Contents

p?gesvx

Uses the
LU
factorization to compute the solution to the system of linear equations with a square matrix
A
and multiple right-hand sides, and provides error bounds on the solution.

Syntax

void
psgesvx
(
char
*fact
,
char
*trans
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
float
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
float
*af
,
MKL_INT
*iaf
,
MKL_INT
*jaf
,
MKL_INT
*descaf
,
MKL_INT
*ipiv
,
char
*equed
,
float
*r
,
float
*c
,
float
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
float
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
float
*rcond
,
float
*ferr
,
float
*berr
,
float
*work
,
MKL_INT
*lwork
,
MKL_INT
*iwork
,
MKL_INT
*liwork
,
MKL_INT
*info
);
void
pdgesvx
(
char
*fact
,
char
*trans
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
double
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
double
*af
,
MKL_INT
*iaf
,
MKL_INT
*jaf
,
MKL_INT
*descaf
,
MKL_INT
*ipiv
,
char
*equed
,
double
*r
,
double
*c
,
double
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
double
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
double
*rcond
,
double
*ferr
,
double
*berr
,
double
*work
,
MKL_INT
*lwork
,
MKL_INT
*iwork
,
MKL_INT
*liwork
,
MKL_INT
*info
);
void
pcgesvx
(
char
*fact
,
char
*trans
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
MKL_Complex8
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex8
*af
,
MKL_INT
*iaf
,
MKL_INT
*jaf
,
MKL_INT
*descaf
,
MKL_INT
*ipiv
,
char
*equed
,
float
*r
,
float
*c
,
MKL_Complex8
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex8
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
float
*rcond
,
float
*ferr
,
float
*berr
,
MKL_Complex8
*work
,
MKL_INT
*lwork
,
float
*rwork
,
MKL_INT
*lrwork
,
MKL_INT
*info
);
void
pzgesvx
(
char
*fact
,
char
*trans
,
MKL_INT
*n
,
MKL_INT
*nrhs
,
MKL_Complex16
*a
,
MKL_INT
*ia
,
MKL_INT
*ja
,
MKL_INT
*desca
,
MKL_Complex16
*af
,
MKL_INT
*iaf
,
MKL_INT
*jaf
,
MKL_INT
*descaf
,
MKL_INT
*ipiv
,
char
*equed
,
double
*r
,
double
*c
,
MKL_Complex16
*b
,
MKL_INT
*ib
,
MKL_INT
*jb
,
MKL_INT
*descb
,
MKL_Complex16
*x
,
MKL_INT
*ix
,
MKL_INT
*jx
,
MKL_INT
*descx
,
double
*rcond
,
double
*ferr
,
double
*berr
,
MKL_Complex16
*work
,
MKL_INT
*lwork
,
double
*rwork
,
MKL_INT
*lrwork
,
MKL_INT
*info
);
Include Files
  • mkl_scalapack.h
Description
The
p?gesvx
function
uses the
LU
factorization to compute the solution to a real or complex system of linear equations
A
X
=
B
, where
A
denotes the
n
-by-
n
submatrix
A
(
ia:ia+n-1
,
ja:ja+n-1
)
,
B
denotes the
n
-by-
nrhs
submatrix
B
(
ib:ib+n-1
,
jb:jb+nrhs-1
)
and
X
denotes the
n
-by-
nrhs
submatrix
X
(
ix:ix+n-1
,
jx:jx+nrhs-1
)
.
Error bounds on the solution and a condition estimate are also provided.
In the following description,
af
stands for the subarray
of
af
from row
iaf
and column
jaf
to row
iaf+n-1
and column
jaf+n-1
.
The
function
p?gesvx
performs the following steps:
  1. If
    fact
    =
    'E'
    , real scaling factors
    R
    and
    C
    are computed to equilibrate the system:
    trans
    =
    'N'
    :
    diag(
    R
    )*
    A
    *diag(
    C
    ) *diag(
    C
    )-1*
    X
    = diag(
    R
    )*B
    trans
    =
    'T'
    :
    (diag(
    R
    )*
    A
    *diag(
    C
    ))
    T
    *diag(
    R
    )-1*
    X
    = diag(
    C
    )*B
    trans
    =
    'C'
    :
    (diag(
    R
    )*
    A
    *diag(
    C
    ))
    H
    *diag(
    R
    )-1*
    X
    = diag(
    C
    )*B
    Whether or not the system will be equilibrated depends on the scaling of the matrix
    A
    , but if equilibration is used,
    A
    is overwritten by
    diag(
    R
    )*
    A
    *diag(
    C
    )
    and
    B
    by
    diag(
    R
    )*
    B
    (if
    trans
    ='
    N
    ')
    or
    diag(
    c
    )*
    B
    (if
    trans
    =
    'T'
    or
    'C'
    ).
