NAG CL Interface
f08cec (dgeqlf)
1
Purpose
f08cec computes a factorization of a real by matrix .
2
Specification
void |
f08cec (Nag_OrderType order,
Integer m,
Integer n,
double a[],
Integer pda,
double tau[],
NagError *fail) |
|
The function may be called by the names: f08cec, nag_lapackeig_dgeqlf or nag_dgeqlf.
3
Description
f08cec forms the factorization of an arbitrary rectangular real by matrix.
If
, the factorization is given by:
where
is an
by
lower triangular matrix and
is an
by
orthogonal matrix. If
the factorization is given by
where
is an
by
lower trapezoidal matrix and
is again an
by
orthogonal matrix. In the case where
the factorization can be expressed as
where
consists of the first
columns of
, and
the remaining
columns.
The matrix
is not formed explicitly but is represented as a product of
elementary reflectors (see
Section 3.4.6 in the
F08 Chapter Introduction for details). Functions are provided to work with
in this representation (see
Section 9).
Note also that for any
, the information returned in the last
columns of the array
a represents a
factorization of the last
columns of the original matrix
.
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:
– double
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the by matrix .
On exit: if
, the lower triangle of the subarray
contains the
by
lower triangular matrix
.
If
, the elements on and below the
th superdiagonal contain the
by
lower trapezoidal matrix
. The remaining elements, with the array
tau, represent the orthogonal matrix
as a product of elementary reflectors (see
Section 3.4.6 in the
F08 Chapter Introduction).
-
5:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
-
6:
– double
Output
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On exit: the scalar factors of the elementary reflectors (see
Section 9).
-
7:
– 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: .
On entry, .
Constraint: .
- NE_INT_2
-
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 factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
8
Parallelism and Performance
f08cec 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 if or if .
To form the orthogonal matrix
f08cec may be followed by a call to
f08cfc:
nag_lapackeig_dorgql(order,m,m,MIN(m,n),&a,pda,tau,&fail)
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
f08cec.
When
, it is often only the first
columns of
that are required, and they may be formed by the call:
nag_lapackeig_dorgql(order,m,n,n,&a,pda,tau,&fail)
To apply
to an arbitrary real rectangular matrix
,
f08cec may be followed by a call to
f08cgc. For example,
nag_lapackeig_dormql(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)
forms
, where
is
by
.
The complex analogue of this function is
f08csc.
10
Example
This example solves the linear least squares problems
for
and
, where
is the
th column of the matrix
,
The solution is obtained by first obtaining a factorization of the matrix .
10.1
Program Text
10.2
Program Data
10.3
Program Results