Parallelization of pipelined algorithms for sets of linear banded systems

Solution of independent sets of linear banded systems is a core part of implicit numerical algorithms. In this study we propose a novel pipelined Thomas algorithm with low parallelization penalty. We introduce two-step pipelined algorithms (PAs) formally and show that the idle processor time is invariant with respect to the order of backward and forward steps, Therefore. the parallelization efficiency of the PA cannot be improved directly. However. the processor idle time can be used if some Lines have been computed by the time processors become idle. We develop the immediate backward pipelined Thomas algorithm (IB-PTA). The backward step is computed immediately after the forward step has been completed for the first portion of lines. The advantage of the IB-PTA over the basic PTA is the presence of solved lines, which are available for other computations, by the time processors become idle. Implementation of the IB-PTA is based on a proposed static processor schedule that switches between forward and backward computations and controls communication between processors, Computations are performed on the Gray T3E MIMD computer. Combination of the proposed IB-PTA with the "burn from two ends" algorithm shows low; parallelization penalty. (C) 1999 Academic Press.