OPM: a parallel method to solve banded systems of equations

studied in this report. We can see, in general, that low accuracies will require less overlapping and, in those cases, the algorithm will run faster. As the diagonal dominance increases, the amount of overlapping will be lower and the timings for different accuracies within the same diagonal dominance will get closer and closer. In this report we presented the Overlapped Partitions Method, a method for the solution of banded systems with strictly diagonal dominant coefficient matrices. The method is based on the solution of the independent systems that are obtained after the partition of the coefficient matrix. Some extra equations are added to each independent system in order to control the amount of error genereated by OPM. In the report, a formula is given to decide on the amount of overlapping depending on the maximum error allowed and the diagonal dominance of the system. In order to compare the method with other already existing methods, we studied the case of tridiagonal systems. We compared the method to the Cyclic Reduction and the Tricyclic Reduction algorithms. The tests were made on a processor of the vector multiprocessor Convex C-3480. OPM shows to be faster than CR for the cases tested in the report. For example, OPM is approximately a 20% faster than CR for the case of ρ=0.9 (weakest diagonal dominance tested) and N=50,000, while for the same diagonal dominance and N=100,000, OPM is a 45% faster. OPM also shows to be faster than TR in some of the cases tested here. For matrices with diagonal dominance ρ<0.8, OPM is faster than TR. Only for the case of matrices smaller than 70,000 and ρ=0.8, and for all the cases when ρ=0.9, TR shows to be faster than OPM. TR, though, is an algorithm that will not be comparatively so fast on scalar parallel computers. The reason is that it benefits directly from the features of vector processors as we said before. As a consequence, the implementation of OPM on scalar parallel computers will be compared to CR and other parallel algorithms suitable for this type of computers. There, OPM will have the advantage of not needing communication among processors. Also, if Gaussian elimination is used for the solution of the independent overlapped partitions, OPM will have a lower operation count for most of the diagonal dominances. Consequently, OPM will run faster than CR for most of the cases on scalar parallel …