A Parallel Preex Algorithm for Almost Toeplitz Tridiagonal Systems

A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiago-nal systems that are diicult to solve eeciently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel preex (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is eecient on parallel machines. It consists of a preex communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study has been conducted to provide a simple truncation formula. Experimental results have been measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel preex algorithm is a good algorithm for symmetric, almost symmetric Toeplitz tridiagonal systems and for the compact scheme on high-performance computers.