning is proposed. In this algorithm, several steps of the ADI iteration are computed simultaneously. This means that several tridiagonal systems that are traditionally solved sequentially are now solved concurrently. The high performance of this algorithm is achieved by increasing the degree of parallelism and reducing memory contention. The algorithm can easily be implemented in a multiprocessor architecture. Experiments have been conducted on the Myrias SPS-2 computer with 64 processors and good performance of this algorithm is observed.