Scalable Stable Solvers for Non-symmetric Narrow-Banded Linear Systems

Banded linear systems with large bandwidths can be solved by similar methods as full linear systems. In particular, parallel algorithms based on torus-wrap mapping and Gaussian elimination with partial pivoting have been used with success. These algorithms are not suitable, however , if the bandwidth is small, say, between 1 and 100. As the bandwidth limits the amount of parallelism available at any elimination step severe load imbalance results if large numbers of processors are used for narrow banded systems. Most of the parallel algorithms suggested so far for solving narrow banded systems do require some sort of diagonal dominance for stability reasons. Thus they can break down or, worse, they can generate results with very large errors. New algorithms which have similar properties as the partial pivoting algorithm for uniprocessors have been studied in 3, 4]. However, in order to achieve stability, computational overhead is required as in the uniprocessor case. As it is a priori often hard to determine if pivoting is needed it is suggested to use partial pivoting in any case. (Notice that this is the strategy of the dense linear solvers in LAPACK.) Of course, this is not the optimal choice if pivoting is not needed. The loss in eeciency is studied here for the case of narrow banded linear systems. Practical experiments have been carried out on the Fujitsu AP1000 and on the Intel Paragon.