A Parallel Algorithm for the Reduction of a Nonsymmetric Matrix to Block Upper-Hessenberg Form

In this paper, we present an algorithm for the reduction to block upper-Hessenberg form which can be used to solve the nonsymmetric eigenvalue problem on message-passing multicomputers. On such multicomputers, a nonsymmetric matrix can be distributed across processing nodes logically configured into a two-dimensional mesh using the block-cyclic data distribution. Based on the matrix partitioning and mapping, the algorithm employs both Householder reflectors and Givens rotations within each reduction step. We analyze the arithmetic and communication complexities and describe the implementation details of the algorithm on message-passing multicomputers. We discuss two different implementations synchronous and asynchronous and present performance results on the Intel iPSC/860 and DELTA. We conclude with an evaluation of the algorithm’s communication cost, and suggest areas for further improvement.