733 lines
27 KiB
Fortran
733 lines
27 KiB
Fortran
*> \brief \b SGERFSX
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SGERFSX + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgerfsx.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgerfsx.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgerfsx.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
|
* R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
|
|
* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
|
|
* WORK, IWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER TRANS, EQUED
|
|
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
|
|
* $ N_ERR_BNDS
|
|
* REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* INTEGER IPIV( * ), IWORK( * )
|
|
* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
|
* $ X( LDX , * ), WORK( * )
|
|
* REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
|
|
* $ ERR_BNDS_NORM( NRHS, * ),
|
|
* $ ERR_BNDS_COMP( NRHS, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SGERFSX improves the computed solution to a system of linear
|
|
*> equations and provides error bounds and backward error estimates
|
|
*> for the solution. In addition to normwise error bound, the code
|
|
*> provides 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 arguments EQUED, R
|
|
*> and C below. In this case, the solution and error bounds returned
|
|
*> are for the original unequilibrated system.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \verbatim
|
|
*> Some optional parameters are bundled in the PARAMS array. These
|
|
*> settings determine how refinement is performed, but often the
|
|
*> defaults are acceptable. If the defaults are acceptable, users
|
|
*> can pass NPARAMS = 0 which prevents the source code from accessing
|
|
*> the PARAMS argument.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] TRANS
|
|
*> \verbatim
|
|
*> TRANS is CHARACTER*1
|
|
*> Specifies the form of the system of equations:
|
|
*> = 'N': A * X = B (No transpose)
|
|
*> = 'T': A**T * X = B (Transpose)
|
|
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] EQUED
|
|
*> \verbatim
|
|
*> EQUED is CHARACTER*1
|
|
*> Specifies the form of equilibration that was done to A
|
|
*> before calling this routine. This is needed to compute
|
|
*> the solution and error bounds correctly.
|
|
*> = 'N': No equilibration
|
|
*> = 'R': Row equilibration, i.e., A has been premultiplied by
|
|
*> diag(R).
|
|
*> = 'C': Column equilibration, i.e., A has been postmultiplied
|
|
*> by diag(C).
|
|
*> = 'B': Both row and column equilibration, i.e., A has been
|
|
*> replaced by diag(R) * A * diag(C).
|
|
*> The right hand side B has been changed accordingly.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] N
|
|
*> \verbatim
|
|
*> N is INTEGER
|
|
*> The order of the matrix A. N >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NRHS
|
|
*> \verbatim
|
|
*> NRHS is INTEGER
|
|
*> The number of right hand sides, i.e., the number of columns
|
|
*> of the matrices B and X. NRHS >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] A
|
|
*> \verbatim
|
|
*> A is REAL array, dimension (LDA,N)
|
|
*> The original N-by-N matrix A.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDA
|
|
*> \verbatim
|
|
*> LDA is INTEGER
|
|
*> The leading dimension of the array A. LDA >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] AF
|
|
*> \verbatim
|
|
*> AF is REAL array, dimension (LDAF,N)
|
|
*> The factors L and U from the factorization A = P*L*U
|
|
*> as computed by SGETRF.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDAF
|
|
*> \verbatim
|
|
*> LDAF is INTEGER
|
|
*> The leading dimension of the array AF. LDAF >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] IPIV
|
|
*> \verbatim
|
|
*> IPIV is INTEGER array, dimension (N)
|
|
*> The pivot indices from SGETRF; for 1<=i<=N, row i of the
|
|
*> matrix was interchanged with row IPIV(i).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] R
|
|
*> \verbatim
|
|
*> R is REAL array, dimension (N)
|
|
*> The row scale factors for A. If EQUED = 'R' or 'B', A is
|
|
*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
|
|
*> is not accessed.
|
|
*> If R is accessed, each element of R 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] C
|
|
*> \verbatim
|
|
*> C is REAL array, dimension (N)
|
|
*> The column scale factors for A. If EQUED = 'C' or 'B', A is
|
|
*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
|
|
*> is not accessed.
