NAG Library Routine Document
F08CWF (ZUNGRQ)
1 Purpose
F08CWF (ZUNGRQ) generates all or part of the complex
by
unitary matrix
from an
factorization computed by
F08CVF (ZGERQF).
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 zungrq.
3 Description
F08CWF (ZUNGRQ) is intended to be used following a call to
F08CVF (ZGERQF), which performs an
factorization of a complex matrix
and represents the unitary matrix
as a product of
elementary reflectors of order
.
This routine may be used to generate explicitly as a square matrix, or to form only its trailing rows.
Usually
is determined from the
factorization of a
by
matrix
with
. The whole of
may be computed by:
CALL ZUNGRQ(N,N,P,A,LDA,TAU,WORK,LWORK,INFO)
(note that the matrix
must have at least
rows), or its trailing
rows as:
CALL ZUNGRQ(P,N,P,A,LDA,TAU,WORK,LWORK,INFO)
The rows of
returned by the last call form an orthonormal basis for the space spanned by the rows of
; thus
F08CVF (ZGERQF) followed by F08CWF (ZUNGRQ) can be used to orthogonalize the rows of
.
The information returned by
F08CVF (ZGERQF) also yields the
factorization of the trailing
rows of
, where
. The unitary matrix arising from this factorization can be computed by:
CALL ZUNGRQ(N,N,K,A,LDA,TAU,WORK,LWORK,INFO)
or its leading
columns by:
CALL ZUNGRQ(K,N,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: , the number of rows of the matrix .
Constraint:
.
- 2: N – INTEGERInput
On entry: , the number of columns of the matrix .
Constraint:
.
- 3: K – INTEGERInput
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
- 4: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: details of the vectors which define the elementary reflectors, as returned by
F08CVF (ZGERQF).
On exit: the by matrix .
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08CWF (ZUNGRQ) is called.
Constraint:
.
- 6: TAU() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
TAU
must be at least
.
On entry:
must contain the scalar factor of the elementary reflector
, as returned by
F08CVF (ZGERQF).
- 7: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
, the real part of
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 F08CWF (ZUNGRQ) is called.
If
, 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, , where is the optimal block size.
Constraint:
or .
- 9: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
The computed matrix
differs from an exactly unitary matrix by a matrix
such that
and
is the
machine precision.
The total number of floating point operations is approximately ; when this becomes .
The real analogue of this routine is
F08CJF (DORGRQ).
9 Example
This example generates the first four rows of the matrix
of the
factorization of
as returned by
F08CVF (ZGERQF), where
Note that the block size (NB) of 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 (f08cwfe.f90)
9.2 Program Data
Program Data (f08cwfe.d)
9.3 Program Results
Program Results (f08cwfe.r)