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dsytd2.f
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*>
\
brief
\
b
DSYTD2
reduces
a
symmetric
matrix
to
real
symmetric
tridiagonal
form
by
an
orthogonal
similarity
transformation
(
unblocked
algorithm
)
.
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DSYTD2
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytd2.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytd2.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DSYTD2
(
UPLO
,
N
,
A
,
LDA
,
D
,
E
,
TAU
,
INFO
)
*
*
..
Scalar
Arguments
..
*
CHARACTER
UPLO
*
INTEGER
INFO
,
LDA
,
N
*
..
*
..
Array
Arguments
..
*
DOUBLE PRECISION
A
(
LDA
,
*
),
D
(
*
),
E
(
*
),
TAU
(
*
)
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DSYTD2
reduces
a
real
symmetric
matrix
A
to
symmetric
tridiagonal
*>
form
T
by
an
orthogonal
similarity
transformation
:
Q
**
T
*
A
*
Q
=
T
.
*>
\
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
.
N
>=
0.
*>
\
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
diagonal
and
first
superdiagonal
*>
of
A
are
overwritten
by
the
corresponding
elements
of
the
*>
tridiagonal
matrix
T
,
and
the
elements
above
the
first
*>
superdiagonal
,
with
the
array
TAU
,
represent
the
orthogonal
*>
matrix
Q
as
a
product
of
elementary
reflectors
;
if
UPLO
*>
=
'L'
,
the
diagonal
and
first
subdiagonal
of
A
are
over
-
*>
written
by
the
corresponding
elements
of
the
tridiagonal
*>
matrix
T
,
and
the
elements
below
the
first
subdiagonal
,
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
>=
max
(
1
,
N
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
D
*>
\
verbatim
*>
D
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
The
diagonal
elements
of
the
tridiagonal
matrix
T
:
*>
D
(
i
)
=
A
(
i
,
i
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
E
*>
\
verbatim
*>
E
is
DOUBLE PRECISION
array
,
dimension
(
N
-
1
)
*>
The
off
-
diagonal
elements
of
the
tridiagonal
matrix
T
:
*>
E
(
i
)
=
A
(
i
,
i
+
1
)
if
UPLO
=
'U'
,
E
(
i
)
=
A
(
i
+
1
,
i
)
if
UPLO
=
'L'
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
TAU
*>
\
verbatim
*>
TAU
is
DOUBLE PRECISION
array
,
dimension
(
N
-
1
)
*>
The
scalar
factors
of
the
elementary
reflectors
(
see
Further
*>
Details
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
INFO
*>
\
verbatim
*>
INFO
is
INTEGER
*>
=
0
:
successful
exit
*>
<
0
:
if
INFO
=
-
i
,
the
i
-
th
argument
had
an
illegal
value
.
*>
\
endverbatim
*
*
Authors
:
*
========
*
*>
\
author
Univ
.
of
Tennessee
*>
\
author
Univ
.
of
California
Berkeley
*>
\
author
Univ
.
of
Colorado
Denver
*>
\
author
NAG
Ltd
.
*
*>
\
date
September
2012
*
*>
\
ingroup
doubleSYcomputational
*
*>
\
par
Further
Details
:
*
=====================
*>
*>
\
verbatim
*>
*>
If
UPLO
=
'U'
,
the
matrix
Q
is
represented
as
a
product
of
elementary
*>
reflectors
*>
*>
Q
=
H
(
n
-
1
)
.
.
.
H
(
2
)
H
(
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
+
1
:
n
)
=
0
and
v
(
i
)
=
1
;
v
(
1
:
i
-
1
)
is
stored
on
exit
in
*>
A
(
1
:
i
-
1
,
i
+
1
),
and
tau
in
TAU
(
i
)
.
*>
*>
If
UPLO
=
'L'
,
the
matrix
Q
is
represented
as
a
product
of
elementary
*>
reflectors
*>
*>
Q
=
H
(
1
)
H
(
2
)
.
