NAG CL Interface
f08cwc (zungrq)
1
Purpose
f08cwc generates all or part of the complex
by
unitary matrix
from an
factorization computed by
f08cvc.
2
Specification
void |
f08cwc (Nag_OrderType order,
Integer m,
Integer n,
Integer k,
Complex a[],
Integer pda,
const Complex tau[],
NagError *fail) |
|
The function may be called by the names: f08cwc, nag_lapackeig_zungrq or nag_zungrq.
3
Description
f08cwc is intended to be used following a call to
f08cvc, which performs an
factorization of a complex matrix
and represents the unitary matrix
as a product of
elementary reflectors of order
.
This function may be used to generate explicitly as a square matrix, or to form only its trailing rows.
Usually
is determined from the
factorization of a
by
matrix
with
. The whole of
may be computed by:
nag_lapackeig_zungrq(order,n,n,p,a,pda,tau,info)
(note that the matrix
must have at least
rows), or its trailing
rows as:
nag_lapackeig_zungrq(order,p,n,p,a,pda,tau,info)
The rows of
returned by the last call form an orthonormal basis for the space spanned by the rows of
; thus
f08cvc followed by
f08cwc can be used to orthogonalize the rows of
.
The information returned by
f08cvc also yields the
factorization of the trailing
rows of
, where
. The unitary matrix arising from this factorization can be computed by:
nag_lapackeig_zungrq(order,n,n,k,a,pda,tau,info)
or its leading
columns by:
nag_lapackeig_zungrq(order,k,n,k,a,pda,tau,info)
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:
– Complex
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
f08cvc.
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 Complex
Input
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On entry:
must contain the scalar factor of the elementary reflector
, as returned by
f08cvc.
-
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 unitary matrix by a matrix
such that
and
is the
machine precision.
8
Parallelism and Performance
f08cwc 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 this becomes .
The real analogue of this function is
f08cjc.
10
Example
This example generates the first four rows of the matrix
of the
factorization of
as returned by
f08cvc, where
10.1
Program Text
10.2
Program Data
10.3
Program Results