nag_dorghr (f08nfc) (PDF version)
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NAG Library Manual
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NAG Library Function Document
nag_dorghr (f08nfc)
▸
▿
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_dorghr (f08nfc)
generates the real orthogonal matrix
Q
which was determined by
nag_dgehrd (f08nec)
when reducing a real general matrix
A
to Hessenberg form.
2
Specification
#include <nag.h>
#include <nagf08.h>
void
nag_dorghr (
Nag_OrderType
order
,
Integer
n
,
Integer
ilo
,
Integer
ihi
,
double
a
[],
Integer
pda
,
const double
tau
[],
NagError *
fail
)
3
Description
nag_dorghr (f08nfc)
is intended to be used following a call to
nag_dgehrd (f08nec)
, which reduces a real general matrix
A
to upper Hessenberg form
H
by an orthogonal similarity transformation:
A
=
Q
H
Q
T
.
nag_dgehrd (f08nec)
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_dgebal (f08nhc)
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_dgehrd (f08nec)
.
Constraints
:
if
n
>
0
,
1
≤
ilo
≤
ihi
≤
n
;
if
n
=
0
,
ilo
=
1
and
ihi
=
0
.
5:
a
[
dim
]
–
double
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_dgehrd (f08nec)
.
On exit
: the
n
by
n
orthogonal 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 double
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_dgehrd (f08nec)
.
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 orthogonal matrix by a matrix
E
such that
E
2
=
O
ε
,
where
ε
is the
machine precision
.
8
Parallelism and Performance
nag_dorghr (f08nfc)
is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dorghr (f08nfc)
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 floating-point operations is approximately
4
3
q
3
, where
q
=
i
hi
-
i
lo
.
The complex analogue of this function is
nag_zunghr (f08ntc)
.
10
Example
This example computes the Schur factorization of the matrix
A
, where
A
=
0.35
0.45
-
0.14
-
0.17
0.09
0.07
-
0.54
0.35
-
0.44
-
0.33
-
0.03
0.17
0.25
-
0.32
-
0.13
0.11
.
Here
A
is general and must first be reduced to Hessenberg form by
nag_dgehrd (f08nec)
. The program then calls
nag_dorghr (f08nfc)
to form
Q
, and passes this matrix to
nag_dhseqr (f08pec)
which computes the Schur factorization of
A
.
10.1
Program Text
Program Text (f08nfce.c)
10.2
Program Data
Program Data (f08nfce.d)
10.3
Program Results
Program Results (f08nfce.r)
nag_dorghr (f08nfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017