|
|
*> If C is accessed, each element of 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] B
|
|
*> \verbatim
|
|
*> B is REAL array, dimension (LDB,NRHS)
|
|
*> The right hand side matrix B.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDB
|
|
*> \verbatim
|
|
*> LDB is INTEGER
|
|
*> The leading dimension of the array B. LDB >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] X
|
|
*> \verbatim
|
|
*> X is REAL array, dimension (LDX,NRHS)
|
|
*> On entry, the solution matrix X, as computed by SGETRS.
|
|
*> On exit, the improved solution matrix X.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDX
|
|
*> \verbatim
|
|
*> LDX is INTEGER
|
|
*> The leading dimension of the array X. LDX >= max(1,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RCOND
|
|
*> \verbatim
|
|
*> RCOND is REAL
|
|
*> Reciprocal scaled condition number. This is an estimate of the
|
|
*> reciprocal Skeel condition number of the matrix A after
|
|
*> equilibration (if done). If this is less than the machine
|
|
*> precision (in particular, if it is zero), 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.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] BERR
|
|
*> \verbatim
|
|
*> BERR is REAL array, dimension (NRHS)
|
|
*> Componentwise relative backward error. This is the
|
|
*> componentwise relative backward error of each solution vector X(j)
|
|
*> (i.e., the smallest relative change in any element of A or B that
|
|
*> makes X(j) an exact solution).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] N_ERR_BNDS
|
|
*> \verbatim
|
|
*> N_ERR_BNDS is INTEGER
|
|
*> 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 below.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] ERR_BNDS_NORM
|
|
*> \verbatim
|
|
*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
|
|
*> For each right-hand side, this array 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 ith solution vector:
|
|
*> max_j (abs(XTRUE(j,i) - X(j,i)))
|
|
*> ------------------------------
|
|
*> max_j abs(X(j,i))
|
|
*>
|
|
*> The array is indexed by the type of error information as described
|
|
*> below. There currently are up to three pieces of information
|
|
*> returned.
|
|
*>
|
|
*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
|
|
*> right-hand side.
|
|
*>
|
|
*> The second index in ERR_BNDS_NORM(:,err) contains the following
|
|
*> three fields:
|
|
*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
|
|
*> reciprocal condition number is less than the threshold
|
|
*> sqrt(n) * slamch('Epsilon').
|
|
*>
|
|
*> 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('Epsilon'). 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('Epsilon') to determine if the error
|
|
*> estimate is "guaranteed". These reciprocal condition
|
|
*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
|
|
*> appropriately scaled matrix Z.
|
|
*> 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.
|
|
*>
|
|
*> See Lapack Working Note 165 for further details and extra
|
|
*> cautions.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] ERR_BNDS_COMP
|
|
*> \verbatim
|
|
*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
|
|
*> For each right-hand side, this array 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 ith solution vector:
|
|
*> abs(XTRUE(j,i) - X(j,i))
|
|
*> max_j ----------------------
|
|
*> abs(X(j,i))
|
|
*>
|
|
*> The array is indexed by the right-hand side i (on which the
|
|
*> componentwise relative error depends), and the type of error
|
|
*> information as described below. There currently are up to three
|
|
*> pieces of information returned for each right-hand side. If
|
|
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
|
|
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
|
|
*> the first (:,N_ERR_BNDS) entries are returned.
|
|
*>
|
|
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
|
|
*> right-hand side.
|
|
*>
|
|
*> The second index in ERR_BNDS_COMP(:,err) contains the following
|
|
*> three fields:
|
|
*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
|
|
*> reciprocal condition number is less than the threshold
|
|
*> sqrt(n) * slamch('Epsilon').
|
|
*>
|
|
*> 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('Epsilon'). 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('Epsilon') to determine if the error
|
|
*> estimate is "guaranteed". These reciprocal condition
|
|
*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
|
|
*> appropriately scaled matrix Z.
|
|
*> 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.
|
|
*>
|
|
*> See Lapack Working Note 165 for further details and extra
|
|
*> cautions.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NPARAMS
|
|
*> \verbatim
|
|
*> NPARAMS is INTEGER
|
|
*> Specifies the number of parameters set in PARAMS. If .LE. 0, the
|
|
*> PARAMS array is never referenced and default values are used.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] PARAMS
|
|
*> \verbatim
|
|
*> PARAMS is REAL array, dimension NPARAMS
|
|
*> Specifies algorithm parameters. If an entry is .LT. 0.0, then
|
|
*> that entry will be filled with default value used for that
|
|
*> parameter. Only positions up to NPARAMS are accessed; defaults
|
|
*> are used for higher-numbered parameters.
