Contents

# p?porfs

Improves the computed solution to a system of linear equations with symmetric/Hermitian positive definite distributed matrix and provides error bounds and backward error estimates for the solution.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?porfs
function
improves the computed solution to the system of linear equations
sub(
A
)*sub(
X
) = sub(
B
),
where sub(
A
) =
A
(
ia
:
ia
+
n
-1,
ja
:
ja
+
n
-1) is a real symmetric or complex Hermitian positive definite distributed matrix and
sub(
B
) =
B
(
ib
:
ib
+
n
-1,
jb
:
jb
+
nrhs
-1),
sub(
X
) =
X
(
ix
:
ix
+
n
-1,
jx
:
jx
+
nrhs
-1)
are right-hand side and solution submatrices, respectively. This
function
also provides error bounds and backward error estimates for the solution.
Input Parameters
uplo
(global) Must be
'U'
or
'L'
.
Specifies whether the upper or lower triangular part of the symmetric/Hermitian matrix sub(
A
) is stored.
If
uplo
=
'U'
, sub(
A
) is upper triangular. If
uplo
=
'L'
, sub(
A
) is lower triangular.
n
(global) The order of the distributed matrix sub(
A
)
(
n
0)
.
nrhs
(global) The number of right-hand sides, i.e., the number of columns of the matrices sub(
B
) and sub(
X
)
(
nrhs
0)
.
a
,
af
,
b
,
x
(local)
Pointers into the local memory to arrays of local sizes
a
:
lld_a
*
LOCc
(
ja
+
n
-1),
af
:
lld_af
*
LOCc
(
jaf
+
n
-1),
b
:
lld_b
*
LOCc
(
jb
+
nrhs
-1),
x
:
lld_x
*
LOCc
(
jx
+
nrhs
-1).
The array
a
contains the local pieces of the
n
-by-
n
symmetric/Hermitian distributed matrix sub(
A
).
If
uplo
=
'U'
n
-by-
n
upper triangular part of sub(
A
) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced.
If
uplo
=
'L'
n
-by-
n
lower triangular part of sub(
A
) contains the lower triangular part of the distributed matrix, and its strictly upper triangular part is not referenced.
The array
af
contains the factors
L
or
U
from the Cholesky factorization sub(
A
) =
L*L
H
or sub(
A
) =
U
H
*U
, as computed by
p?potrf
.
On entry, the array
b
contains the local pieces of the distributed matrix of right hand sides sub(
B
).
On entry, the array
x
contains the local pieces of the solution vectors sub(
X
).
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
.
iaf
,
jaf
(global) The row and column indices in the global matrix
AF
indicating the first row and the first column of the matrix sub(
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 matrix sub(
B
), respectively.
descb
(global and local) array of size
dlen_
. The array descriptor for the distributed matrix
B
.
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
.
work
(local)
The array
work
of size
lwork
is a workspace array.
lwork
(local) The size of the array
work
.
For real flavors:
lwork
must be at least
lwork
3*
LOCr
(
n
+mod(
ia
-1,
mb_a
))
For complex flavors:
lwork
must be at least
lwork
2*
LOCr
(
n
+mod(
ia
-1,
mb_a
))
mod(
x
,
y
)
is the integer remainder of
x
/
y
.
iwork
(local) Workspace array of size
liwork
. Used in real flavors only.
liwork
(local or global) The size of the array
iwork
; used in real flavors only. Must be at least
liwork
LOCr
(
n
+mod(
ib
-1,
mb_b
))
.
rwork
(local)
Workspace array of size
lrwork
. Used in complex flavors only.
lrwork
(local or global) The size of the array
rwork
; used in complex flavors only. Must be at least
lrwork
LOCr
(
n
+mod(
ib
-1,
mb_b
)))
.
Output Parameters
x
On exit, contains the improved solution vectors.
ferr
,
berr
Arrays of size
LOCc
(
jb
+
nrhs
-1) each.
The array
ferr
contains the estimated forward error bound for each solution vector of sub(
X
).
If
XTRUE
is the true solution corresponding to sub(
X
),
ferr
is an estimated upper bound for the magnitude of the largest element in (sub(
X
) -
XTRUE
) divided by the magnitude of the largest element in sub(
X
). The estimate is as reliable as the estimate for
rcond
, and is almost always a slight overestimate of the true error.
This array is tied to the distributed matrix
X
.
The array
berr
contains the component-wise relative backward error of each solution vector (that is, the smallest relative change in any entry of sub(
A
) or sub(
B
) that makes sub(
X
) an exact solution). This array is tied to the distributed matrix
X
.
work
[0]
On exit,
work
[0]
contains the minimum value of
lwork
required for optimum performance.
iwork
[0]
On exit,
iwork
[0]
contains the minimum value of
liwork
required for optimum performance (for real flavors).
rwork
[0]
On exit,
rwork
[0]
contains the minimum value of
lrwork
required for optimum performance (for complex flavors).
info
(global) If
info
=0
, the execution is successful.
info
< 0
:
If the
i
-th argument is an array and the
j-
th entry
, indexed
j
- 1,
info
= -(
i
*100+
j
); if the
i-
th argument is a scalar and had an illegal value, then
info
=
-i
.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserverd for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804