Contents

# ?gesvj

Computes the singular value decomposition of a real matrix using Jacobi plane rotations.

## Syntax

Include Files
• mkl.h
Description
The routine computes the singular value decomposition (SVD) of a real or complex
m
-by-
n
matrix
A
, where
m
n
.
The SVD of
A
is written as
A
=
U
*
Σ
*
V
T
for real flavors, or
A
=
U
*
Σ
*
V
H
for complex flavors,
where
Σ
is an
m
-by-
n
diagonal matrix,
U
is an
m
-by-
n
orthonormal matrix, and
V
is an
n
-by-
n
orthogonal/unitary matrix. The diagonal elements of
Σ
are the singular values of
A
; the columns of
U
and
V
are the left and right singular vectors of
A
, respectively. The matrices
U
and
V
are computed and stored in the arrays
u
and
v
, respectively. The diagonal of
Σ
is computed and stored in the array
sva
.
The
?gesvj
routine can sometimes compute tiny singular values and their singular vectors much more accurately than other SVD routines.
The
n
-by-
n
orthogonal matrix
V
is obtained as a product of Jacobi plane rotations. The rotations are implemented as fast scaled rotations of Anda and Park [AndaPark94]. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac ([Drmac08-4]) is used. Pivot strategy uses column interchanges of de Rijk ([deRijk98]). The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [Demmel92]. The condition number that determines the accuracy in the full rank case is essentially
where
κ
(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [Drmac08-1], [Drmac08-2].
The computational range for the nonzero singular values is the machine number interval (
UNDERFLOW
,
OVERFLOW
). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digit.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
joba
Must be
'L'
,
'U'
or
'G'
.
Specifies the structure of
A
:
If
joba
=
'L'
, the input matrix
A
is lower triangular.
If
joba
=
'U'
, the input matrix
A
is upper triangular.
If
joba
=
'G'
, the input matrix
A
is a general
m
-by-
n
,
m
n
.
jobu
Must be
'U'
,
'C'
or
'N'
.
Specifies whether to compute the left singular vectors (columns of
U
):
If
jobu
=
'U'
, the left singular vectors corresponding to the nonzero singular values are computed and returned in the leading columns of
A
. See more details in the description of
a
. The default numerical orthogonality threshold is set to approximately
TOL=CTOL*EPS, CTOL=sqrt(
m
), EPS =
?lamch
(
'E'
)
If
jobu
=
'C'
, analogous to
jobu
=
'U'
, except that you can control the level of numerical orthogonality of the computed left singular vectors.
TOL
can be set to
TOL=CTOL*EPS
, where
CTOL
is given on input in the array
stat
. No
CTOL
smaller than ONE is allowed.
CTOL
greater than
1 / EPS
is meaningless. The option
'C'
can be used if
m
*EPS
is satisfactory orthogonality of the computed left singular vectors, so
CTOL=
m
could save a few sweeps of Jacobi rotations. See the descriptions of
a
and
stat[0]
.
If
jobu
=
'N'
,
u
is not computed. However, see the description of
a
.
jobv
Must be
'V'
,
'A'
or
'N'
.
Specifies whether to compute the right singular vectors, that is, the matrix
V
:
If
jobv
=
'V'
, the matrix
V
is computed and returned in the array
v
.
If
jobv
=
'A'
, the Jacobi rotations are applied to the
mv
-by
n
array
v
. In other words, the right singular vector matrix
V
is not computed explicitly, instead it is applied to an
mv
-by
n
matrix initially stored in the first
mv
rows of
V
.
If
jobv
=
'N'
, the matrix
V
is not computed and the array
v
is not referenced.
m
The number of rows of the input matrix
A
.
1/slamch('E')>
m
0
for
sgesvj
.
1/dlamch('E')>
m
0
for
dgesvj
.
n
The number of columns in the input matrix
A
;
m
n
0.
a
,
v
Array
a
(size at least
lda
*
n
for column major layout and
lda
*
m
for row major layout)
is an array containing the
m
-by-
n
matrix
A
.
Array
v
(size at least max(1,
ldv
*
n
)) contains, if
jobv
= 'A'
the
mv
-by-
n
matrix to be post-multiplied by Jacobi rotations
.
lda
The leading dimension of the array
a
. Must be at least
max(1,
m
)
for column major layout and at least max(1,