  2. If
    fact
    =
    'N'
    or
    'E'
    , the
    LU
    decomposition is used to factor the matrix
    A
    (after equilibration if
    fact
    =
    'E'
    )
    as
    A
    =
    P
    L
    U
    , where
    P
    is a permutation matrix,
    L
    is a unit lower triangular matrix, and
    U
    is upper triangular.
  3. The factored form of
    A
    is used to estimate the condition number of the matrix
    A
    . If the reciprocal of the condition number is less than relative machine precision, steps 4 - 6 are skipped.
  4. The system of equations is solved for
    X
    using the factored form of
    A
    .
  5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  6. If equilibration was used, the matrix
    X
    is premultiplied by diag(
    C
    ) (if
    trans
    =
    'N'
    )
    or diag(
    R
    ) (if
    trans
    =
    'T'
    or
    'C'
    ) so that it solves the original system before equilibration.
Input Parameters
fact
(global) Must be
'F'
,
'N'
, or
'E'
.
Specifies whether or not the factored form of the matrix
A
is supplied on entry, and if not, whether the matrix
A
should be equilibrated before it is factored.
If
fact
=
'F'
then, on entry,
af
and
ipiv
contain the factored form of
A
. If
equed
is not
'N'
, the matrix
A
has been equilibrated with scaling factors given by
r
and
c
. Arrays
a
,
af
, and
ipiv
are not modified.
If
fact
=
'N'
, the matrix
A
is copied to
af
and factored.
If
fact
=
'E'
, the matrix
A
is equilibrated if necessary, then copied to
af
and factored.
trans
(global) Must be
'N'
,
'T'
, or
'C'
.
Specifies the form of the system of equations:
If
trans
=
'N'
, the system has the form
A
*X
=
B
(No transpose);
If
trans
=
'T'
, the system has the form
A
T
*
X
=
B
(Transpose);
If
trans
=
'C'
, the system has the form
A
H
*X
=
B
(Conjugate transpose);
n
(global) The number of linear equations; the order of the submatrix
A
(
n
0)
.
nrhs
(global) The number of right hand sides; the number of columns of the distributed submatrices
B
and
X
(
nrhs
0)
.
a
,
af
,
b
,
work
(local)
Pointers into the local memory to arrays of local size
a
:
lld_a
*
LOCc
(
ja
+
n
-1)
,
af
:
lld_af
*
LOCc
(
ja
+
n
-1)
,
b
:
lld_b
*
LOCc
(
jb+nrhs-1
)
,
work
:
lwork
.
The array
a
contains the matrix
A
. If
fact
=
'F'
and
equed
is not
'N'
, then
A
must have been equilibrated by the scaling factors in
r
and/or
c
.
The array
af
is an input argument if
fact
=
'F'
. In this case it contains on entry the factored form of the matrix
A
, that is, the factors
L
and
U
from the factorization
A
=
P
*
L
*
U
as computed by
p?getrf
. If
equed
is not
'N'
, then
af
is the factored form of the equilibrated matrix
A
.
The array
b
contains on entry the matrix
B
whose columns are the right-hand sides for the systems of equations.
work
is a workspace array. The size of
work
is (
lwork
).
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
(
ia:ia+n-1
,
ja:ja+n-1
)
, respectively.
desca
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
A
.
iaf
,
jaf
(global) The row and column indices in the global matrix
AF
indicating the first row and the first column of the subarray
af
, respectively.
descaf
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
AF
.
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
(
ib:ib+n-1
,
jb:jb+nrhs-1
)
, respectively.
descb
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
B
.
ipiv
(local) Array of size
LOCr
(
m_a
)+
mb_a
.
The array
ipiv
is an input argument if
fact
=
'F'
.
On entry, it contains the pivot indices from the factorization
A
=
P
*
L
*
U
as computed by
p?getrf
; (local) row
i
of the matrix was interchanged with the (global) row
ipiv
[
i
- 1]
.
This array must be aligned with
A
(
ia:ia+n-1
,
*
)
.
equed
(global) Must be
'N'
,
'R'
,
'C'
, or
'B'
.
equed
is an input argument if
fact
=
'F'
. It specifies the form of equilibration that was done:
If
equed
=
'N'
, no equilibration was done (always true if
fact
=
'N'
);
If
equed
=
'R'
, row equilibration was done, that is,
A
has been premultiplied by diag(
r
);
If
equed
=
'C'
, column equilibration was done, that is,
A
has been postmultiplied by diag(
c
);
If
equed
=
'B'
, both row and column equilibration was done;
A
has been replaced by
diag(
r
)*
A
*diag(
c
)
.
r
,
c
(local)
Arrays of size
LOCr
(m_a)
and
LOCc
(n_a)
, respectively.
