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
A
(if
fact
=
'E'
). See the description of
a
for the form of the equilibrated matrix.
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 componentwise 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
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors.
err
=2
"Guaranteed" error bound. The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold
sqrt
(
n
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors. This error bound should only be trusted if the previous boolean is true.
err
=3
Reciprocal condition number. Estimated normwise reciprocal condition number. Compared with the threshold
sqrt
(
n
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors to determine if the error estimate is "guaranteed". These reciprocal condition numbers for some appropriately scaled matrix
Z
are:
Let
z
=
s
*
a
, where
s
scales each row by a power of the radix so all absolute row sums of
z
are approximately 1.
The information for right-hand side
i
, where 1
i
nrhs
, and type of error
err
is stored in
err_bnds_norm
[(
err
-1)*
nrhs
+
i
- 1]
.
err_bnds_comp
Array of size
nrhs
*
n_err_bnds
. For each right-hand side, contains information about various error bounds and condition numbers corresponding to the componentwise relative error
, which is defined as follows:
Componentwise 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 for each right-hand side. If componentwise accuracy is not requested (
params[2]
= 0.0), then
err_bnds_comp
is not accessed.
err
=1
"Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold
sqrt
(
n
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors.
err
=2
"Guaranteed" error bpound. The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold
sqrt
(
n
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors. This error bound should only be trusted if the previous boolean is true.
err
=3
Reciprocal condition number. Estimated componentwise reciprocal condition number. Compared with the threshold
sqrt
(
n
)*
slamch
(
ε
)
for single precision flavors and
sqrt
(
n
)*
dlamch
(
ε
)
for double precision flavors to determine if the error estimate is "guaranteed". These reciprocal condition numbers for some appropriately scaled matrix
Z
are:
Let
z
=
s
*(
a
*diag(
x
))
, where
x
is the solution for the current right-hand side and
s
scales each row of
a
*diag(
x
)
by a power of the radix so all absolute row sums of
z
are approximately 1.
The information for right-hand side
i
, where 1
i
nrhs
, and type of error
err
is stored in
err_bnds_comp
[(
err
-1)*
nrhs
+
i
- 1]
.
equed
If
fact
'F'
, then
equed
is an output argument. It specifies the form of equilibration that was done (see the description of
equed
in
Input Arguments
section).
params
If an entry is less than 0.0, that entry is filled with the default value used for that parameter, otherwise the entry is not modified.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful. The solution to every right-hand side is guaranteed.
If
info
=
-i
, parameter
i
had an illegal value.
If 0 <
info
n
:
U
info
,
info
is exactly zero. The factorization has been completed, but the factor
U
is exactly singular, so the solution and error bounds could not be computed;
rcond
= 0 is returned.
If
info
=
n
+
j
: The solution corresponding to the
j
-th right-hand side is not guaranteed. The solutions corresponding to other right-hand sides
k
with
k
>
j
may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested
params[2]
= 0.0
, then the
j
-th right-hand side is the first with a normwise error bound that is not guaranteed (the smallest
j
such that for column major layout
err_bnds_norm
[
j
- 1] = 0.0
or
err_bnds_comp
[
j
- 1] = 0.0
; or for row major layout
err_bnds_norm
[(
j
- 1)*
n_err_bnds
] = 0.0
or
err_bnds_comp
[(
j
- 1)*
n_err_bnds
] = 0.0
). See the definition of
err_bnds_norm
and
err_bnds_comp
for
err
= 1. To get information about all of the right-hand sides, check
err_bnds_norm
or
err_bnds_comp
.
1

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 reservered 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