P is given as a sequence of Householder transformation matrices
the
m-k+1th transformation matrix,
Pk, being used to introduce zeros into the
kth row of
A.
Pk has the form
where
γk is a scalar for which
Reγk=1.0,
ζk is a real scalar,
wk is a
k-1 element vector and
zk is an
n-m element vector.
γk and
uk are chosen to annihilate the elements in the
kth row of
A.
The scalar
γk and the vector
uk are returned in the
kth element of
THETA and in the
kth row of
A, such that
θk, given by
is in
THETAk, the elements of
wk are in
Ak1,…,Akk-1 and the elements of
zk are in
Akm+1,…,Akn. The elements of
R are returned in the upper triangular part of
A.
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1965)
The Algebraic Eigenvalue Problem Oxford University Press, Oxford
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The computed factors
R and
P satisfy the relation
where
ε is the
machine precision (see
X02AJF),
c is a modest function of
m and
n, and
. denotes the spectral (two) norm.
The first
k rows of the unitary matrix
PH can be obtained by calling
F01RKF, which overwrites the
k rows of
PH on the first
k rows of the array
A.
PH is obtained by the call:
IFAIL = 0
CALL F01RKF('Separate',M,N,K,A,LDA,THETA,WORK,IFAIL)
WORK must be a
maxm-1,k-m,1 element array. If
K is larger than
M, then
A must have been declared to have at least
K rows.
Operations involving the matrix
R can readily be performed by the Level 2 BLAS routines
F06SFF (ZTRMV) and
F06SJF (ZTRSV), (see
Chapter F06), but note that no test for near singularity of
R is incorporated into
F06SFF (ZTRMV). If
R is singular, or nearly singular then
F02XUF can be used to determine the singular value decomposition of
R.