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dlasv2.f
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*>
\
brief
\
b
DLASV2
computes
the
singular
value
decomposition
of
a
2
-
by
-
2
triangular
matrix
.
*
*
===========
DOCUMENTATION
===========
*
*
Online
html
documentation
available
at
*
http
:
//
www
.
netlib
.
org
/
lapack
/
explore
-
html
/
*
*>
\
htmlonly
*>
Download
DLASV2
+
dependencies
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasv2.f"
>
*>
[
TGZ
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasv2.f"
>
*>
[
ZIP
]
</
a
>
*>
<
a
href
=
"http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasv2.f"
>
*>
[
TXT
]
</
a
>
*>
\
endhtmlonly
*
*
Definition
:
*
===========
*
*
SUBROUTINE
DLASV2
(
F
,
G
,
H
,
SSMIN
,
SSMAX
,
SNR
,
CSR
,
SNL
,
CSL
)
*
*
..
Scalar
Arguments
..
*
DOUBLE PRECISION
CSL
,
CSR
,
F
,
G
,
H
,
SNL
,
SNR
,
SSMAX
,
SSMIN
*
..
*
*
*>
\
par
Purpose
:
*
=============
*>
*>
\
verbatim
*>
*>
DLASV2
computes
the
singular
value
decomposition
of
a
2
-
by
-
2
*>
triangular
matrix
*>
[
F
G
]
*>
[
0
H
].
*>
On
return
,
abs
(
SSMAX
)
is
the
larger
singular
value
,
abs
(
SSMIN
)
is
the
*>
smaller
singular
value
,
and
(
CSL
,
SNL
)
and
(
CSR
,
SNR
)
are
the
left
and
*>
right
singular
vectors
for
abs
(
SSMAX
),
giving
the
decomposition
*>
*>
[
CSL
SNL
]
[
F
G
]
[
CSR
-
SNR
]
=
[
SSMAX
0
]
*>
[
-
SNL
CSL
]
[
0
H
]
[
SNR
CSR
]
[
0
SSMIN
].
*>
\
endverbatim
*
*
Arguments
:
*
==========
*
*>
\
param
[
in
]
F
*>
\
verbatim
*>
F
is
DOUBLE PRECISION
*>
The
(
1
,
1
)
element
of
the
2
-
by
-
2
matrix
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
G
*>
\
verbatim
*>
G
is
DOUBLE PRECISION
*>
The
(
1
,
2
)
element
of
the
2
-
by
-
2
matrix
.
*>
\
endverbatim
*>
*>
\
param
[
in
]
H
*>
\
verbatim
*>
H
is
DOUBLE PRECISION
*>
The
(
2
,
2
)
element
of
the
2
-
by
-
2
matrix
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
SSMIN
*>
\
verbatim
*>
SSMIN
is
DOUBLE PRECISION
*>
abs
(
SSMIN
)
is
the
smaller
singular
value
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
SSMAX
*>
\
verbatim
*>
SSMAX
is
DOUBLE PRECISION
*>
abs
(
SSMAX
)
is
the
larger
singular
value
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
SNL
*>
\
verbatim
*>
SNL
is
DOUBLE PRECISION
*>
\
endverbatim
*>
*>
\
param
[
out
]
CSL
*>
\
verbatim
*>
CSL
is
DOUBLE PRECISION
*>
The
vector
(
CSL
,
SNL
)
is
a
unit
left
singular
vector
for
the
*>
singular
value
abs
(
SSMAX
)
.
*>
\
endverbatim
*>
*>
\
param
[
out
]
SNR
*>
\
verbatim
*>
SNR
is
DOUBLE PRECISION
*>
\
endverbatim
*>
*>
\
param
[
out
]
CSR
*>
\
verbatim
*>
CSR
is
DOUBLE PRECISION
*>
The
vector
(
CSR
,
SNR
)
is
a
unit
right
singular
vector
for
the
*>
singular
value
abs
(
SSMAX
)
.
*>
\
endverbatim
*
*
Authors
:
*
========
*
*>
\
author
Univ
.
of
Tennessee
*>
\
author
Univ
.
of
California
Berkeley
*>
\
author
Univ
.
of
Colorado
Denver
*>
\
author
NAG
Ltd
.
