The traditional algorithm for `QR`
factorization is based on the use of
elementary Householder
matrices of the general form

where `v` is a column vector and

is a scalar. This leads to an algorithm with very good vector performance, especially if coded to use Level 2 BLAS.

The key to developing a block form of this algorithm is to represent a product
of `b` elementary Householder matrices of order `n` as a block
form of a Householder matrix . This can be done in various ways.
LAPACK uses the following form [68]:

where `V` is an `n`-by-`n` matrix whose columns are the individual vectors

associated with the Householder matrices

, and `T` is an upper triangular matrix of order `b`.
Extra work is required to compute the elements of `T`, but once again this
is compensated for by the greater speed of applying the block form.
Table 3.6
summarizes results obtained with the LAPACK routine SGEQRF /DGEQRF .

------------------------------------------------- No. of Block Values of n processors size 100 1000 ------------------------------------------------- CONVEX C-4640 1 64 81 521 CONVEX C-4640 4 64 94 1204 CRAY C90 1 128 384 859 CRAY C90 16 128 390 7641 DEC 3000-500X Alpha 1 32 50 86 IBM POWER2 model 590 1 32 108 208 IBM RISC Sys/6000-550 1 32 30 61 SGI POWER CHALLENGE 1 64 61 190 SGI POWER CHALLENGE 4 64 39 342 -------------------------------------------------

Tue Nov 29 14:03:33 EST 1994