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dlatrd.f
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*>
\
brief
\
b
DLATRD
reduces
the
first
nb
rows
and
columns
of
a
symmetric
/
Hermitian
matrix
A
to
real
tridiagonal
form
by
an
orthogonal
similarity
transformation
.
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DLATRD
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DLATRD
(
UPLO
,
N
,
NB
,
A
,
LDA
,
E
,
TAU
,
W
,
LDW
)
*
*
..
Scalar
Arguments
..
*
CHARACTER
UPLO
*
INTEGER
LDA
,
LDW
,
N
,
NB
*
..
*
..
Array
Arguments
..
*
DOUBLE PRECISION
A
(
LDA
,
*
),
E
(
*
),
TAU
(
*
),
W
(
LDW
,
*
)
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DLATRD
reduces
NB
rows
and
columns
of
a
real
symmetric
matrix
A
to
*>
symmetric
tridiagonal
form
by
an
orthogonal
similarity
*>
transformation
Q
**
T
*
A
*
Q
,
and
returns
the
matrices
V
and
W
which
are
*>
needed
to
apply
the
transformation
to
the
unreduced
part
of
A
.
*>
*>
If
UPLO
=
'U'
,
DLATRD
reduces
the
last
NB
rows
and
columns
of
a
*>
matrix
,
of
which
the
upper
triangle
is
supplied
;
*>
if
UPLO
=
'L'
,
DLATRD
reduces
the
first
NB
rows
and
columns
of
a
*>
matrix
,
of
which
the
lower
triangle
is
supplied
.
*>
*>
This
is
an
auxiliary
routine
called
by
DSYTRD
.
*>
\
endverbatim
*
*
Arguments
:
*
==========
*
*>
\
param
[
in
]
UPLO
*>
\
verbatim
*>
UPLO
is
CHARACTER
*
1
*>
Specifies
whether
the
upper
or
lower
triangular
part
of
the
*>
symmetric
matrix
A
is
stored
:
*>
=
'U'
:
Upper
triangular
*>
=
'L'
:
Lower
triangular
*>
\
endverbatim
*>
*>
\
param
[
in
]
N
*>
\
verbatim
*>
N
is
INTEGER
*>
The
order
of
the
matrix
A
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
NB
*>
\
verbatim
*>
NB
is
INTEGER
*>
The
number
of
rows
and
columns
to
be
reduced
.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
A
*>
\
verbatim
*>
A
is
DOUBLE PRECISION
array
,
dimension
(
LDA
,
N
)
*>
On
entry
,
the
symmetric
matrix
A
.
If
UPLO
=
'U'
,
the
leading
*>
n
-
by
-
n
upper
triangular
part
of
A
contains
the
upper
*>
triangular
part
of
the
matrix
A
,
and
the
strictly
lower
*>
triangular
part
of
A
is
not
referenced
.
If
UPLO
=
'L'
,
the
*>
leading
n
-
by
-
n
lower
triangular
part
of
A
contains
the
lower
*>
triangular
part
of
the
matrix
A
,
and
the
strictly
upper
*>
triangular
part
of
A
is
not
referenced
.
*>
On
exit
:
*>
if
UPLO
=
'U'
,
the
last
NB
columns
have
been
reduced
to
*>
tridiagonal
form
,
with
the
diagonal
elements
overwriting
*>
the
diagonal
elements
of
A
;
the
elements
above
the
diagonal
*>
with
the
array
TAU
,
represent
the
orthogonal
matrix
Q
as
a
*>
product
of
elementary
reflectors
;
*>
if
UPLO
=
'L'
,
the
first
NB
columns
have
been
reduced
to
*>
tridiagonal
form
,
with
the
diagonal
elements
overwriting
*>
the
diagonal
elements
of
A
;
the
elements
below
the
diagonal
*>
with
the
array
TAU
,
represent
the
orthogonal
matrix
Q
as
a
*>
product
of
elementary
reflectors
.
*>
See
Further
Details
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
LDA
*>
\
verbatim
*>
LDA
is
INTEGER
*>
The
leading
dimension
of
the
array
A
.
