NAG Library Routine Document
F08CVF (ZGERQF)
1 Purpose
F08CVF (ZGERQF) computes an RQ factorization of a complex m by n matrix A.
2 Specification
| INTEGER |
M, N, LDA, LWORK, INFO |
| COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name zgerqf.
3 Description
F08CVF (ZGERQF) forms the
RQ factorization of an arbitrary rectangular real
m by
n matrix. If
m≤n, the factorization is given by
where
R is an
m by
m lower triangular matrix and
Q is an
n by
n unitary matrix. If
m>n the factorization is given by
where
R is an
m by
n upper trapezoidal matrix and
Q is again an
n by
n unitary matrix. In the case where
m<n the factorization can be expressed as
where
Q1 consists of the first
n-m rows of
Q and
Q2 the remaining
m rows.
The matrix
Q is not formed explicitly, but is represented as a product of
minm,n elementary reflectors (see the
F08 Chapter Introduction for details). Routines are provided to work with
Q in this representation (see
Section 8).
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/lugGolub 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 A.
Constraint:
M≥0.
- 2: N – INTEGERInput
-
On entry:
n, the number of columns of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the m by n matrix A.
On exit: if
m≤n, the upper triangle of the subarray
A1:m,n-m+1:n contains the
m by
m upper triangular matrix
R.
If
m≥n, the elements on and above the
m-nth subdiagonal contain the
m by
n upper trapezoidal matrix
R; the remaining elements, with the array
TAU, represent the
unitary
matrix
Q as a product of
minm,n elementary reflectors (see
Section 3.3.6 in the F08 Chapter Introduction).
- 4: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08CVF (ZGERQF) is called.
Constraint:
LDA≥max1,M.
- 5: TAU(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAU
must be at least
max1,minM,N.
On exit: the scalar factors of the elementary reflectors.
- 6: 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.
- 7: LWORK – INTEGERInput
-
On entry:
the dimension of the array
WORK as declared in the (sub)program from which F08CVF (ZGERQF) 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.
- 8: 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 factorization is the exact factorization of a nearby matrix
A+E, where
and
ε is the
machine precision.
8 Further Comments
The total number of floating point operations is approximately 23m23n-m if m≤n, or 23n23m-n if m>n.
To form the unitary matrix
Q F08CVF (ZGERQF) may be followed by a call to
F08CWF (ZUNGRQ):
CALL ZUNGRQ(N,N,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)
but note that the first dimension of the array
A must be at least
N, which may be larger than was required by F08CVF (ZGERQF). When
m≤n, it is often only the first
m rows of
Q that are required and they may be formed by the call:
CALL ZUNGRQ(M,N,M,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an arbitrary real rectangular matrix
C, F08CVF (ZGERQF) may be followed by a call to
F08CXF (ZUNMRQ). For example:
CALL ZUNMRQ('Left','C',N,P,MIN(M,N),A,LDA,TAU,C,LDC, &
WORK,LWORK,INFO)
forms
C=QHC, where
C is
n by
p.
The real analogue of this routine is
F08CHF (DGERQF).
9 Example
This example finds the minimum norm solution to the underdetermined equations
where
and
The solution is obtained by first obtaining an RQ factorization of the matrix A.
Note that the block size (NB) of 64 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 (f08cvfe.f90)
9.2 Program Data
Program Data (f08cvfe.d)
9.3 Program Results
Program Results (f08cvfe.r)