.
.
H
(
n
-
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
(
1
:
i
)
=
0
and
v
(
i
+
1
)
=
1
;
v
(
i
+
2
:
n
)
is
stored
on
exit
in
A
(
i
+
2
:
n
,
i
),
*>
and
tau
in
TAU
(
i
)
.
*>
*>
The
contents
of
A
on
exit
are
illustrated
by
the
following
examples
*>
with
n
=
5
:
*>
*>
if
UPLO
=
'U'
:
if
UPLO
=
'L'
:
*>
*>
(
d
e
v2
v3
v4
)
(
d
)
*>
(
d
e
v3
v4
)
(
e
d
)
*>
(
d
e
v4
)
(
v1
e
d
)
*>
(
d
e
)
(
v1
v2
e
d
)
*>
(
d
)
(
v1
v2
v3
e
d
)
*>
*>
where
d
and
e
denote
diagonal
and
off
-
diagonal
elements
of
T
,
and
vi
*>
denotes
an
element
of
the
vector
defining
H
(
i
)
.
*>
\
endverbatim
*>
*
=====================================================================
SUBROUTINE
DSYTD2
(
UPLO
,
N
,
A
,
LDA
,
D
,
E
,
TAU
,
INFO
)
*
*
--
LAPACK
computational
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
INFO
,
LDA
,
N
*
..
*
..
Array
Arguments
..
DOUBLE PRECISION
A
(
LDA
,
*
),
D
(
*
),
E
(
*
),
TAU
(
*
)
*
..
*
*
=====================================================================
*
*
..
Parameters
..
DOUBLE PRECISION
ONE
,
ZERO
,
HALF
PARAMETER
(
ONE
=
1.0
D0
,
ZERO
=
0.0
D0
,
$
HALF
=
1.0
D0
/
2.0
D0
)
*
..
*
..
Local
Scalars
..
LOGICAL
UPPER
INTEGER
I
DOUBLE PRECISION
ALPHA
,
TAUI
*
..
*
..
External
Subroutines
..
EXTERNAL
DAXPY
,
DLARFG
,
DSYMV
,
DSYR2
,
XERBLA
*
..
*
..
External
Functions
..
LOGICAL
LSAME
DOUBLE PRECISION
DDOT
EXTERNAL
LSAME
,
DDOT
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
MAX
,
MIN
*
..
*
..
Executable
Statements
..
*
*
Test
the
input
parameters
*
INFO
=
0
UPPER
=
LSAME
(
UPLO
,
'U'
)
IF
(
.NOT.
UPPER
.AND.
.NOT.
LSAME
(
UPLO
,
'L'
)
)
THEN
INFO
=
-
1
ELSE IF
(
N
.LT.
0
)
THEN
INFO
=
-
2
ELSE IF
(
LDA
.LT.
MAX
(
1
,
N
)
)
THEN
INFO
=
-
4
END IF
IF
(
INFO
.NE.
0
)
THEN
CALL
XERBLA
(
'DSYTD2'
,
-
INFO
)
RETURN
END IF
*
*
Quick
return if
possible
*
IF
(
N
.LE.
0
)
$
RETURN
*
IF
(
UPPER
)
THEN
*
*
Reduce
the
upper
triangle
of
A
*
DO
10
I
=
N
-
1
,
1
,
-
1
*
*
Generate
elementary
reflector
H
(
i
)
=
I
-
tau
*
v
*
v
**
T
*
to
annihilate
A
(
1
:
i
-
1
,
i
+
1
)
*
CALL
DLARFG
(
I
,
A
(
I
,
I
+
1
),
A
(
1
,
I
+
1
),
1
,
TAUI
)
E
(
I
)
=
A
(
I
,
I
+
1
)
*
IF
(
TAUI
.NE.
