LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ sgesvxx()

subroutine sgesvxx ( character  FACT,
character  TRANS,
integer  N,
integer  NRHS,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
character  EQUED,
real, dimension( * )  R,
real, dimension( * )  C,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( ldx , * )  X,
integer  LDX,
real  RCOND,
real  RPVGRW,
real, dimension( * )  BERR,
integer  N_ERR_BNDS,
real, dimension( nrhs, * )  ERR_BNDS_NORM,
real, dimension( nrhs, * )  ERR_BNDS_COMP,
integer  NPARAMS,
real, dimension( * )  PARAMS,
real, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

SGESVXX computes the solution to system of linear equations A * X = B for GE matrices

Download SGESVXX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    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.
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 (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.
     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.
Parameters
[in]FACT
          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.
[in]TRANS
          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)
[in]N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
[in]NRHS
          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices B and X.  NRHS >= 0.
[in,out]A
          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).
[in]LDA
          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).
[in,out]AF
          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).
[in]LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
[in,out]IPIV
          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.
[in,out]EQUED
          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.
[in,out]R
          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.
[in,out]C
          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.
[in,out]B
          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.
[in]LDB
          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).
[out]X
          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'.
[in]LDX
          LDX is INTEGER
     The leading dimension of the array X.  LDX >= max(1,N).
[out]RCOND
          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.
[out]RPVGRW
          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).
[out]BERR
          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).
[in]N_ERR_BNDS
          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.
[out]ERR_BNDS_NORM
          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.
[out]ERR_BNDS_COMP
          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.
[in]NPARAMS
          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.
[in,out]PARAMS
          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)
[out]WORK
          WORK is REAL array, dimension (4*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[out]INFO
          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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
April 2012

Definition at line 545 of file sgesvxx.f.

