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dlaed2.f
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*>
\
brief
\
b
DLAED2
used
by
sstedc
.
Merges
eigenvalues
and
deflates
secular
equation
.
Used
when
the
original
matrix
is
tridiagonal
.
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DLAED2
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DLAED2
(
K
,
N
,
N1
,
D
,
Q
,
LDQ
,
INDXQ
,
RHO
,
Z
,
DLAMDA
,
W
,
*
Q2
,
INDX
,
INDXC
,
INDXP
,
COLTYP
,
INFO
)
*
*
..
Scalar
Arguments
..
*
INTEGER
INFO
,
K
,
LDQ
,
N
,
N1
*
DOUBLE PRECISION
RHO
*
..
*
..
Array
Arguments
..
*
INTEGER
COLTYP
(
*
),
INDX
(
*
),
INDXC
(
*
),
INDXP
(
*
),
*
$
INDXQ
(
*
)
*
DOUBLE PRECISION
D
(
*
),
DLAMDA
(
*
),
Q
(
LDQ
,
*
),
Q2
(
*
),
*
$
W
(
*
),
Z
(
*
)
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DLAED2
merges
the
two
sets
of
eigenvalues
together
into
a
single
*>
sorted
set
.
Then
it
tries
to
deflate
the
size
of
the
problem
.
*>
There
are
two
ways
in
which
deflation
can
occur
:
when
two
or
more
*>
eigenvalues
are
close
together
or
if
there
is
a
tiny
entry
in
the
*>
Z
vector
.
For
each
such
occurrence
the
order
of
the
related
secular
*>
equation
problem
is
reduced
by
one
.
*>
\
endverbatim
*
*
Arguments
:
*
==========
*
*>
\
param
[
out
]
K
*>
\
verbatim
*>
K
is
INTEGER
*>
The
number
of
non
-
deflated
eigenvalues
,
and
the
order
of
the
*>
related
secular
equation
.
0
<=
K
<=
N
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
N
*>
\
verbatim
*>
N
is
INTEGER
*>
The
dimension
of
the
symmetric
tridiagonal
matrix
.
N
>=
0.
*>
\
endverbatim
*>
*>
\
param
[
in
]
N1
*>
\
verbatim
*>
N1
is
INTEGER
*>
The
location
of
the
last
eigenvalue
in
the
leading
sub
-
matrix
.
*>
min
(
1
,
N
)
<=
N1
<=
N
/
2.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
D
*>
\
verbatim
*>
D
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
On
entry
,
D
contains
the
eigenvalues
of
the
two
submatrices
to
*>
be
combined
.
*>
On
exit
,
D
contains
the
trailing
(
N
-
K
)
updated
eigenvalues
*>
(
those
which
were
deflated
)
sorted
into
increasing
order
.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
Q
*>
\
verbatim
*>
Q
is
DOUBLE PRECISION
array
,
dimension
(
LDQ
,
N
)
*>
On
entry
,
Q
contains
the
eigenvectors
of
two
submatrices
in
*>
the
two
square
blocks
with
corners
at
(
1
,
1
),
(
N1
,
N1
)
*>
and
(
N1
+
1
,
N1
+
1
),
(
N
,
N
)
.
*>
On
exit
,
Q
contains
the
trailing
(
N
-
K
)
updated
eigenvectors
*>
(
those
which
were
deflated
)
in
its
last
N
-
K
columns
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
LDQ
*>
\
verbatim
*>
LDQ
is
INTEGER
*>
The
leading
dimension
of
the
array
Q
.
LDQ
>=
max
(
1
,
N
)
.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
INDXQ
*>
\
verbatim
*>
INDXQ
is
INTEGER
array
,
dimension
(
N
)
*>
The
permutation
which
separately
sorts
the
two
sub
-
problems
*>
in
D
into
ascending
order
.
Note
that
elements
in
the
second
*>
half
of
this
permutation
must
first
have
N1
added
to
their
*>
values
.
Destroyed
on
exit
.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
RHO
*>
\
verbatim
*>
RHO
is
DOUBLE PRECISION
*>
On
entry
,
the
off
-
diagonal
element
associated
with
the
rank
-
1
*>
cut
which
originally
split
the
two
submatrices
which
are
now
*>
being
recombined
.
