Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?hesvxx

Uses extra precise iterative refinement to compute the solution to the system of linear equations with a Hermitian indefinite coefficient matrix A applying the diagonal pivoting factorization.

Syntax

lapack_int LAPACKE_chesvxx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
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*
s
,
lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
rpvgrw
,
float*
berr
,
lapack_int
n_err_bnds
,
float*
err_bnds_norm
,
float*
err_bnds_comp
,
lapack_int
nparams
,
const float*
params
);
lapack_int LAPACKE_zhesvxx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
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*
s
,
lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
rpvgrw
,
double*
berr
,
lapack_int
n_err_bnds
,
double*
err_bnds_norm
,
double*
err_bnds_comp
,
lapack_int
nparams
,
const double*
params
);
Include Files
  • mkl.h
Description
The routine uses the diagonal pivoting factorization to compute the solution to a complex/double complex system of linear equations
A*X
 =
B
, where
A
 is an
n
-by-
n
 Hermitian matrix, the columns of matrix
B
 are individual right-hand sides, and the columns of
X
 are the corresponding solutions.
Both normwise and maximum componentwise error bounds are also provided on request. The routine returns a solution with a small guaranteed error (
O(eps)
, where
eps
 is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also calculated and returned.
The routine accepts user-provided factorizations and equilibration factors; see definitions of the
fact
 and
equed
 options. Solving with refinement and using a factorization from a previous call of the routine also produces a solution with
O(eps)
 errors or warnings but that may not be true for general user-provided factorizations and equilibration factors if they differ from what the routine would itself produce.
The routine
?hesvxx
 performs the following steps:
  1. If
    fact
     =
    'E'
    , scaling factors are computed to equilibrate the system:
    diag
    (
    s
    )*
    A
    *
    diag
    (
    s
    ) *inv(
    diag
    (
    s
    ))*
    X
     =
    diag
    (
    s
    )*
    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
    (
    s
    )*
    A
    *
    diag
    (
    s
    )
     and
    B
     by
    diag
    (
    s
    )*
    B
    .
  2. If
    fact
     =
    'N'
     or
    'E'
    , the LU decomposition is used to factor the matrix
    A
     (after equilibration if
    fact
     =
    'E'
    ) as
    A
     =
    U
    *
    D
    *
    U
    T
    , if
    uplo
     =
    'U'
    ,
    or
    A
     =
    L
    *
    D
    *
    L
    T
    , if
    uplo
     =
    'L'
    ,
    where
    U
     or
    L
     is a product of permutation and unit upper (lower) triangular matrices, and
    D
     is a symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
  3. If some
    D
    (
    i
    ,
    i
    )=0
    , so that
    D
     is exactly singular, the routine returns with
    info
     =
    i
    . Otherwise, the factored form of
    A
     is used to estimate the condition number of the matrix
    A
     (see the
    rcond
     parameter). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for
    X
     and compute error bounds.
  4. The system of equations is solved for
    X
     using the factored form of
    A
    .
  5. By default, unless
    params[0]
     is set to zero, the routine applies iterative refinement to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision.
  6. If equilibration was used, the matrix
    X
     is premultiplied by
    diag
    (
    r
    )
     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
s
. Parameters
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, 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 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
,
af
,
b
Arrays:
a
(size max(
lda
*
n
))
,
af
(size max(
ldaf
*
n
))
,
b
, (size max(
ldb
*
nrhs
) for column major layout and max(
ldb
*
n
) for row major layout),
.
The array
a
 contains the Hermitian matrix
A
 as specified by
uplo
. If
uplo
 =
'U'
, the leading
n
-by-
n
 upper triangular part of
a
 contains the upper triangular part of the matrix
A
 and the strictly lower triangular part of
a
 is not referenced. If
uplo
 =
'L'
, the leading
n
-by-
n
 lower triangular part of
a
 contains the lower triangular part of the matrix
A
 and the strictly upper triangular part of
a
 is not referenced.
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
 and
L
 from the factorization
A
 =
U
*
D
*
U
T
 or
A
 =
L
*
D
*
L
T
 as computed by
?hetrf
.
The array
b
 contains the matrix
B
 whose columns are the right-hand sides for the systems of equations.
lda
The leading dimension of the array
