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dlaed9.f
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*>
\
brief
\
b
DLAED9
used
by
sstedc
.
Finds
the
roots
of
the
secular
equation
and
updates
the
eigenvectors
.
Used
when
the
original
matrix
is
dense
.
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DLAED9
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed9.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed9.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed9.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DLAED9
(
K
,
KSTART
,
KSTOP
,
N
,
D
,
Q
,
LDQ
,
RHO
,
DLAMDA
,
W
,
*
S
,
LDS
,
INFO
)
*
*
..
Scalar
Arguments
..
*
INTEGER
INFO
,
K
,
KSTART
,
KSTOP
,
LDQ
,
LDS
,
N
*
DOUBLE PRECISION
RHO
*
..
*
..
Array
Arguments
..
*
DOUBLE PRECISION
D
(
*
),
DLAMDA
(
*
),
Q
(
LDQ
,
*
),
S
(
LDS
,
*
),
*
$
W
(
*
)
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DLAED9
finds
the
roots
of
the
secular
equation
,
as
defined
by
the
*>
values
in
D
,
Z
,
and
RHO
,
between
KSTART
and
KSTOP
.
It
makes
the
*>
appropriate
calls
to
DLAED4
and
then
stores
the
new
matrix
of
*>
eigenvectors
for
use
in
calculating
the
next
level
of
Z
vectors
.
*>
\
endverbatim
*
*
Arguments
:
*
==========
*
*>
\
param
[
in
]
K
*>
\
verbatim
*>
K
is
INTEGER
*>
The
number
of
terms
in
the
rational
function
to
be
solved
by
*>
DLAED4
.
K
>=
0.
*>
\
endverbatim
*>
*>
\
param
[
in
]
KSTART
*>
\
verbatim
*>
KSTART
is
INTEGER
*>
\
endverbatim
*>
*>
\
param
[
in
]
KSTOP
*>
\
verbatim
*>
KSTOP
is
INTEGER
*>
The
updated
eigenvalues
Lambda
(
I
),
KSTART
<=
I
<=
KSTOP
*>
are
to
be
computed
.
1
<=
KSTART
<=
KSTOP
<=
K
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
N
*>
\
verbatim
*>
N
is
INTEGER
*>
The
number
of
rows
and
columns
in
the
Q
matrix
.
*>
N
>=
K
(
delation
may
result
in
N
>
K
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
D
*>
\
verbatim
*>
D
is
DOUBLE PRECISION
array
,
dimension
(
N
)
*>
D
(
I
)
contains
the
updated
eigenvalues
*>
for
KSTART
<=
I
<=
KSTOP
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
Q
*>
\
verbatim
*>
Q
is
DOUBLE PRECISION
array
,
dimension
(
LDQ
,
N
)
*>
\
endverbatim
*>
*>
\
param
[
in
]
LDQ
*>
\
verbatim
*>
LDQ
is
INTEGER
*>
The
leading
dimension
of
the
array
Q
.
LDQ
>=
max
(
1
,
N
)
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
RHO
*>
\
verbatim
*>
RHO
is
DOUBLE PRECISION
*>
The
value
of
the
parameter
in
the
rank
one
update
equation
.
*>
RHO
>=
0
required
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
DLAMDA
*>
\
verbatim
*>
DLAMDA
is
DOUBLE PRECISION
array
,
dimension
(
K
)
*>
The
first
K
elements
of
this
array
contain
the
old
roots
*>
of
the
deflated
updating
problem
.
These
are
the
poles
*>
of
the
secular
equation
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
W
*>
\
verbatim
*>
W
is
DOUBLE PRECISION
array
,
dimension
(
K
)
*>
The
first
K
elements
of
this
array
contain
the
components
*>
of
the
deflation
-
adjusted
updating
vector
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
S
*>
\
verbatim
*>
S
is
DOUBLE PRECISION
array
,
dimension
(
LDS
,
K
)
*>
Will
contain
the
eigenvectors
of
the
repaired
matrix
which
*>
will
be
stored
for
subsequent
Z
vector
calculation
and
*>
multiplied
by
the
previously
accumulated
eigenvectors
*>
to
update
the
system
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
LDS
*>
\
verbatim
*>
LDS
is
INTEGER
*>
The
leading
dimension
of
S
.
LDS
>=
max
(
1
,
K
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
INFO
*>
\
verbatim
*>
INFO
is
INTEGER
*>
=
0
:
successful
exit
.
*>
<
0
:
if
INFO
=
-
i
,
the
i
-
th
argument
had
an
illegal
value
.
*>
>
0
:
if
INFO
=
1
,
an
eigenvalue
did
not
converge
*>
\
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
*
*
=====================================================================
SUBROUTINE
DLAED9
(
K
,
KSTART
,
KSTOP
,
N
,
D
,
Q
,
LDQ
,
RHO
,
DLAMDA
,
W
,
$
S
,
LDS
,
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
,
KSTART
,
KSTOP
,
LDQ
,
LDS
,
N
DOUBLE PRECISION
RHO
*
..
*
..
Array
Arguments
..
DOUBLE PRECISION
D
(
*
),
DLAMDA
(
*
),
Q
(
LDQ
,
*
),
S
(
LDS
,
*
),
$
W
(
*
)
*
..
*
*
=====================================================================
*
*
..
Local
Scalars
..
INTEGER
I
,
J
DOUBLE PRECISION
TEMP
*
..
*
..
External
Functions
..
DOUBLE PRECISION
DLAMC3
,
DNRM2
EXTERNAL
DLAMC3
,
DNRM2
*
..
