NAG CL Interface
f08cfc (dorgql)
1
Purpose
f08cfc generates all or part of the real
by
orthogonal matrix
from a
factorization computed by
f08cec.
2
Specification
void |
f08cfc (Nag_OrderType order,
Integer m,
Integer n,
Integer k,
double a[],
Integer pda,
const double tau[],
NagError *fail) |
|
The function may be called by the names: f08cfc, nag_lapackeig_dorgql or nag_dorgql.
3
Description
f08cfc is intended to be used after a call to
f08cec, which performs a
factorization of a real matrix
. The orthogonal matrix
is represented as a product of elementary reflectors.
This function may be used to generate explicitly as a square matrix, or to form only its trailing columns.
Usually
is determined from the
factorization of an
by
matrix
with
. The whole of
may be computed by:
nag_lapackeig_dorgql(order,m,m,p,a,pda,tau,&fail)
(note that the array
a must have at least
columns) or its trailing
columns by:
nag_lapackeig_dorgql(order,m,p,p,a,pda,tau,&fail)
The columns of
returned by the last call form an orthonormal basis for the space spanned by the columns of
; thus
f08cec followed by
f08cfc can be used to orthogonalize the columns of
.
The information returned by
f08cec also yields the
factorization of the trailing
columns of
, where
. The orthogonal matrix arising from this factorization can be computed by:
nag_lapackeig_dorgql(order,m,m,k,a,pda,tau,&fail)
or its trailing
columns by:
nag_lapackeig_dorgql(order,m,k,k,a,pda,tau,&fail)
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:
– Nag_OrderType
Input
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
-
5:
– double
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
On entry: details of the vectors which define the elementary reflectors, as returned by
f08cec.
On exit: the by matrix .
-
6:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
-
7:
– const double
Input
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On entry: further details of the elementary reflectors, as returned by
f08cec.
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The computed matrix
differs from an exactly orthogonal matrix by a matrix
such that
where
is the
machine precision.
8
Parallelism and Performance
f08cfc 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 function. 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 , the number is approximately .
The complex analogue of this function is
f08ctc.
10
Example
This example generates the first four columns of the matrix
of the
factorization of
as returned by
f08cec, where
10.1
Program Text
10.2
Program Data
10.3
Program Results