## Developer Reference

• 0.9
• 09/09/2020
• Public Content
Contents

# ?gbsvxx

Uses extra precise iterative refinement to compute the solution to the system of linear equations with a banded coefficient matrix A and multiple right-hand sides

## Syntax

Include Files
• mkl.h
Description
The routine uses the LU factorization to compute the solution to a real or complex system of linear equations
A*X
=
B
,
A
T
*
X
=
B
, or
A
H
*
X
=
B
, where
A
is an
n
-by-
n
banded matrix, the columns of the matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
Both normwise and maximum componentwise error bounds are also provided on request. The routine returns a solution with a small guaranteed error (
O(eps)
, where
eps
is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also calculated and returned.
The routine accepts user-provided factorizations and equilibration factors; see definitions of the
fact
and
equed
options. Solving with refinement and using a factorization from a previous call of the routine also produces a solution with
O(eps)
errors or warnings but that may not be true for general user-provided factorizations and equilibration factors if they differ from what the routine would itself produce.
The routine
?gbsvxx
performs the following steps:
1. If
fact
=
'E'
, scaling factors
r
and
c
are computed to equilibrate the system:
trans
=
'N'
:
diag
(
r
)*
A
*
diag
(
c
)*inv(
diag
(
c
))*
X
=
diag
(
r
)*
B
trans
=
'T'
:
(
diag
(
r
)*
A
*
diag
(
c
))
T
*inv(
diag
(
r
))*
X
=
diag
(
c
)*
B
trans
=
'C'
:
(
diag
(
r
)*
A
*
diag
(
c
))
H
*inv(
diag
(
r
))*
X
=
diag
(
c
)*
B
Whether or not the system will be equilibrated depends on the scaling of the matrix
A
, but if equilibration is used,
A
is overwritten by
diag
(
r
)*
A
*
diag
(
c
)
and
B
by
diag
(
r
)*
B
(if
trans
=
'N'
)
or
diag
(
c
)*
B
(if
trans
=
'T'
or
'C'
).
2. If
fact
=
'N'
or
'E'
, the
LU
decomposition is used to factor the matrix
A
(after equilibration if
fact
=
'E'
) as
A
=
P*L*U
, where
P
is a permutation matrix,
L
is a unit lower triangular matrix, and
U
is upper triangular.
3. If some
U
i
,
i
= 0, so that
U
is exactly singular, then the routine returns with
info
=
i
. Otherwise, the factored form of
A
is used to estimate the condition number of the matrix
A
(see the
rcond
parameter). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for
X
and compute error bounds.
4. The system of equations is solved for
X
using the factored form of
A
.
5. By default, unless
params
is set to zero, the routine applies iterative refinement to improve the computed solution matrix and calculate error bounds. Refinement calculates the residual to at least twice the working precision.
6. If equilibration was used, the matrix
X
is premultiplied by
diag
(
c
)
(if
trans
=
'N'
) or
diag
(
r
)
(if
trans
=
'T'
or
'C'
) so that it solves the original system before equilibration.
Input Parameters
matrix_layout
Specifies whether two-dimensional array storage is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
fact
Must be
'F'
,
'N'
, or
'E'
.
Specifies whether or not the factored form of the matrix
A
is supplied on entry, and if not, whether the matrix
A
should be equilibrated before it is factored.
If
fact
=
'F'
, on entry,
afb
and
ipiv
contain the factored form of
A
. If
equed
is not
'N'
, the matrix
A
has been equilibrated with scaling factors given by
r
and
c
. Parameters
ab
,
afb
, and
ipiv
are not modified.
If
fact
=
'N'
, the matrix
A
will be copied to
afb
and factored.
If
fact
=
'E'
, the matrix
A
will be equilibrated, if necessary, copied to
afb
and factored.
trans
Must be
'N'
,
'T'
, or
'C'
.
Specifies the form of the system of equations:
If
trans
=
'N'
, the system has the form
A
*
X
=
B
(No transpose).
If
trans
=
'T'
, the system has the form
A
T
*
X
=
B
(Transpose).
If
trans
=
'C'
, the system has the form
A
H
*
X
=
B
(Conjugate Transpose = Transpose for real flavors, Conjugate Transpose for complex flavors).
n
The number of linear equations; the order of the matrix
A
;
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; the number of columns of the matrices
B
and
X
;
nrhs
0.
ab
,
afb
,
b
Arrays:
ab
(max(
ldab
*
n
))
,
afb
(max(
ldafb
*
n
))
,
b
(max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout)
.
The array
ab
contains the matrix
A
in band storage.
If
fact
=
'F'
and
equed
is not
'N'
, then
AB
must have been equilibrated by the scaling factors in
r
and/or
c
.
The array
afb
is an input argument if
fact
=
'F'
. It contains the factored form of the banded matrix
A
, that is, the factors
L
and
U
from the factorization
A
=
P*L*U
as computed by
?gbtrf
.
U
is stored as an upper triangular banded matrix with
kl
+
ku
superdiagonals.
L
is stored as lower triangular band matrix with
kl
subdiagonals. If
equed
is not
'N'
, then
afb
is the factored form of the equilibrated matrix
A
.
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
kl
+
ku
+1
.
ldafb
The leading dimension of the array
afb
;
ldafb
2*
kl
+
ku
+1
.
ipiv
Array, size at least
max(1,
n
)
. The array
ipiv
is an input argument if
fact
=
'F'
. It contains the pivot indices from the factorization
A
=
P*L*U
as computed by
?gbtrf
; row
i
of the matrix was interchanged with row
ipiv
[
i
-1]
.
equed
Must be
'N'
,
'R'
,
'C'
, or
'B'
.
equed
is an input argument if
fact
=
'F'
. It specifies the form of equilibration that was done:
If
equed
=
'N'
, no equilibration was done (always true if
fact
=
'N'
).
If
equed
=
'R'
, row equilibration was done, that is,
A
has been premultiplied by
diag
(
r
).
If
equed
=
'C'