dd/d23/sgesvxx_8f_source.html

 *> \brief <b> SGESVXX computes the solution to system of linear equations A * X = B for GE matrices</b>
 *
 *  =========== DOCUMENTATION ===========
 *
 * Online html documentation available at
 *            http://www.netlib.org/lapack/explore-html/
 *
 *> \htmlonly
 *> Download SGESVXX + dependencies
 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgesvxx.f">
 *> [TGZ]</a>
 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgesvxx.f">
 *> [ZIP]</a>
 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgesvxx.f">
 *> [TXT]</a>
 *> \endhtmlonly
 *
 *  Definition:
 *  ===========
 *
 *       SUBROUTINE SGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
 *                           EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
 *                           BERR, N_ERR_BNDS, ERR_BNDS_NORM,
 *                           ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
 *                           INFO )
 *
 *       .. Scalar Arguments ..
 *       CHARACTER          EQUED, FACT, TRANS
 *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
 *      $                   N_ERR_BNDS
 *       REAL               RCOND, RPVGRW
 *       ..
 *       .. 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
 *>
 *>    SGESVXX uses the LU factorization to compute the solution to a
 *>    real system of linear equations  A * X = B,  where A is an
 *>    N-by-N matrix and X and B are N-by-NRHS matrices.
 *>
 *>    If requested, both normwise and maximum componentwise error bounds
 *>    are returned. SGESVXX will return a solution with a tiny
 *>    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 also are
 *>    calculated and returned.
 *>
 *>    SGESVXX accepts user-provided factorizations and equilibration
 *>    factors; see the definitions of the FACT and EQUED options.
 *>    Solving with refinement and using a factorization from a previous
 *>    SGESVXX call will also produce a solution with either O(eps)
 *>    errors or warnings, but we cannot make that claim for general
 *>    user-provided factorizations and equilibration factors if they
 *>    differ from what SGESVXX would itself produce.
 *> \endverbatim
 *
 *> \par Description:
 *  =================
 *>
 *> \verbatim
 *>
 *>    The following steps are performed:
 *>
 *>    1. If FACT = 'E', real scaling factors are computed to equilibrate
 *>    the system:
 *>
 *>      TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
 *>      TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
 *>      TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
 *>    or diag(C)*B (if TRANS = 'T' or 'C').
 *>
 *>    2. If FACT = 'N' or 'E', the LU decomposition is used to factor
 *>    the matrix A (after equilibration if FACT = 'E') as
 *>
 *>      A = P * L * U,
 *>
 *>    where P is a permutation matrix, L is a unit lower triangular
 *>    matrix, and U is upper triangular.
 *>
 *>    3. If some U(i,i)=0, so that U is exactly singular, then the
 *>    routine returns with INFO = i. Otherwise, the factored form of A
 *>    is used to estimate the condition number of the matrix A (see
 *>    argument RCOND). 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 as described below.
 *>
 *>    4. The system of equations is solved for X using the factored form
 *>    of A.
 *>
 *>    5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
 *>    the routine will use iterative refinement to try 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
 *>    that it solves the original system before equilibration.
 *> \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] FACT
 *> \verbatim
 *>          FACT is CHARACTER*1
 *>     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.
 *>       = '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 R and C.
 *>               A, AF, and IPIV are not modified.
 *>       = 'N':  The matrix A will be copied to AF and factored.
 *>       = 'E':  The matrix A will be equilibrated if necessary, then
 *>               copied to AF and factored.
 *> \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] N
 *> \verbatim
 *>          N is INTEGER
 *>     The number of linear equations, i.e., 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,out] A
 *> \verbatim
 *>          A is REAL array, dimension (LDA,N)
