Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?posvxx

Uses extra precise iterative refinement to compute the solution to the system of linear equations with a symmetric or Hermitian positive-definite coefficient matrix A applying the Cholesky factorization.

Syntax

lapack_int LAPACKE_sposvxx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
float*
a
,
lapack_int
lda
,
float*
af
,
lapack_int
ldaf
,
char*
equed
,
float*
s
,
float*
b
,
lapack_int
ldb
,
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_dposvxx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
double*
a
,
lapack_int
lda
,
double*
af
,
lapack_int
ldaf
,
char*
equed
,
double*
s
,
double*
b
,
lapack_int
ldb
,
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
);
lapack_int LAPACKE_cposvxx
(
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
,
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_zposvxx
(
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
,
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 Cholesky factorization
A
=
U
T
*U
 (real flavors) /
A
=
U
H
*U
 (complex flavors) or
A
=
L*L
T
 (real flavors) /
A
=
L*L
H
 (complex flavors) to compute the solution to a real or complex system of linear equations
A*X
 =
B
, where
A
 is an
n
-by-
n
 real symmetric/Hermitian positive definite 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
?posvxx
 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 Cholesky decomposition is used to factor the matrix
    A
     (after equilibration if
    fact
     =
    'E'
    ) as
    A
     =
    U
    T
    *U
     (real),
    A
     =
    U
    H
    *U
     (complex), if
    uplo
     =
    'U'
    ,
    or
    A
     =
    L*L
    T
     (real),
    A
     =
    L*L
    H
     (complex), if
    uplo
     =
    'L'
    ,
    where
    U
     is an upper triangular matrix and
    L
     is a lower triangular matrix.
  3. If the leading
    i
    -by-
    i
     principal minor is not positive-definite, 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
    (
    s
    )
     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
 contains the factored form of
A
. If
equed
 is not
'N'
, the matrix
A
 has been equilibrated with scaling factors given by
s
. Parameters
a
 and
af
 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(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout)
.
The array
a
 contains the matrix
A
 as specified by
uplo
 . If
fact
 =
'F'
 and
equed
 =
'Y'
, then
A
 must have been equilibrated by the scaling factors in
s
, and
a
 must contain the equilibrated matrix
diag
(
s
)*
A
*
diag
(
s
)
.
The array
af
 is an input argument if
fact
 =
'F'
. It contains the triangular factor
U
 or
L
 from the Cholesky factorization of
A
 in the same storage format as
A
. If
equed
 is not
'N'
, then
af
 is the factored form of the equilibrated matrix
diag
(
s
)*
A
*
diag
(
s
)
.
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
)
.
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
. This array is an input argument if
fact
 =
'F'
 only; otherwise it is an output argument.
If
equed
 =
'N'
,
s
 is not accessed.
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
10.0
Aggressive
Set to 100.0 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
Array
a
 is not modified on exit if
fact
 =
'F'
 or
'N'
, or if
fact
 =
'E'
 and
equed
 =
'N'
.
If
fact
 =
'E'
 and
equed
 =
'Y'
,
A
 is overwritten by
diag
(
s
)*
A
*
diag
(
s
)
.
af
If
fact
 =
'N'
 or
'E'
, then
af
 is an output argument and on exit returns the triangular factor
U
 or
L
 from the Cholesky factorization
A
=
U
T
*
U
 or
A
=
L
*
L
T
 (real routines),
A
=
U
H
*
U
 or
A
=
L
*
L
H
 (complex routines) of the original matrix
A
 (if
fact
 =
'N'
), or of the equilibrated matrix