NAG Library Routine Document
F08AJF (DORGLQ)
1 Purpose
F08AJF (DORGLQ) generates all or part of the real orthogonal matrix
Q from an
LQ factorization computed by
F08AHF (DGELQF).
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 dorglq.
3 Description
F08AJF (DORGLQ) is intended to be used after a call to
F08AHF (DGELQF), which performs an
LQ 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 rows.
Usually
Q is determined from the
LQ factorization of a
p by
n matrix
A with
p≤n. The whole of
Q may be computed by:
CALL DORGLQ(N,N,P,A,LDA,TAU,WORK,LWORK,INFO)
(note that the array
A must have at least
n rows) or its leading
p rows by:
CALL DORGLQ(P,N,P,A,LDA,TAU,WORK,LWORK,INFO)
The rows of
Q returned by the last call form an orthonormal basis for the space spanned by the rows of
A; thus
F08AHF (DGELQF) followed by F08AJF (DORGLQ) can be used to orthogonalize the rows of
A.
The information returned by the
LQ factorization routines also yields the
LQ factorization of the leading
k rows of
A, where
k<p. The orthogonal matrix arising from this factorization can be computed by:
CALL DORGLQ(N,N,K,A,LDA,TAU,WORK,LWORK,INFO)
or its leading
k rows by:
CALL DORGLQ(K,N,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 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:
N≥M.
- 3: K – INTEGERInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint:
M≥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
F08AHF (DGELQF).
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 F08AJF (DORGLQ) 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
F08AHF (DGELQF).
- 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 F08AJF (DORGLQ) 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≥M×nb, where nb is the optimal block size.
Constraint:
LWORK≥max1,M 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 m=k, the number is approximately
23
m2
3n-m
.
The complex analogue of this routine is
F08AWF (ZUNGLQ).
9 Example
This example forms the leading
4 rows of the orthogonal matrix
Q from the
LQ factorization of the matrix
A, where
The rows of
Q form an orthonormal basis for the space spanned by the rows of
A.
9.1 Program Text
Program Text (f08ajfe.f90)
9.2 Program Data
Program Data (f08ajfe.d)
9.3 Program Results
Program Results (f08ajfe.r)