*
*>
\
date
September
2012
*
*>
\
ingroup
auxOTHERauxiliary
*
*>
\
par
Further
Details
:
*
=====================
*>
*>
\
verbatim
*>
*>
Any
input
parameter
may
be
aliased
with
any
output
parameter
.
*>
*>
Barring
over
/
underflow
and
assuming
a
guard
digit
in
subtraction
,
all
*>
output
quantities
are
correct
to
within
a
few
units
in
the
last
*>
place
(
ulps
)
.
*>
*>
In
IEEE
arithmetic
,
the
code
works
correctly
if
one
matrix
element
is
*>
infinite
.
*>
*>
Overflow
will
not
occur
unless
the
largest
singular
value
itself
*>
overflows
or
is
within
a
few
ulps
of
overflow
.
(
On
machines
with
*>
partial
overflow
,
like
the
Cray
,
overflow
may
occur
if
the
largest
*>
singular
value
is
within
a
factor
of
2
of
overflow
.
)
*>
*>
Underflow
is
harmless
if
underflow
is
gradual
.
Otherwise
,
results
*>
may
correspond
to
a
matrix
modified
by
perturbations
of
size
near
*>
the
underflow
threshold
.
*>
\
endverbatim
*>
*
=====================================================================
SUBROUTINE
DLASV2
(
F
,
G
,
H
,
SSMIN
,
SSMAX
,
SNR
,
CSR
,
SNL
,
CSL
)
*
*
--
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
..
DOUBLE PRECISION
CSL
,
CSR
,
F
,
G
,
H
,
SNL
,
SNR
,
SSMAX
,
SSMIN
*
..
*
*
=====================================================================
*
*
..
Parameters
..
DOUBLE PRECISION
ZERO
PARAMETER
(
ZERO
=
0.0
D0
)
DOUBLE PRECISION
HALF
PARAMETER
(
HALF
=
0.5
D0
)
DOUBLE PRECISION
ONE
PARAMETER
(
ONE
=
1.0
D0
)
DOUBLE PRECISION
TWO
PARAMETER
(
TWO
=
2.0
D0
)
DOUBLE PRECISION
FOUR
PARAMETER
(
FOUR
=
4.0
D0
)
*
..
*
..
Local
Scalars
..
LOGICAL
GASMAL
,
SWAP
INTEGER
PMAX
DOUBLE PRECISION
A
,
CLT
,
CRT
,
D
,
FA
,
FT
,
GA
,
GT
,
HA
,
HT
,
L
,
M
,
$
MM
,
R
,
S
,
SLT
,
SRT
,
T
,
TEMP
,
TSIGN
,
TT
*
..
*
..
Intrinsic
Functions
..
INTRINSIC
ABS
,
SIGN
,
SQRT
*
..
*
..
External
Functions
..
DOUBLE PRECISION
DLAMCH
EXTERNAL
DLAMCH
*
..
*
..
Executable
Statements
..
*
FT
=
F
FA
=
ABS
(
FT
)
HT
=
H
HA
=
ABS
(
H
)
*
*
PMAX
points
to
the
maximum
absolute
element
of
matrix
*
PMAX
=
1
if
F
largest
in
absolute
values
*
PMAX
=
2
if
G
largest
in
absolute
values
*
PMAX
=
3
if
H
largest
in
absolute
values
*
PMAX
=
1
SWAP
=
(
HA
.GT.
FA
)
IF
(
SWAP
)
THEN
PMAX
=
3
TEMP
=
FT
FT
=
HT
HT
=
TEMP
TEMP
=
FA
FA
=
HA
HA
=
TEMP
*
*
Now
FA
.ge.
HA
*
END IF
GT
=
G
GA
=
ABS
(
GT
)
IF
(
GA
.EQ.
ZERO
)
THEN
*
*
Diagonal
matrix
*
SSMIN
=
HA
SSMAX
=
FA
CLT
=
ONE
CRT
=
ONE
SLT
=
ZERO
SRT
=
ZERO
ELSE
GASMAL
=
.TRUE.