*>
On
exit
,
RHO
has
been
modified
to
the
value
required
by
*>
DLAED3
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
Z
*>
\
verbatim
*>
Z
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
On
entry
,
Z
contains
the
updating
vector
(
the
last
*>
row
of
the
first
sub
-
eigenvector
matrix
and
the
first
row
of
*>
the
second
sub
-
eigenvector
matrix
)
.
*>
On
exit
,
the
contents
of
Z
have
been
destroyed
by
the
updating
*>
process
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
DLAMDA
*>
\
verbatim
*>
DLAMDA
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
A
copy
of
the
first
K
eigenvalues
which
will
be
used
by
*>
DLAED3
to
form
the
secular
equation
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
W
*>
\
verbatim
*>
W
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
The
first
k
values
of
the
final
deflation
-
altered
z
-
vector
*>
which
will
be
passed
to
DLAED3
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
Q2
*>
\
verbatim
*>
Q2
is
DOUBLE PRECISION
array
,
dimension
(
N1
**
2
+
(
N
-
N1
)
**
2
)
*>
A
copy
of
the
first
K
eigenvectors
which
will
be
used
by
*>
DLAED3
in
a
matrix
multiply
(
DGEMM
)
to
solve
for
the
new
*>
eigenvectors
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
INDX
*>
\
verbatim
*>
INDX
is
INTEGER
array
,
dimension
(
N
)
*>
The
permutation
used
to
sort
the
contents
of
DLAMDA
into
*>
ascending
order
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
INDXC
*>
\
verbatim
*>
INDXC
is
INTEGER
array
,
dimension
(
N
)
*>
The
permutation
used
to
arrange
the
columns
of
the
deflated
*>
Q
matrix
into
three
groups
:
the
first
group
contains
non
-
zero
*>
elements
only
at
and
above
N1
,
the
second
contains
*>
non
-
zero
elements
only
below
N1
,
and
the
third
is
dense
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
INDXP
*>
\
verbatim
*>
INDXP
is
INTEGER
array
,
dimension
(
N
)
*>
The
permutation
used
to
place
deflated
values
of
D
at
the
end
*>
of
the
array
.
INDXP
(
1
:
K
)
points
to
the
nondeflated
D
-
values
*>
and
INDXP
(
K
+
1
:
N
)
points
to
the
deflated
eigenvalues
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
COLTYP
*>
\
verbatim
*>
COLTYP
is
INTEGER
array
,
dimension
(
N
)
*>
During
execution
,
a
label
which
will
indicate
which
of
the
*>
following
types
a
column
in
the
Q2
matrix
is
:
*>
1
:
non
-
zero
in
the
upper
half
only
;
*>
2
:
dense
;
*>
3
:
non
-
zero
in
the
lower
half
only
;
*>
4
:
deflated
.
*>
On
exit
,
COLTYP
(
i
)
is
the
number
of
columns
of
type
i
,
*>
for
i
=
1
to
4
only
.
*>
\
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
auxOTHERcomputational
*
*>
\
par
Contributors
:
*
==================
*>
*>
Jeff
Rutter
,
Computer
Science
Division
,
University
of
California
*>
at
Berkeley
,
USA
\
n
*>
Modified
by
Francoise
Tisseur
,
University
of
Tennessee
*>
*
=====================================================================
SUBROUTINE
DLAED2
(
K
,
N
,
N1
,
D
,
Q
,
LDQ
,
INDXQ
,
RHO
,
Z
,
DLAMDA
,
W
,
$
Q2
,
INDX
,
INDXC
,
INDXP
,
COLTYP
,
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
..
INTEGER
INFO
,
K
,
LDQ
,
N
,
N1
DOUBLE PRECISION
RHO
*
..
*
..
Array
Arguments
..
INTEGER
COLTYP
(
*
),
INDX
(
*
),
INDXC
(
*
),
INDXP
(
*
),
$
INDXQ
(
*
)
DOUBLE PRECISION
D
(
*
),
DLAMDA
(
*
),
Q
(
LDQ
,
*
),
Q2
(
*
),
$
W
(
*
),
Z
(
*
)
*
..
*
*
=====================================================================
*
*
..
Parameters
..
DOUBLE PRECISION
MONE
,
ZERO
,
ONE
,
TWO
,
EIGHT
PARAMETER
(
MONE
=
-
1.0
D0
,
ZERO
=
0.0
D0
,
ONE
=
1.0
D0
,
$
TWO
=
2.0
D0
,
EIGHT
=
8.0
D0
)
*
..
*
..
Local
Arrays
..
INTEGER
CTOT
(
4
),
PSM
(
4
)
*
..
