On block Householder algorithms for the reduction of a matrix to Hessenberg form

Abstract

A block algorithm is presented for the Householder reduction of a matrix to Hessenberg form using the Bischof-Van Loan expression of a product of elementary matrices. Results of performance measurements on an IBM 3090 VG include a comparison with an alternate formulation considered for LAPACK. The algorithms based on the straightforward application of the Bischof-Van Loan formulations consistently appear to produce the best performance in all experiments conducted. It is likely that such behavior would be observed with other machines, but this conjecture remains to be tested. The algorithmic derivations presented are general enough to apply to other computational schemes based on similarity transformations, including those for the solution of the Hessenberg eigenvalue problem (QR).<>