An Implementation of the Block Householder Method

Abstract

When large matrix problems are treated, the locality of storage reference is very important. Usually higher locality of storage reference is attained by means of block algorithms. This paper introduces an implementation of block Householder transformation based on the block reflector (Schreiber, 1988) or “GGT” representation rather than on the method using “WYT” representations or compact “WYT” or “YTYT”(Bischof, 1993, etc.). This version of block Householder transformation can be regarded as a most natural extension of the original non-blocked Householder transformation, with the matrix elements of the algorithm changed from numbers to small matrices. Thus, an algorithm that uses the non-blocked version of Householder transformation can be converted into the corresponding block algorithm in the most natural manner. To demonstrate the implementation of the Householder method based on the block reflector described in this paper, block tridiagonalization of a dense real symmetric matrix is carried out to calculate the required number of eigenpairs, following the idea of the two-step reduction method(Bischof, 1996, etc.).