LDA
>=
(
1
,
N
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
E
*>
\
verbatim
*>
E
is
DOUBLE PRECISION
array
,
dimension
(
N
-
1
)
*>
If
UPLO
=
'U'
,
E
(
n
-
nb
:
n
-
1
)
contains
the
superdiagonal
*>
elements
of
the
last
NB
columns
of
the
reduced
matrix
;
*>
if
UPLO
=
'L'
,
E
(
1
:
nb
)
contains
the
subdiagonal
elements
of
*>
the
first
NB
columns
of
the
reduced
matrix
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
TAU
*>
\
verbatim
*>
TAU
is
DOUBLE PRECISION
array
,
dimension
(
N
-
1
)
*>
The
scalar
factors
of
the
elementary
reflectors
,
stored
in
*>
TAU
(
n
-
nb
:
n
-
1
)
if
UPLO
=
'U'
,
and
in
TAU
(
1
:
nb
)
if
UPLO
=
'L'
.
*>
See
Further
Details
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
W
*>
\
verbatim
*>
W
is
DOUBLE PRECISION
array
,
dimension
(
LDW
,
NB
)
*>
The
n
-
by
-
nb
matrix
W
required
to
update
the
unreduced
part
*>
of
A
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
LDW
*>
\
verbatim
*>
LDW
is
INTEGER
*>
The
leading
dimension
of
the
array
W
.
LDW
>=
max
(
1
,
N
)
.
*>
\
endverbatim
*
*
Authors
:
*
========
*
*>
\
author
Univ
.
of
Tennessee
*>
\
author
Univ
.
of
California
Berkeley
*>
\
author
Univ
.
of
Colorado
Denver
*>
\
author
NAG
Ltd
.
*
*>
\
date
September
2012
*
*>
\
ingroup
doubleOTHERauxiliary
*
*>
\
par
Further
Details
:
*
=====================
*>
*>
\
verbatim
*>
*>
If
UPLO
=
'U'
,
the
matrix
Q
is
represented
as
a
product
of
elementary
*>
reflectors
*>
*>
Q
=
H
(
n
)
H
(
n
-
1
)
.
.
.
H
(
n
-
nb
+
1
)
.
*>
*>
Each
H
(
i
)
has
the
form
*>
*>
H
(
i
)
=
I
-
tau
*
v
*
v
**
T
*>
*>
where
tau
is
a
real
scalar
,
and
v
is
a
real
vector
with
*>
v
(
i
:
n
)
=
0
and
v
(
i
-
1
)
=
1
;
v
(
1
:
i
-
1
)
is
stored
on
exit
in
A
(
1
:
i
-
1
,
i
),
*>
and
tau
in
TAU
(
i
-
1
)
.
*>
*>
If
UPLO
=
'L'
,
the
matrix
Q
is
represented
as
a
product
of
elementary
*>
reflectors
*>
*>
Q
=
H
(
1
)
H
(
2
)
.
.
.
H
(
nb
)
.
*>
*>
Each
H
(
i
)
has
the
form
*>
*>
H
(
i
)
=
I
-
tau
*
v
*
v
**
T
*>
*>
where
tau
is
a
real
scalar
,
and
v
is
a
real
vector
with
*>
v
(
1
:
i
)
=
0
and
v
(
i
+
1
)
=
1
;
v
(
i
+
1
:
n
)
is
stored
on
exit
in
A
(
i
+
1
:
n
,
i
),
*>
and
tau
in
TAU
(
i
)
.
*>
*>
The
elements
of
the
vectors
v
together
form
the
n
-
by
-
nb
matrix
V
*>
which
is
needed
,
with
W
,
to
apply
the
transformation
to
the
unreduced
*>
part
of
the
matrix
,
using
a
symmetric
rank
-
2
k
update
of
the
form
:
*>
A
:
=
A
-
V
*
W
**
T
-
W
*
V
**
T
.
*>
*>
The
contents
of
A
on
exit
are
illustrated
by
the
following
examples
*>
with
n
=
5
and
nb
=
2
:
*>
*>
if
UPLO
=
'U'
:
if
UPLO
=
'L'
:
*>
*>
(
a
a
a
v4
v5
)
(
d
)
*>
(
a
a
v4
v5
)
(
1
d
)
*>
(
a
1
v5
)
(
v1
1
a
)
*>
(
d
1
)
(
v1
v2
a
a
)
*>
(
d
)
(
v1
v2
a
a
a
)
*>
*>
where
d
denotes
a
diagonal
element
of
the
reduced
matrix
,
a
denotes
*>
an
element
of
the
original
matrix
that
is
unchanged
,
and
vi
denotes
*>
an
element
of
the
vector
defining
H
(
i
)
.