*
..
Local
Scalars
..
INTEGER
CT
,
I
,
IMAX
,
IQ1
,
IQ2
,
J
,
JMAX
,
JS
,
K2
,
N1P1
,
$
N2
,
NJ
,
PJ
DOUBLE PRECISION
C
,
EPS
,
S
,
T
,
TAU
,
TOL
*
..
*
..
External
Functions
..
INTEGER
IDAMAX
DOUBLE PRECISION
DLAMCH
,
DLAPY2
EXTERNAL
IDAMAX
,
DLAMCH
,
DLAPY2
*
..
*
..
External
Subroutines
..
EXTERNAL
DCOPY
,
DLACPY
,
DLAMRG
,
DROT
,
DSCAL
,
XERBLA
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
ABS
,
MAX
,
MIN
,
SQRT
*
..
*
..
Executable
Statements
..
*
*
Test
the
input
parameters
.
*
INFO
=
0
*
IF
(
N
.LT.
0
)
THEN
INFO
=
-
2
ELSE IF
(
LDQ
.LT.
MAX
(
1
,
N
)
)
THEN
INFO
=
-
6
ELSE IF
(
MIN
(
1
,
(
N
/
2
)
)
.GT.
N1
.OR.
(
N
/
2
)
.LT.
N1
)
THEN
INFO
=
-
3
END IF
IF
(
INFO
.NE.
0
)
THEN
CALL
XERBLA
(
'DLAED2'
,
-
INFO
)
RETURN
END IF
*
*
Quick
return if
possible
*
IF
(
N
.EQ.
0
)
$
RETURN
*
N2
=
N
-
N1
N1P1
=
N1
+
1
*
IF
(
RHO
.LT.
ZERO
)
THEN
CALL
DSCAL
(
N2
,
MONE
,
Z
(
N1P1
),
1
)
END IF
*
*
Normalize
z
so
that
norm
(
z
)
=
1.
Since
z
is
the
concatenation
of
*
two
normalized
vectors
,
norm2
(
z
)
=
sqrt
(
2
)
.
*
T
=
ONE
/
SQRT
(
TWO
)
CALL
DSCAL
(
N
,
T
,
Z
,
1
)
*
*
RHO
=
ABS
(
norm
(
z
)
**
2
*
RHO
)
*
RHO
=
ABS
(
TWO
*
RHO
)
*
*
Sort
the
eigenvalues
into
increasing
order
*
DO
10
I
=
N1P1
,
N
INDXQ
(
I
)
=
INDXQ
(
I
)
+
N1
10
CONTINUE
*
*
re
-
integrate
the
deflated
parts
from
the
last
pass
*
DO
20
I
=
1
,
N
DLAMDA
(
I
)
=
D
(
INDXQ
(
I
)
)
20
CONTINUE
CALL
DLAMRG
(
N1
,
N2
,
DLAMDA
,
1
,
1
,
INDXC
)
DO
30
I
=
1
,
N
INDX
(
I
)
=
INDXQ
(
INDXC
(
I
)
)
30
CONTINUE
*
*
Calculate
the
allowable
deflation
tolerance
*
IMAX
=
IDAMAX
(
N
,
Z
,
1
)
JMAX
=
IDAMAX
(
N
,
D
,
1
)
EPS
=
DLAMCH
(
'Epsilon'
)
TOL
=
EIGHT
*
EPS
*
MAX
(
ABS
(
D
(
JMAX
)
),
ABS
(
Z
(
IMAX
)
)
)
*
*
If
the
rank
-
1
modifier
is
small
enough
,
no
more
needs
to
be
done
*
except
to
reorganize
Q
so
that
its
columns
correspond
with
the
*
elements
in
D
.
*
IF
(
RHO
*
ABS
(
Z
(
IMAX
)
)
.LE.
TOL
)
THEN
K
=
0
IQ2
=
1
DO
40
J
=
1
,
N
I
=
INDX
(
J
)
CALL
DCOPY
(
N
,
Q
(
1
,
I
),
1
,
Q2
(
IQ2
),
1
)
DLAMDA
(
J
)
=
D
(
I
)
IQ2
=
IQ2
+
N
40
CONTINUE
CALL
DLACPY
(
'A'
,
N
,
N
,
Q2
,
N
,
Q
,
LDQ
)
CALL
DCOPY
(
N
,
DLAMDA
,
1
,
D
,
1
)
GO
TO
190
END IF
*
*
If
there
are
multiple
eigenvalues
then
the
problem
deflates
.
