NAG CL Interface
f08cvc (zgerqf)
1
Purpose
f08cvc computes an RQ factorization of a complex by matrix .
2
Specification
void |
f08cvc (Nag_OrderType order,
Integer m,
Integer n,
Complex a[],
Integer pda,
Complex tau[],
NagError *fail) |
|
The function may be called by the names: f08cvc, nag_lapackeig_zgerqf or nag_zgerqf.
3
Description
f08cvc 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
unitary matrix. If
the factorization is given by
where
is an
by
upper trapezoidal matrix and
is again an
by
unitary matrix. In the case where
the factorization can be expressed as
where
consists of the first
rows of
and
the remaining
rows.
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).
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:
– Complex
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 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
tau, represent the unitary 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:
– Complex
Output
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On exit: the scalar factors of the elementary reflectors.
-
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
f08cvc 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 unitary matrix
f08cvc may be followed by a call to
f08cwc:
nag_lapackeig_zungrq(order, n, n, minmn, a, pda, tau, &fail)
where
,
but note that the first dimension of the array
a must be at least
n, which may be larger than was required by
f08cvc. When
, it is often only the first
rows of
that are required and they may be formed by the call:
nag_lapackeig_zungrq(order, m, n, m, a, pda, tau, c, pdc, &fail)
To apply
to an arbitrary real rectangular matrix
,
f08cvc may be followed by a call to
f08cxc. For example:
nag_lapackeig_zunmrq(Nag_LeftSide, Nag_ConjTrans, n, p, minmn, a, pda, tau, c, pdc, &fail)
forms
, where
is
by
.
The real analogue of this function is
f08chc.
10
Example
This example finds the minimum norm solution to the underdetermined equations
where
and
The solution is obtained by first obtaining an factorization of the matrix .
10.1
Program Text
10.2
Program Data
10.3
Program Results