*>
\
endverbatim
*>
*
=====================================================================
SUBROUTINE
DLATRD
(
UPLO
,
N
,
NB
,
A
,
LDA
,
E
,
TAU
,
W
,
LDW
)
*
*
--
LAPACK
auxiliary
routine
(
version
3.4.2
)
--
*
--
LAPACK
is
a
software
package
provided
by
Univ
.
of
Tennessee
,
--
*
--
Univ
.
of
California
Berkeley
,
Univ
.
of
Colorado
Denver
and
NAG
Ltd
..
--
*
September
2012
*
*
..
Scalar
Arguments
..
CHARACTER
UPLO
INTEGER
LDA
,
LDW
,
N
,
NB
*
..
*
..
Array
Arguments
..
DOUBLE PRECISION
A
(
LDA
,
*
),
E
(
*
),
TAU
(
*
),
W
(
LDW
,
*
)
*
..
*
*
=====================================================================
*
*
..
Parameters
..
DOUBLE PRECISION
ZERO
,
ONE
,
HALF
PARAMETER
(
ZERO
=
0.0
D
+
0
,
ONE
=
1.0
D
+
0
,
HALF
=
0.5
D
+
0
)
*
..
*
..
Local
Scalars
..
INTEGER
I
,
IW
DOUBLE PRECISION
ALPHA
*
..
*
..
External
Subroutines
..
EXTERNAL
DAXPY
,
DGEMV
,
DLARFG
,
DSCAL
,
DSYMV
*
..
*
..
External
Functions
..
LOGICAL
LSAME
DOUBLE PRECISION
DDOT
EXTERNAL
LSAME
,
DDOT
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
MIN
*
..
*
..
Executable
Statements
..
*
*
Quick
return if
possible
*
IF
(
N
.LE.
0
)
$
RETURN
*
IF
(
LSAME
(
UPLO
,
'U'
)
)
THEN
*
*
Reduce
last
NB
columns
of
upper
triangle
*
DO
10
I
=
N
,
N
-
NB
+
1
,
-
1
IW
=
I
-
N
+
NB
IF
(
I
.LT.
N
)
THEN
*
*
Update
A
(
1
:
i
,
i
)
*
CALL
DGEMV
(
'No transpose'
,
I
,
N
-
I
,
-
ONE
,
A
(
1
,
I
+
1
),
$
LDA
,
W
(
I
,
IW
+
1
),
LDW
,
ONE
,
A
(
1
,
I
),
1
)
CALL
DGEMV
(
'No transpose'
,
I
,
N
-
I
,
-
ONE
,
W
(
1
,
IW
+
1
),
$
LDW
,
A
(
I
,
I
+
1
),
LDA
,
ONE
,
A
(
1
,
I
),
1
)
END IF
IF
(
I
.GT.
1
)
THEN
*
*
Generate
elementary
reflector
H
(
i
)
to
annihilate
*
A
(
1
:
i
-
2
,
i
)
*
CALL
DLARFG
(
I
-
1
,
A
(
I
-
1
,
I
),
A
(
1
,
I
),
1
,
TAU
(
I
-
1
)
)
E
(
I
-
1
)
=
A
(
I
-
1
,
I
)
A
(
I
-
1
,
I
)
=
ONE
*
*
Compute
W
(
1
:
i
-
1
,
i
)
*
CALL
DSYMV
(
'Upper'
,
I
-
1
,
ONE
,
A
,
LDA
,
A
(
1
,
I
),
1
,
$
ZERO
,
W
(
1
,
IW
),
1
)
IF
(
I
.LT.
