NAG FL Interface
f08cjf (dorgrq)
1
Purpose
f08cjf generates all or part of the real
by
orthogonal matrix
from an
factorization computed by
f08chf.
2
Specification
Fortran Interface
Integer, Intent (In) |
:: |
m, n, k, lda, lwork |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (In) |
:: |
tau(*) |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
|
C Header Interface
#include <nag.h>
void |
f08cjf_ (const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08cjf_ (const Integer &m, const Integer &n, const Integer &k, double a[], const Integer &lda, const double tau[], double work[], const Integer &lwork, Integer &info) |
}
|
The routine may be called by the names f08cjf, nagf_lapackeig_dorgrq or its LAPACK name dorgrq.
3
Description
f08cjf is intended to be used following a call to
f08chf, which performs an
factorization of a real matrix
and represents the orthogonal matrix
as a product of
elementary reflectors of order
.
This routine may be used to generate explicitly as a square matrix, or to form only its trailing rows.
Usually
is determined from the
factorization of a
by
matrix
with
. The whole of
may be computed by:
Call dorgrq(n,n,p,a,lda,tau,work,lwork,info)
(note that the matrix
must have at least
rows), or its trailing
rows as:
Call dorgrq(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
f08chf followed by
f08cjf can be used to orthogonalize the rows of
.
The information returned by
f08chf also yields the
factorization of the trailing
rows of
, where
. The orthogonal matrix arising from this factorization can be computed by:
Call dorgrq(n,n,k,a,lda,tau,work,lwork,info)
or its leading
columns by:
Call dorgrq(k,n,k,a,lda,tau,work,lwork,info)
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input/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
f08chf.
On exit: the by matrix .
-
5:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08cjf is called.
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Input
-
Note: the dimension of the array
tau
must be at least
.
On entry:
must contain the scalar factor of the elementary reflector
, as returned by
f08chf.
-
7:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
8:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08cjf 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:
– 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 matrix
differs from an exactly orthogonal matrix by a matrix
such that
and
is the
machine precision.
8
Parallelism and Performance
f08cjf 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 total number of floating-point operations is approximately ; when this becomes .
The complex analogue of this routine is
f08cwf.
10
Example
This example generates the first four rows of the matrix
of the
factorization of
as returned by
f08chf, where
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