 *>     On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
 *>     not 'N', then A must have been equilibrated by the scaling
 *>     factors in R and/or C.  A is not modified if FACT = 'F' or
 *>     'N', or if FACT = 'E' and EQUED = 'N' on exit.
 *>
 *>     On exit, if EQUED .ne. 'N', A is scaled as follows:
 *>     EQUED = 'R':  A := diag(R) * A
 *>     EQUED = 'C':  A := A * diag(C)
 *>     EQUED = 'B':  A := diag(R) * A * diag(C).
 *> \endverbatim
 *>
 *> \param[in] LDA
 *> \verbatim
 *>          LDA is INTEGER
 *>     The leading dimension of the array A.  LDA >= max(1,N).
 *> \endverbatim
 *>
 *> \param[in,out] AF
 *> \verbatim
 *>          AF is REAL array, dimension (LDAF,N)
 *>     If FACT = 'F', then AF is an input argument and on entry
 *>     contains the factors L and U from the factorization
 *>     A = P*L*U as computed by SGETRF.  If EQUED .ne. 'N', then
 *>     AF is the factored form of the equilibrated matrix A.
 *>
 *>     If FACT = 'N', then AF is an output argument and on exit
 *>     returns the factors L and U from the factorization A = P*L*U
 *>     of the original matrix A.
 *>
 *>     If FACT = 'E', then AF is an output argument and on exit
 *>     returns the factors L and U from the factorization A = P*L*U
 *>     of the equilibrated matrix A (see the description of A for
 *>     the form of the equilibrated matrix).
 *> \endverbatim
 *>
 *> \param[in] LDAF
 *> \verbatim
 *>          LDAF is INTEGER
 *>     The leading dimension of the array AF.  LDAF >= max(1,N).
 *> \endverbatim
 *>
 *> \param[in,out] IPIV
 *> \verbatim
 *>          IPIV is INTEGER array, dimension (N)
 *>     If FACT = 'F', then IPIV is an input argument and on entry
 *>     contains the pivot indices from the factorization A = P*L*U
 *>     as computed by SGETRF; row i of the matrix was interchanged
 *>     with row IPIV(i).
 *>
 *>     If FACT = 'N', then IPIV is an output argument and on exit
 *>     contains the pivot indices from the factorization A = P*L*U
 *>     of the original matrix A.
 *>
 *>     If FACT = 'E', then IPIV is an output argument and on exit
 *>     contains the pivot indices from the factorization A = P*L*U
 *>     of the equilibrated matrix A.
 *> \endverbatim
 *>
 *> \param[in,out] EQUED
 *> \verbatim
 *>          EQUED is CHARACTER*1
 *>     Specifies the form of equilibration that was done.
 *>       = 'N':  No equilibration (always true if FACT = 'N').
 *>       = '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).
 *>     EQUED is an input argument if FACT = 'F'; otherwise, it is an
 *>     output argument.
 *> \endverbatim
 *>
 *> \param[in,out] 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.  R is an input argument if FACT = 'F';
 *>     otherwise, R is an output argument.  If FACT = 'F' and
 *>     EQUED = 'R' or 'B', each element of R must be positive.
 *>     If R is output, each element of R is a power of the radix.
 *>     If R is input, 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,out] 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.  C is an input argument if FACT = 'F';
 *>     otherwise, C is an output argument.  If FACT = 'F' and
 *>     EQUED = 'C' or 'B', each element of C must be positive.
 *>     If C is output, each element of C is a power of the radix.
 *>     If C is input, 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,out] B
 *> \verbatim
 *>          B is REAL array, dimension (LDB,NRHS)
 *>     On entry, the N-by-NRHS right hand side matrix B.
 *>     On exit,
 *>     if EQUED = 'N', B is not modified;
 *>     if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
 *>        diag(R)*B;
 *>     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
 *>        overwritten by diag(C)*B.
 *> \endverbatim
 *>
 *> \param[in] LDB
 *> \verbatim
 *>          LDB is INTEGER
 *>     The leading dimension of the array B.  LDB >= max(1,N).
 *> \endverbatim
 *>
 *> \param[out] X
 *> \verbatim
 *>          X is REAL array, dimension (LDX,NRHS)
 *>     If INFO = 0, the N-by-NRHS solution matrix X to the original
 *>     system of equations.  Note that A and B are modified on exit
 *>     if EQUED .ne. 'N', and the solution to the equilibrated system is
 *>     inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