N
)
THEN
CALL
DGEMV
(
'Transpose'
,
I
-
1
,
N
-
I
,
ONE
,
W
(
1
,
IW
+
1
),
$
LDW
,
A
(
1
,
I
),
1
,
ZERO
,
W
(
I
+
1
,
IW
),
1
)
CALL
DGEMV
(
'No transpose'
,
I
-
1
,
N
-
I
,
-
ONE
,
$
A
(
1
,
I
+
1
),
LDA
,
W
(
I
+
1
,
IW
),
1
,
ONE
,
$
W
(
1
,
IW
),
1
)
CALL
DGEMV
(
'Transpose'
,
I
-
1
,
N
-
I
,
ONE
,
A
(
1
,
I
+
1
),
$
LDA
,
A
(
1
,
I
),
1
,
ZERO
,
W
(
I
+
1
,
IW
),
1
)
CALL
DGEMV
(
'No transpose'
,
I
-
1
,
N
-
I
,
-
ONE
,
$
W
(
1
,
IW
+
1
),
LDW
,
W
(
I
+
1
,
IW
),
1
,
ONE
,
$
W
(
1
,
IW
),
1
)
END IF
CALL
DSCAL
(
I
-
1
,
TAU
(
I
-
1
),
W
(
1
,
IW
),
1
)
ALPHA
=
-
HALF
*
TAU
(
I
-
1
)
*
DDOT
(
I
-
1
,
W
(
1
,
IW
),
1
,
$
A
(
1
,
I
),
1
)
CALL
DAXPY
(
I
-
1
,
ALPHA
,
A
(
1
,
I
),
1
,
W
(
1
,
IW
),
1
)
END IF
*
10
CONTINUE
ELSE
*
*
Reduce
first
NB
columns
of
lower
triangle
*
DO
20
I
=
1
,
NB
*
*
Update
A
(
i
:
n
,
i
)
*
CALL
DGEMV
(
'No transpose'
,
N
-
I
+
1
,
I
-
1
,
-
ONE
,
A
(
I
,
1
),
$
LDA
,
W
(
I
,
1
),
LDW
,
ONE
,
A
(
I
,
I
),
1
)
CALL
DGEMV
(
'No transpose'
,
N
-
I
+
1
,
I
-
1
,
-
ONE
,
W
(
I
,
1
),
$
LDW
,
A
(
I
,
1
),
LDA
,
ONE
,
A
(
I
,
I
),
1
)
IF
(
I
.LT.
N
)
THEN
*
*
Generate
elementary
reflector
H
(
i
)
to
annihilate
*
A
(
i
+
2
:
n
,
i
)
*
CALL
DLARFG
(
N
-
I
,
A
(
I
+
1
,
I
),
A
(
MIN
(
I
+
2
,
N
),
I
),
1
,
$
TAU
(
I
)
)
E
(
I
)
=
A
(
I
+
1
,
I
)
A
(
I
+
1
,
I
)
=
ONE
*
*
Compute
W
(
i
+
1
:
n
,
i
)
*
CALL
DSYMV
(
'Lower'
,
N
-
I
,
ONE
,
A
(
I
+
1
,
I
+
1
),
LDA
,
$
A
(
I
+
1
,
I
),
1
,
ZERO
,
W
(
I
+
1
,
I
),
1
)
CALL
DGEMV
(
'Transpose'
,
N
-
I
,
I
-
1
,
ONE
,
W
(
I
+
1
,
1
),
LDW
,
$
A
(
I
+
1
,
I
),
1
,
ZERO
,
W
(
1
,
I
),
1
)
CALL
DGEMV
(
'No transpose'
,
N
-
I
,
I
-
1
,
-
ONE
,
A
(
I
+
1
,
1
),
$
LDA
,
W
(
1
,
I
),
1
,
ONE
,
W
(
I
+
1
,
I
),
1
)
CALL
DGEMV
(
'Transpose'
,
N
-
I
,
I
-
1
,
ONE
,
A
(
I
+
1
,
1
),
LDA
,
$
A
(
I
+
1
,
I
),
1
,
ZERO
,
W
(
1
,
I
),
1
)
CALL
DGEMV
(
'No transpose'
,
N
-
I
,
I
-
1
,
-
ONE
,
W
(
I
+
1
,
1
),
$
LDW
,
W
(
1
,
I
),
1
,
ONE
,
W
(
I
+
1
,
I
),
1
)
CALL
DSCAL
(
N
-
I
,
TAU
(
I
),
W
(
I
+
1
,
I
),
1
)
ALPHA
=
-
HALF
*
TAU
(
I
)
*
DDOT
(
N
-
I
,
W
(
I
+
1
,
I
),
1
,
$
A
(
I
+
1
,
I
),
1
)
CALL
DAXPY
(
N
-
I
,
ALPHA
,
A
(
I
+
1
,
I
),
1
,
W
(
I
+
1
,
I
),
1
)
END IF
*
20
CONTINUE
END IF
*
RETURN
*
*
End
of
DLATRD
*
END
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