Contents

# ?tbtrs

Solves a system of linear equations with a band triangular coefficient matrix, with multiple right-hand sides.

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the following systems of linear equations with a band triangular matrix
A
, with multiple right-hand sides stored in
B
:
A*X
=
B
if
trans
=
'N'
,
A
T
*X
=
B
if
trans
=
'T'
,
A
H
*X
=
B
if
trans
=
'C'
(for complex matrices only).
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
or
'L'
.
Indicates whether
A
is upper or lower triangular:
If
uplo
=
'U'
, then
A
is upper triangular.
If
uplo
=
'L'
, then
A
is lower triangular.
trans
Must be
'N'
or
'T'
or
'C'
.
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
.
diag
Must be
'N'
or
'U'
.
If
diag
=
'N'
, then
A
is not a unit triangular matrix.
If
diag
=
'U'
, then
A
is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array
ab
.
n
The order of
A
; the number of rows in
B
;
n
0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
nrhs
The number of right-hand sides;
nrhs
0.
ab
The array
ab
contains the matrix
A
in band storage form.
The size of
ab
must be
max(1,
ldab
*
n
)
b
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
The size of
b
is max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
ldab
ab
;
ldab
kd
+ 1.
ldb
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
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)ε|A|`
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
2
n
*
kd
for real flavors and
8
n
*
kd
for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?tbcon
.
To estimate the error in the solution, call
?tbrfs
.

#### Product and Performance Information

1

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Notice revision #20110804