a
;
lda
≥
 max(1,
n
)
.
ldaf
The leading dimension of the array
af
;
ldaf
≥
 max(1,
n
)
.
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
[
k
-1]
 > 0, rows and columns
k
 and
ipiv
[
k
-1]
 were interchanged and
D
k
,
k
) is a 1-by-1 diagonal block.
If
uplo
 =
'U'
 and
ipiv
[
i
] =
ipiv
[
i
 - 1] =
m
 < 0
,
D
 has a 2-by-2 diagonal block in rows and columns
i
 and
i
 + 1, and the
i
-th row and column of
A
 were interchanged with the
m
-th row and column.
If
uplo
 =
'L'
 and
ipiv
[
i
] =
ipiv
[
i
 - 1] =
m
 < 0
,
D
 has a 2-by-2 diagonal block in rows and columns
i
 and
i
 + 1, and the (
i
 + 1)-st row and column of
A
 were interchanged with the
m
-th row and column.
equed
Must be
'N'
 or
'Y'
.
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
 =
'Y'
, both row and column equilibration was done, that is,
A
 has been replaced by
diag
(
s
)*
A
*
diag
(
s
)
.
s
Array, size (
n
). The array
s
 contains the scale factors for
A
. If
equed
 =
'Y'
,
A
 is multiplied on the left and right by
diag(
s
)
.
This array is an input argument if
fact
 =
'F'
 only; otherwise it is an output argument.
If
fact
 =
'F'
 and
equed
 =
'Y'
, each element of
s
 must be positive.
Each element of
s
 should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
ldb
The leading dimension of the array
b
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
ldx
The leading dimension of the output array
x
;
ldx
≥
 max(1,
n
) for column major layout and
ldx
≥
nrhs
 for row major layout
.
n_err_bnds
Number of error bounds to return for each right hand side and each type (normwise or componentwise). See
err_bnds_norm
 and
err_bnds_comp
 descriptions in the
Output Arguments
 section below.
nparams
Specifies the number of parameters set in
params
. If
≤
 0, the
params
 array is never referenced and default values are used.
params
Array, size
max(1,
nparams
)
. Specifies algorithm parameters. If an entry is less than 0.0, that entry is filled with the default value used for that parameter. Only positions up to
nparams
 are accessed; defaults are used for higher-numbered parameters. If defaults are acceptable, you can pass
nparams
 = 0, which prevents the source code from accessing the
params
 argument.
params
[0]
 : Whether to perform iterative refinement or not. Default: 1.0 (for single precision flavors), 1.0D+0 (for double precision flavors).
=0.0
No refinement is performed and no error bounds are computed.
=1.0
Use the extra-precise refinement algorithm.
(Other values are reserved for future use.)
params
[1]
 : Maximum number of residual computations allowed for refinement.
Default
10
Aggressive
Set to 100 to permit convergence using approximate factorizations or factorizations other than
LU
. If the factorization uses a technique other than Gaussian elimination, the guarantees in
err_bnds_norm
 and
err_bnds_comp
 may no longer be trustworthy.
params
[2]
 : Flag determining if the code will attempt to find a solution with a small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence).
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
, the array
x
 contains the solution
n
-by-
nrhs
 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
(
s
))*
X
.
a
If
fact
 =
'E'
 and
equed
 =
'Y'
, overwritten by
diag
(
s
)*
A
*
diag
(
s
)
.
af
If
fact
 =
'N'
,
af
 is an output argument and on exit returns 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
.
b
If
equed
 =
'N'
,
B
 is not modified.
If
equed
 =
'Y'
,
B
 is overwritten by
diag
(
s
)*
B
.
s
This array is an output argument if
fact
≠
'F'
. Each element of this array is a power of the radix. See the description of
s
 in
Input Arguments
 section.
rcond
Reciprocal scaled condition number. An estimate of the reciprocal Skeel 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. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned.
rpvgrw
Contains the reciprocal pivot growth factor:
If this is much less than 1, the stability of the
LU
 factorization of the (equlibrated) matrix
A
 could be poor. This also means that the solution
X
, estimated condition numbers, and error bounds could be unreliable. If factorization fails with
0 <
info
≤
n
, this parameter contains the reciprocal pivot growth factor for the leading
info
 columns of
A
.
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.
err_bnds_norm
Array of size
nrhs
*
n_err_bnds
. For each right-hand side, contains information about various error bounds and condition numbers corresponding to the normwise relative error
, which is defined as follows:
Normwise relative error in the
i
-th solution vector
The array is indexed by the type of error information as described below. There are currently up to three pieces of information returned.
err
=1
"Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold
sqrt
(
n
)*