Here
*
the
number
of
equal
eigenvalues
are
found
.
As
each
equal
*
eigenvalue
is
found
,
an
elementary
reflector
is
computed
to
rotate
*
the
corresponding
eigensubspace
so
that
the
corresponding
*
components
of
Z
are
zero
in
this
new
basis
.
*
DO
50
I
=
1
,
N1
COLTYP
(
I
)
=
1
50
CONTINUE
DO
60
I
=
N1P1
,
N
COLTYP
(
I
)
=
3
60
CONTINUE
*
*
K
=
0
K2
=
N
+
1
DO
70
J
=
1
,
N
NJ
=
INDX
(
J
)
IF
(
RHO
*
ABS
(
Z
(
NJ
)
)
.LE.
TOL
)
THEN
*
*
Deflate
due
to
small
z
component
.
*
K2
=
K2
-
1
COLTYP
(
NJ
)
=
4
INDXP
(
K2
)
=
NJ
IF
(
J
.EQ.
N
)
$
GO
TO
100
ELSE
PJ
=
NJ
GO
TO
80
END IF
70
CONTINUE
80
CONTINUE
J
=
J
+
1
NJ
=
INDX
(
J
)
IF
(
J
.GT.
N
)
$
GO
TO
100
IF
(
RHO
*
ABS
(
Z
(
NJ
)
)
.LE.
TOL
)
THEN
*
*
Deflate
due
to
small
z
component
.
*
K2
=
K2
-
1
COLTYP
(
NJ
)
=
4
INDXP
(
K2
)
=
NJ
ELSE
*
*
Check
if
eigenvalues
are
close
enough
to
allow
deflation
.
*
S
=
Z
(
PJ
)
C
=
Z
(
NJ
)
*
*
Find
sqrt
(
a
**
2
+
b
**
2
)
without
overflow
or
*
destructive
underflow
.
*
TAU
=
DLAPY2
(
C
,
S
)
T
=
D
(
NJ
)
-
D
(
PJ
)
C
=
C
/
TAU
S
=
-
S
/
TAU
IF
(
ABS
(
T
*
C
*
S
)
.LE.
TOL
)
THEN
*
*
Deflation
is
possible
.
*
Z
(
NJ
)
=
TAU
Z
(
PJ
)
=
ZERO
IF
(
COLTYP
(
NJ
)
.NE.
COLTYP
(
PJ
)
)
$
COLTYP
(
NJ
)
=
2
COLTYP
(
PJ
)
=
4
CALL
DROT
(
N
,
Q
(
1
,
PJ
),
1
,
Q
(
1
,
NJ
),
1
,
C
,
S
)
T
=
D
(
PJ
)
*
C
**
2
+
D
(
NJ
)
*
S
**
2
D
(
NJ
)
=
D
(
PJ
)
*
S
**
2
+
D
(
NJ
)
*
C
**
2
D
(
PJ
)
=
T
K2
=
K2
-
1
I
=
1
90
CONTINUE
IF
(
K2
+
I
.LE.
N
)
THEN
IF
(
D
(
PJ
)
.LT.
D
(
INDXP
(
K2
+
I
)
)
)
THEN
INDXP
(
K2
+
I
-
1
)
=
INDXP
(
K2
+
I
)
INDXP
(
K2
+
I
)
=
PJ
I
=
I
+
1
GO
TO
90
ELSE
INDXP
(
K2
+
I
-
1
)
=
PJ
END IF
ELSE
INDXP
(
K2
+
I
-
1
)
=
PJ
END IF
PJ
=
NJ
ELSE
K
=
K
+
1
DLAMDA
(
K
)
=
D
(
PJ
)
W
(
K
)
=
Z
(
PJ
)
INDXP
(
K
)
=
PJ
PJ
=
NJ
END IF
END IF
GO
TO
80
100
CONTINUE
*
*
Record
the
last
eigenvalue
.
*
K
=
K
+
1
DLAMDA
(
K
)
=
D
(
PJ
)
W
(
K
)
=
Z
(
PJ
)
INDXP
(
K
)
=
PJ
*
*
Count
up
the
total
number
of
the
various
types
of
columns
,
then
*
form
a
permutation
which
positions
the
four
column
types
into
*
four
uniform
groups
(
although
one
or
more
of
these
groups
may
be
*
empty
)
.