ZERO
)
THEN
*
*
Apply
H
(
i
)
from
both
sides
to
A
(
1
:
i
,
1
:
i
)
*
A
(
I
,
I
+
1
)
=
ONE
*
*
Compute
x
:
=
tau
*
A
*
v
storing
x
in
TAU
(
1
:
i
)
*
CALL
DSYMV
(
UPLO
,
I
,
TAUI
,
A
,
LDA
,
A
(
1
,
I
+
1
),
1
,
ZERO
,
$
TAU
,
1
)
*
*
Compute
w
:
=
x
-
1
/
2
*
tau
*
(
x
**
T
*
v
)
*
v
*
ALPHA
=
-
HALF
*
TAUI
*
DDOT
(
I
,
TAU
,
1
,
A
(
1
,
I
+
1
),
1
)
CALL
DAXPY
(
I
,
ALPHA
,
A
(
1
,
I
+
1
),
1
,
TAU
,
1
)
*
*
Apply
the
transformation
as
a
rank
-
2
update
:
*
A
:
=
A
-
v
*
w
**
T
-
w
*
v
**
T
*
CALL
DSYR2
(
UPLO
,
I
,
-
ONE
,
A
(
1
,
I
+
1
),
1
,
TAU
,
1
,
A
,
$
LDA
)
*
A
(
I
,
I
+
1
)
=
E
(
I
)
END IF
D
(
I
+
1
)
=
A
(
I
+
1
,
I
+
1
)
TAU
(
I
)
=
TAUI
10
CONTINUE
D
(
1
)
=
A
(
1
,
1
)
ELSE
*
*
Reduce
the
lower
triangle
of
A
*
DO
20
I
=
1
,
N
-
1
*
*
Generate
elementary
reflector
H
(
i
)
=
I
-
tau
*
v
*
v
**
T
*
to
annihilate
A
(
i
+
2
:
n
,
i
)
*
CALL
DLARFG
(
N
-
I
,
A
(
I
+
1
,
I
),
A
(
MIN
(
I
+
2
,
N
),
I
),
1
,
$
TAUI
)
E
(
I
)
=
A
(
I
+
1
,
I
)
*
IF
(
TAUI
.NE.
ZERO
)
THEN
*
*
Apply
H
(
i
)
from
both
sides
to
A
(
i
+
1
:
n
,
i
+
1
:
n
)
*
A
(
I
+
1
,
I
)
=
ONE
*
*
Compute
x
:
=
tau
*
A
*
v
storing
y
in
TAU
(
i
:
n
-
1
)
*
CALL
DSYMV
(
UPLO
,
N
-
I
,
TAUI
,
A
(
I
+
1
,
I
+
1
),
LDA
,
$
A
(
I
+
1
,
I
),
1
,
ZERO
,
TAU
(
I
),
1
)
*
*
Compute
w
:
=
x
-
1
/
2
*
tau
*
(
x
**
T
*
v
)
*
v
*
ALPHA
=
-
HALF
*
TAUI
*
DDOT
(
N
-
I
,
TAU
(
I
),
1
,
A
(
I
+
1
,
I
),
$
1
)
CALL
DAXPY
(
N
-
I
,
ALPHA
,
A
(
I
+
1
,
I
),
1
,
TAU
(
I
),
1
)
*
*
Apply
the
transformation
as
a
rank
-
2
update
:
*
A
:
=
A
-
v
*
w
**
T
-
w
*
v
**
T
*
CALL
DSYR2
(
UPLO
,
N
-
I
,
-
ONE
,
A
(
I
+
1
,
I
),
1
,
TAU
(
I
),
1
,
$
A
(
I
+
1
,
I
+
1
),
LDA
)
*
A
(
I
+
1
,
I
)
=
E
(
I
)
END IF
D
(
I
)
=
A
(
I
,
I
)
TAU
(
I
)
=
TAUI
20
CONTINUE
D
(
N
)
=
A
(
N
,
N
)
END IF
*
RETURN
*
*
End
of
DSYTD2
*
END
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