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*>
\
brief
\
b
DGEBRD
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DGEBRD
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DGEBRD
(
M
,
N
,
A
,
LDA
,
D
,
E
,
TAUQ
,
TAUP
,
WORK
,
LWORK
,
*
INFO
)
*
*
..
Scalar
Arguments
..
*
INTEGER
INFO
,
LDA
,
LWORK
,
M
,
N
*
..
*
..
Array
Arguments
..
*
DOUBLE PRECISION
A
(
LDA
,
*
),
D
(
*
),
E
(
*
),
TAUP
(
*
),
*
$
TAUQ
(
*
),
WORK
(
*
)
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DGEBRD
reduces
a
general
real
M
-
by
-
N
matrix
A
to
upper
or
lower
*>
bidiagonal
form
B
by
an
orthogonal
transformation
:
Q
**
T
*
A
*
P
=
B
.
*>
*>
If
m
>=
n
,
B
is
upper
bidiagonal
;
if
m
<
n
,
B
is
lower
bidiagonal
.
*>
\
endverbatim
*
*
Arguments
:
*
==========
*
*>
\
param
[
in
]
M
*>
\
verbatim
*>
M
is
INTEGER
*>
The
number
of
rows
in
the
matrix
A
.
M
>=
0.
*>
\
endverbatim
*>
*>
\
param
[
in
]
N
*>
\
verbatim
*>
N
is
INTEGER
*>
The
number
of
columns
in
the
matrix
A
.
N
>=
0.
*>
\
endverbatim
*>
*>
\
param
[
in
,
out
]
A
*>
\
verbatim
*>
A
is
DOUBLE PRECISION
array
,
dimension
(
LDA
,
N
)
*>
On
entry
,
the
M
-
by
-
N
general
matrix
to
be
reduced
.
*>
On
exit
,
*>
if
m
>=
n
,
the
diagonal
and
the
first
superdiagonal
are
*>
overwritten
with
the
upper
bidiagonal
matrix
B
;
the
*>
elements
below
the
diagonal
,
with
the
array
TAUQ
,
represent
*>
the
orthogonal
matrix
Q
as
a
product
of
elementary
*>
reflectors
,
and
the
elements
above
the
first
superdiagonal
,
*>
with
the
array
TAUP
,
represent
the
orthogonal
matrix
P
as
*>
a
product
of
elementary
reflectors
;
*>
if
m
<
n
,
the
diagonal
and
the
first
subdiagonal
are
*>
overwritten
with
the
lower
bidiagonal
matrix
B
;
the
*>
elements
below
the
first
subdiagonal
,
with
the
array
TAUQ
,
*>
represent
the
orthogonal
matrix
Q
as
a
product
of
*>
elementary
reflectors
,
and
the
elements
above
the
diagonal
,
*>
with
the
array
TAUP
,
represent
the
orthogonal
matrix
P
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
,
M
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
D
*>
\
verbatim
*>
D
is
DOUBLE PRECISION
array
,
dimension
(
min
(
M
,
N
))
*>
The
diagonal
elements
of
the
bidiagonal
matrix
B
:
*>
D
(
i
)
=
A
(
i
,
i
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
E
*>
\
verbatim
*>
E
is
DOUBLE PRECISION
array
,
dimension
(
min
(
M
,
N
)
-
1
)
*>
The
off
-
diagonal
elements
of
the
bidiagonal
matrix
B
:
*>
if
m
>=
n
,
E
(
i
)
=
A
(
i
,
i
+
1
)
for
i
=
1
,
2
,
...
,
n
-
1
;
*>
if
m
<
n
,
E
(
i
)
=
A
(
i
+
1
,
i
)
for
i
=
1
,
2
,
...
,
m
-
1.
*>
\
endverbatim
*>
*>
\
param
[
out
]
TAUQ
*>
\
verbatim
*>
TAUQ
is
DOUBLE PRECISION
array dimension
(
min
(
M
,
N
))
*>
The
scalar
factors
of
the
elementary
reflectors
which
*>
represent
the
orthogonal
matrix
Q
.