|
|
*>
|
|
*> PARAMS(LA_LINRX_ITREF_I = 1) : 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(LA_LINRX_ITHRESH_I = 2) : 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(LA_LINRX_CWISE_I = 3) : Flag determining if the code
|
|
*> will attempt to find a solution with small componentwise
|
|
*> relative error in the double-precision algorithm. Positive
|
|
*> is true, 0.0 is false.
|
|
*> Default: 1.0 (attempt componentwise convergence)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is REAL array, dimension (4*N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] IWORK
|
|
*> \verbatim
|
|
*> IWORK is INTEGER array, dimension (N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] INFO
|
|
*> \verbatim
|
|
*> INFO is INTEGER
|
|
*> = 0: Successful exit. The solution to every right-hand side is
|
|
*> guaranteed.
|
|
*> < 0: If INFO = -i, the i-th argument had an illegal value
|
|
*> > 0 and <= 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.
|
|
*> = N+J: The solution corresponding to the Jth 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(3) = 0.0) then
|
|
*> the Jth right-hand side is the first with a normwise error
|
|
*> bound that is not guaranteed (the smallest J such
|
|
*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
|
|
*> the Jth right-hand side is the first with either a normwise or
|
|
*> componentwise error bound that is not guaranteed (the smallest
|
|
*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
|
|
*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
|
|
*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
|
|
*> about all of the right-hand sides check ERR_BNDS_NORM or
|
|
*> ERR_BNDS_COMP.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \date November 2011
|
|
*
|
|
*> \ingroup realGEcomputational
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE SGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
|
|
$ R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
|
|
$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
|
|
$ WORK, IWORK, INFO )
|
|
*
|
|
* -- LAPACK computational routine (version 3.4.0) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* November 2011
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER TRANS, EQUED
|
|
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
|
|
$ N_ERR_BNDS
|
|
REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IPIV( * ), IWORK( * )
|
|
REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
|
|
$ X( LDX , * ), WORK( * )
|
|
REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
|
|
$ ERR_BNDS_NORM( NRHS, * ),
|
|
$ ERR_BNDS_COMP( NRHS, * )
|
|
* ..
|
|
*
|
|
* ==================================================================
|
|
*
|
|
* .. Parameters ..
|
|
REAL ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
|
REAL ITREF_DEFAULT, ITHRESH_DEFAULT,
|
|
$ COMPONENTWISE_DEFAULT
|
|
REAL RTHRESH_DEFAULT, DZTHRESH_DEFAULT
|
|
PARAMETER ( ITREF_DEFAULT = 1.0 )
|
|
PARAMETER ( ITHRESH_DEFAULT = 10.0 )
|
|
PARAMETER ( COMPONENTWISE_DEFAULT = 1.0 )
|
|
PARAMETER ( RTHRESH_DEFAULT = 0.5 )
|
|
PARAMETER ( DZTHRESH_DEFAULT = 0.25 )
|
|
INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
|
|
$ LA_LINRX_CWISE_I
|
|
PARAMETER ( LA_LINRX_ITREF_I = 1,
|
|
$ LA_LINRX_ITHRESH_I = 2 )
|
|
PARAMETER ( LA_LINRX_CWISE_I = 3 )
|
|
INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
|
|
$ LA_LINRX_RCOND_I
|
|
PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
|
|
PARAMETER ( LA_LINRX_RCOND_I = 3 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
CHARACTER(1) NORM
|
|
LOGICAL ROWEQU, COLEQU, NOTRAN
|
|
INTEGER J, TRANS_TYPE, PREC_TYPE, REF_TYPE
|
|
INTEGER N_NORMS
|
|
REAL ANORM, RCOND_TMP
|
|
REAL ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
|
|
LOGICAL IGNORE_CWISE
|
|
INTEGER ITHRESH
|
|
REAL RTHRESH, UNSTABLE_THRESH
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL XERBLA, SGECON, SLA_GERFSX_EXTENDED
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, SQRT
|
|
* ..
|
|
* .. External Functions ..
