A PARALLEL ALGORITHM FOR BANDED LINEAR SYSTEM

Abstract

A direct parallel method called Alternate Quadrant Interlocking Factorization (AQIF); A = WZ, is introduced (Rao, Parallel Algorithms and Applications, 4, 1-20, 1994) for the general solution of the linear system Ax = b. The matrices W and Z are closed under multiplication and inversion. In this paper AQIF is used with partition method for the solution of the banded linear system. The AQIF of the coefficient matrix in each block has the properties that when A is banded with the semibandwidth β, the space generated by ei, en−I+1 1≤i≤β, is invariant under the transformation W, so is invariant under the transformation W −1, where ej denotes n dimensional unit vector with I in jth position and 0's elsewhere and the solution process with the coefficient matrix Z proceeds from the first and last unknowns towards middle ones. These properties of AQIF help us to decouple the partitioned systems for the parallel execution once ‘reduced system’ is solved.