Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

void
pslabrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nb
,
float
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*d
,
float
*e
,
float
*tauq
,
float
*taup
,
float
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
float
*y
,
MKL_INT
*iy
,
MKL_INT
*jy
,
MKL_INT
*descy
,
float
*work
);
void
pdlabrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nb
,
double
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*d
,
double
*e
,
double
*tauq
,
double
*taup
,
double
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
double
*y
,
MKL_INT
*iy
,
MKL_INT
*jy
,
MKL_INT
*descy
,
double
*work
);
void
pclabrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nb
,
MKL_Complex8
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*d
,
float
*e
,
MKL_Complex8
*tauq
,
MKL_Complex8
*taup
,
MKL_Complex8
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
MKL_Complex8
*y
,
MKL_INT
*iy
,
MKL_INT
*jy
,
MKL_INT
*descy
,
MKL_Complex8
*work
);
void
pzlabrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nb
,
MKL_Complex16
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*d
,
double
*e
,
MKL_Complex16
*tauq
,
MKL_Complex16
*taup
,
MKL_Complex16
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
MKL_Complex16
*y
,
MKL_INT
*iy
,
MKL_INT
*jy
,
MKL_INT
*descy
,
MKL_Complex16
*work
);
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
:
Equation
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
)
.