The array
r
contains the row scale factors for
A
, and the array
c
contains the column scale factors for
A
. These arrays are input arguments if
fact
=
'F'
only; otherwise they are output arguments. If
equed
=
'R'
or
'B'
,
A
is multiplied on the left by diag(
r
); if
equed
=
'N'
or
'C'
,
r
is not accessed.
If
fact
=
'F'
and
equed
=
'R'
or
'B'
, each element of
r
must be positive.
If
equed
=
'C'
or
'B'
,
A
is multiplied on the right by diag(
c
); if
equed
=
'N'
or
'R'
,
c
is not accessed.
If
fact
=
'F'
and
equed
=
'C'
or
'B'
, each element of
c
must be positive. Array
r
is replicated in every process column, and is aligned with the distributed matrix
A
. Array
c
is replicated in every process row, and is aligned with 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 submatrix
X
(
ix:ix+n-1
,
jx:jx+nrhs-1
)
, respectively.
descx
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
X
.
lwork
(local or global) The size of the array
work
; must be at least
max(
p?gecon
(
lwork
),
p?gerfs
(
lwork
))+
LOCr
(n_a)
.
iwork
(local,
psgesvx/pdgesvx
only). Workspace array. The size of
iwork
is (
liwork
).
liwork
(local,
psgesvx/pdgesvx
only). The size of the array
iwork
, must be at least
LOCr
(n_a)
.
rwork
(local)
Workspace array, used in complex flavors only.
The size of
rwork
is (
lrwork
).
lrwork
(local or global,
pcgesvx/pzgesvx
only). The size of the array
rwork
;must be at least
2
*
LOCc
(n_a)
.
Output Parameters
x
(local)
Pointer into the local memory to an array of local size
lld_x
*
LOCc
(
jx+nrhs-1
)
.
If
info
= 0
, the array
x
contains the solution matrix
X
to the
original
system of equations. Note that
A
and
B
are modified on exit if
equed
'N'
, and the solution to the
equilibrated
system is:
diag(
C
)-1*
X
, if
trans
=
'N'
and
equed
=
'C'
or
'B'
; and
diag(
R
)-1*
X
, if
trans
=
'T'
or
'C'
and
equed
=
'R'
or
'B'
.
a
Array
a
is not modified on exit if
fact
=
'F'
or
'N'
, or if
fact
=
'E'
and
equed
=
'N'
.
If
equed
'N'
,
A
is scaled on exit as follows:
equed
=
'R'
:
A
= diag(
R
)*
A
equed
=
'C'
:
A
=
A
*diag(
c
)
equed
=
'B'
:
A
= diag(
R
)*
A
*diag(
c
)
af
If
fact
=
'N'
or
'E'
, then
af
is an output argument and on exit returns the factors
L
and
U
from the factorization
A
=
P
*L
*
U
of the original matrix
A
(if
fact
=
'N'
) or of the equilibrated matrix
A
(if
fact
=
'E'
). See the description of
a
for the form of the equilibrated matrix.
b
Overwritten by
diag(
R
)*
B
if
trans
=
'N'
and
equed
=
'R'
or
'B'
;
overwritten by
diag(
c
)*
B
if
trans
=
'T'
and
equed
=
'C'
or
'B'
; not changed if
equed
=
'N'
.
r
,
c
These arrays are output arguments if
fact
'F'
.
See the description of
r
,
c
in
Input Arguments
section.
rcond
(global).
An estimate of the reciprocal condition number of the matrix
A
after equilibration (if done). The
function
sets
rcond
=0
if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime
rcond
is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.
ferr
,
berr
(local)
Arrays of size
LOCc
(
n_b
)
each. Contain the component-wise forward and relative backward errors, respectively, for each solution vector.
Arrays
ferr
and
berr
are both replicated in every process row, and are aligned with the matrices
B
and
X
.
ipiv
If
fact
=
'N'
or
'E'
, then
ipiv
is an output argument and on exit contains the pivot indices from the factorization
A
=
P
*
L
*
U
of the original matrix
A
(if
fact
=
'N'
) or of the equilibrated matrix
A
(if
fact
=
'E'
).
equed
If
fact
'F'
, then
equed
is an output argument. It specifies the form of equilibration that was done (see the description of
equed
in
Input Arguments
section).
work
[0]
If
info
=0
, on exit
work
[0]
returns the minimum value of
lwork
required for optimum performance.
iwork
[0]
If
info
=0
, on exit
iwork
[0]
returns the minimum value of
liwork
required for optimum performance.
rwork
[0]
If
info
=0
, on exit
rwork