NAG CL Interface
f08avc (zgelqf)
1
Purpose
f08avc computes the factorization of a complex by matrix.
2
Specification
void |
f08avc (Nag_OrderType order,
Integer m,
Integer n,
Complex a[],
Integer pda,
Complex tau[],
NagError *fail) |
|
The function may be called by the names: f08avc, nag_lapackeig_zgelqf or nag_zgelqf.
3
Description
f08avc forms the factorization of an arbitrary rectangular complex by matrix. No pivoting is performed.
If
, the factorization is given by:
where
is an
by
lower triangular matrix (with real diagonal elements) and
is an
by
unitary matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
consists of the first
rows of
, and
the remaining
rows.
If
,
is trapezoidal, and the factorization can be written
where
is lower triangular and
is rectangular.
The
factorization of
is essentially the same as the
factorization of
, since
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
rows of the array
a represents an
factorization of the first
rows of the original matrix
.
4
References
None.
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
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: if
, the elements above the diagonal are overwritten by details of the unitary matrix
and the lower triangle is overwritten by the corresponding elements of the
by
lower triangular matrix
.
If , the strictly upper triangular part is overwritten by details of the unitary matrix and the remaining elements are overwritten by the corresponding elements of the by lower trapezoidal matrix .
The diagonal elements of are real.
-
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: further details of the unitary matrix .
-
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
f08avc 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 real floating-point operations is approximately if or if .
To form the unitary matrix
f08avc may be followed by a call to
f08awc:
nag_lapackeig_zunglq(order,n,n,MIN(m,n),&a,pda,tau,&fail)
but note that the first dimension of the array
a, specified by the argument
pda, must be at least
n, which may be larger than was required by
f08avc.
When
, it is often only the first
rows of
that are required, and they may be formed by the call:
nag_lapackeig_zunglq(order,m,n,m,&a,pda,tau,&fail)
To apply
to an arbitrary complex rectangular matrix
,
f08avc may be followed by a call to
f08axc. For example,
nag_lapackeig_zunmlq(order,Nag_LeftSide,Nag_ConjTrans,m,p,MIN(m,n),&a,pda,
tau,&c,pdc,&fail)
forms the matrix product , where is by .
The real analogue of this function is
f08ahc.
10
Example
This example finds the minimum norm solutions of the under-determined systems of linear equations
where
and
are the columns of the matrix
,
and
10.1
Program Text
10.2
Program Data
10.3
Program Results