A Comparison of Parallel Solvers for Diagonally Dominant and General Narrow-Banded Linear Systems II

Abstract

Abstract. We continue the comparison of parallel algorithms for solving diagonally dominant and general narrow-banded linear systems of equations that we started in 2 . The solvers compared are the banded system solvers of ScaLAPACK 6 and those investigated by Arbenz and Hegland 1, 5 . We present the numerical experiments that we conducted on the IBM SP 2.

Incomplete Cyclic Reduction of Banded and Strictly Diagonally Dominant Linear Systems

“…Normally, we would illustrate the cyclic reduction algorithm using explicit block permutations, but here we use "in-place" computations in order to estimate B (1) . Now, let…”

“…We inherited the communication pattern from the ScaLAPACK implementation, but choose the memory layout of the Arbenz-Hegland implementation [1]. In reality we rewrote the ScaLAPACK implementation from scratch inserting a parameter controlling the number of reduction steps and replacing the blocking receive instructions from BLACS with nonblocking ones.…”

“…The ScaLAPACK implementation of block cyclic reduction of narrow banded linear systems is described in detail in the paper by Arbenz, Cleary, Dongarra and Hegland [1].…”

Parallel Solution of Narrow Banded Diagonally Dominant Linear Systems

“…The ScaLAPACK routine PDDBTRF can be used to obtain a factorization of a narrow banded matrix A which is diagonally dominant by rows, [1]. Mathematically, the algorithm is block cyclic reduction applied to a special partitioning of the matrix, which is designed to exploit the banded structure.…”

“…This paper builds on the analysis of the truncated SPIKE algorithm by Mikkelsen and Manguoglu [5] and it requires a good understanding of the routines PDDBTRF/PDDBTRS [2,1] from ScaLAPACK, as well as the work of Heller [3]. The truncated SPIKE algorithm (introduced by Polizzi and Sameh [6,7]) also applies to systems which are banded and strictly diagonally dominant by rows.…”