nag_dorgtr (f08ffc) (PDF version)
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NAG Library Manual
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NAG Library Function Document
nag_dorgtr (f08ffc)
▸
▿
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_dorgtr (f08ffc)
generates the real orthogonal matrix
Q
, which was determined by
nag_dsytrd (f08fec)
when reducing a symmetric matrix to tridiagonal form.
2
Specification
#include <nag.h>
#include <nagf08.h>
void
nag_dorgtr (
Nag_OrderType
order
,
Nag_UploType
uplo
,
Integer
n
,
double
a
[],
Integer
pda
,
const double
tau
[],
NagError *
fail
)
3
Description
nag_dorgtr (f08ffc)
is intended to be used after a call to
nag_dsytrd (f08fec)
, which reduces a real symmetric matrix
A
to symmetric tridiagonal form
T
by an orthogonal similarity transformation:
A
=
Q
T
Q
T
.
nag_dsytrd (f08fec)
represents the orthogonal matrix
Q
as a product of
n
-
1
elementary reflectors.
This function may be used to generate
Q
explicitly as a square matrix.
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:
uplo
–
Nag_UploType
Input
On entry
: this
must
be the same argument
uplo
as supplied to
nag_dsytrd (f08fec)
.
Constraint
:
uplo
=
Nag_Upper
or
Nag_Lower
.
3:
n
–
Integer
Input
On entry
:
n
, the order of the matrix
Q
.
Constraint
:
n
≥
0
.
4:
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_dsytrd (f08fec)
.
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
]
.
5:
pda
–
Integer
Input
On entry
: the stride separating row or column elements (depending on the value of
order
) of the matrix
A
in the array
a
.
Constraint
:
pda
≥
max
1
,
n
.
6:
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_dsytrd (f08fec)
.
7:
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_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_dorgtr (f08ffc)
is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dorgtr (f08ffc)
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
n
3
.
The complex analogue of this function is
nag_zungtr (f08ftc)
.
10
Example
This example computes all the eigenvalues and eigenvectors of the matrix
A
, where
A
=
2.07
3.87
4.20
-
1.15
3.87
-
0.21
1.87
0.63
4.20
1.87
1.15
2.06
-
1.15
0.63
2.06
-
1.81
.
Here
A
is symmetric and must first be reduced to tridiagonal form by
nag_dsytrd (f08fec)
. The program then calls
nag_dorgtr (f08ffc)
to form
Q
, and passes this matrix to
nag_dsteqr (f08jec)
which computes the eigenvalues and eigenvectors of
A
.
10.1
Program Text
Program Text (f08ffce.c)
10.2
Program Data
Program Data (f08ffce.d)
10.3
Program Results
Program Results (f08ffce.r)
nag_dorgtr (f08ffc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017