NAG Library Routine Document
F08AWF (ZUNGLQ)
1 Purpose
F08AWF (ZUNGLQ) generates all or part of the complex unitary matrix
from an
factorization computed by
F08AVF (ZGELQF).
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 zunglq.
3 Description
F08AWF (ZUNGLQ) is intended to be used after a call to
F08AVF (ZGELQF), which performs an
factorization of a complex matrix
. The unitary matrix
is represented as a product of elementary reflectors.
This routine may be used to generate explicitly as a square matrix, or to form only its leading rows.
Usually
is determined from the
factorization of a
by
matrix
with
. The whole of
may be computed by:
CALL ZUNGLQ(N,N,P,A,LDA,TAU,WORK,LWORK,INFO)
(note that the array
A must have at least
rows) or its leading
rows by:
CALL ZUNGLQ(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
F08AVF (ZGELQF) followed by F08AWF (ZUNGLQ) can be used to orthogonalize the rows of
.
The information returned by the
factorization routines also yields the
factorization of the leading
rows of
, where
. The unitary matrix arising from this factorization can be computed by:
CALL ZUNGLQ(N,N,K,A,LDA,TAU,WORK,LWORK,INFO)
or its leading
rows by:
CALL ZUNGLQ(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: , 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
F08AVF (ZGELQF).
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 F08AWF (ZUNGLQ) is called.
Constraint:
.
- 6: TAU() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
TAU
must be at least
.
On entry: further details of the elementary reflectors, as returned by
F08AVF (ZGELQF).
- 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 F08AWF (ZUNGLQ) 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
where
is the
machine precision.
The total number of real floating point operations is approximately ; when , the number is approximately .
The real analogue of this routine is
F08AJF (DORGLQ).
9 Example
This example forms the leading
rows of the unitary matrix
from the
factorization of the matrix
, where
The rows of
form an orthonormal basis for the space spanned by the rows of
.
9.1 Program Text
Program Text (f08awfe.f90)
9.2 Program Data
Program Data (f08awfe.d)
9.3 Program Results
Program Results (f08awfe.r)