More and more scientists and engineers are becoming interested in using supercomputers. Earlier barriers to using these machines are disappearing as software for their use improves. Meanwhile, new parallel supercomputer architectures are emerging that may provide rapid growth in performance. These systems may use a large number of processors with an intricate memory system that is both parallel and hierarchical; they will require even more advanced software. Compilers that restructure user programs to exploit the machine organization seem to be essential. A wide range of algorithms and applications is being developed in an effort to provide high parallel processing performance in many fields. The Cedar supercomputer, presently operating with eight processors in parallel, uses advanced system and applications software developed at the University of Illinois during the past 12 years. This software should allow the number of processors in Cedar to be doubled annually, providing rapid performance advances in the next decade.
We present an algorithm for solving banded positive defimte linear systems on a multiprocessor computer whose number of processors p is much less than the order of the system n. Assuming that the banded matrix, of bandwidth 2m + 1, is stored in the global memory by diagonals as several onedlmensmnal arrays, we consider the time required by several alignment networks for allocating the appropriate data to the local memory of each processor. We demonstrate that the time required in this preprocessmg stage does not exceed that required by the algorithm provided we use a shuffle exchange, a plpelmed shuffle exchange, or a crossbar switch. Once the data are allocated in the local memories, the algorithm requires only a "nearest neighbor" alignment network to achieve a total time of O(m'~n/p). The total cost of the algorithm is minimized when p ~ ~.
In this paper, we give an overview of the Cedar mutliprocessor and present recent performance results. These include the performance of some computational kernels and the Perfect Benchmarks@ . We also present a methodology f o r judging parallel system performance and apply this methodology to Cedar, Cray YMP-8, and Thinking Machines CM-5.
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