 *>     inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
 *> \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] RPVGRW
 *> \verbatim
 *>          RPVGRW is REAL
 *>     Reciprocal pivot growth.  On exit, this contains the reciprocal
 *>     pivot growth factor norm(A)/norm(U). The "max absolute element"
 *>     norm is used.  If this is much less than 1, then the stability of
 *>     the LU factorization of the (equilibrated) 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, then this contains the reciprocal pivot growth factor
 *>     for the leading INFO columns of A.  In SGESVX, this quantity is
 *>     returned in WORK(1).
 *> \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 April 2012
 *
 *> \ingroup realGEsolve
 *
 *  =====================================================================
       SUBROUTINE sgesvxx( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
      $                    EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
      $                    BERR, N_ERR_BNDS, ERR_BNDS_NORM,
      $                    ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
      $                    INFO )
 *
 *  -- LAPACK driver routine (version 3.7.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     April 2012
 *
 *     .. Scalar Arguments ..
       CHARACTER          EQUED, FACT, TRANS
       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
      $                   n_err_bnds
       REAL               RCOND, RPVGRW
 *     ..
 *     .. 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 )
       INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
       INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
       INTEGER            CMP_ERR_I, PIV_GROWTH_I
       parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
      $                   berr_i = 3 )
       parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
       parameter( cmp_rcond_i = 7, cmp_err_i = 8,
      $                   piv_growth_i = 9 )
 *     ..
 *     .. Local Scalars ..
       LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
       INTEGER            INFEQU, J
       REAL               AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND,
      $                   smlnum
 *     ..
 *     .. External Functions ..
       EXTERNAL           lsame, slamch, sla_gerpvgrw
       LOGICAL            LSAME
       REAL               SLAMCH, SLA_GERPVGRW
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           sgeequb, sgetrf, sgetrs, slacpy, slaqge,
      $                   xerbla, slascl2, sgerfsx
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          max, min
 *     ..
 *     .. Executable Statements ..
 *
       info = 0
       nofact = lsame( fact, 'N' )
       equil = lsame( fact, 'E' )
       notran = lsame( trans, 'N' )
       smlnum = slamch( 'Safe minimum' )
       bignum = one / smlnum
       IF( nofact .OR. equil ) THEN
          equed = 'N'
          rowequ = .false.
          colequ = .false.
       ELSE
          rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
          colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
       END IF
 *
 *     Default is failure.  If an input parameter is wrong or
 *     factorization fails, make everything look horrible.  Only the
 *     pivot growth is set here, the rest is initialized in SGERFSX.
 *
       rpvgrw = zero
 *
 *     Test the input parameters.  PARAMS is not tested until SGERFSX.
 *
       IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
      $     lsame( fact, 'F' ) ) THEN
          info = -1
       ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
      $        lsame( trans, 'C' ) ) 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( lsame( fact, 'F' ) .AND. .NOT.
      $        ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
          info = -10
       ELSE
          IF( rowequ ) THEN
             rcmin = bignum
             rcmax = zero
             DO 10 j = 1, n
                rcmin = min( rcmin, r( j ) )
                rcmax = max( rcmax, r( j ) )
  10         CONTINUE
             IF( rcmin.LE.zero ) THEN
                info = -11
             ELSE IF( n.GT.0 ) THEN
                rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
             ELSE
                rowcnd = one
             END IF