See
Further
Details
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
TAUP
*>
\
verbatim
*>
TAUP
is
DOUBLE PRECISION
array
,
dimension
(
min
(
M
,
N
))
*>
The
scalar
factors
of
the
elementary
reflectors
which
*>
represent
the
orthogonal
matrix
P
.
See
Further
Details
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
WORK
*>
\
verbatim
*>
WORK
is
DOUBLE PRECISION
array
,
dimension
(
MAX
(
1
,
LWORK
))
*>
On
exit
,
if
INFO
=
0
,
WORK
(
1
)
returns
the
optimal
LWORK
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
LWORK
*>
\
verbatim
*>
LWORK
is
INTEGER
*>
The
length
of
the
array
WORK
.
LWORK
>=
max
(
1
,
M
,
N
)
.
*>
For
optimum
performance
LWORK
>=
(
M
+
N
)
*
NB
,
where
NB
*>
is
the
optimal
blocksize
.
*>
*>
If
LWORK
=
-
1
,
then
a
workspace
query
is
assumed
;
the
routine
*>
only
calculates
the
optimal
size
of
the
WORK
array
,
returns
*>
this
value
as
the
first
entry
of
the
WORK
array
,
and
no
error
*>
message
related
to
LWORK
is
issued
by
XERBLA
.
*>
\
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
November
2011
*
*>
\
ingroup
doubleGEcomputational
*
*>
\
par
Further
Details
:
*
=====================
*>
*>
\
verbatim
*>
*>
The
matrices
Q
and
P
are
represented
as
products
of
elementary
*>
reflectors
:
*>
*>
If
m
>=
n
,
*>
*>
Q
=
H
(
1
)
H
(
2
)
.
.
.
H
(
n
)
and
P
=
G
(
1
)
G
(
2
)
.
.
.
G
(
n
-
1
)
*>
*>
Each
H
(
i
)
and
G
(
i
)
has
the
form
:
*>
*>
H
(
i
)
=
I
-
tauq
*
v
*
v
**
T
and
G
(
i
)
=
I
-
taup
*
u
*
u
**
T
*>
*>
where
tauq
and
taup
are
real
scalars
,
and
v
and
u
are
real
vectors
;
*>
v
(
1
:
i
-
1
)
=
0
,
v
(
i
)
=
1
,
and
v
(
i
+
1
:
m
)
is
stored
on
exit
in
A
(
i
+
1
:
m
,
i
);
*>
u
(
1
:
i
)
=
0
,
u
(
i
+
1
)
=
1
,
and
u
(
i
+
2
:
n
)
is
stored
on
exit
in
A
(
i
,
i
+
2
:
n
);
*>
tauq
is
stored
in
TAUQ
(
i
)
and
taup
in
TAUP
(
i
)
.
*>
*>
If
m
<
n
,
*>
*>
Q
=
H
(
1
)
H
(
2
)
.
.
.
H
(
m
-
1
)
and
P
=
G
(
1
)
G
(
2
)
.
.
.
G
(
m
)
*>
*>
Each
H
(
i
)
and
G
(
i
)
has
the
form
:
*>
*>
H
(
i
)
=
I
-
tauq
*
v
*
v
**
T
and
G
(
i
)
=
I
-
taup
*
u
*
u
**
T
*>
*>
where
tauq
and
taup
are
real
scalars
,
and
v
and
u
are
real
vectors
;
*>
v
(
1
:
i
)
=
0
,
v
(
i
+
1
)
=
1
,
and
v
(
i
+
2
:
m
)
is
stored
on
exit
in
A
(
i
+
2
:
m
,
i
);
*>
u
(
1
:
i
-
1
)
=
0
,
u
(
i
)
=
1
,
and
u
(
i
+
1
:
n
)
is
stored
on
exit
in
A
(
i
,
i
+
1
:
n
);
*>
tauq
is
stored
in
TAUQ
(
i
)
and
taup
in
TAUP
(
i
)
.