545 *
546 * -- LAPACK driver routine (version 3.7.0) --
547 * -- LAPACK is a software package provided by Univ. of Tennessee, --
548 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
549 * April 2012
550 *
551 * .. Scalar Arguments ..
552  CHARACTER equed, fact, trans
553  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
554  $ n_err_bnds
555  REAL rcond, rpvgrw
556 * ..
557 * .. Array Arguments ..
558  INTEGER ipiv( * ), iwork( * )
559  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
560  $ x( ldx , * ),work( * )
561  REAL r( * ), c( * ), params( * ), berr( * ),
562  $ err_bnds_norm( nrhs, * ),
563  $ err_bnds_comp( nrhs, * )
564 * ..
565 *
566 * ==================================================================
567 *
568 * .. Parameters ..
569  REAL zero, one
570  parameter( zero = 0.0e+0, one = 1.0e+0 )
571  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
572  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
573  INTEGER cmp_err_i, piv_growth_i
574  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
575  $ berr_i = 3 )
576  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
577  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
578  $ piv_growth_i = 9 )
579 * ..
580 * .. Local Scalars ..
581  LOGICAL colequ, equil, nofact, notran, rowequ
582  INTEGER infequ, j
583  REAL amax, bignum, colcnd, rcmax, rcmin, rowcnd,
584  $ smlnum
585 * ..
586 * .. External Functions ..
587  EXTERNAL lsame, slamch, sla_gerpvgrw
588  LOGICAL lsame
589  REAL slamch, sla_gerpvgrw
590 * ..
591 * .. External Subroutines ..
592  EXTERNAL sgeequb, sgetrf, sgetrs, slacpy, slaqge,
594 * ..
595 * .. Intrinsic Functions ..
596  INTRINSIC max, min
597 * ..
598 * .. Executable Statements ..
599 *
600  info = 0
601  nofact = lsame( fact, 'N' )
602  equil = lsame( fact, 'E' )
603  notran = lsame( trans, 'N' )
604  smlnum = slamch( 'Safe minimum' )
605  bignum = one / smlnum
606  IF( nofact .OR. equil ) THEN
607  equed = 'N'
608  rowequ = .false.
609  colequ = .false.
610  ELSE
611  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
612  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
613  END IF
614 *
615 * Default is failure. If an input parameter is wrong or
616 * factorization fails, make everything look horrible. Only the
617 * pivot growth is set here, the rest is initialized in SGERFSX.
618 *
619  rpvgrw = zero
620 *
621 * Test the input parameters. PARAMS is not tested until SGERFSX.
622 *
623  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
624  $ lsame( fact, 'F' ) ) THEN
625  info = -1
626  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
627  $ lsame( trans, 'C' ) ) THEN
628  info = -2
629  ELSE IF( n.LT.0 ) THEN
630  info = -3
631  ELSE IF( nrhs.LT.0 ) THEN
632  info = -4
633  ELSE IF( lda.LT.max( 1, n ) ) THEN
634  info = -6
635  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
636  info = -8
637  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
638  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
639  info = -10
640  ELSE
641  IF( rowequ ) THEN
642  rcmin = bignum
643  rcmax = zero
644  DO 10 j = 1, n
645  rcmin = min( rcmin, r( j ) )
646  rcmax = max( rcmax, r( j ) )
647  10 CONTINUE
648  IF( rcmin.LE.zero ) THEN
649  info = -11
650  ELSE IF( n.GT.0 ) THEN
651  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
652  ELSE
653  rowcnd = one
654  END IF
655  END IF
656  IF( colequ .AND. info.EQ.0 ) THEN
657  rcmin = bignum
658  rcmax = zero
659  DO 20 j = 1, n
660  rcmin = min( rcmin, c( j ) )
661  rcmax = max( rcmax, c( j ) )
662  20 CONTINUE
663  IF( rcmin.LE.zero ) THEN
664  info = -12
665  ELSE IF( n.GT.0 ) THEN
666  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
667  ELSE
668  colcnd = one
669  END IF
670  END IF
671  IF( info.EQ.0 ) THEN
672  IF( ldb.LT.max( 1, n ) ) THEN
673  info = -14
674  ELSE IF( ldx.LT.max( 1, n ) ) THEN
675  info = -16
676  END IF
677  END IF
678  END IF
679 *
680  IF( info.NE.0 ) THEN
681  CALL xerbla( 'SGESVXX', -info )
682  RETURN
683  END IF
684 *
685  IF( equil ) THEN
686 *
687 * Compute row and column scalings to equilibrate the matrix A.
688 *
689  CALL sgeequb( n, n, a, lda, r, c, rowcnd, colcnd, amax,
690  $ infequ )
691  IF( infequ.EQ.0 ) THEN
692 *
693 * Equilibrate the matrix.
694 *
695  CALL slaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
696  $ equed )
697  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
698  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
699  END IF
700 *
701 * If the scaling factors are not applied, set them to 1.0.
702 *
703  IF ( .NOT.rowequ ) THEN
704  DO j = 1, n
705  r( j ) = 1.0
706  END DO
707  END IF
708  IF ( .NOT.colequ ) THEN
709  DO j = 1, n
710  c( j ) = 1.0
711  END DO
712  END IF
713  END IF
714 *
715 * Scale the right-hand side.
716 *
717  IF( notran ) THEN
718  IF( rowequ ) CALL slascl2( n, nrhs, r, b, ldb )
719  ELSE
720  IF( colequ ) CALL slascl2( n, nrhs, c, b, ldb )
721  END IF
722 *
723  IF( nofact .OR. equil ) THEN
724 *
725 * Compute the LU factorization of A.
726 *
727  CALL slacpy( 'Full', n, n, a, lda, af, ldaf )
728  CALL sgetrf( n, n, af, ldaf, ipiv, info )
729 *
730 * Return if INFO is non-zero.
731 *
732  IF( info.GT.0 ) THEN
733 *
734 * Pivot in column INFO is exactly 0
735 * Compute the reciprocal pivot growth factor of the
736 * leading rank-deficient INFO columns of A.
737 *
738  rpvgrw = sla_gerpvgrw( n, info, a, lda, af, ldaf )
739  RETURN
740  END IF
741  END IF
742 *
743 * Compute the reciprocal pivot growth factor RPVGRW.
744 *
745  rpvgrw = sla_gerpvgrw( n, n, a, lda, af, ldaf )
746 *
747 * Compute the solution matrix X.
748 *
749  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
750  CALL sgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
751 *
752 * Use iterative refinement to improve the computed solution and
753 * compute error bounds and backward error estimates for it.
754 *
755  CALL sgerfsx( trans, equed, n, nrhs, a, lda, af, ldaf,
756  $ ipiv, r, c, b, ldb, x, ldx, rcond, berr,
757  $ n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params,
758  $ work, iwork, info )
759 *
760 * Scale solutions.
761 *
762  IF ( colequ .AND. notran ) THEN
763  CALL slascl2 ( n, nrhs, c, x, ldx )
764  ELSE IF ( rowequ .AND. .NOT.notran ) THEN
765  CALL slascl2 ( n, nrhs, r, x, ldx )
766  END IF
767 *
768  RETURN
769 *
770 * End of SGESVXX
771 
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:123
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a vector.
Definition: slascl2.f:92
real function sla_gerpvgrw(N, NCOLS, A, LDA, AF, LDAF)
SLA_GERPVGRW
Definition: sla_gerpvgrw.f:99
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
subroutine sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQUB
Definition: sgeequb.f:148
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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
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
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