Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

p?gels

Solves overdetermined or underdetermined linear systems involving a matrix of full rank.

Syntax

void
psgels
(
char
*trans
,
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
float
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
float
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pdgels
(
char
*trans
,
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
double
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
double
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pcgels
(
char
*trans
,
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
MKL_Complex8
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex8
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex8
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
void
pzgels
(
char
*trans
,
MKL_INT
*m
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
MKL_Complex16
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex16
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex16
*work
,
MKL_INT
*lwork
,
MKL_INT
*info
);
Include Files
  • mkl_scalapack.h
Description
The
p?gels
function
solves overdetermined or underdetermined real/ complex linear systems involving an
m
-by-
n
 matrix
sub(
A
) =
A
(
ia
:
ia
+
m
-1,
ja
:
ja
+
n
-1)
, or its transpose/ conjugate-transpose, using a
QTQ
 or
LQ
 factorization of sub(
A
). It is assumed that sub(
A
) has full rank.
The following options are provided:
  1. If
    trans
     =
    'N'
     and
    m
    ≥
    n
    : find the least squares solution of an overdetermined system, that is, solve the least squares problem
    minimize ||sub(
    B
    ) - sub(
    A
    )*
    X
    ||
  2. If
    trans
     =
    'N'
     and
    m
     <
    n
    : find the minimum norm solution of an underdetermined system
    sub(
    A
    )*
    X
     = sub(
    B
    )
    .
  3. If
    trans
     =
    'T'
     and
    m
    ≥
    n
    : find the minimum norm solution of an undetermined system
    sub(
    A
    )
    T
    *X
     = sub(
    B
    )
    .
  4. If
    trans
     =
    'T'
     and
    m
     <
    n
    : find the least squares solution of an overdetermined system, that is, solve the least squares problem
    minimize ||sub(
    B
    ) - sub(
    A
    )
    T
    *
    X
    ||
    ,
    where
    sub(
    B
    ) denotes
    B
    (
    ib
    :
    ib
    +
    m
    -1,
    jb
    :
    jb
    +
    nrhs
    -1)
     when
    trans
     =
    'N'
     and
    B
    (
    ib
    :
    ib
    +
    n
    -1,
    jb
    :
    jb
    +
    nrhs
    -1)
     otherwise. Several right hand side vectors
    b
     and solution vectors
    x
     can be handled in a single call; when
    trans
     =
    'N'
    , the solution vectors are stored as the columns of the
    n
    -by-
    nrhs
     right hand side matrix sub(
    B
    ) and the
    m
    -by-
    nrhs
     right hand side matrix sub(
    B
    ) otherwise.
Input Parameters
trans
(global) Must be
'N'
, or
'T'
.
If
trans
 =
'N'
, the linear system involves matrix sub(
A
);
If
trans
 =
'T'
, the linear system involves the transposed matrix
A
T
 (for real flavors only).
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)
.
nrhs
(global) The number of right-hand sides; the number of columns in the distributed submatrices sub(
B
) and
X
.
(
nrhs
≥
 0)
.
a
(local)
Pointer into the local memory to an array of size
lld_a
*
LOCc
(
ja
+
n
-1)
. On entry, contains the
m
-by-
n
 matrix
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
.
b
(local)
Pointer into the local memory to an array of local size
lld_b
*
LOCc
(
jb
+
nrhs
-1)
. On entry, this array contains the local pieces of the distributed matrix
B
 of right-hand side vectors, stored columnwise; sub(
B
) is
m
-by-
nrhs
 if
trans
=
'N'
, and
n
-by-
nrhs
 otherwise.
ib
,
jb
(global) The row and column indices in the global matrix
B
 indicating the first row and the first column of the submatrix
B
, respectively.
descb
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
B
.
work
(local)
Workspace array with size
lwork
.
lwork
(local or global) .
The size of the array
work
lwork
 is local input and must be at least
lwork
 ≥
ltau
 +
max
(
lwf
,
lws
)
, where if
m
 >
n
, then
ltau
 =
numroc
(
ja
+
min
(
m
,
n
)-1,
nb_a
,
MYCOL
,
csrc_a
,
NPCOL
)
,
lwf
 =
nb_a
*(
mpa
0 +
nqa
0 +
nb_a
)
lws
 =
max
((
nb_a
*(
nb_a
-1))/2, (
nrhsqb
0 +
mpb
0)*
nb_a
) +
nb_a
*
nb_a
else
ltau
 =
numroc
(
ia
+
min
(
m
,
n
)-1,
mb_a
,
MYROW
,
rsrc_a
,
NPROW
)
,
lwf
 =
mb_a
 * (
mpa
0 +
nqa
0 +
mb_a
)
lws
 =
max
((
mb_a
*(
mb_a
-1))/2, (
npb
0 +
max
(
nqa
0 +
numroc
(
numroc
(
n
+
iroffb
,
mb_a
, 0, 0,
NPROW
),
mb_a
, 0, 0,
lcmp
),
nrhsqb
0))*
mb_a
) +
mb_a
*
mb_a
end if
,
where
lcmp
 =
lcm
/
NPROW
 with
lcm
 =
ilcm
(
NPROW
,
NPCOL
)
,
iroffa
 =
mod
(
ia
-1,
mb_a
)
,
icoffa
 =
mod
(
ja
-1,
nb_a
)
,
iarow
 =
indxg2p
(
ia
,
mb_a
,
MYROW
,
rsrc_a
,
NPROW
)
,
iacol
=
indxg2p
(
ja
,
nb_a
,
MYROW
,
rsrc_a
,
NPROW
)
mpa
0 =
numroc
(
m
+
iroffa
,
mb_a
,
MYROW
,
iarow
,
NPROW
)
,
nqa
0 =
numroc
(
n
+
icoffa
,
nb_a
,
MYCOL
,
iacol
,
NPCOL
)
,
iroffb
 =
mod
(
ib
-1,
mb_b
)
,
icoffb
 =
mod
(
jb
-1,
nb_b
)
,
ibrow
 =
indxg2p
(
ib
,
mb_b
,
MYROW
,
rsrc_b
,
NPROW
)
,
ibcol
 =
indxg2p
(
jb
,
nb_b
,
MYCOL
,
csrc_b
,
NPCOL
)
,
mpb
0 =
numroc
(
m
+
iroffb
,
mb_b
,
MYROW
,
icrow
,
NPROW
)
,
nqb
0 =
numroc
(
n
+
icoffb
,
nb_b
,
MYCOL
,
ibcol
,
NPCOL
)
,
mod(
x
,
y
)
 is the integer remainder of
x
/
y
.
ilcm
,
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
, sub(
A
) is overwritten by the details of its
QR
 factorization as returned by
p?geqrf
; if
m
 <
n
, sub(
A
) is overwritten by details of its
LQ
 factorization as returned by
p?gelqf
.
b
On exit, sub(
B
) is overwritten by the solution vectors, stored columnwise: if
trans
 =
'N'
 and
m
≥
n
, rows 1 to
n
 of sub(
B
) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements
n
+1
 to
m
 in that column;
If
trans
 =
'N'
 and
m
 <
n
, rows 1 to
n
 of sub(
B
) contain the minimum norm solution vectors;
If
trans
 =
'T'
 and
m
≥
n
, rows 1 to
m
 of sub(
B
) contain the minimum norm solution vectors; if
trans
 =
'T'
 and
m
 <
n
, rows 1 to
m
 of sub(
B
) contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements
m
+1
 to
n
 in that column.
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
-entry 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
.