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

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