Parallel Algorithms for the AT&TKorbx® System

Abstract

The AT&T KORBX system employs a sophisticated compiler and vector/parallel processors to solve linear-programming problems using Karmarkar-type algorithms. The main computational effort of the Karmarkar-type linear-programming methods involve the repeated solution of large symmetric sets of linear equations. For this reason, algorithms that optimize the KORBX system performance for the solution of sparse and dense sets of linear equations are presented. These algorithms involve both Cholesky factorization and forward-and-backward substitution for the solution of linear equations and exploit data locality, concurrency, and vectorization. In Cholesky factorization, block-operation methods are efficient and instrumental for the parallel solution of the problem. Forward-and-backward solvers involve the solution of triangular sets of linear equations. For the solution of sparse triangular systems of linear equations, our algorithms schedule the operations among the processors and take advantage of the concurrency and vectorization capabilities of the KORBX system multiprocessor.