A Parallel Prefix 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 tridiagonal systems that are di cult to solve e ciently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel pre x (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is e cient on parallel machines. It consists of a pre x 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 pre x algorithm is a good algorithm for symmetric, almost symmetric Toeplitz tridiagonal systems and for the compact scheme on high-performance computers. Note: Most of the detailied proofs given in Section 4 are for review purposes. They will be removed in the nal paper. This research was supported by the National Aeronautics and Space Administration under NASA contract NAS119480 while the rst author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001. A SIMPLE PARALLEL PREFIX ALGORITHM FOR COMPACT FINITE-DIFFERENCE SCHEMES