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QR Factorization
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 [90]:
where V is an nbyb 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 3.10
summarizes results obtained with the LAPACK routine DGEQRF.
Table 3.10:
Speed in megaflops of DGEQRF for square matrices of order n

No. of 
Block 
Values of n 

processors 
size 
100 
1000 
Dec Alpha Miata 
1 
28 
141 
363 
Compaq AlphaServer DS20 
1 
28 
326 
444 
IBM Power 3 
1 
32 
244 
559 
IBM PowerPC 
1 
52 
45 
127 
Intel Pentium II 
1 
40 
113 
250 
Intel Pentium III 
1 
40 
135 
297 
SGI Origin 2000 
1 
64 
173 
451 
SGI Origin 2000 
4 
64 
55 
766 
Sun Ultra 2 
1 
64 
20 
230 
Sun Enterprise 450 
1 
64 
48 
329 
Next: Eigenvalue Problems
Up: Examples of Block Algorithms
Previous: Factorizations for Solving Linear
Contents
Index
Susan Blackford
19991001