Contents

# ?sysvx

Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a real or complex symmetric coefficient matrix A, and provides error bounds on the solution.

## Syntax

Include Files
• mkl.h
Description
The routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of linear equations
A*X
=
B
, where
A
is a
n
-by-
n
symmetric 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
?sysvx
performs the following steps:
1. If
fact
=
'N'
, the diagonal pivoting method is used to factor the matrix
A
. The form of the factorization is
A
=
U*D*U
T
or
A
=
L*D*L
T
, where
U
(or
L
) is a product of permutation and unit upper (lower) triangular matrices, and
D
is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
2. If some
d
i
,
i
= 0, so that
D
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.
3. The system of equations is solved for
X
using the factored form of
A
.
4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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'
or
'N'
.
Specifies whether or not the factored form of the matrix
A
has been supplied on entry.
If
fact
=
'F'
: on entry,
af
and
ipiv
contain the factored form of
A
. Arrays
a
,
af
, and
ipiv
will not be modified.
If
fact
=
'N'
, the matrix
A
will be copied to
af
and factored.
uplo
Must be
'U'
or
'L'
.
Indicates whether the upper or lower triangular part of
A
is stored:
If
uplo
=
'U'
, the upper triangle of
A
is stored.
If
uplo
=
'L'
, the lower triangle of
A
is stored.
n
The order of matrix
A
;
n
0.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0
.
a
,
af
,
b
Arrays:
a
(size max(1,
lda
*
n
))
,
af
(size max(1,
ldaf
*
n
))
,
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
.
The array
a
contains the upper or the lower triangular part of the symmetric matrix
A
(see
uplo
).
The array
af
is an input argument if
fact
=
'F'
. It contains the block diagonal matrix
D
and the multipliers used to obtain the factor
U
or
L
from the factorization
A
=
U*D*U
T
or
A
=
L*D*L
T
as computed by
?sytrf
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
lda
a
;
lda
max(1,
n
)
.
ldaf
af
;
ldaf
max(1,
n
)
.
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 details of the interchanges and the block structure of
D
, as determined by
?sytrf
.
If
ipiv
[
i
-1] =
k
> 0
, then
d
i
i
is a 1-by-1 diagonal block, and the
i
-th row and column of
A
was interchanged with the
k
-th row and column.
If
uplo
=
'U'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0
, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1
, and (
i
)-th row and column of
A
was interchanged with the
m
-th row and column.
If
uplo
=
'L'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0
, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1
, and (
i
+1
)-th row and column of
A
was interchanged with the
m
-th row and column.
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 system of equations.
af
,
ipiv
These arrays are output arguments if
fact
=
'N'
.
See the description of
af
,
ipiv
in
Input Arguments
section.
rcond
An estimate of the reciprocal condition number of the matrix
A
. 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.
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
d
i
i
is exactly zero. The factorization has been completed, but the block diagonal matrix
D
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
D
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

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