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version 1.7, 2010/12/21 13:53:26 version 1.8, 2011/11/21 20:42:52
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   *> \brief <b> DGESVX 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 DGESVX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
   *                          EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
   *                          WORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          EQUED, FACT, TRANS
   *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * ), IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   *      $                   BERR( * ), C( * ), FERR( * ), R( * ),
   *      $                   WORK( * ), X( LDX, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGESVX 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.
   *>
   *> Error bounds on the solution and a condition estimate are also
   *> provided.
   *> \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.  If the
   *>    reciprocal of the condition number is less than machine precision,
   *>    INFO = N+1 is returned as a warning, but 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. Iterative refinement is applied to improve the computed solution
   *>    matrix and calculate error bounds and backward error estimates
   *>    for it.
   *>
   *> 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:
   *  ==========
   *
   *> \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  (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 DOUBLE PRECISION 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 or output) DOUBLE PRECISION 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 DGETRF.  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 or output) 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 DGETRF; 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 or output) 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 or output) DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] C
   *> \verbatim
   *>          C is or output) DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
   *>          If INFO = 0 or INFO = N+1, 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 DOUBLE PRECISION
   *>          The estimate of the reciprocal 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.  This condition is
   *>          indicated by a return code of INFO > 0.
   *> \endverbatim
   *>
   *> \param[out] FERR
   *> \verbatim
   *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
   *>          The estimated forward error bound for each solution vector
   *>          X(j) (the j-th column of the solution matrix X).
   *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
   *>          is an estimated upper bound for the magnitude of the largest
   *>          element in (X(j) - XTRUE) divided by the magnitude of the
   *>          largest element in X(j).  The estimate is as reliable as
   *>          the estimate for RCOND, and is almost always a slight
   *>          overestimate of the true error.
   *> \endverbatim
   *>
   *> \param[out] BERR
   *> \verbatim
   *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
   *>          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[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (4*N)
   *>          On exit, WORK(1) contains the reciprocal pivot growth
   *>          factor norm(A)/norm(U). The "max absolute element" norm is
   *>          used. If WORK(1) 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, condition
   *>          estimator RCOND, and forward error bound FERR could be
   *>          unreliable. If factorization fails with 0<INFO<=N, then
   *>          WORK(1) contains the reciprocal pivot growth factor for the
   *>          leading INFO columns of A.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   *>          > 0:  if INFO = i, and i is
   *>                <= N:  U(i,i) 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+1: U is nonsingular, but RCOND is less than machine
   *>                       precision, meaning that the matrix is singular
   *>                       to working precision.  Nevertheless, the
   *>                       solution and error bounds are computed because
   *>                       there are a number of situations where the
   *>                       computed solution can be more accurate than the
   *>                       value of RCOND would suggest.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEsolve
   *
   *  =====================================================================
       SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,        SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
      $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,       $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
      $                   WORK, IWORK, INFO )       $                   WORK, IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          EQUED, FACT, TRANS        CHARACTER          EQUED, FACT, TRANS
Line 19 Line 366
      $                   WORK( * ), X( LDX, * )       $                   WORK( * ), X( LDX, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGESVX 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.  
 *  
 *  Error bounds on the solution and a condition estimate are also  
 *  provided.  
 *  
 *  Description  
 *  ===========  
 *  
 *  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.  If the  
 *     reciprocal of the condition number is less than machine precision,  
 *     INFO = N+1 is returned as a warning, but 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. Iterative refinement is applied to improve the computed solution  
 *     matrix and calculate error bounds and backward error estimates  
 *     for it.  
 *  
 *  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.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  FACT    (input) 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.  
 *  
 *  TRANS   (input) 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  (Transpose)  
 *  
 *  N       (input) INTEGER  
 *          The number of linear equations, i.e., the order of the  
 *          matrix A.  N >= 0.  
 *  
 *  NRHS    (input) INTEGER  
 *          The number of right hand sides, i.e., the number of columns  
 *          of the matrices B and X.  NRHS >= 0.  
 *  
 *  A       (input/output) DOUBLE PRECISION 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).  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  AF      (input or output) DOUBLE PRECISION 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 DGETRF.  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).  
 *  
 *  LDAF    (input) INTEGER  
 *          The leading dimension of the array AF.  LDAF >= max(1,N).  
 *  
 *  IPIV    (input or output) 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 DGETRF; 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.  
 *  
 *  EQUED   (input or output) 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.  
 *  
 *  R       (input or output) DOUBLE PRECISION 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.  
 *  
 *  C       (input or output) DOUBLE PRECISION 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.  
 *  
 *  B       (input/output) DOUBLE PRECISION 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.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *          If INFO = 0 or INFO = N+1, 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'.  
 *  
 *  LDX     (input) INTEGER  
 *          The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *  RCOND   (output) DOUBLE PRECISION  
 *          The estimate of the reciprocal 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.  This condition is  
 *          indicated by a return code of INFO > 0.  
 *  
 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)  
 *          The estimated forward error bound for each solution vector  
 *          X(j) (the j-th column of the solution matrix X).  
 *          If XTRUE is the true solution corresponding to X(j), FERR(j)  
 *          is an estimated upper bound for the magnitude of the largest  
 *          element in (X(j) - XTRUE) divided by the magnitude of the  
 *          largest element in X(j).  The estimate is as reliable as  
 *          the estimate for RCOND, and is almost always a slight  
 *          overestimate of the true error.  
 *  
 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)  
 *          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).  
 *  
 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (4*N)  
 *          On exit, WORK(1) contains the reciprocal pivot growth  
 *          factor norm(A)/norm(U). The "max absolute element" norm is  
 *          used. If WORK(1) 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, condition  
 *          estimator RCOND, and forward error bound FERR could be  
 *          unreliable. If factorization fails with 0<INFO<=N, then  
 *          WORK(1) contains the reciprocal pivot growth factor for the  
 *          leading INFO columns of A.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (N)  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value  
 *          > 0:  if INFO = i, and i is  
 *                <= N:  U(i,i) 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+1: U is nonsingular, but RCOND is less than machine  
 *                       precision, meaning that the matrix is singular  
 *                       to working precision.  Nevertheless, the  
 *                       solution and error bounds are computed because  
 *                       there are a number of situations where the  
 *                       computed solution can be more accurate than the  
 *                       value of RCOND would suggest.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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