The stable parallel solution of general narrow banded linear systems

A list of technical reports, including some abstracts and copies of some full reports may be found at: A parallel architecture for query processing over a terabyte of text. June 1996. Abstract. We propose a stable algorithm for the parallel solution of banded and periodically banded linear systems. While most of the known parallel algorithms are stable only for symmetric positive deenite or diagonally dominant systems, the new algorithm incorporates pivoting without sacriicing eeciency. The principle ingredient of the algorithm is a bidiagonal cyclic reduction that admits pivoting. We report on numerical experiments conducted on various multiprocessor computers. 1. Introduction Methods will be discussed for the solution of linear systems Ax = b (1.1) with n unknowns and a banded matrix A. The system matrix A has upper band-width k u if ij = 0 for j > i + k u and lower bandwidth k l if ij = 0 for i > j + k l. If k u + k l is small compared to n then A is said to be narrow banded. This is the case which shall be considered here. Frequently occurring problems include bidiagonal systems (k u + k l = 1), tridiagonal systems (k u + k l = 2) and systems obtained from the discretization of 2-dimensional partial diierential equations (k u + k l =