          END IF
          IF( colequ .AND. info.EQ.0 ) THEN
             rcmin = bignum
             rcmax = zero
             DO 20 j = 1, n
                rcmin = min( rcmin, c( j ) )
                rcmax = max( rcmax, c( j ) )
  20         CONTINUE
             IF( rcmin.LE.zero ) THEN
                info = -12
             ELSE IF( n.GT.0 ) THEN
                colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
             ELSE
                colcnd = one
             END IF
          END IF
          IF( info.EQ.0 ) THEN
             IF( ldb.LT.max( 1, n ) ) THEN
                info = -14
             ELSE IF( ldx.LT.max( 1, n ) ) THEN
                info = -16
             END IF
          END IF
       END IF
 *
       IF( info.NE.0 ) THEN
          CALL xerbla( 'SGESVXX', -info )
          RETURN
       END IF
 *
       IF( equil ) THEN
 *
 *     Compute row and column scalings to equilibrate the matrix A.
 *
          CALL sgeequb( n, n, a, lda, r, c, rowcnd, colcnd, amax,
      $        infequ )
          IF( infequ.EQ.0 ) THEN
 *
 *     Equilibrate the matrix.
 *
             CALL slaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
      $           equed )
             rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
             colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
          END IF
 *
 *     If the scaling factors are not applied, set them to 1.0.
 *
          IF ( .NOT.rowequ ) THEN
             DO j = 1, n
                r( j ) = 1.0
             END DO
          END IF
          IF ( .NOT.colequ ) THEN
             DO j = 1, n
                c( j ) = 1.0
             END DO
          END IF
       END IF
 *
 *     Scale the right-hand side.
 *
       IF( notran ) THEN
          IF( rowequ ) CALL slascl2( n, nrhs, r, b, ldb )
       ELSE
          IF( colequ ) CALL slascl2( n, nrhs, c, b, ldb )
       END IF
 *
       IF( nofact .OR. equil ) THEN
 *
 *        Compute the LU factorization of A.
 *
          CALL slacpy( 'Full', n, n, a, lda, af, ldaf )
          CALL sgetrf( n, n, af, ldaf, ipiv, info )
 *
 *        Return if INFO is non-zero.
 *
          IF( info.GT.0 ) THEN
 *
 *           Pivot in column INFO is exactly 0
 *           Compute the reciprocal pivot growth factor of the
 *           leading rank-deficient INFO columns of A.
 *
             rpvgrw = sla_gerpvgrw( n, info, a, lda, af, ldaf )
             RETURN
          END IF
       END IF
 *
 *     Compute the reciprocal pivot growth factor RPVGRW.
 *
       rpvgrw = sla_gerpvgrw( n, n, a, lda, af, ldaf )
 *
 *     Compute the solution matrix X.
 *
       CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
       CALL sgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
 *
 *     Use iterative refinement to improve the computed solution and
 *     compute error bounds and backward error estimates for it.
 *
       CALL 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 )
 *
 *     Scale solutions.
 *
       IF ( colequ .AND. notran ) THEN
          CALL slascl2 ( n, nrhs, c, x, ldx )
       ELSE IF ( rowequ .AND. .NOT.notran ) THEN
          CALL slascl2 ( n, nrhs, r, x, ldx )
       END IF
 *
       RETURN
 *
 *     End of SGESVXX

       END
sgetrs
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:123

slascl2
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:92

sla_gerpvgrw
real function sla_gerpvgrw(N, NCOLS, A, LDA, AF, LDAF)
SLA_GERPVGRW
Definition: sla_gerpvgrw.f:99

sgerfsx
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)
SGERFSX
Definition: sgerfsx.f:416

sgetrf
subroutine sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:110

xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62

sgeequb
subroutine sgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQUB
Definition: sgeequb.f:148

lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

slamch
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69

slaqge
subroutine slaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ...
Definition: slaqge.f:144

slacpy
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105

sgesvxx
subroutine sgesvxx(FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
 SGESVXX computes the solution to system of linear equations A * X = B for GE matrices ...
Definition: sgesvxx.f:545