NAG Library Routine Document
F08CFF (DORGQL)
1 Purpose
F08CFF (DORGQL) generates all or part of the real
m by
m orthogonal matrix
Q from a
QL factorization computed by
F08CEF (DGEQLF).
2 Specification
INTEGER |
M, N, K, LDA, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name dorgql.
3 Description
F08CFF (DORGQL) is intended to be used after a call to
F08CEF (DGEQLF), which performs a
QL factorization of a real matrix
A. The orthogonal matrix
Q is represented as a product of elementary reflectors.
This routine may be used to generate Q explicitly as a square matrix, or to form only its trailing columns.
Usually
Q is determined from the
QL factorization of an
m by
p matrix
A with
m≥p. The whole of
Q may be computed by:
CALL DORGQL(M,M,P,A,LDA,TAU,WORK,LWORK,INFO)
(note that the array
A must have at least
m columns) or its trailing
p columns by:
CALL DORGQL(M,P,P,A,LDA,TAU,WORK,LWORK,INFO)
The columns of
Q returned by the last call form an orthonormal basis for the space spanned by the columns of
A; thus
F08CEF (DGEQLF) followed by F08CFF (DORGQL) can be used to orthogonalize the columns of
A.
The information returned by
F08CEF (DGEQLF) also yields the
QL factorization of the trailing
k columns of
A, where
k<p. The orthogonal matrix arising from this factorization can be computed by:
CALL DORGQL(M,M,K,A,LDA,TAU,WORK,LWORK,INFO)
or its trailing
k columns by:
CALL DORGQL(M,K,K,A,LDA,TAU,WORK,LWORK,INFO)
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: M – INTEGERInput
On entry: m, the number of rows of the matrix Q.
Constraint:
M≥0.
- 2: N – INTEGERInput
On entry: n, the number of columns of the matrix Q.
Constraint:
M≥N≥0.
- 3: K – INTEGERInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint:
N≥K≥0.
- 4: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: details of the vectors which define the elementary reflectors, as returned by
F08CEF (DGEQLF).
On exit: the m by n matrix Q.
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08CFF (DORGQL) is called.
Constraint:
LDA≥max1,M.
- 6: TAU(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
TAU
must be at least
max1,K.
On entry: further details of the elementary reflectors, as returned by
F08CEF (DGEQLF).
- 7: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 8: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08CFF (DORGQL) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Suggested value:
for optimal performance, LWORK≥N×nb, where nb is the optimal block size.
Constraint:
LWORK≥max1,N.
- 9: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
The computed matrix
Q differs from an exactly orthogonal matrix by a matrix
E such that
where
ε is the
machine precision.
8 Further Comments
The total number of floating point operations is approximately
4mnk-2
m+n
k2
+
43
k3
; when n=k, the number is approximately
23
n2
3m-n
.
The complex analogue of this routine is
F08CTF (ZUNGQL).
9 Example
This example generates the first four columns of the matrix
Q of the
QL factorization of
A as returned by
F08CEF (DGEQLF), where
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
9.1 Program Text
Program Text (f08cffe.f90)
9.2 Program Data
Program Data (f08cffe.d)
9.3 Program Results
Program Results (f08cffe.r)