dgesvxx.f

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      SUBROUTINE DGESVXX( 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 3.2.2)                          --
*     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
*     -- Jason Riedy of Univ. of California Berkeley.                 --
*     -- June 2010                                                    --
*
*     -- LAPACK is a software package provided by Univ. of Tennessee, --
*     -- Univ. of California Berkeley and NAG Ltd.                    --
*
      IMPLICIT NONE
*     ..
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, TRANS
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
     $                   N_ERR_BNDS
      DOUBLE PRECISION   RCOND, RPVGRW
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   X( LDX , * ),WORK( * )
      DOUBLE PRECISION   R( * ), C( * ), PARAMS( * ), BERR( * ),
     $                   ERR_BNDS_NORM( NRHS, * ),
     $                   ERR_BNDS_COMP( NRHS, * )
*     ..
*
*     Purpose
*     =======
*
*     DGESVXX uses the LU factorization to compute the solution to a
*     double precision 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. DGESVXX 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.
*
*     DGESVXX 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
*     DGESVXX 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 DGESVXX would itself produce.
*
*     Description
*     ===========
*
*     The following steps are performed:
*
*     1. If FACT = 'E', double precision 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.
*
*     Arguments
*     =========
*
*     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.
*
*     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  (Conjugate Transpose = 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.
*     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.
*
*     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.
*     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.
*
*     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, 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
*     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.
*
*     RPVGRW  (output) DOUBLE PRECISION
*     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 DGESVX, this quantity is
*     returned in WORK(1).
*
*     BERR    (output) DOUBLE PRECISION 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).
*
*     N_ERR_BNDS (input) 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.
*
*     ERR_BNDS_NORM  (output) DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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.
*
*     ERR_BNDS_COMP  (output) DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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.
*
*     NPARAMS (input) INTEGER
*     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
*     PARAMS array is never referenced and default values are used.
*
*     PARAMS  (input / output) DOUBLE PRECISION 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: 1.0D+0
*            = 0.0 : No refinement is performed, and no error bounds are
*                    computed.
*            = 1.0 : Use the extra-precise refinement algorithm.
*              (other values are reserved for future use)
*
*       PARAMS(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)
*
*     WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)
*
*     IWORK   (workspace) INTEGER array, dimension (N)
*
*     INFO    (output) 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.
*
*     ==================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+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
      DOUBLE PRECISION   AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      EXTERNAL           LSAME, DLAMCH, DLA_RPVGRW
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLA_RPVGRW
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEEQUB, DGETRF, DGETRS, DLACPY, DLAQGE,
     $                   XERBLA, DLASCL2, DGERFSX
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      NOTRAN = LSAME( TRANS, 'N' )
      SMLNUM = DLAMCH( '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 DGERFSX.
*
      RPVGRW = ZERO
*
*     Test the input parameters.  PARAMS is not tested until DGERFSX.
*
      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( 'DGESVXX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*     Compute row and column scalings to equilibrate the matrix A.
*
         CALL DGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $        INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*     Equilibrate the matrix.
*
            CALL DLAQGE( 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 ) = 1.0D+0
            END DO
         END IF
         IF ( .NOT.COLEQU ) THEN
            DO J = 1, N
               C( J ) = 1.0D+0
            END DO
         END IF
      END IF
*
*     Scale the right-hand side.
*
      IF( NOTRAN ) THEN
         IF( ROWEQU ) CALL DLASCL2( N, NRHS, R, B, LDB )
      ELSE
         IF( COLEQU ) CALL DLASCL2( N, NRHS, C, B, LDB )
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the LU factorization of A.
*
         CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
         CALL DGETRF( 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 = DLA_RPVGRW( N, INFO, A, LDA, AF, LDAF )
            RETURN
         END IF
      END IF
*
*     Compute the reciprocal pivot growth factor RPVGRW.
*
      RPVGRW = DLA_RPVGRW( N, N, A, LDA, AF, LDAF )
*
*     Compute the solution matrix X.
*
      CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL DGETRS( 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 DGERFSX( 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 DLASCL2 ( N, NRHS, C, X, LDX )
      ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
         CALL DLASCL2 ( N, NRHS, R, X, LDX )
      END IF
*
      RETURN
*
*     End of DGESVXX

      END