Contents

# ?gtrfs

Refines the solution of a system of linear equations with a 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
or
A
T
*X
=
B
or
A
H
*X
=
B
with a 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
|/|
a
i
j
|
β
|
a
i
j
|, |
δ
b
i
|/|
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
?gttrf
• call the solver routine
?gttrs
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
or
'T'
or
'C'
.
Indicates the form of the equations:
If
trans
=
'N'
, the system has the form
A*X
=
B
.
If
trans
=
'T'
, the system has the form
A
T
*X
=
B
.
If
trans
=
'C'
, the system has the form
A
H
*X
=
B
.
n
The order of the matrix
A
;
n
0.
nrhs
The number of right-hand sides, that is, the number of columns of the matrix
B
;
nrhs
0.
dl
Array
dl
of size
n
-1 contains the subdiagonal elements of
A
.
d
Array
d
of size
n
contains the diagonal elements of
A
.
du
Array
du
of size
n
-1 contains the superdiagonal elements of
A
.
dlf
Array
dlf
of size
n
-1 contains the (
n
- 1) multipliers that define the matrix
L
from the
LU
factorization of
A
as computed by
?gttrf
.
df
Array
df
of size
n
contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
duf
Array
duf
of size
n
-1 contains the (
n
- 1) elements of the first superdiagonal of
U
.
du2
Array
du2
of size
n
-2 contains the (
n
- 2) elements of the second superdiagonal of
U
.
b
Array
b
(size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout) contains the right-hand side matrix
B
.
x
Array
x
(size max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) contains the solution matrix
X
, as computed by
?gttrs
.
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
.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
array, as returned by
?gttrf
.
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

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 reserved 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