Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
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

lapack_int LAPACKE_chpsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_float*
ap
,
lapack_complex_float*
afp
,
lapack_int*
ipiv
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zhpsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_double*
ap
,
lapack_complex_double*
afp
,
lapack_int*
ipiv
,
const lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
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
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 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
 had an illegal value.
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.