NAG Library Routine Document
F08ATF (ZUNGQR)
1 Purpose
F08ATF (ZUNGQR) generates all or part of the complex unitary matrix
Q from a
QR factorization computed by
F08ASF (ZGEQRF),
F08BSF (ZGEQPF) or
F08BTF (ZGEQP3).
2 Specification
INTEGER |
M, N, K, LDA, LWORK, INFO |
COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name zungqr.
3 Description
F08ATF (ZUNGQR) is intended to be used after a call to
F08ASF (ZGEQRF),
F08BSF (ZGEQPF) or
F08BTF (ZGEQP3), which perform a
QR factorization of a complex matrix
A. The unitary 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 ZUNGQR(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 ZUNGQR(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
F08ASF (ZGEQRF) followed by F08ATF (ZUNGQR) 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 unitary matrix arising from this factorization can be computed by:
CALL ZUNGQR(M,M,K,A,LDA,TAU,WORK,LWORK,INFO)
or its leading
k columns by:
CALL ZUNGQR(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 unitary 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,*) – COMPLEX (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
F08ASF (ZGEQRF),
F08BSF (ZGEQPF) or
F08BTF (ZGEQP3).
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 F08ATF (ZUNGQR) is called.
Constraint:
LDA≥max1,M.
- 6: TAU(*) – COMPLEX (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
F08ASF (ZGEQRF),
F08BSF (ZGEQPF) or
F08BTF (ZGEQP3).
- 7: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
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 F08ATF (ZUNGQR) 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 unitary matrix by a matrix
E such that
where
ε is the
machine precision.
8 Further Comments
The total number of real floating point operations is approximately
16mnk-8
m+n
k2
+
163
k3
; when n=k, the number is approximately
83
n2
3m-n
.
The real analogue of this routine is
F08AFF (DORGQR).
9 Example
This example forms the leading
4 columns of the unitary 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 (f08atfe.f90)
9.2 Program Data
Program Data (f08atfe.d)
9.3 Program Results
Program Results (f08atfe.r)