Contents

# ?bdsvdx

Computes the SVD of a bidiagonal matrix.

## Syntax

Include Files
• mkl.h
Description
?bdsvdx
computes the singular value decomposition (SVD) of a real
n
-by-
n
(upper or lower) bidiagonal matrix
B
,
B
=
U
*
S
*
VT
, where
S
is a diagonal matrix with non-negative diagonal elements (the singular values of
B
), and
U
and
VT
are orthogonal matrices of left and right singular vectors, respectively.
Given an upper bidiagonal
B
with diagonal
d
= [
d
1
d
2
...
d
n
] and superdiagonal
e
= [
e
1
e
2
...
e
n
- 1
],
?bdsvdx
computes the singular value decompositon of
B
through the eigenvalues and eigenvectors of the
n
*2-by-
n
*2 tridiagonal matrix
If (
s
,
u
,
v
) is a singular triplet of
B
with ||
u
|| = ||
v
|| = 1, then (
±
s
,
q
), ||
q
|| = 1, are eigenpairs of TGK, with
, and
.
Given a TGK matrix, one can either
1. compute -
s
, -
v
and change signs so that the singular values (and corresponding vectors) are already in descending order (as in
?gesvd
/
?gesdd
) or
2. compute
s
,
v
and reorder the values (and corresponding vectors).
?bdsvdx
implements (1) by calling
?stevx
(bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
= 'U':
B
is upper bidiagonal;
= 'L': B is lower bidiagonal.
jobz
= 'N': Compute singular values only;
= 'V': Compute singular values and singular vectors.
range
= 'A': Find all singular values.
= 'V': all singular values in the half-open interval [
vl
,
vu
) are found.
= 'I': the
il
-th through
iu
-th singular values are found.
n
The order of the bidiagonal matrix.
n
>= 0.
d
Array, size
n
.
The
n
diagonal elements of the bidiagonal matrix
B
.
e
Array, size (max(1,
n
- 1))
The (
n
- 1) superdiagonal elements of the bidiagonal matrix
B
in elements 1 to
n
- 1.
vl
vl
0.
vu
If
range
='V', the lower and upper bounds of the interval to be searched for singular values.
vu
>
vl
.
Not referenced if
range
= 'A' or 'I'.
il
,
iu
If
range
='I', the indices (in ascending order) of the smallest and largest singular values to be returned.
1
il
iu
min(
m
,
n
), if min(
m
,
n
) > 0.
Not referenced if
range
= 'A' or 'V'.
ldz
The leading dimension of the array
z
.
ldz
1, and if
jobz
= 'V',
ldz
max(2,
n
*2).
Output Parameters
ns
The total number of singular values found. 0
ns
n
.
If
range
= 'A',
ns
=
n
, and if
range
= 'I',
ns
=
iu
-
il
+ 1.
s
Array, size (
n
)
The first
ns
elements contain the selected singular values in ascending order.
z
Array, size
2*
n
*
k
If
jobz
= 'V', then if
info
= 0 the first
ns
columns of
z
contain the singular vectors of the matrix
B
corresponding to the selected singular values, with
U
in rows 1 to
n
and
V
in rows
n
+1 to
n
*2, i.e.
z
=
If
jobz
= 'N', then
z
is not referenced.
Make sure that at least
k
=
ns
+1 columns are supplied in the array
z
; if
range
= 'V', the exact value of
ns
is not known in advance and an upper bound must be used.
superb
Array, size (12*
n
).
If
jobz
= 'V', then if
info
= 0, the first
ns
elements of
iwork
are zero. If
info
> 0, then
iwork
contains the indices of the eigenvectors that failed to converge in
?stevx
.
Return Values
This function returns a value
info
.
= 0: successful exit.
< 0: if
info
= -
i
, the
i
-th argument had an illegal value.
> 0:
if
info
=
i
, then
i
eigenvectors failed to converge in
?stevx
. The indices of the eigenvectors (as returned by
?stevx
) are stored in the array
iwork
.
if
info
=
n
*2 + 1, an internal error occurred.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804