*
..
External
Subroutines
..
EXTERNAL
DCOPY
,
DLAED4
,
XERBLA
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
MAX
,
SIGN
,
SQRT
*
..
*
..
Executable
Statements
..
*
*
Test
the
input
parameters
.
*
INFO
=
0
*
IF
(
K
.LT.
0
)
THEN
INFO
=
-
1
ELSE IF
(
KSTART
.LT.
1
.OR.
KSTART
.GT.
MAX
(
1
,
K
)
)
THEN
INFO
=
-
2
ELSE IF
(
MAX
(
1
,
KSTOP
)
.LT.
KSTART
.OR.
KSTOP
.GT.
MAX
(
1
,
K
)
)
$
THEN
INFO
=
-
3
ELSE IF
(
N
.LT.
K
)
THEN
INFO
=
-
4
ELSE IF
(
LDQ
.LT.
MAX
(
1
,
K
)
)
THEN
INFO
=
-
7
ELSE IF
(
LDS
.LT.
MAX
(
1
,
K
)
)
THEN
INFO
=
-
12
END IF
IF
(
INFO
.NE.
0
)
THEN
CALL
XERBLA
(
'DLAED9'
,
-
INFO
)
RETURN
END IF
*
*
Quick
return if
possible
*
IF
(
K
.EQ.
0
)
$
RETURN
*
*
Modify
values
DLAMDA
(
i
)
to
make
sure
all
DLAMDA
(
i
)
-
DLAMDA
(
j
)
can
*
be
computed
with
high
relative
accuracy
(
barring
over
/
underflow
)
.
*
This
is
a
problem
on
machines
without
a
guard
digit
in
*
add
/
subtract
(
Cray
XMP
,
Cray
YMP
,
Cray
C
90
and
Cray
2
)
.
*
The
following
code
replaces
DLAMDA
(
I
)
by
2
*
DLAMDA
(
I
)
-
DLAMDA
(
I
),
*
which
on
any
of
these
machines
zeros
out
the
bottommost
*
bit
of
DLAMDA
(
I
)
if
it
is
1
;
this
makes
the
subsequent
*
subtractions
DLAMDA
(
I
)
-
DLAMDA
(
J
)
unproblematic
when
cancellation
*
occurs
.
On
binary
machines
with
a
guard
digit
(
almost
all
*
machines
)
it
does
not
change
DLAMDA
(
I
)
at
all
.
On
hexadecimal
*
and
decimal
machines
with
a
guard
digit
,
it
slightly
*
changes
the
bottommost
bits
of
DLAMDA
(
I
)
.
It
does
not
account
*
for
hexadecimal
or
decimal
machines
without
guard
digits
*
(
we
know
of
none
)
.
We
use
a
subroutine call
to
compute
*
2
*
DLAMBDA
(
I
)
to
prevent
optimizing
compilers
from
eliminating
*
this
code
.
*
DO
10
I
=
1
,
N
DLAMDA
(
I
)
=
DLAMC3
(
DLAMDA
(
I
),
DLAMDA
(
I
)
)
-
DLAMDA
(
I
)
10
CONTINUE
*
DO
20
J
=
KSTART
,
KSTOP
CALL
DLAED4
(
K
,
J
,
DLAMDA
,
W
,
Q
(
1
,
J
),
RHO
,
D
(
J
),
INFO
)
*
*
If
the
zero
finder
fails
,
the
computation
is
terminated
.
*
IF
(
INFO
.NE.
0
)
$
GO
TO
120
20
CONTINUE
*
IF
(
K
.EQ.
1
.OR.
K
.EQ.
2
)
THEN
DO
40
I
=
1
,
K
DO
30
J
=
1
,
K
S
(
J
,
I
)
=
Q
(
J
,
I
)
30
CONTINUE
40
CONTINUE
GO
TO
120
END IF
*
*
Compute
updated
W
.
*
CALL
DCOPY
(
K
,
W
,
1
,
S
,
1
)
*
*
Initialize
W
(
I
)
=
Q
(
I
,
I
)
*
CALL
DCOPY
(
K
,
Q
,
LDQ
+
1
,
W
,
1
)
DO
70
J
=
1
,
K
DO
50
I
=
1
,
J
-
1
W
(
I
)
=
W
(
I
)
*
(
Q
(
I
,
J
)
/
(
DLAMDA
(
I
)
-
DLAMDA
(
J
)
)
)
50
CONTINUE
DO
60
I
=
J
+
1
,
K
W
(
I
)
=
W
(
I
)
*
(
Q
(
I
,
J
)
/
(
DLAMDA
(
I
)
-
DLAMDA
(
J
)
)
)
60
CONTINUE
70
CONTINUE
DO
80
I
=
1
,
K
W
(
I
)
=
SIGN
(
SQRT
(
-
W
(
I
)
),
S
(
I
,
1
)
)
80
CONTINUE
*
*
Compute
eigenvectors
of
the
modified
rank
-
1
modification
.
*
DO
110
J
=
1
,
K
DO
90
I
=
1
,
K
Q
(
I
,
J
)
=
W
(
I
)
/
Q
(
I
,
J
)
90
CONTINUE
TEMP
=
DNRM2
(
K
,
Q
(
1
,
J
),
1
)
DO
100
I
=
1
,
K
S
(
I
,
J
)
=
Q
(
I
,
J
)
/
TEMP
100
CONTINUE
110
CONTINUE
*
120
CONTINUE
RETURN
*
*
End
of
DLAED9
*
END
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