Contents

?gerfsx

Uses extra precise iterative refinement to improve the solution to the system of linear equations with a general coefficient matrix A and provides error bounds and backward error estimates.

Syntax

lapack_int LAPACKE_sgerfsx
(
int
matrix_layout
,
char
trans
,
char
equed
,
lapack_int
n
,
lapack_int
nrhs
,
const float*
a
,
lapack_int
lda
,
const float*
af
,
lapack_int
ldaf
,
const lapack_int*
ipiv
,
const float*
r
,
const float*
c
,
const float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
berr
,
lapack_int
n_err_bnds
,
float*
err_bnds_norm
,
float*
err_bnds_comp
,
lapack_int
nparams
,
float*
params
);
lapack_int LAPACKE_dgerfsx
(
int
matrix_layout
,
char
trans
,
char
equed
,
lapack_int
n
,
lapack_int
nrhs
,
const double*
a
,
lapack_int
lda
,
const double*
af
,
lapack_int
ldaf
,
const lapack_int*
ipiv
,
const double*
r
,
const double*
c
,
const double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
berr
,
lapack_int
n_err_bnds
,
double*
err_bnds_norm
,
double*
err_bnds_comp
,
lapack_int
nparams
,
double*
params
);
lapack_int LAPACKE_cgerfsx
(
int
matrix_layout
,
char
trans
,
char
equed
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_float*
a
,
lapack_int
lda
,
const lapack_complex_float*
af
,
lapack_int
ldaf
,
const lapack_int*
ipiv
,
const float*
r
,
const float*
c
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
berr
,
lapack_int
n_err_bnds
,
float*
err_bnds_norm
,
float*
err_bnds_comp
,
lapack_int
nparams
,
float*
params
);
lapack_int LAPACKE_zgerfsx
(
int
matrix_layout
,
char
trans
,
char
equed
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_double*
a
,
lapack_int
lda
,
const lapack_complex_double*
af
,
lapack_int
ldaf
,
const lapack_int*
ipiv
,
const double*
r
,
const double*
c
,
const lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
berr
,
lapack_int
n_err_bnds
,
double*
err_bnds_norm
,
double*
err_bnds_comp
,
lapack_int
nparams
,
double*
params
);
Include Files
  • mkl.h
Description
The routine improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. In addition to a normwise error bound, the code provides a maximum componentwise error bound, if possible. See comments for
err_bnds_norm
and
err_bnds_comp
for details of the error bounds.
The original system of linear equations may have been equilibrated before calling this routine, as described by the parameters
equed
,
r
, and
c
below. In this case, the solution and error bounds returned are for the original unequilibrated system.
Input Parameters
matrix_layout
Specifies whether two-dimensional array storage is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
,
'T'
, or
'C'
.
Specifies the form of the system of equations:
If
trans
=
'N'
, the system has the form
A
*
X
=
B
(No transpose).
If
trans
=
'T'
, the system has the form
A
T
*
X
=
B
(Transpose).
If
trans
=
'C'
, the system has the form
A
H
*
X
=
B
(Conjugate transpose for complex flavors, Transpose for real flavors).
equed
Must be
'N'
,
'R'
,
'C'
, or
'B'
.
Specifies the form of equilibration that was done to
A
before calling this routine.
If
equed
=
'N'
, no equilibration was done.
If
equed
=
'R'
, row equilibration was done, that is,
A
has been premultiplied by
diag
(
r
).
If
equed
=
'C'
, column equilibration was done, that is,
A
has been postmultiplied by
diag
(
c
).
If
equed
=
'B'
, both row and column equilibration was done, that is,
A
has been replaced by
diag
(
r
)*
A
*
diag
(
c
)
. The right-hand side
B
has been changed accordingly.
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(1,
lda
*
n
))
,
af
(size max(1,
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 original
n
-by-
n
matrix
A
.
The array
af
contains the factored form of the matrix
A
, that is, the factors
L
and
U
from the factorization
A
=
P*L*U
as computed by
?getrf
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
lda
The leading dimension of
a
;
lda
max(1,
n
)
.
ldaf
The leading dimension of
af
;
ldaf
max(1,
n
)
.
ipiv
Array, size at least
max(1,
n
)
. Contains the pivot indices as computed by
?getrf
; for row
1
i
n
, row
i
of the matrix was interchanged with row
ipiv
(
i
)
.
r
,
c
Arrays:
r
(size
n
),
c
(size
n
). The array
r
contains the row scale factors for
A
, and the array
c
contains the column scale factors for
A
.
equed
=
'R'
or
'B'
,
A
is multiplied on the left by
diag
(
r
); if
equed
=
'N'
or
'C'
,
r
is not accessed.
If
equed
=
'R'
or
'B'
, each element of
r
must be positive.
If
equed
=
'C'
or
'B'
,
A
is multiplied on the right by
diag
(
c
); if
equed
=
'N'
or
'R'
,
c
is not accessed.
If
equed
=
'C'
or
'B'
, each element of
c
must be positive.
Each element of
r
or
c
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
.
x
Array,
of size max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
.
The solution matrix
X
as computed by
?getrs
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
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
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
=0.0
No refinement is performed and no error bounds are computed.
=1.0
Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support double precision.
(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
The improved solution matrix
X
.
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.
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:
  • Column major layout:
    err_bnds_norm
    [(
    err
    - 1)*
    nrhs
    +
    i
    - 1]
    .
  • Row major layout:
    err_bnds_norm
    [
    err
    - 1 + (
    i
    - 1)*
    n_err_bnds
    ]
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. If
n_err_bnds
< 3, then at most the first
n_err_bnds
columns of the
err_bnds_comp
array are 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 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:
  • Column major layout:
    err_bnds_comp
    [(
    err
    - 1)*
    nrhs
    +
    i
    - 1]
    .
  • Row major layout:
    err_bnds_comp
    [
    err
    - 1 + (
    i
    - 1)*
    n_err_bnds
    ]
params
Output parameter only if the input contains erroneous values, namely, in
params[0]
,
params[1]
,
params[2]
. In such a case, the corresponding elements of
params
are filled with default values on output.
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 reserverd 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