Efficient implementation of Jacobi algorithms and Jacobi sets on distributed memory architectures

One-sided methods for implementing Jacobi diagonalization algorithms have been recently proposed for both distributed memory and vector machines. These methods are naturally well suited to distributed memory and vector architectures because of their inherent parallelism and their abundance of vector operations. Also, one-sided methods require substantially less message passing than the two-sided methods, and thus can achieve higher efficiency. We describe in detail the use of the one-sided Jacobi rotation as opposed to the rotation used in the “Hestenes” algorithm; we perceive this difference to have been widely misunderstood. Furthermore the one-sided algorithm generalizes to other problems such as the nonsymmetric eigenvalue problem while the Hestenes algorithm does not. We discuss two new implementations for Jacobi sets for a ring connected array of processors and show their isomorphism to the round-robin ordering. Moreover, we show that two implementations produce Jacobi sets in identical orders up to a relabeling. These orderings are optimal in the sense that they complete each sweep in a minimum number of stages with minimal communication. We present implementation results of one-sided Jacobi algorithms using these orderings on the NCUBE/seven hypercube as well as the Intel iPSC/2 hypercube. Finally, we mention how other orderings, and can be, implemented. The number of nonisomorphic Jacobi sets has recently been shown to become infinite with increasing n.