On a Block Implementation of Hessenberg Multishift QR Iteration

The usual QR algorithm for finding the eigenvalues of a Hessenberg matrix H is based on vector-vector operations, e.g. adding a multiple of one row to another. The opportunities for parallelism in such an algorithm are limited. In this report, we describe a reorganization of the QR algorithm to permit either matrix-vector or matrix-matrix operations to be performed, both of which yield more efficient implementations on vector and parallel machines. The idea is to chase a k by k bulge rather than a 1 by 1 or 2 by 2 bulge as in the standard QR algorithm. We report our preliminary numerical experiments on the CONVEX C-1 and CYBER 205 vector machines.