Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 0.10
  • 10/21/2020
  • Public Content
Contents

p?gebrd

Reduces a general matrix to bidiagonal form.

Syntax

void
psgebrd
(
MKL_INT
*m
,
MKL_INT
*n
,
float
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*d
,
float
*e
,
float
*tauq
,
float
*taup
,
float
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pdgebrd
(
MKL_INT
*m
,
MKL_INT
*n
,
double
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*d
,
double
*e
,
double
*tauq
,
double
*taup
,
double
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pcgebrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_Complex8
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*d
,
float
*e
,
MKL_Complex8
*tauq
,
MKL_Complex8
*taup
,
MKL_Complex8
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pzgebrd
(
MKL_INT
*m
,
MKL_INT
*n
,
MKL_Complex16
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*d
,
double
*e
,
MKL_Complex16
*tauq
,
MKL_Complex16
*taup
,
MKL_Complex16
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
Include Files
  • mkl_scalapack.h
Description
The
p?gebrd
function
reduces 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
B
 by an orthogonal/unitary transformation:
Q'
*sub(
A
)*
P
 =
B
.
If
m
≥
n
,
B
 is upper bidiagonal; if
m
 <
n
,
B
 is lower bidiagonal.
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)
.
a
(local)
Real pointer into the local memory to an array of size
lld_a
*
LOCc
(
ja
+
n
-1)
. On entry, this array contains the 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 submatrix
A
, respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
work
(local)
Workspace array of size
lwork
.
lwork
(local or global) size of
work
, must be at least:
lwork
≥
nb
*(
mpa
0 +
nqa
0+1)+
nqa
0
where
nb
 =
mb_a
 =
nb_a
,
iroffa
 =
mod
(
ia
-1,
nb
)
,
icoffa
 =
mod
(
ja
-1,
nb
)
,
iarow
 =
indxg2p
(
ia
,
nb
,
MYROW
,
rsrc_a
,
NPROW
)
,
iacol
 =
indxg2p
 (
ja
,
nb
,
MYCOL
,
csrc_a
,
NPCOL
)
,
mpa
0 =
numroc
(
m
 +
iroffa
,
nb
,
MYROW
, iarow,
NPROW
)
,
nqa
0 =
numroc
(
n
 +
icoffa
,
nb
,
MYCOL
,
iacol
,
NPCOL
)
,
mod(
x
,
y
)
 is the integer remainder of
x
/
y
.
indxg2p
 and
numroc
 are ScaLAPACK tool functions;
MYROW
,
MYCOL
,
NPROW
 and
NPCOL
 can be determined by calling the
function
blacs_gridinfo
.
If
lwork
 = -1
, then
lwork
 is global input and a workspace query is assumed; the
function
only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.
Output Parameters
a
On exit, if
m
≥
n
, the diagonal and the first superdiagonal of sub(
A
) are overwritten with the upper bidiagonal matrix
B
; the elements below the diagonal, with the array
tauq
, represent the orthogonal/unitary matrix
Q
 as a product of elementary reflectors, and the elements above the first superdiagonal, with the array
taup
, represent the orthogonal matrix
P
 as a product of elementary reflectors. If
m
 <
n
, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix
B
; the elements below the first subdiagonal, with the array
tauq
, represent the orthogonal/unitary matrix
Q
 as a product of elementary reflectors, and the elements above the diagonal, with the array
taup
, represent the orthogonal matrix
P
 as a product of elementary reflectors.
See
Application Notes
 below.
d
(local)
Array of size
LOCc
(
ja
+
min
(
m
,
n
)-1)
 if
m
≥
n
 and
LOCr
(
ia
+
min
(
m
,
n
)-1)
 otherwise. The distributed diagonal elements of the bidiagonal matrix
B
:
d
[
i
] =
A
(
i
+1,
i
+1), 0 ≤
i
 < size (
d
)
.
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
(
i
+1,
i
+2)
 for
i
 = 0,1,...,
n
-2
;
if
m
 <
n
,
e
[
i
] =
A
(
i
+2,
i
+1)
 for
i
 = 0,1,...,
m
-2
.
e
 is tied to the distributed matrix
A
.
tauq
,
taup
(local)
Arrays of size
LOCc
(
ja
+
min
(
m
,
n
)-1)
 for
tauq
 and
LOCr
(
ia
+
min
(
m
,
n
)-1)
 for
taup
. Contain the scalar factors of the elementary reflectors that represent the orthogonal/unitary matrices
Q
 and
P
, respectively.
tauq
 and
taup
 are tied to the distributed matrix
A
.
See
Application Notes
 below.
work
[0]
On exit
work
[0]
 contains the minimum value of
lwork
 required for optimum performance.
info
(global)
= 0
: the execution is successful.
< 0
: if the
i
-th argument is an array and the
j-
th entry
, indexed
j
 - 1,
 had an illegal value, then
info
 = -(
i
*100+
j
); if the
i-
th argument is a scalar and had an illegal value, then
info
 =
-i
.
Application Notes
The matrices
Q
 and
P
 are represented as products of elementary reflectors:
If
m
≥
n
,
Q
 =
H
(1)*
H
(2)*...*
H
(
n
), and
P
 =
G
(1)*
G
(2)*...*
G
(
n
-1).
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;
v
(1:
i
-1) = 0,
v
(
i
) = 1, and
v
(
i
+1:
m
) is stored on exit in
A
(
ia
+
i
:
ia
+
m
-1,
ja
+
i
-1);
u
(1:
i
) = 0,
u
(
i
+1) = 1, and
u
(
i
+2:
n
) is stored on exit in
A
 (
ia
+
i
-1,
ja
+
i
+1:
ja
+
n
-1);
tauq
 is stored in
tauq
[
ja
+
i
-2]
 and
taup
 in
taup
[
ia
+
i
-2]
.
If
m
 <
n
,
Q
 =
H
(1)*
H
(2)*...*
H
(
m
-1), and
P
 =
G
(1)*
G
(2)*...*
G
(
m
)
Each
H
 (
i
) and
G
(
i
) has the form:
H
(
i
)=
i
-
tauq
*
v
*
v'
 and
G
(
i
)=
i
-
taup
*
u
*
u'
here
tauq
 and
taup
 are real/complex scalars, and
v
 and
u
 are real/complex vectors;
v
(1:
i
) = 0,
v
(
i
+1) = 1, and
v
(
i
+2:
m
) is stored on exit in
A
 (
ia
+
i
:
ia
+
m
-1,
ja
+
i
-1);
u
(1:
i
-1) = 0,
u
(
i
) = 1, and
u
(
i
+1:
n
) is stored on exit in
A
(
ia
+
i
-1,
ja
+
i
+1:
ja
+
n
-1);
tauq
 is stored in
tauq
[
ja
+
i
-2]
 and
taup
 in
taup
[
ia
+
i
-2]
.
The contents of sub(
A
) on exit are illustrated by the following examples:
m
 = 6
 and
n
 = 5
(
m
 >
n
)
:
Equation
m
 = 5
 and
n
 = 6
(
m
 <
n
)
:
Equation
where
d
 and
e
 denote diagonal and off-diagonal elements of