NAG CL Interface
f08bfc (dgeqp3)
1
Purpose
f08bfc computes the factorization, with column pivoting, of a real by matrix.
2
Specification
void |
f08bfc (Nag_OrderType order,
Integer m,
Integer n,
double a[],
Integer pda,
Integer jpvt[],
double tau[],
NagError *fail) |
|
The function may be called by the names: f08bfc, nag_lapackeig_dgeqp3 or nag_dgeqp3.
3
Description
f08bfc forms the factorization, with column pivoting, of an arbitrary rectangular real by matrix.
If
, the factorization is given by:
where
is an
by
upper triangular matrix,
is an
by
orthogonal matrix and
is an
by
permutation matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
consists of the first
columns of
, and
the remaining
columns.
If
,
is trapezoidal, and the factorization can be written
where
is upper triangular and
is rectangular.
The matrix
is not formed explicitly but is represented as a product of
elementary reflectors (see 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 first
columns of the array
a represents a
factorization of the first
columns of the permuted matrix
.
The function allows specified columns of to be moved to the leading columns of at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the th stage the pivot column is chosen to be the column which maximizes the -norm of elements to over columns to .
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
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: if
, the elements below the diagonal are overwritten by details of the orthogonal matrix
and the upper triangle is overwritten by the corresponding elements of the
by
upper triangular matrix
.
If , the strictly lower triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the by upper trapezoidal matrix .
-
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:
– Integer
Input/Output
-
Note: the dimension,
dim, of the array
jpvt
must be at least
.
On entry: if , the th column of is moved to the beginning of before the decomposition is computed and is fixed in place during the computation. Otherwise, the th column of is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix . More precisely, if , the th column of is moved to become the th column of ; in other words, the columns of are the columns of in the order .
-
7:
– double
Output
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On exit: the scalar factors of the elementary reflectors.
-
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: .
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
f08bfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bfc 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
f08bfc may be followed by a call to
f08afc:
nag_lapackeig_dorgqr(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
f08bfc.
When
, it is often only the first
columns of
that are required, and they may be formed by the call:
nag_lapackeig_dorgqr(order,m,n,n,&a,pda,tau,&fail)
To apply
to an arbitrary real rectangular matrix
,
f08bfc may be followed by a call to
f08agc. For example,
nag_lapackeig_dormqr(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)
forms
, where
is
by
.
To compute a
factorization without column pivoting, use
f08aec.
The complex analogue of this function is
f08btc.
10
Example
This example solves the linear least squares problems
for the basic solutions
and
, where
and
is the
th column of the matrix
. The solution is obtained by first obtaining a
factorization with column pivoting of the matrix
. A tolerance of
is used to estimate the rank of
from the upper triangular factor,
.
10.1
Program Text
10.2
Program Data
10.3
Program Results