NAG Library Routine Document
F08CHF (DGERQF)
1 Purpose
F08CHF (DGERQF) computes an RQ factorization of a real m by n matrix A.
2 Specification
INTEGER |
M, N, LDA, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name dgerqf.
3 Description
F08CHF (DGERQF) 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 orthogonal 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 orthogonal 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/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: 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,*) – REAL (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:mn-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 orthogonal 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 F08CHF (DGERQF) is called.
Constraint:
LDA≥max1,M.
- 5: TAU(*) – REAL (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) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
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 F08CHF (DGERQF) 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 orthogonal matrix
Q F08CHF (DGERQF) may be followed by a call to
F08CJF (DORGRQ):
CALL DORGRQ(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 F08CHF (DGERQF). 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 DORGRQ(M,N,M,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an arbitrary real rectangular matrix
C, F08CHF (DGERQF) may be followed by a call to
F08CKF (DORMRQ). For example:
CALL DORMRQ('Left','Transpose',N,P,MIN(M,N),A,LDA,TAU,C,LDC, &
WORK,LWORK,INFO)
forms
C=QTC, where
C is
n by
p.
The complex analogue of this routine is
F08CVF (ZGERQF).
9 Example
This example finds the minimum norm solution to the underdetermined equations
where
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 (f08chfe.f90)
9.2 Program Data
Program Data (f08chfe.d)
9.3 Program Results
Program Results (f08chfe.r)