*>
*>
The
contents
of
A
on
exit
are
illustrated
by
the
following
examples
:
*>
*>
m
=
6
and
n
=
5
(
m
>
n
):
m
=
5
and
n
=
6
(
m
<
n
):
*>
*>
(
d
e
u1
u1
u1
)
(
d
u1
u1
u1
u1
u1
)
*>
(
v1
d
e
u2
u2
)
(
e
d
u2
u2
u2
u2
)
*>
(
v1
v2
d
e
u3
)
(
v1
e
d
u3
u3
u3
)
*>
(
v1
v2
v3
d
e
)
(
v1
v2
e
d
u4
u4
)
*>
(
v1
v2
v3
v4
d
)
(
v1
v2
v3
e
d
u5
)
*>
(
v1
v2
v3
v4
v5
)
*>
*>
where
d
and
e
denote
diagonal
and
off
-
diagonal
elements
of
B
,
vi
*>
denotes
an
element
of
the
vector
defining
H
(
i
),
and
ui
an
element
of
*>
the
vector
defining
G
(
i
)
.
*>
\
endverbatim
*>
*
=====================================================================
SUBROUTINE
DGEBRD
(
M
,
N
,
A
,
LDA
,
D
,
E
,
TAUQ
,
TAUP
,
WORK
,
LWORK
,
$
INFO
)
*
*
--
LAPACK
computational
routine
(
version
3.4.0
)
--
*
--
LAPACK
is
a
software
package
provided
by
Univ
.
of
Tennessee
,
--
*
--
Univ
.
of
California
Berkeley
,
Univ
.
of
Colorado
Denver
and
NAG
Ltd
..
--
*
November
2011
*
*
..
Scalar
Arguments
..
INTEGER
INFO
,
LDA
,
LWORK
,
M
,
N
*
..
*
..
Array
Arguments
..
DOUBLE PRECISION
A
(
LDA
,
*
),
D
(
*
),
E
(
*
),
TAUP
(
*
),
$
TAUQ
(
*
),
WORK
(
*
)
*
..
*
*
=====================================================================
*
*
..
Parameters
..
DOUBLE PRECISION
ONE
PARAMETER
(
ONE
=
1.0
D
+
0
)
*
..
*
..
Local
Scalars
..
LOGICAL
LQUERY
INTEGER
I
,
IINFO
,
J
,
LDWRKX
,
LDWRKY
,
LWKOPT
,
MINMN
,
NB
,
$
NBMIN
,
NX
DOUBLE PRECISION
WS
*
..
*
..
External
Subroutines
..
EXTERNAL
DGEBD2
,
DGEMM
,
DLABRD
,
XERBLA
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
DBLE
,
MAX
,
MIN
*
..
*
..
External
Functions
..
INTEGER
ILAENV
EXTERNAL
ILAENV
*
..
*
..
Executable
Statements
..
*
*
Test
the
input
parameters
*
INFO
=
0
NB
=
MAX
(
1
,
ILAENV
(
1
,
'DGEBRD'
,
' '
,
M
,
N
,
-
1
,
-
1
)
)
LWKOPT
=
(
M
+
N
)
*
NB
WORK
(
1
)
=
DBLE
(
LWKOPT
)
LQUERY
=
(
LWORK
.EQ.
-
1
)
IF
(
M
.LT.
0
)
THEN
INFO
=
-
1
ELSE IF
(
N
.LT.
0
)
THEN
INFO
=
-
2
ELSE IF
(
LDA
.LT.
MAX
(
1
,
M
)
)
THEN
INFO
=
-
4
ELSE IF
(
LWORK
.LT.
MAX
(
1
,
M
,
N
)
.AND.
.NOT.
LQUERY
)
THEN
INFO
=
-
10
END IF
IF
(
INFO
.LT.
0
)
THEN
CALL
XERBLA
(
'DGEBRD'
,
-
INFO
)
RETURN
ELSE IF
(
LQUERY
)
THEN
RETURN
END IF
*
*
Quick
return if
possible
*
MINMN
=
MIN
(
M
,
N
)
IF
(
MINMN
.EQ.
0
)
THEN
WORK
(
1
)
=
1
RETURN
END IF
*
WS
=
MAX
(
M
,
N
)
LDWRKX
=
M
LDWRKY
=
N
*
IF
(
NB
.GT.
