Contents

# ?ptrfs

Refines the solution of a system of linear equations with a symmetric (Hermitian) positive-definite tridiagonal coefficient matrix and estimates its error.

## Syntax

Include Files
• mkl.h
Description
The routine performs an iterative refinement of the solution to a system of linear equations
A*X
=
B
with a symmetric (Hermitian) positive definite tridiagonal matrix
A
, with multiple right-hand sides. For each computed solution vector
x
, the routine computes the component-wise backward error
β
. This error is the smallest relative perturbation in elements of
A
and
b
such that
x
is the exact solution of the perturbed system:
|
δ
a
i
j
|
β
|
a
i
j
|, |
δ
b
i
|
β
|
b
i
|
such that
(
A
+
δ
A
)
x
= (
b
+
δ
b
)
.
Finally, the routine estimates the component-wise forward error in the computed solution
||
x
-
x
e
||
/||
x
||
(here
x
e
is the exact solution).
Before calling this routine:
• call the factorization routine
?pttrf
• call the solver routine
?pttrs
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Used for complex flavors only. Must be
'U'
or
'L'
.
Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix
A
is stored and how
A
is factored:
If
uplo
=
'U'
, the array
e
stores the superdiagonal of
A
, and
A
is factored as
U
H
*D*U
.
If
uplo
=
'L'
, the array
e
stores the subdiagonal of
A
, and
A
is factored as
L*D*L
H
.
n
The order of the matrix
A
;
n
0.
nrhs
The number of right-hand sides;
nrhs
0.
d
The array
d
(size
n
) contains the
n
diagonal elements of the tridiagonal matrix
A
.
df
The array
df
(size
n
) contains the
n
diagonal elements of the diagonal matrix
D
from the factorization of
A
as computed by
?pttrf
.
e
,
ef
,
b
,
x
The array
e
(size
n
-1) contains the (
n
- 1) off-diagonal elements of the tridiagonal matrix
A
(see
uplo
).
The array
ef
(size
n
-1) contains the (
n
- 1) off-diagonal elements of the unit bidiagonal factor
U
or
L
from the factorization computed by
?pttrf
(see
uplo
).
The array
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
The array
x
of size max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout contains the solution matrix
X
as computed by
?pttrs
.
ldb
The leading dimension of
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
The leading dimension of
x
;
ldx
max(1,
n
) for column major layout and
ldx
nrhs
for row major layout
.
Output Parameters
x
The refined solution matrix
X
.
ferr
,
berr
Arrays, size at least
max(1,
nrhs
)
. Contain the component-wise forward and backward errors, respectively, for each solution vector.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
had an illegal value.

#### Product and Performance Information

1

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