A MULTILEVEL PARALLEL SOLVER FOR BLOCK TRIDIAGONAL AND BANDED LINEAR-SYSTEMS

This paper describes an efficient algorithm for the parallel solution of systems of linear equations with a block tridiagonal coefficient matrix. The algorithm comprises a multilevel LU-factorization based on block cyclic reduction and a corresponding solution algorithm. The paper includes a general presentation of the parallel multilevel LU-factorization and solution algorithms, but the main emphasis is on implementation principles for a message passing computer with hypercube topology. Problem partitioning, processor allocation and communication requirement are discussed for the general block tridiagonal algorithm. Band matrices can be cast into block tridiagonal form, and this special but important problem is dealt with in detail. It is demonstrated how the efficiency of the general block tridiagonal multilevel algorithm can be improved by introducing the equivalent of two-way Gaussian elimination for the first and the last partitioning and by carefully balancing the load of the processors. The presentation of the multilevel band solver is accompanied by detailed complexity analyses. The properties of the parallel band solver were evaluated by implementing the algorithm on an Intel iPSC hypercube parallel computer and solving a larger number of banded linear equations using 2 to 32 processors. The results of the evaluation include speed-up over a sequential processor, and the measure values are in good agreement with the theoretical values resulting from complexity analysis. It is found that the maximum asymptotic speed-up of the multilevel LU-factorization using p processors and load balancing is approximated well by the expression (p +6)/4. Finally, the multilevel parallel solver is compared with solvers based on row and column interleaved organization. © 1990.