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?gesvx

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

Syntax

lapack_int LAPACKE_sgesvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
float*
a
,
lapack_int
lda
,
float*
af
,
lapack_int
ldaf
,
lapack_int*
ipiv
,
char*
equed
,
float*
r
,
float*
c
,
float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
,
float*
rpivot
);
lapack_int LAPACKE_dgesvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
double*
a
,
lapack_int
lda
,
double*
af
,
lapack_int
ldaf
,
lapack_int*
ipiv
,
char*
equed
,
double*
r
,
double*
c
,
double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
,
double*
rpivot
);
lapack_int LAPACKE_cgesvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_float*
a
,
lapack_int
lda
,
lapack_complex_float*
af
,
lapack_int
ldaf
,
lapack_int*
ipiv
,
char*
equed
,
float*
r
,
float*
c
,
lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
,
float*
rpivot
);
lapack_int LAPACKE_zgesvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_double*
a
,
lapack_int
lda
,
lapack_complex_double*
af
,
lapack_int
ldaf
,
lapack_int*
ipiv
,
char*
equed
,
double*
r
,
double*
c
,
lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
,
double*
rpivot
);
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
, where
A
is an
n
-by-
n
matrix, 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
?gesvx
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 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
    . 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 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,
af
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
.
a
,
af
, and
ipiv
are not modified.
If
fact
=
'N'
, the matrix
A
will be copied to
af
and factored.
If
fact
=
'E'
, the matrix
A
will be equilibrated if necessary, then copied to
af
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.
nrhs
The number of right hand sides; the number of columns of the matrices
B
and
X
;
nrhs
0.
a
The array
a
(size max(1,
lda
*
n
))
contains the matrix
A
. If
fact
=
'F'
and
equed
is not
'N'
, then
A
must have been equilibrated by the scaling factors in
r
and/or
c
.
af
The array
af
af
(size max(1,
ldaf
*
n
))
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
?getrf
. If
equed
is not
'N'
, then
af
is the factored form of the equilibrated matrix
A
.
b
The array
b
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
lda
The leading dimension of
a
;
lda
max(1,
n
)
.
ldaf
The leading dimension of
af
;
ldaf
max(1,
n
)
.
ldb
The leading dimension of
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
?getrf
; 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:
diag
(
C
)
-1
*
X
, if
trans
=
'N'
and
equed
=
'C'
or
'B'
;
diag
(
R
)
-1
*
X
, if
trans
=
'T'
or
'C'
and
equed
=
'R'
or
'B'
. The second dimension of
x
must be at least
max(1,
nrhs
)
.
a
Array
a
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
)
.
af
If
fact
=
'N'
or
'E'
, then
af
is an output argument and on exit returns the factors
L
and
U
from the factorization
A
=
PLU
of the original matrix
A
(if
fact
=
'N'
) or of the equilibrated matrix
A
(if
fact
=
'E'
). See the description of
a
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
=
P*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
had an illegal value.
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
=
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.