*
DO
110
J
=
1
,
4
CTOT
(
J
)
=
0
110
CONTINUE
DO
120
J
=
1
,
N
CT
=
COLTYP
(
J
)
CTOT
(
CT
)
=
CTOT
(
CT
)
+
1
120
CONTINUE
*
*
PSM
(
*
)
=
Position
in
SubMatrix
(
of
types
1
through
4
)
*
PSM
(
1
)
=
1
PSM
(
2
)
=
1
+
CTOT
(
1
)
PSM
(
3
)
=
PSM
(
2
)
+
CTOT
(
2
)
PSM
(
4
)
=
PSM
(
3
)
+
CTOT
(
3
)
K
=
N
-
CTOT
(
4
)
*
*
Fill
out
the
INDXC
array
so
that
the
permutation
which
it
induces
*
will
place
all
type
-
1
columns
first
,
all
type
-
2
columns
next
,
*
then
all
type
-
3
's, and finally all type-4'
s
.
*
DO
130
J
=
1
,
N
JS
=
INDXP
(
J
)
CT
=
COLTYP
(
JS
)
INDX
(
PSM
(
CT
)
)
=
JS
INDXC
(
PSM
(
CT
)
)
=
J
PSM
(
CT
)
=
PSM
(
CT
)
+
1
130
CONTINUE
*
*
Sort
the
eigenvalues
and
corresponding
eigenvectors
into
DLAMDA
*
and
Q2
respectively
.
The
eigenvalues
/
vectors
which
were
not
*
deflated
go
into
the
first
K
slots
of
DLAMDA
and
Q2
respectively
,
*
while
those
which
were
deflated
go
into
the
last
N
-
K
slots
.
*
I
=
1
IQ1
=
1
IQ2
=
1
+
(
CTOT
(
1
)
+
CTOT
(
2
)
)
*
N1
DO
140
J
=
1
,
CTOT
(
1
)
JS
=
INDX
(
I
)
CALL
DCOPY
(
N1
,
Q
(
1
,
JS
),
1
,
Q2
(
IQ1
),
1
)
Z
(
I
)
=
D
(
JS
)
I
=
I
+
1
IQ1
=
IQ1
+
N1
140
CONTINUE
*
DO
150
J
=
1
,
CTOT
(
2
)
JS
=
INDX
(
I
)
CALL
DCOPY
(
N1
,
Q
(
1
,
JS
),
1
,
Q2
(
IQ1
),
1
)
CALL
DCOPY
(
N2
,
Q
(
N1
+
1
,
JS
),
1
,
Q2
(
IQ2
),
1
)
Z
(
I
)
=
D
(
JS
)
I
=
I
+
1
IQ1
=
IQ1
+
N1
IQ2
=
IQ2
+
N2
150
CONTINUE
*
DO
160
J
=
1
,
CTOT
(
3
)
JS
=
INDX
(
I
)
CALL
DCOPY
(
N2
,
Q
(
N1
+
1
,
JS
),
1
,
Q2
(
IQ2
),
1
)
Z
(
I
)
=
D
(
JS
)
I
=
I
+
1
IQ2
=
IQ2
+
N2
160
CONTINUE
*
IQ1
=
IQ2
DO
170
J
=
1
,
CTOT
(
4
)
JS
=
INDX
(
I
)
CALL
DCOPY
(
N
,
Q
(
1
,
JS
),
1
,
Q2
(
IQ2
),
1
)
IQ2
=
IQ2
+
N
Z
(
I
)
=
D
(
JS
)
I
=
I
+
1
170
CONTINUE
*
*
The
deflated
eigenvalues
and
their
corresponding
vectors
go
back
*
into
the
last
N
-
K
slots
of
D
and
Q
respectively
.
*
IF
(
K
.LT.
N
)
THEN
CALL
DLACPY
(
'A'
,
N
,
CTOT
(
4
),
Q2
(
IQ1
),
N
,
$
Q
(
1
,
K
+
1
),
LDQ
)
CALL
DCOPY
(
N
-
K
,
Z
(
K
+
1
),
1
,
D
(
K
+
1
),
1
)
END IF
*
*
Copy
CTOT
into
COLTYP
for
referencing
in
DLAED3
.
*
DO
180
J
=
1
,
4
COLTYP
(
J
)
=
CTOT
(
J
)
180
CONTINUE
*
190
CONTINUE
RETURN
*
*
End
of
DLAED2
*
END
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