1
.AND.
NB
.LT.
MINMN
)
THEN
*
*
Set
the
crossover
point
NX
.
*
NX
=
MAX
(
NB
,
ILAENV
(
3
,
'DGEBRD'
,
' '
,
M
,
N
,
-
1
,
-
1
)
)
*
*
Determine
when
to
switch
from
blocked
to
unblocked
code
.
*
IF
(
NX
.LT.
MINMN
)
THEN
WS
=
(
M
+
N
)
*
NB
IF
(
LWORK
.LT.
WS
)
THEN
*
*
Not
enough
work
space
for
the
optimal
NB
,
consider
using
*
a
smaller
block
size
.
*
NBMIN
=
ILAENV
(
2
,
'DGEBRD'
,
' '
,
M
,
N
,
-
1
,
-
1
)
IF
(
LWORK
.GE.
(
M
+
N
)
*
NBMIN
)
THEN
NB
=
LWORK
/
(
M
+
N
)
ELSE
NB
=
1
NX
=
MINMN
END IF
END IF
END IF
ELSE
NX
=
MINMN
END IF
*
DO
30
I
=
1
,
MINMN
-
NX
,
NB
*
*
Reduce
rows
and
columns
i
:
i
+
nb
-
1
to
bidiagonal
form
and
return
*
the
matrices
X
and
Y
which
are
needed
to
update
the
unreduced
*
part
of
the
matrix
*
CALL
DLABRD
(
M
-
I
+
1
,
N
-
I
+
1
,
NB
,
A
(
I
,
I
),
LDA
,
D
(
I
),
E
(
I
),
$
TAUQ
(
I
),
TAUP
(
I
),
WORK
,
LDWRKX
,
$
WORK
(
LDWRKX
*
NB
+
1
),
LDWRKY
)
*
*
Update
the
trailing
submatrix
A
(
i
+
nb
:
m
,
i
+
nb
:
n
),
using
an
update
*
of
the
form
A
:
=
A
-
V
*
Y
**
T
-
X
*
U
**
T
*
CALL
DGEMM
(
'No transpose'
,
'Transpose'
,
M
-
I
-
NB
+
1
,
N
-
I
-
NB
+
1
,
$
NB
,
-
ONE
,
A
(
I
+
NB
,
I
),
LDA
,
$
WORK
(
LDWRKX
*
NB
+
NB
+
1
),
LDWRKY
,
ONE
,
$
A
(
I
+
NB
,
I
+
NB
),
LDA
)
CALL
DGEMM
(
'No transpose'
,
'No transpose'
,
M
-
I
-
NB
+
1
,
N
-
I
-
NB
+
1
,
$
NB
,
-
ONE
,
WORK
(
NB
+
1
),
LDWRKX
,
A
(
I
,
I
+
NB
),
LDA
,
$
ONE
,
A
(
I
+
NB
,
I
+
NB
),
LDA
)
*
*
Copy
diagonal
and
off
-
diagonal
elements
of
B
back
into
A
*
IF
(
M
.GE.
N
)
THEN
DO
10
J
=
I
,
I
+
NB
-
1
A
(
J
,
J
)
=
D
(
J
)
A
(
J
,
J
+
1
)
=
E
(
J
)
10
CONTINUE
ELSE
DO
20
J
=
I
,
I
+
NB
-
1
A
(
J
,
J
)
=
D
(
J
)
A
(
J
+
1
,
J
)
=
E
(
J
)
20
CONTINUE
END IF
30
CONTINUE
*
*
Use
unblocked
code
to
reduce
the
remainder
of
the
matrix
*
CALL
DGEBD2
(
M
-
I
+
1
,
N
-
I
+
1
,
A
(
I
,
I
),
LDA
,
D
(
I
),
E
(
I
),
$
TAUQ
(
I
),
TAUP
(
I
),
WORK
,
IINFO
)
WORK
(
1
)
=
WS
RETURN
*
*
End
of
DGEBRD
*
END
Event Timeline
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