|
|
EXTERNAL LSAME, BLAS_FPINFO_X, ILATRANS, ILAPREC
|
|
EXTERNAL SLAMCH, SLANGE, SLA_GERCOND
|
|
REAL SLAMCH, SLANGE, SLA_GERCOND
|
|
LOGICAL LSAME
|
|
INTEGER BLAS_FPINFO_X
|
|
INTEGER ILATRANS, ILAPREC
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Check the input parameters.
|
|
*
|
|
INFO = 0
|
|
TRANS_TYPE = ILATRANS( TRANS )
|
|
REF_TYPE = INT( ITREF_DEFAULT )
|
|
IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN
|
|
IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0 ) THEN
|
|
PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT
|
|
ELSE
|
|
REF_TYPE = PARAMS( LA_LINRX_ITREF_I )
|
|
END IF
|
|
END IF
|
|
*
|
|
* Set default parameters.
|
|
*
|
|
ILLRCOND_THRESH = REAL( N ) * SLAMCH( 'Epsilon' )
|
|
ITHRESH = INT( ITHRESH_DEFAULT )
|
|
RTHRESH = RTHRESH_DEFAULT
|
|
UNSTABLE_THRESH = DZTHRESH_DEFAULT
|
|
IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0
|
|
*
|
|
IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN
|
|
IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0 ) THEN
|
|
PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH
|
|
ELSE
|
|
ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) )
|
|
END IF
|
|
END IF
|
|
IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN
|
|
IF ( PARAMS( LA_LINRX_CWISE_I ).LT.0.0 ) THEN
|
|
IF ( IGNORE_CWISE ) THEN
|
|
PARAMS( LA_LINRX_CWISE_I ) = 0.0
|
|
ELSE
|
|
PARAMS( LA_LINRX_CWISE_I ) = 1.0
|
|
END IF
|
|
ELSE
|
|
IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0
|
|
END IF
|
|
END IF
|
|
IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN
|
|
N_NORMS = 0
|
|
ELSE IF ( IGNORE_CWISE ) THEN
|
|
N_NORMS = 1
|
|
ELSE
|
|
N_NORMS = 2
|
|
END IF
|
|
*
|
|
NOTRAN = LSAME( TRANS, 'N' )
|
|
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
|
|
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
|
|
*
|
|
* Test input parameters.
|
|
*
|
|
IF( TRANS_TYPE.EQ.-1 ) THEN
|
|
INFO = -1
|
|
ELSE IF( .NOT.ROWEQU .AND. .NOT.COLEQU .AND.
|
|
$ .NOT.LSAME( EQUED, 'N' ) ) THEN
|
|
INFO = -2
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -13
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -15
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SGERFSX', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible.
|
|
*
|
|
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
|
|
RCOND = 1.0
|
|
DO J = 1, NRHS
|
|
BERR( J ) = 0.0
|
|
IF ( N_ERR_BNDS .GE. 1 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_TRUST_I) = 1.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
|
|
END IF
|
|
IF ( N_ERR_BNDS .GE. 2 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_ERR_I) = 0.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0
|
|
END IF
|
|
IF ( N_ERR_BNDS .GE. 3 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_RCOND_I) = 1.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0
|
|
END IF
|
|
END DO
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Default to failure.
|
|
*
|
|
RCOND = 0.0
|
|
DO J = 1, NRHS
|
|
BERR( J ) = 1.0
|
|
IF ( N_ERR_BNDS .GE. 1 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
|
|
END IF
|
|
IF ( N_ERR_BNDS .GE. 2 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
|
|
END IF
|
|
IF ( N_ERR_BNDS .GE. 3 ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0
|
|
END IF
|
|
END DO
|
|
*
|
|
* Compute the norm of A and the reciprocal of the condition
|
|
* number of A.
