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

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.