nag_zunghr (f08ntc) (PDF version)
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NAG Library Function Document
nag_zunghr (f08ntc)
▸
▿
Contents
1
Purpose
2
Specification
3
Description
4
References
5
Arguments
6
Error Indicators and Warnings
7
Accuracy
8
Parallelism and Performance
9
Further Comments
▸
▿
10
Example
10.1
Program Text
10.2
Program Data
10.3
Program Results
1
Purpose
nag_zunghr (f08ntc)
generates the complex unitary matrix
Q
which was determined by
nag_zgehrd (f08nsc)
when reducing a complex general matrix
A
to Hessenberg form.
2
Specification
#include <nag.h>
#include <nagf08.h>
void
nag_zunghr (
Nag_OrderType
order
,
Integer
n
,
Integer
ilo
,
Integer
ihi
,
Complex
a
[],
Integer
pda
,
const Complex
tau
[],
NagError *
fail
)
3
Description
nag_zunghr (f08ntc)
is intended to be used following a call to
nag_zgehrd (f08nsc)
, which reduces a complex general matrix
A
to upper Hessenberg form
H
by a unitary similarity transformation:
A
=
Q
H
Q
H
.
nag_zgehrd (f08nsc)
represents the matrix
Q
as a product of
i
hi
-
i
lo
elementary reflectors. Here
i
lo
and
i
hi
are values determined by
nag_zgebal (f08nvc)
when balancing the matrix; if the matrix has not been balanced,
i
lo
=
1
and
i
hi
=
n
.
This function may be used to generate
Q
explicitly as a square matrix.
Q
has the structure:
Q
=
I
0
0
0
Q
22
0
0
0
I
where
Q
22
occupies rows and columns
i
lo
to
i
hi
.
4
References
Golub G H and Van Loan C F (1996)
Matrix Computations
(3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
1:
order
–
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
order
=
Nag_RowMajor
. 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
:
order
=
Nag_RowMajor
or
Nag_ColMajor
.
2:
n
–
Integer
Input
On entry
:
n
, the order of the matrix
Q
.
Constraint
:
n
≥
0
.
3:
ilo
–
Integer
Input
4:
ihi
–
Integer
Input
On entry
: these
must
be the same arguments
ilo
and
ihi
, respectively, as supplied to
nag_zgehrd (f08nsc)
.
Constraints
:
if
n
>
0
,
1
≤
ilo
≤
ihi
≤
n
;
if
n
=
0
,
ilo
=
1
and
ihi
=
0
.
5:
a
[
dim
]
–
Complex
Input/Output
Note:
the dimension,
dim
, of the array
a
must be at least
max
1
,
pda
×
n
.
On entry
: details of the vectors which define the elementary reflectors, as returned by
nag_zgehrd (f08nsc)
.
On exit
: the
n
by
n
unitary matrix
Q
.
If
order
=
Nag_ColMajor
, the
i
,
j
th element of the matrix is stored in
a
[
j
-
1
×
pda
+
i
-
1
]
.
If
order
=
Nag_RowMajor
, the
i
,
j
th element of the matrix is stored in
a
[
i
-
1
×
pda
+
j
-
1
]
.
6:
pda
–
Integer
Input
On entry
: the stride separating row or column elements (depending on the value of
order
) in the array
a
.
Constraint
:
pda
≥
max
1
,
n
.
7:
tau
[
dim
]
–
const Complex
Input
Note:
the dimension,
dim
, of the array
tau
must be at least
max
1
,
n
-
1
.
On entry
: further details of the elementary reflectors, as returned by
nag_zgehrd (f08nsc)
.
8:
fail
–
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
value
had an illegal value.
NE_INT
On entry,
n
=
value
.
Constraint:
n
≥
0
.
On entry,
pda
=
value
.
Constraint:
pda
>
0
.
NE_INT_2
On entry,
pda
=
value
and
n
=
value
.
Constraint:
pda
≥
max
1
,
n
.
NE_INT_3
On entry,
n
=
value
,
ilo
=
value
and
ihi
=
value
.
Constraint: if
n
>
0
,
1
≤
ilo
≤
ihi
≤
n
;
if
n
=
0
,
ilo
=
1
and
ihi
=
0
.
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
Q
differs from an exactly unitary matrix by a matrix
E
such that
E
2
=
O
ε
,
where
ε
is the
machine precision
.
8
Parallelism and Performance
nag_zunghr (f08ntc)
is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zunghr (f08ntc)
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.
9
Further Comments
The total number of real floating-point operations is approximately
16
3
q
3
, where
q
=
i
hi
-
i
lo
.
The real analogue of this function is
nag_dorghr (f08nfc)
.
10
Example
This example computes the Schur factorization of the matrix
A
, where
A
=
-
3.97
-
5.04
i
-
4.11
+
3.70
i
-
0.34
+
1.01
i
1.29
-
0.86
i
0.34
-
1.50
i
1.52
-
0.43
i
1.88
-
5.38
i
3.36
+
0.65
i
3.31
-
3.85
i
2.50
+
3.45
i
0.88
-
1.08
i
0.64
-
1.48
i
-
1.10
+
0.82
i
1.81
-
1.59
i
3.25
+
1.33
i
1.57
-
3.44
i
.
Here
A
is general and must first be reduced to Hessenberg form by
nag_zgehrd (f08nsc)
. The program then calls
nag_zunghr (f08ntc)
to form
Q
, and passes this matrix to
nag_zhseqr (f08psc)
which computes the Schur factorization of
A
.
10.1
Program Text
Program Text (f08ntce.c)
10.2
Program Data
Program Data (f08ntce.d)
10.3
Program Results
Program Results (f08ntce.r)
nag_zunghr (f08ntc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017