## Developer Reference

• 2021.1
• 12/04/2020
• Public Content
Contents

# p?labrd

Reduces the first
nb
rows and columns of a general rectangular matrix A to real bidiagonal form by an orthogonal/unitary transformation, and returns auxiliary matrices that are needed to apply the transformation to the unreduced part of A.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?labrd
function
reduces the first
nb
rows and columns of a real/complex general
m
-by-
n
distributed matrix sub(
A
) =
A
(
ia
:
ia
+
m
-1
,
ja
:
ja
+
n
-1)
to upper or lower bidiagonal form by an orthogonal/unitary transformation
Q'* A * P
, and returns the matrices
X
and
Y
necessary to apply the transformation to the unreduced part of sub(
A
).
If
m
n
,
sub(
A
)
is reduced to upper bidiagonal form; if
m
<
n
,
sub(
A
)
is reduced to lower bidiagonal form.
This is an auxiliary
function
called by
p?gebrd
.
Input Parameters
m
(global) The number of rows in the distributed matrix
sub(
A
)
.
(
m
0)
.
n
(global) The number of columns in the distributed matrix sub(
A
).
(
n
0)
.
nb
(global)
The number of leading rows and columns of sub(
A
) to be reduced.
a
(local).
Pointer into the local memory to an array of size
lld_a
*
LOC
c
(
ja
+
n
-1)
.
On entry, this array contains the local pieces of the general distributed matrix sub(
A
).
ia
,
ja
(global) The row and column indices in the global matrix
A
indicating the first row and the first column of the matrix sub(
A
), respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
ix
,
jx
(global) The row and column indices in the global matrix
X
indicating the first row and the first column of the matrix sub(
X
), respectively.
descx
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
X
.
iy
,
jy
(global) The row and column indices in the global matrix
Y
indicating the first row and the first column of the matrix sub(
Y
), respectively.
descy
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
Y
.
work
(local).
Workspace array of size
lwork
.
lwork
nb_a
+
nq
,
with
nq
=
numroc
(
n
+
mod
(
ia
-1,
nb_y
),
nb_y
,
mycol
,
iacol
,
npcol
)
iacol
=
indxg2p
(
ja
,
nb_a
,
mycol
,
csrc_a
,
npcol
)
indxg2p
and
numroc
are ScaLAPACK tool functions;
myrow
,
mycol
,
nprow
, and
npcol
can be determined by calling the
function
blacs_gridinfo
.
Output Parameters
a
(local)
On exit, the first
nb
rows and columns of the matrix are overwritten; the rest of the distributed matrix
sub(
A
)
is unchanged.
If
m
n
, elements on and below the diagonal in the first
nb
columns, with the array
tauq
, represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors; and elements above the diagonal in the first
nb
rows, with the array
taup
, represent the orthogonal/unitary matrix
P
as a product of elementary reflectors.
If
m
<
n
, elements below the diagonal in the first
nb
columns, with the array
tauq
, represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors, and elements on and above the diagonal in the first
nb
rows, with the array
taup
, represent the orthogonal/unitary matrix
P
as a product of elementary reflectors.
See
Application Notes
below.
d
(local).
Array of size
LOCr
(
ia
+min(
m,n
)-1)
if
m
n
;
LOCc
(
ja
+min(
m,n
)-1)
otherwise. The distributed diagonal elements of the bidiagonal distributed matrix
B
:
d
[
i
] =
A
(
ia
+
i
,
ja
+
i
),
i
= 0, 1, ..., size (
d
)-1
d
is tied to the distributed matrix
A
.
e
(local).
Array of size
LOCr
(
ia
+min(
m,n
)-1)
if
m
n
;
LOCc
(
ja
+min(
m,n
)-2)
otherwise. The distributed off-diagonal elements of the bidiagonal distributed matrix
B
:
if
m
n
,
e
[
i
] =
A
(
ia
+
i
,
ja
+
i+1
) for
i
= 0, 1, ...,
n
-2
;
if
m
<
n
,
e
[
i
] =
A
(
ia
+
i+1
,
ja
+
i
) for
i
= 0, 1, ...,
m
-2
.
e
is tied to the distributed matrix
A
.
tauq
,
taup
(local).
Array size
LOCc
(
ja
+min(
m
,
n
)-1) for
tauq
, size
LOCr
(
ia
+min(
m
,
n
)-1) for
taup
. The scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix
Q
for
tauq
,
P
for
taup
.
tauq
and
taup
are tied to the distributed matrix
A
.
See
Application Notes
below.
x
(local)
Pointer into the local memory to an array of size
lld_x
*
nb
. On exit, the local pieces of the distributed
m
-by-
nb
matrix
X
(
ix
:
ix
+
m
-1
,
jx
:
jx
+
nb
-1)
required to update the unreduced part of sub(
A
).
y
(local).
Pointer into the local memory to an array of size
lld_y
*
nb
. On exit, the local pieces of the distributed
n
-by-
nb
matrix
Y
(
iy
:
iy
+
n
-1
,
jy
:
jy
+
nb
-1)
required to update the unreduced part of sub(
A
).
Application Notes
The matrices
Q
and
P
are represented as products of elementary reflectors:
Q
=
H
(1)*
H
(2)*...*
H
(
nb
)
, and
P
=
G
(1)*
G
(2)*...*
G
(
nb
)
Each
H
(i)
and
G
(i)
has the form:
H
(i) =
I
-
tauq
*
v
*
v'
, and
G
(i) =
I
-
taup
*
u
*
u'
,
where
tauq
and
taup
are real/complex scalars, and
v
and
u
are real/complex vectors.
If
m
n
,
v
(1:
i
-1 ) = 0,
v
(
i
) = 1
, and
v
(
i
:
m
)
is stored on exit in
A
(
ia
+
i
-1:
ia
+
m
-1,
ja
+
i
-1)
;
u
(1:
i
) = 0,
u
(
i
+1 ) = 1
, and
u
(
i
+1:
n
)
is stored on exit in
A
(
ia
+
i
-1,
ja
+
i
:
ja
+
n
-1)
;
tauq
is stored in
tauq
[
ja
+
i
-2]
and
taup
in
taup
[
ia
+
i
-2]
.
If
m
<
n
,
v
(1:
i
) = 0
,
v
(
i
+1 ) = 1
, and
v
(
i+
1:
m
)
is stored on exit in
A
(
ia
+
i+
1:
ia
+
m
-1,
ja
+
i
-1)
;
u
(1:
i
-1 ) = 0
,
u
(
i
) = 1
, and
u
(
i
:
n
)
is stored on exit in
A
(
ia
+
i
-1
,
ja
+
i
:
ja
+
n
-1)
;
tauq
is stored in
tauq
[
ja
+
i
-2]
and
taup
in
taup
[
ia
+
i
-2]
. The elements of the vectors
v
and
u
together form the
m
-by-
nb
matrix
V
and the
nb
-by-
n
matrix
U'
which are necessary, with
X
and
Y
, to apply the transformation to the unreduced part of the matrix, using a block update of the form:
sub(
A
):= sub(
A
) -
V*Y'
-
X*U'
.
The contents of
sub(
A
)
on exit are illustrated by the following examples with
nb
= 2
:
where
a
denotes an element of the original matrix which is unchanged,
vi
denotes an element of the vector defining
H
(
i
)
, and
ui
an element of the vector defining
G
(
i
)
.

#### Product and Performance Information

1

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