Contents

# ?gbsvx

Computes the solution to the real or complex system of linear equations with a band coefficient matrix A and multiple right-hand sides, and provides error bounds on the solution.

## 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 a band matrix of order
n
with
kl
subdiagonals and
ku
superdiagonals, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine
?gbsvx
performs the following steps:
1. If
fact
=
'E'
, real 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 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 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
=
L*U
, where
L
is a product of permutation and unit lower triangular matrices with
kl
subdiagonals, and
U
is upper triangular with
kl
+
ku
superdiagonals.
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
. If the reciprocal of the condition number is less than machine precision,
info
=
n
+ 1
is returned as a warning, but the routine still goes on to solve for
X
and compute error bounds as described below.
4. The system of equations is solved for
X
using the factored form of
A
.
5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
fact
Must be
'F'
,
'N'
, or
'E'
.
Specifies whether 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
is equilibrated with scaling factors given by
r
and
c
.
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, then 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
(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 (see Matrix Storage Schemes). If
fact
=
'F'
and
equed
is not
'N'
, then
A
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 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 band 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
ab
;
ldab
kl
+
ku
+1.
ldafb
afb
;
ldafb
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 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'
, column equilibration was done, that is,
A
has been postmultiplied by
diag
(
c
)
.
if
equed
=
'B'
, both row and column equilibration was done, that is,
A
has been replaced by
diag
(
r
)*
A
*
diag
(
c
)
.
r
,
c
Arrays:
r
(size
n
),
c
(size
n
)
.
The array
r
contains the row scale factors for
A
, and the array
c
contains the column scale factors for
A
. These arrays are input arguments if
fact
=
'F'
only; otherwise they are output arguments.
If
equed
=
'R'
or
'B'
,
A
is multiplied on the left by
diag
(
r
)
; if
equed
=
'N'
or
'C'
,
r
is not accessed.
If
fact
=
'F'
and
equed
=
'R'
or
'B'
, each element of
r
must be positive.
If
equed
=
'C'
or
'B'
,
A
is multiplied on the right by
diag
(
c
)
; if
equed
=
'N'
or
'R'
,
c
is not accessed.
If
fact
=
'F'
and
equed
=
'C'
or
'B'
, each element of
c
must be positive.
ldx
The leading dimension of the output array
x
;
ldx
max(1,
n
) for column major layout and
ldx
nrhs
for row major layout
.
Output Parameters
x
Array, size
max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
.
If
info
= 0
or
info
=
n
+1
, the array
x
contains the solution matrix
X
to the
original
system of equations. Note that
A
and
B
are modified on exit if
equed
'N'
, and the solution to the equilibrated system is:
inv(
diag
(
c
))*
X
, if
trans
=
'N'
and
equed
=
'C'
or
'B'
;
inv(
diag
(
r
))*
X
, if
trans
=
'T'
or
'C'
and
equed
=
'R'
or
'B'
.
ab
Array
ab
is not modified on exit if
fact
=
'F'
or
'N'
, or if
fact
=
'E'
and
equed
=
'N'
.
If
equed
'N'
,
A
is scaled on exit as follows:
equed
=
'R'
:
A
=
diag
(
r
)*
A
equed
=
'C'
:
A
=
A
*
diag
(
c
)
equed
=
'B'
:
A
=
diag
(
r
)*
A
*
diag
(
c
)
.
afb
If
fact
=
'N'
or
'E'
, then
afb
is an output argument and on exit returns details of the
LU
factorization of the original matrix
A
(if
fact
=
'N'
) or of the equilibrated matrix
A
(if
fact
=
'E'
). See the description of
ab
for the form of the equilibrated matrix.
b
Overwritten by
diag
(
r
)*
b
if
trans
=
'N'
and
equed
=
'R'
or
'B'
;
overwritten by
diag
(
c
)*
b
if
trans
=
'T'
or
'C'
and
equed
=
'C'
or
'B'
;
not changed if
equed
=
'N'
.
r
,
c
These arrays are output arguments if
fact
'F'
. See the description of
r
,
c
in
Input Arguments
section.
rcond
An estimate of the reciprocal condition number of the matrix
A
after equilibration (if done).
If
rcond
is less than the machine precision (in particular, if
rcond
=0), the matrix is singular to working precision. This condition is indicated by a return code of
info
>0.
ferr
Array, size at least
max(1,
nrhs
)
. Contains the estimated forward error bound for each solution vector
x
j
(the
j
-th column of the solution matrix
X
). If
xtrue
is the true solution corresponding to
x
j
,
ferr
[
j
-1]
is an estimated upper bound for the magnitude of the largest element in (
x
j
-
xtrue
) divided by the magnitude of the largest element in
x
j
. The estimate is as reliable as the estimate for
rcond
, and is almost always a slight overestimate of the true error.
berr
Array, size at least
max(1,
nrhs
)
. Contains the component-wise relative backward error for each solution vector
x
j
, that is, the smallest relative change in any element of
A
or
B
that makes
x
j
an exact solution.
ipiv
If
fact
=
'N'
or
'E'
, then
ipiv
is an output argument and on exit contains the pivot indices from the factorization
A
=
L*U
of the original matrix
A
(if
fact
=
'N'
) or of the equilibrated matrix
A
(if
fact
=
'E'
).
equed
If
fact
'F'
, then
equed
is an output argument. It specifies the form of equilibration that was done (see the description of
equed
in
Input Arguments
section).
rpivot
On exit,
rpivot
contains the reciprocal pivot growth factor: If
rpivot
is much less than 1, then the stability of the
LU
factorization of the (equilibrated) matrix
A
could be poor. This also means that the solution
x
, condition estimator
rcond
, and forward error bound
ferr
could be unreliable. If factorization fails with
0 <
info
n
, then
rpivot
contains the reciprocal pivot growth factor for the leading
info
columns of
A
.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
If
info
=
i
, and
i
n
, then
U
i
,
i
is exactly zero. The factorization has been completed, but the factor
U
is exactly singular, so the solution and error bounds could not be computed;
rcond
= 0 is returned. If
info
=
i
, and
i
=
n
+1
, then
U
is nonsingular, but
rcond
is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of
rcond
would suggest.

#### Product and Performance Information

1

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Notice revision #20110804