IF
(
GA
.GT.
FA
)
THEN
PMAX
=
2
IF
(
(
FA
/
GA
)
.LT.
DLAMCH
(
'EPS'
)
)
THEN
*
*
Case
of
very
large
GA
*
GASMAL
=
.FALSE.
SSMAX
=
GA
IF
(
HA
.GT.
ONE
)
THEN
SSMIN
=
FA
/
(
GA
/
HA
)
ELSE
SSMIN
=
(
FA
/
GA
)
*
HA
END IF
CLT
=
ONE
SLT
=
HT
/
GT
SRT
=
ONE
CRT
=
FT
/
GT
END IF
END IF
IF
(
GASMAL
)
THEN
*
*
Normal
case
*
D
=
FA
-
HA
IF
(
D
.EQ.
FA
)
THEN
*
*
Copes
with
infinite
F
or
H
*
L
=
ONE
ELSE
L
=
D
/
FA
END IF
*
*
Note
that
0
.le.
L
.le.
1
*
M
=
GT
/
FT
*
*
Note
that
abs
(
M
)
.le.
1
/
macheps
*
T
=
TWO
-
L
*
*
Note
that
T
.ge.
1
*
MM
=
M
*
M
TT
=
T
*
T
S
=
SQRT
(
TT
+
MM
)
*
*
Note
that
1
.le.
S
.le.
1
+
1
/
macheps
*
IF
(
L
.EQ.
ZERO
)
THEN
R
=
ABS
(
M
)
ELSE
R
=
SQRT
(
L
*
L
+
MM
)
END IF
*
*
Note
that
0
.le.
R
.le.
1
+
1
/
macheps
*
A
=
HALF
*
(
S
+
R
)
*
*
Note
that
1
.le.
A
.le.
1
+
abs
(
M
)
*
SSMIN
=
HA
/
A
SSMAX
=
FA
*
A
IF
(
MM
.EQ.
ZERO
)
THEN
*
*
Note
that
M
is
very
tiny
*
IF
(
L
.EQ.
ZERO
)
THEN
T
=
SIGN
(
TWO
,
FT
)
*
SIGN
(
ONE
,
GT
)
ELSE
T
=
GT
/
SIGN
(
D
,
FT
)
+
M
/
T
END IF
ELSE
T
=
(
M
/
(
S
+
T
)
+
M
/
(
R
+
L
)
)
*
(
ONE
+
A
)
END IF
L
=
SQRT
(
T
*
T
+
FOUR
)
CRT
=
TWO
/
L
SRT
=
T
/
L
CLT
=
(
CRT
+
SRT
*
M
)
/
A
SLT
=
(
HT
/
FT
)
*
SRT
/
A
END IF
END IF
IF
(
SWAP
)
THEN
CSL
=
SRT
SNL
=
CRT
CSR
=
SLT
SNR
=
CLT
ELSE
CSL
=
CLT
SNL
=
SLT
CSR
=
CRT
SNR
=
SRT
END IF
*
*
Correct
signs
of
SSMAX
and
SSMIN
*
IF
(
PMAX
.EQ.
1
)
$
TSIGN
=
SIGN
(
ONE
,
CSR
)
*
SIGN
(
ONE
,
CSL
)
*
SIGN
(
ONE
,
F
)
IF
(
PMAX
.EQ.
2
)
$
TSIGN
=
SIGN
(
ONE
,
SNR
)
*
SIGN
(
ONE
,
CSL
)
*
SIGN
(
ONE
,
G
)
IF
(
PMAX
.EQ.
3
)
$
TSIGN
=
SIGN
(
ONE
,
SNR
)
*
SIGN
(
ONE
,
SNL
)
*
SIGN
(
ONE
,
H
)
SSMAX
=
SIGN
(
SSMAX
,
TSIGN
)
SSMIN
=
SIGN
(
SSMIN
,
TSIGN
*
SIGN
(
ONE
,
F
)
*
SIGN
(
ONE
,
H
)
)
RETURN
*
*
End
of
DLASV2
*
END
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