NAG Library Routine Document
F08ZFF (DGGRQF)
1 Purpose
F08ZFF (DGGRQF) computes a generalized RQ factorization of a real matrix pair A,B, where A is an m by n matrix and B is a p by n matrix.
2 Specification
SUBROUTINE F08ZFF ( |
M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO) |
INTEGER |
M, P, N, LDA, LDB, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), TAUA(min(M,N)), B(LDB,*), TAUB(min(P,N)), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name dggrqf.
3 Description
F08ZFF (DGGRQF) forms the generalized
RQ factorization of an
m by
n matrix
A and a
p by
n matrix
B
where
Q is an
n by
n orthogonal matrix,
Z is a
p by
p orthogonal matrix and
R and
T are of the form
with
R12 or
R21 upper triangular,
with
T11 upper triangular.
In particular, if
B is square and nonsingular, the generalized
RQ factorization of
A and
B implicitly gives the
RQ factorization of
AB-1 as
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
Anderson E, Bai Z and Dongarra J (1992) Generalized
QR factorization and its applications
Linear Algebra Appl. (Volume 162–164) 243–271
Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model
Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press
Paige C C (1990) Some aspects of generalized
QR factorizations
. In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press
5 Parameters
- 1: M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint:
M≥0.
- 2: P – INTEGERInput
On entry: p, the number of rows of the matrix B.
Constraint:
P≥0.
- 3: N – INTEGERInput
On entry: n, the number of columns of the matrices A and B.
Constraint:
N≥0.
- 4: 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
R12.
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
TAUA, represent the orthogonal matrix
Q as a product of
minm,n elementary reflectors (see
Section 3.3.6 in the F08 Chapter Introduction).
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08ZFF (DGGRQF) is called.
Constraint:
LDA≥max1,M.
- 6: TAUA(minM,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Q.
- 7: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the p by n matrix B.
On exit: the elements on and above the diagonal of the array contain the
minp,n by
n upper trapezoidal matrix
T (
T is upper triangular if
p≥n); the elements below the diagonal, with the array
TAUB, represent the orthogonal matrix
Z as a product of elementary reflectors (see
Section 3.3.6 in the F08 Chapter Introduction).
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08ZFF (DGGRQF) is called.
Constraint:
LDB≥max1,P.
- 9: TAUB(minP,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Z.
- 10: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 11: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08ZFF (DGGRQF) 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≥maxN,M,P×maxnb1,nb2,nb3, where
nb1 is the optimal
block size for the
RQ factorization of an
m by
n matrix by
F08CHF (DGERQF),
nb2 is the optimal
block size for the
QR factorization of a
p by
n matrix by
F08AEF (DGEQRF), and
nb3 is the optimal
block size for a call of
F08CKF (DORMRQ).
Constraint:
LWORK≥max1,N,M,P or LWORK=-1.
- 12: 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 generalized
RQ factorization is the exact factorization for nearby matrices
A+E and
B+F, where
and
ε is the
machine precision.
8 Further Comments
The orthogonal matrices
Q and
Z may be formed explicitly by calls to
F08CJF (DORGRQ) and
F08AFF (DORGQR) respectively.
F08CKF (DORMRQ) may be used to multiply
Q by another matrix and
F08AGF (DORMQR) may be used to multiply
Z by another matrix.
The complex analogue of this routine is
F08ZTF (ZGGRQF).
9 Example
This example solves the linear equality constrained least squares problem
where
The constraints
Bx=d correspond to
x1=x3 and
x2=x4.
The solution is obtained by first computing a generalized RQ factorization of the matrix pair B,A. The example illustrates the general solution process.
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 (f08zffe.f90)
9.2 Program Data
Program Data (f08zffe.d)
9.3 Program Results
Program Results (f08zffe.r)