A parallel overlapping preconditioner is applied to ICCG method and the effect of the parallel preconditioning on the convergence of the method is investigated by solving large scale block tridiagonal linear systems arising from the discretization of Poisson's equation. Compared with the original ICCG method, the parallel preconditioned ICCG method can solve the problems in high parallelism with slight increasing the number of iterations. Furthermore, the speedup and the efficiency are evaluated for the parallel preconditioned ICCG method by substituting the experimental results into formulae of complexity. For example, when a domain of simulation is discretized on a 250250 rectangular grid and the preconditioner is divided into 249 smaller ones, its speedup is 146.3 with the efficiency 0.59.

Many numerical simulation problems of natural phenomena are formulated by large tridiagonal and block tridiagonal linear systems. In this paper, an efficient parallel algorithm to solve a tridiagonal linear system is proposed. The algorithm named bi-recurrence algorithm has an inherent parallelism which is suitable for parallel processing. Its time complexity is 8N - 4 for a tridiagonal linear system of order N. The complexity is little more than the Gaussian elimination algorithm. For parallel implementation with two processors, the time complexity is 4N - 1. Based on the bi-recurrence algorithm, a VLSI oriented tridiagonal solver is designed, which has an architecture of 1-D linear systolic array with three processing cells. The systolic tridiagonal solver completes finding the solution of a tridiagonal linear system in 3N + 6 units of time. A highly parallel systolic tridiagonal solver is also presented. The solver is characterized by highly parallel computability which originates in the divide-and-conquer strategy and high cost performance which originates in the systolic architecture. This solver completes finding the solution in 10(N/p) + 6p + 23 time units, where p is the number of partitions of the system.