Contents

# ?gbtrs

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

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the following systems of linear equations:
A*X
=
B
if
trans
=
'N'
,
A
T
*X
=
B
if
trans
=
'T'
,
A
H
*X
=
B
if
trans
=
'C'
(for complex matrices only).
Here
A
is an
LU
-factored general band matrix of order
n
with
kl
non-zero subdiagonals and
ku
nonzero superdiagonals. Before calling this routine, call
?gbtrf
to compute the
LU
factorization of
A
.
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'
.
n
The order of
A
; the number of rows in
B
;
n
0.
kl
The number of subdiagonals within the band of
A
;
kl
0.
ku
The number of superdiagonals within the band of
A
;
ku
0.
nrhs
The number of right-hand sides;
nrhs
0.
ab
Array
ab
size max(1,
ldab
*
n
)
The array
ab
contains elements of the LU factors of the matrix
A
as returned by gbtrf .
b
Array
b
size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
ldab
The leading dimension of the array
ab
;
ldab
2*
kl
+
ku
+1.
ldb
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
array, as returned by
?gbtrf
.
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

(

kl

+

ku

+ 1)

ε

P

|

L

||

U

|```
c
(
k
)
is a modest linear function of
k
, 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 is 2
n
(
ku
+ 2
kl
) for real flavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that
kl
and
ku
are much less than min(
m
,
n
).
To estimate the condition number
κ
(
A
)
, call
?gbcon
.
To refine the solution and estimate the error, call
?gbrfs
.

#### 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