|
|
*
|
|
IF( NOTRAN ) THEN
|
|
NORM = 'I'
|
|
ELSE
|
|
NORM = '1'
|
|
END IF
|
|
ANORM = SLANGE( NORM, N, N, A, LDA, WORK )
|
|
CALL SGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
|
|
*
|
|
* Perform refinement on each right-hand side
|
|
*
|
|
IF ( REF_TYPE .NE. 0 ) THEN
|
|
|
|
PREC_TYPE = ILAPREC( 'D' )
|
|
|
|
IF ( NOTRAN ) THEN
|
|
CALL SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N,
|
|
$ NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B,
|
|
$ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
|
|
$ ERR_BNDS_COMP, WORK(N+1), WORK(1), WORK(2*N+1),
|
|
$ WORK(1), RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH,
|
|
$ IGNORE_CWISE, INFO )
|
|
ELSE
|
|
CALL SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N,
|
|
$ NRHS, A, LDA, AF, LDAF, IPIV, ROWEQU, R, B,
|
|
$ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
|
|
$ ERR_BNDS_COMP, WORK(N+1), WORK(1), WORK(2*N+1),
|
|
$ WORK(1), RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH,
|
|
$ IGNORE_CWISE, INFO )
|
|
END IF
|
|
END IF
|
|
|
|
ERR_LBND = MAX( 10.0, SQRT( REAL( N ) ) ) * SLAMCH( 'Epsilon' )
|
|
IF ( N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1 ) THEN
|
|
*
|
|
* Compute scaled normwise condition number cond(A*C).
|
|
*
|
|
IF ( COLEQU .AND. NOTRAN ) THEN
|
|
RCOND_TMP = SLA_GERCOND( TRANS, N, A, LDA, AF, LDAF, IPIV,
|
|
$ -1, C, INFO, WORK, IWORK )
|
|
ELSE IF ( ROWEQU .AND. .NOT. NOTRAN ) THEN
|
|
RCOND_TMP = SLA_GERCOND( TRANS, N, A, LDA, AF, LDAF, IPIV,
|
|
$ -1, R, INFO, WORK, IWORK )
|
|
ELSE
|
|
RCOND_TMP = SLA_GERCOND( TRANS, N, A, LDA, AF, LDAF, IPIV,
|
|
$ 0, R, INFO, WORK, IWORK )
|
|
END IF
|
|
DO J = 1, NRHS
|
|
*
|
|
* Cap the error at 1.0.
|
|
*
|
|
IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
|
|
$ .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0 )
|
|
$ ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
|
|
*
|
|
* Threshold the error (see LAWN).
|
|
*
|
|
IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
|
|
ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0
|
|
IF ( INFO .LE. N ) INFO = N + J
|
|
ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND )
|
|
$ THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND
|
|
ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0
|
|
END IF
|
|
*
|
|
* Save the condition number.
|
|
*
|
|
IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
|
|
ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP
|
|
END IF
|
|
END DO
|
|
END IF
|
|
|
|
IF ( N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2 ) THEN
|
|
*
|
|
* Compute componentwise condition number cond(A*diag(Y(:,J))) for
|
|
* each right-hand side using the current solution as an estimate of
|
|
* the true solution. If the componentwise error estimate is too
|
|
* large, then the solution is a lousy estimate of truth and the
|
|
* estimated RCOND may be too optimistic. To avoid misleading users,
|
|
* the inverse condition number is set to 0.0 when the estimated
|
|
* cwise error is at least CWISE_WRONG.
|
|
*
|
|
CWISE_WRONG = SQRT( SLAMCH( 'Epsilon' ) )
|
|
DO J = 1, NRHS
|
|
IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG )
|
|
$ THEN
|
|
RCOND_TMP = SLA_GERCOND( TRANS, N, A, LDA, AF, LDAF,
|
|
$ IPIV, 1, X(1,J), INFO, WORK, IWORK )
|
|
ELSE
|
|
RCOND_TMP = 0.0
|
|
END IF
|
|
*
|
|
* Cap the error at 1.0.
|
|
*
|
|
IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
|
|
$ .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0 )
|
|
$ ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
|
|
*
|
|
* Threshold the error (see LAWN).
|
|
*
|
|
IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
|
|
ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
|
|
ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0
|
|
IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0
|
|
$ .AND. INFO.LT.N + J ) INFO = N + J
|
|
ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I )
|
|
$ .LT. ERR_LBND ) THEN
|
|
ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND
|
|
ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
|
|
END IF
|
|
*
|
|
* Save the condition number.
|
|
*
|
|
IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
|
|
ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP
|
|
END IF
|
|
END DO
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SGERFSX
|
|
*
|
|
END
|