n
) for row major layout
.
mv
If
jobv
=
'A'
, the product of Jacobi rotations in
?gesvj
is applied to the first
mv
rows of
v
. See the description of
jobv
.
0
mv
ldv
.
ldv
The leading dimension of the array
v
;
ldv
1
.
jobv
=
'V'
,
ldv
max(1,
n
)
.
jobv
=
'A'
,
ldv
max(1,
mv
)
for column major layout and
ldv
max(1,
n
)
for row major layout
.
stat
Array size 6. If
jobu
= 'C'
,
stat
[0] =
CTOL
, where
CTOL
defines the threshold for convergence. The process stops if all columns of
A
are mutually orthogonal up to
CTOL
*
EPS
, where
EPS
=
?lamch
('E')
. It is required that
CTOL
1 - that is, it is not allowed to force the routine to obtain orthogonality below
ε
.
Output Parameters
a
On exit:
If
jobu
=
'U'
or
jobu
=
'C'
:
• if
info
= 0
A
contain left singular vectors corresponding to the computed singular values of
a
that are above the underflow threshold
?lamch
(
'S'
), that is, non-zero singular values. The number of the computed non-zero singular values is returned in
stat[1]
. Also see the descriptions of
sva
and
stat
. The computed columns of
u
are mutually numerically orthogonal up to approximately
TOL=sqrt(
m
)*EPS
(default); or
TOL=CTOL*EPS
jobu
=
'C'
, see the description of
jobu
.
• if
info
> 0
, the procedure
?gesvj
did not converge in the given number of iterations (sweeps). In that case, the computed columns of
u
may not be orthogonal up to
TOL
. The output
u
(stored in
a
),
sigma
(given by the computed singular values in
sva
(1:n)
) and
v
is still a decomposition of the input matrix
A
in the sense that the residual
||
A
-scale*
U
*
sigma
*
V
T
||
2
/ ||
A
||
2
for real flavors or
||
A
-scale*
U
*
sigma
*
V
H
||
2
/ ||
A
||
2
for complex flavors (where
scale =
stat
[0]
) is small.
If
jobu
=
'N'
:
• if
info
= 0
, note that the left singular vectors are 'for free' in the one-sided Jacobi SVD algorithm. However, if only the singular values are needed, the level of numerical orthogonality of
u
is not an issue and iterations are stopped when the columns of the iterated matrix are numerically orthogonal up to approximately
m
*EPS
. Thus, on exit,
a
contains the columns of
u
scaled with the corresponding singular values.
• if
info
> 0
, the procedure
?gesvj
did not converge in the given number of iterations (sweeps).
sva
Array size
n
.
If
info
= 0
, depending on the value
scale
=
stat[0]
, where
scale
is the scaling factor:
• if
scale
= 1
,
sva
[0:
n
- 1]
contains the computed singular values of
a
.
• if
scale
1
, the singular values of
a
are
scale
*
sva
(1:n)
, and this factored representation is due to the fact that some of the singular values of
a
might underflow or overflow.
If
info
> 0
, the procedure
?gesvj
did not converge in the given number of iterations (sweeps) and
scale
*
sva
(1:n)
may not be accurate.
v
On exit:
If
jobv
=
'V'
, contains the
n
-by-
n
matrix of the right singular vectors.
If
jobv
=
'A'
, then
v
contains the product of the computed right singular vector matrix and the initial matrix in the array
v
.
If
jobv
=
'N'
,
v
is not referenced.
stat
On exit,
stat[0]
=
scale
is the scaling factor such that
scale
*
sva
(1:
n
)
are the computed singular values of
A
. See the description of
sva
.
stat[1]
is the number of the computed nonzero singular values.
stat[2]
is the number of the computed singular values that are larger than the underflow threshold.
stat[3]
is the number of sweeps of Jacobi rotations needed for numerical convergence.
stat[4]
=
max_{i
j} |COS(A(:,i),A(:,j))|
in the last sweep. This is useful information in cases when
?gesvj
did not converge, as it can be used to estimate whether the output is still useful and for post festum analysis.
stat[5]
is the largest absolute value over all sines of the Jacobi rotation angles in the last sweep. It can be useful in a post festum analysis.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
If
info
> 0
, the function did not converge in the maximal number (30) of sweeps. The output may still be useful. See the description of
stat
.