NAG Library Function Document
nag_zungrq (f08cwc)
1
Purpose
nag_zungrq (f08cwc) generates all or part of the complex
by
unitary matrix
from an
factorization computed by
nag_zgerqf (f08cvc).
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zungrq (Nag_OrderType order,
Integer m,
Integer n,
Integer k,
Complex a[],
Integer pda,
const Complex tau[],
NagError *fail) |
|
3
Description
nag_zungrq (f08cwc) is intended to be used following a call to
nag_zgerqf (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_zungrq(order,n,n,p,a,pda,tau,info)
(note that the matrix
must have at least
rows), or its trailing
rows as:
nag_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
nag_zgerqf (f08cvc) followed by
nag_zungrq (f08cwc) can be used to orthogonalize the rows of
.
The information returned by
nag_zgerqf (f08cvc) also yields the
factorization of the trailing
rows of
, where
. The unitary matrix arising from this factorization can be computed by:
nag_zungrq(order,n,n,k,a,pda,tau,info)
or its leading
columns by:
nag_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
http://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_OrderTypeInput
-
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: , the number of rows of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of columns of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
- 5:
– ComplexInput/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
nag_zgerqf (f08cvc).
On exit: the
by
matrix
.
If , the th element of the matrix is stored in .
If , the th element of the matrix is stored in .
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
- 7:
– const ComplexInput
-
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
nag_zgerqf (f08cvc).
- 8:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation 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
nag_zungrq (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
nag_dorgrq (f08cjc).
10
Example
This example generates the first four rows of the matrix
of the
factorization of
as returned by
nag_zgerqf (f08cvc), where
10.1
Program Text
Program Text (f08cwce.c)
10.2
Program Data
Program Data (f08cwce.d)
10.3
Program Results
Program Results (f08cwce.r)