Contents

# ?dtsvb

Computes the solution to the system of linear equations with a diagonally dominant tridiagonal coefficient matrix A and multiple right-hand sides.

## Syntax

Include Files
• mkl.h
Description
The
?dtsvb
routine solves a system of linear equations
A
*
X
=
B
for
X
, where
A
is an
n
-by-
n
diagonally dominant tridiagonal matrix, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions. The routine uses the BABE (Burning At Both Ends) algorithm.
Note that the equation
A
T
*
X
=
B
may be solved by interchanging the order of the arguments
du
and
dl
.
Input Parameters
n
The order of
A
, the number of rows in
B
;
n
0.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0.
dl
,
d
,
du
,
b
Arrays:
dl
(size
n
- 1),
d
(size
n
),
du
(size
n
- 1),
b
(max(
ldb
*
nrhs
) for column major layout and max(
ldb
*
n
) for row major layout)
.
The array
dl
contains the
(
n
- 1)
subdiagonal elements of
A
.
The array
d
contains the diagonal elements of
A
.
The array
du
contains the
(
n
- 1)
superdiagonal elements of
A
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
ldb
b
;
ldb
max(1,
n
)
.
Output Parameters
dl
Overwritten by the
(
n
-1)
elements of the subdiagonal of the lower triangular matrices
L
1
,
L
2
from the factorization of
A
(see dttrfb).
d
Overwritten by the
n
diagonal element reciprocals of
U
.
b
Overwritten by the solution matrix
X
.
info
If
info
= 0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
If
info
=
i
,
u
i
i
is exactly zero, and the solution has not been computed. The factorization has not been completed unless
i
=
n
.
Application Notes
A diagonally dominant tridiagonal system is defined such that
|
d
i
| > |
dl
i
-1
| + |
du
i
|
for any
i
:
1 <
i
<
n
, and
|
d
1
| > |
du
1
|, |
d
n
| > |
dl
n
-1
|
The underlying BABE algorithm is designed for diagonally dominant systems. Such systems have no numerical stability issue unlike the canonical systems that use elimination with partial pivoting (see
?gtsv
). The diagonally dominant systems are much faster than the canonical systems.
• The current implementation of BABE has a potential accuracy issue on very small or large data close to the underflow or overflow threshold respectively. Scale the matrix before applying the solver in the case of such input data.
• Applying the
?dtsvb
factorization to non-diagonally dominant systems may lead to an accuracy loss, or false singularity detected due to no pivoting.

#### 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