A new quadrant interlocking factorization for parallel solution of tridiagonal linear systems

Abstract

We introduce a new quadrant interlocking factorization (Q.I.F.) for use with the partition method for the solution of tridiagonal linear systems. A given linear system of size N is partitioned into r blocks each of size n (N = rn n even). The new factorization WZ of the coefficient matrix in each block has the properties that the vector is invariant under the transformation W and the solution process with coefficient matrix Z proceeds from the first and the last unknowns towards the middle ones. These properties of the new factorization help us uncouple the partitioned systems for parallel execution once a core system of order 2r is solved. The scalar count for the resulting algorithm is O(17N) which is the count for cyclic reduction and the partition method of Wang [9]. We also note that for the parallel algorithm based on modified Q.I.F. given in Evans et al. [6] the number of processors required for the algorithm is in terms of semibandwidth and the size of the system; on the other hand, the number of processors required for the present algorithm is in terms of the number of blocks into which the system is partitioned.

Keywords:

• New quadrant interlocking factorization
• partition method
• tridiagonal linear system
• parallel solution