Contents

# ?orbdb/?unbdb

Simultaneously bidiagonalizes the blocks of a partitioned orthogonal/unitary matrix.

## Syntax

Include Files
• mkl.h
Description
The routines
?orbdb
/
?unbdb
simultaneously bidiagonalizes the blocks of an
m
-by-
m
partitioned orthogonal matrix
X
: or unitary matrix: x
11
is
p
-by-
q
.
q
must not be larger than
p
,
m
-
p
, or
m
-
q
. Otherwise,
x
must be transposed and/or permuted in constant time using the
trans
and
signs
options.
The orthogonal/unitary matrices
p
1
,
p
2
,
q
1
, and
q
2
are
p
-by-
p
,
(m-p)
-by-
(m-p)
,
q
-by-
q
,
(m-q)
-by-
(m-q)
, respectively. They are represented implicitly by Housholder vectors.
The bidiagonal matrices
b
11
,
b
12
,
b
21
, and
b
22
are
q
-by-
q
bidiagonal matrices represented implicitly by angles
theta

, ...,
theta
[
q
- 1]
and
phi

, ...,
phi
[
q
- 2]
.
b
11
and
b
12
are upper bidiagonal, while
b
21
and
b
22
are lower bidiagonal. Every entry in each bidiagonal band is a product of a sine or cosine of
theta
with a sine or cosine of
phi
. See [Sutton09] for details.
p
1
,
p
2
,
q
1
, and
q
2
are represented as products of elementary reflectors. .
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
= '
T
':
x
,
u
1
,
u
2
,
v
1
t
,
v
2
t
are stored in row-major order.
otherwise
x
,
u
1
,
u
2
,
v
1
t
,
v
2
t
are stored in column-major order.
signs
= '
O
':
The lower-left block is made nonpositive (the "other" convention).
otherwise
The upper-right block is made nonpositive (the "default" convention).
m
The number of rows and columns of the matrix
X
.
p
The number of rows in
x
11
and
x
12
.
0
p
m
.
q
The number of columns in
x
11
and
x
21
.
0
q
min(
p
,
m
-
p
,
m
-
q
)
.
x11
Array, size
(size max(1,
ldx11
*
q
) for column major layout and max(1,
ldx11
*
p
) for row major layout)
.
On entry, the top-left block of the orthogonal/unitary matrix to be reduced.
ldx11
The leading dimension of the array
X
11
. If
trans
= '
T
',
ldx11
p
for column major layout and
ldx11
q
for row major layout
. Otherwise,
ldx11
q
.
x12
Array, size
(size max(1,
ldx12
*(
m
-
q
)) for column major layout and max(1,
ldx12
*
p
) for row major layout)
.
On entry, the top-right block of the orthogonal/unitary matrix to be reduced.
ldx12
The leading dimension of the array
X
12
. If
trans
= '
N
',
ldx12
p
for column major layout and
ldx12
m
-
q
for row major layout.
. Otherwise,
ldx12
m
-
q
.
x21
Array, size
(size max(1,
ldx21
*
q
) for column major layout and max(1,
ldx21
*(
m
-
p
)) for row major layout)
.
On entry, the bottom-left block of the orthogonal/unitary matrix to be reduced.
ldx21
The leading dimension of the array
X
21
. If
trans
= '
N
',
ldx21
m
-
p
for column major layout and
ldx12
q
for row major layout.
. Otherwise,
ldx21
q
.
x22
Array, size (
(size max(1,
ldx22
*(
m
-
q
)) for column major layout and max(1,
ldx22
*(
m
-
p
)) for row major layout)
.
On entry, the bottom-right block of the orthogonal/unitary matrix to be reduced.
ldx22
The leading dimension of the array
X
21
. If
trans
= '
N
',
ldx22
m
-
p
for column major layout and
ldx22
m
-
q
for row major layout.
. Otherwise,
ldx22
m
-
q
.
Output Parameters
x11
On exit, the form depends on
trans
:
If
trans
='
N
',
the columns of the lower triangle of
x11
specify reflectors for
p
1
, the rows of the upper triangle of
x11
(1:
q
- 1,
q
:q - 1)
specify reflectors for
q
1
otherwise
trans
='
T
',
the rows of the upper triangle of
x11
specify reflectors for
p
1
, the columns of the lower triangle of
x11
(1:
q
- 1,
q
:q - 1)
specify reflectors for
q
1
x12
On exit, the form depends on
trans
:
If
trans
='
N
',
the columns of the upper triangle of
x12
specify the first
p
reflectors for
q
2
otherwise
trans
='
T
',
the columns of the lower triangle of
x12
specify the first
p
reflectors for
q
2
x21
On exit, the form depends on
trans
:
If
trans
='
N
',
the columns of the lower triangle of
x21
specify the reflectors for
p
2
otherwise
trans
='
T
',
the columns of the upper triangle of
x21
specify the reflectors for
p
2
x22
On exit, the form depends on
trans
:
If
trans
='
N
',
the rows of the upper triangle of
x22
(
q
+1:
m
-
p
,
p
+1:
m
-
q
) specify the last
m
-
p
-
q
reflectors for
q
2
otherwise
trans
='
T
',
the columns of the lower triangle of
x22
(
p
+1:
m
-
q
,
q
+1:
m
-
p
) specify the last
m
-
p
-
q
reflectors for
p
2
theta
Array,
size
q
. The entries of bidiagonal blocks
b
11
,
b
12
,
b
21
, and
b
22
can be computed from the angles
theta
and
phi
.
See the Description section for details.
phi
Array,
size
q
-1
. The entries of bidiagonal blocks
b
11
,
b
12
,
b
21
, and
b
22
can be computed from the angles
theta
and
phi
.
See the Description section for details.
taup1
Array,
size
p
.
Scalar factors of the elementary reflectors that define
p
1
.
taup2
Array,
size
m
-
p
.
Scalar factors of the elementary reflectors that define
p
2
.
tauq1
Array,
size
q
.
Scalar factors of the elementary reflectors that define