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
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
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