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

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.