Contents

# p?trrfs

Provides error bounds and backward error estimates for the solution to a system of linear equations with a distributed triangular coefficient matrix.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?trrfs
function
provides error bounds and backward error estimates for the solution to one of the systems of linear equations
sub(
A
)*sub(
X
) = sub(
B
),
sub(
A
)
T
*sub(
X
) = sub(
B
), or
sub(
A
)
H
*sub(
X
) = sub(
B
) ,
where sub(
A
) =
A
(
ia
:
ia
+
n
-1,
ja
:
ja
+
n
-1) is a triangular matrix,
sub(
B
) =
B
(
ib
:
ib
+
n
-1,
jb
:
jb
+
nrhs
-1), and
sub(
X
) =
X
(
ix
:
ix
+
n
-1,
jx
:
jx
+
nrhs
-1).
The solution matrix
X
must be computed by
p?trtrs
or some other means before entering this
function
. The
function
p?trrfs
does not do iterative refinement because doing so cannot improve the backward error.
Input Parameters
uplo
(global) Must be
'U'
or
'L'
.
If
uplo
=
'U'
, sub(
A
) is upper triangular. If
uplo
=
'L'
, sub(
A
) is lower triangular.
trans
(global) Must be
'N'
or
'T'
or
'C'
.
Specifies the form of the system of equations:
If
trans
=
'N'
, the system has the form sub(
A
)*sub(
X
) = sub(
B
) (No transpose);
If
trans
=
'T'
, the system has the form sub(
A
)
T
*sub(
X
) = sub(
B
) (Transpose);
If
trans
=
'C'
, the system has the form sub(
A
)
H
*sub(
X
) = sub(
B
) (Conjugate transpose).
diag
Must be
'N'
or
'U'
.
If
diag
=
'N'
, then sub(
A
) is non-unit triangular.
If
diag
=
'U'
, then sub(
A
) is unit triangular.
n
(global) The order of the distributed matrix sub(
A
)
(
n
0)
.
nrhs
(global) The number of right-hand sides, that is, the number of columns of the matrices sub(
B
) and sub(
X
)
(
nrhs
0)
.
a
,
b
,
x
(local)
Pointers into the local memory to arrays of local sizes
a
:
lld_a
*
LOCc
(
ja
+
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 original triangular 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.
If
diag
=
'U'
, the diagonal elements of sub(
A
) are also not referenced and are assumed to be 1.
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
.
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
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

On exit,
work

contains the minimum value of
lwork
required for optimum performance.
iwork

On exit,
iwork

contains the minimum value of
liwork
required for optimum performance (for real flavors).
rwork

On exit,
rwork

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

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.