NAG Library Routine Document
F08AFF (DORGQR)
1 Purpose
F08AFF (DORGQR) generates all or part of the real orthogonal matrix
Q from a
QR factorization computed by
F08AEF (DGEQRF),
F08BEF (DGEQPF) or
F08BFF (DGEQP3).
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 dorgqr.
3 Description
F08AFF (DORGQR) is intended to be used after a call to
F08AEF (DGEQRF),
F08BEF (DGEQPF) or
F08BFF (DGEQP3).
which perform a
QR 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 leading columns.
Usually
Q is determined from the
QR factorization of an
m by
p matrix
A with
m≥p. The whole of
Q may be computed by:
CALL DORGQR(M,M,P,A,LDA,TAU,WORK,LWORK,INFO)
(note that the array
A must have at least
m columns) or its leading
p columns by:
CALL DORGQR(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
F08AEF (DGEQRF) followed by F08AFF (DORGQR) can be used to orthogonalize the columns of
A.
The information returned by the
QR factorization routines also yields the
QR factorization of the leading
k columns of
A, where
k<p. The orthogonal matrix arising from this factorization can be computed by:
CALL DORGQR(M,M,K,A,LDA,TAU,WORK,LWORK,INFO)
or its leading
k columns by:
CALL DORGQR(M,K,K,A,LDA,TAU,WORK,LWORK,INFO)
4 References
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 order of the orthogonal 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
F08AEF (DGEQRF),
F08BEF (DGEQPF) or
F08BFF (DGEQP3).
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 F08AFF (DORGQR) 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
F08AEF (DGEQRF),
F08BEF (DGEQPF) or
F08BFF (DGEQP3).
- 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 F08AFF (DORGQR) 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 or LWORK=-1.
- 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
F08ATF (ZUNGQR).
9 Example
This example forms the leading
4 columns of the orthogonal matrix
Q from the
QR factorization of the matrix
A, where
The columns of
Q form an orthonormal basis for the space spanned by the columns of
A.
9.1 Program Text
Program Text (f08affe.f90)
9.2 Program Data
Program Data (f08affe.d)
9.3 Program Results
Program Results (f08affe.r)