Contents

# ?gttrs

Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by
?gttrf
.

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the following systems of linear equations with multiple right hand sides:
A*X
=
B
if
trans
=
'N'
,
A
T
*X
=
B
if
trans
=
'T'
,
A
H
*X
=
B
if
trans
=
'C'
(for complex matrices only).
Before calling this routine, you must call
?gttrf
to compute the
LU
factorization of
A
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array
b
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'
, then
A*X
=
B
is solved for
X
.
If
trans
=
'T'
, then
A
T
*X
=
B
is solved for
X
.
If
trans
=
'C'
, then
A
H
*X
=
B
is solved for
X
.
n
The order of
A
;
n
0.
nrhs
The number of right-hand sides, that is, the number of columns in
B
;
nrhs
0
.
dl
,
d
,
du
,
du2
Arrays:
dl
(
n
-1)
,
d
(
n
)
,
du
(
n
-1)
,
du2
(
n
-2)
.
The array
dl
contains the
(
n
- 1)
multipliers that define the matrix
L
from the
LU
factorization of
A
.
The array
d
contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
The array
du
contains the (
n
- 1) elements of the first superdiagonal of
U
.
The array
du2
contains the (
n
- 2) elements of the second superdiagonal of
U
.
b
Array of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
n
*
ldb
) for row major layout. Contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
ldb
b
;
ldb
max(1,
n
)
for column major layout and
ldb
nrhs
for row major layout
.
ipiv
Array, size (
n
). The
ipiv
array, as returned by
?gttrf
.
Output Parameters
b
Overwritten by the solution matrix
X
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, parameter
i
Application Notes
For each right-hand side
b
, the computed solution is the exact solution of a perturbed system of equations
(
A
+
E
)
x
=
b
, where
`|E| ≤c(n)εP|L||U|`
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision.
If
x
0
is the true solution, the computed solution
x
satisfies this error bound:
where
cond(
A
,
x
)
= || |
A
-1
||
A
| |
x
| ||
/ ||
x
||
||
A
-1
||
||
A
||
=
κ
(
A
).
Note that
cond(
A
,
x
)
can be much smaller than
κ
(
A
)
; the condition number of
A
T
and
A
H
might or might not be equal to
κ
(
A
)
.
The approximate number of floating-point operations for one right-hand side vector
b
is
7
n
(including
n
divisions) for real flavors and
34
n
(including
2
n
divisions) for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?gtcon
.
To refine the solution and estimate the error, call
?gtrfs
.

#### 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