NAG FL Interface
f08zff (dggrqf)
1
Purpose
f08zff computes a generalized factorization of a real matrix pair , where is an by matrix and is a by matrix.
2
Specification
Fortran Interface
Subroutine f08zff ( |
m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info) |
Integer, Intent (In) |
:: |
m, p, n, lda, ldb, lwork |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
taua(min(m,n)), taub(min(p,n)), work(max(1,lwork)) |
|
C Header Interface
#include <nag.h>
void |
f08zff_ (const Integer *m, const Integer *p, const Integer *n, double a[], const Integer *lda, double taua[], double b[], const Integer *ldb, double taub[], double work[], const Integer *lwork, Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08zff_ (const Integer &m, const Integer &p, const Integer &n, double a[], const Integer &lda, double taua[], double b[], const Integer &ldb, double taub[], double work[], const Integer &lwork, Integer &info) |
}
|
The routine may be called by the names f08zff, nagf_lapackeig_dggrqf or its LAPACK name dggrqf.
3
Description
f08zff forms the generalized
factorization of an
by
matrix
and a
by
matrix
where
is an
by
orthogonal matrix,
is a
by
orthogonal matrix and
and
are of the form
with
or
upper triangular,
with
upper triangular.
In particular, if
is square and nonsingular, the generalized
factorization of
and
implicitly gives the
factorization of
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
https://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 factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of columns of the matrices and .
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the by matrix .
On exit: if
, the upper triangle of the subarray
contains the
by
upper triangular matrix
.
If
, the elements on and above the
th subdiagonal contain the
by
upper trapezoidal matrix
; the remaining elements, with the array
taua, represent the orthogonal matrix
as a product of
elementary reflectors (see
Section 3.3.6 in the
F08 Chapter Introduction).
-
5:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08zff is called.
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Output
-
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix .
-
7:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the by matrix .
On exit: the elements on and above the diagonal of the array contain the
by
upper trapezoidal matrix
(
is upper triangular if
); the elements below the diagonal, with the array
taub, represent the orthogonal matrix
as a product of elementary reflectors (see
Section 3.3.6 in the
F08 Chapter Introduction).
-
8:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08zff is called.
Constraint:
.
-
9:
– Real (Kind=nag_wp) array
Output
-
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix .
-
10:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
11:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08zff 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 for the
factorization of an
by
matrix by
f08chf,
is the optimal
block size for the
factorization of a
by
matrix by
f08aef, and
is the optimal
block size for a call of
f08ckf.
Constraint:
or .
-
12:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
The computed generalized
factorization is the exact factorization for nearby matrices
and
, where
and
is the
machine precision.
8
Parallelism and Performance
f08zff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zff makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The orthogonal matrices
and
may be formed explicitly by calls to
f08cjf and
f08aff respectively.
f08ckf may be used to multiply
by another matrix and
f08agf may be used to multiply
by another matrix.
The complex analogue of this routine is
f08ztf.
10
Example
This example solves the least squares problem
where
The constraints
correspond to
and
.
The solution is obtained by first computing a generalized factorization of the matrix pair . The example illustrates the general solution process.
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results