Contents

# ?hpsvx

Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a Hermitian coefficient matrix A stored in packed format, 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 complex system of linear equations
A*X
=
B
, where
A
is a
n
-by-
n
Hermitian matrix stored in packed format, 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
?hpsvx
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
H
or
A
=
L*D*L
H
, where
U
(or
L
) is a product of permutation and unit upper (lower) triangular matrices, and
D
is a Hermitian 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,
afp
and
ipiv
contain the factored form of
A
. Arrays
ap
,
afp
, and
ipiv
are not modified.
If
fact
=
'N'
, the matrix
A
is copied to
afp
and factored.
uplo
Must be
'U'
or
'L'
.
Indicates whether the upper or lower triangular part of
A
is stored and how
A
is factored:
If
uplo
=
'U'
, the array
ap
stores the upper triangular part of the Hermitian matrix
A
, and
A
is factored as
U*D*U
H
.
If
uplo
=
'L'
, the array
ap
stores the lower triangular part of the Hermitian matrix
A
, and
A
is factored as
L*D*L
H
.
n
The order of matrix
A
;
n
0.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0
.
ap
,
afp
,
b
Arrays:
ap
(size max(1,
n
*(
n
+1)/2)
,
afp
(size max(1,
n
*(
n
+1)/2)
,
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
.
The array
ap
contains the upper or lower triangle of the Hermitian matrix
A
in packed storage (see Matrix Storage Schemes).
The array
afp
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
H
or
A
=
L*D*L
H
as computed by
?hptrf
, in the same storage format as
A
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
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
?hptrf
.
If
ipiv
[
i
-1] =
k
> 0
, then
d
i
i
is a 1-by-1 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
-1] =
ipiv
[
i
] = -
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.
afp
,
ipiv
These arrays are output arguments if
fact
=
'N'
. See the description of
afp
,
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804