clatrs.f

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*> \brief \b CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/ 
*
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE CLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
*                          CNORM, INFO )
* 
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORMIN, TRANS, UPLO
*       INTEGER            INFO, LDA, N
*       REAL               SCALE
*       ..
*       .. Array Arguments ..
*       REAL               CNORM( * )
*       COMPLEX            A( LDA, * ), X( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CLATRS solves one of the triangular systems
*>
*>    A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
*>
*> with scaling to prevent overflow.  Here A is an upper or lower
*> triangular matrix, A**T denotes the transpose of A, A**H denotes the
*> conjugate transpose of A, x and b are n-element vectors, and s is a
*> scaling factor, usually less than or equal to 1, chosen so that the
*> components of x will be less than the overflow threshold.  If the
*> unscaled problem will not cause overflow, the Level 2 BLAS routine
*> CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
*> then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  Solve A * x = s*b     (No transpose)
*>          = 'T':  Solve A**T * x = s*b  (Transpose)
*>          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*>          NORMIN is CHARACTER*1
*>          Specifies whether CNORM has been set or not.
*>          = 'Y':  CNORM contains the column norms on entry
*>          = 'N':  CNORM is not set on entry.  On exit, the norms will
*>                  be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is COMPLEX array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading n by n
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading n by n lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max (1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is COMPLEX array, dimension (N)
*>          On entry, the right hand side b of the triangular system.
*>          On exit, X is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is REAL
*>          The scaling factor s for the triangular system
*>             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
*>          If SCALE = 0, the matrix A is singular or badly scaled, and
*>          the vector x is an exact or approximate solution to A*x = 0.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*>          CNORM is REAL array, dimension (N)
*>
*>          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*>          contains the norm of the off-diagonal part of the j-th column
*>          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*>          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*>          must be greater than or equal to the 1-norm.
*>
*>          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*>          returns the 1-norm of the offdiagonal part of the j-th column
*>          of A.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup complexOTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  A rough bound on x is computed; if that is less than overflow, CTRSV
*>  is called, otherwise, specific code is used which checks for possible
*>  overflow or divide-by-zero at every operation.
*>
*>  A columnwise scheme is used for solving A*x = b.  The basic algorithm
*>  if A is lower triangular is
*>
*>       x[1:n] := b[1:n]
*>       for j = 1, ..., n
*>            x(j) := x(j) / A(j,j)
*>            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*>       end
*>
*>  Define bounds on the components of x after j iterations of the loop:
*>     M(j) = bound on x[1:j]
*>     G(j) = bound on x[j+1:n]
*>  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*>
*>  Then for iteration j+1 we have
*>     M(j+1) <= G(j) / | A(j+1,j+1) |
*>     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*>            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*>
*>  where CNORM(j+1) is greater than or equal to the infinity-norm of
*>  column j+1 of A, not counting the diagonal.  Hence
*>
*>     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*>                  1<=i<=j
*>  and
*>
*>     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*>                                   1<=i< j
*>
*>  Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTRSV if the
*>  reciprocal of the largest M(j), j=1,..,n, is larger than
*>  max(underflow, 1/overflow).
*>
*>  The bound on x(j) is also used to determine when a step in the
*>  columnwise method can be performed without fear of overflow.  If
*>  the computed bound is greater than a large constant, x is scaled to
*>  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*>  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*>
*>  Similarly, a row-wise scheme is used to solve A**T *x = b  or
*>  A**H *x = b.  The basic algorithm for A upper triangular is
*>
*>       for j = 1, ..., n
*>            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
*>       end
*>
*>  We simultaneously compute two bounds
*>       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
*>       M(j) = bound on x(i), 1<=i<=j
*>
*>  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*>  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*>  Then the bound on x(j) is
*>
*>       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*>
*>            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*>                      1<=i<=j
*>
*>  and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater
*>  than max(underflow, 1/overflow).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE clatrs( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
     $                   cnorm, info )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      CHARACTER          diag, normin, trans, uplo
      INTEGER            info, lda, n
      REAL               scale
*     ..
*     .. Array Arguments ..
      REAL               cnorm( * )
      COMPLEX            a( lda, * ), x( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               zero, half, one, two
      parameter( zero = 0.0e+0, half = 0.5e+0, one = 1.0e+0,
     $                   two = 2.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            notran, nounit, upper
      INTEGER            i, imax, j, jfirst, jinc, jlast
      REAL               bignum, grow, rec, smlnum, tjj, tmax, tscal,
     $                   xbnd, xj, xmax
      COMPLEX            csumj, tjjs, uscal, zdum
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      INTEGER            icamax, isamax
      REAL               scasum, slamch
      COMPLEX            cdotc, cdotu, cladiv
      EXTERNAL           lsame, icamax, isamax, scasum, slamch, cdotc,
     $                   cdotu, cladiv
*     ..
*     .. External Subroutines ..
      EXTERNAL           caxpy, csscal, ctrsv, slabad, sscal, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, aimag, cmplx, conjg, max, min, real
*     ..
*     .. Statement Functions ..
      REAL               cabs1, cabs2
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
      cabs2( zdum ) = abs( REAL( ZDUM ) / 2. ) +
     $                abs( aimag( zdum ) / 2. )
*     ..
*     .. Executable Statements ..
*
      info = 0
      upper = lsame( uplo, 'U' )
      notran = lsame( trans, 'N' )
      nounit = lsame( diag, 'N' )
*
*     Test the input parameters.
*
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
     $         lsame( trans, 'C' ) ) THEN
         info = -2
      ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
         info = -3
      ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
     $         lsame( normin, 'N' ) ) THEN
         info = -4
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -7
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'CLATRS', -info )
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   return
*
*     Determine machine dependent parameters to control overflow.
*
      smlnum = slamch( 'Safe minimum' )
      bignum = one / smlnum
      CALL slabad( smlnum, bignum )
      smlnum = smlnum / slamch( 'Precision' )
      bignum = one / smlnum
      scale = one
*
      IF( lsame( normin, 'N' ) ) THEN
*
*        Compute the 1-norm of each column, not including the diagonal.
*
         IF( upper ) THEN
*
*           A is upper triangular.
*
            DO 10 j = 1, n
               cnorm( j ) = scasum( j-1, a( 1, j ), 1 )
   10       continue
         ELSE
*
*           A is lower triangular.
*
            DO 20 j = 1, n - 1
               cnorm( j ) = scasum( n-j, a( j+1, j ), 1 )
   20       continue
            cnorm( n ) = zero
         END IF
      END IF
*
*     Scale the column norms by TSCAL if the maximum element in CNORM is
*     greater than BIGNUM/2.
*
      imax = isamax( n, cnorm, 1 )
      tmax = cnorm( imax )
      IF( tmax.LE.bignum*half ) THEN
         tscal = one
      ELSE
         tscal = half / ( smlnum*tmax )
         CALL sscal( n, tscal, cnorm, 1 )
      END IF
*
*     Compute a bound on the computed solution vector to see if the
*     Level 2 BLAS routine CTRSV can be used.
*
      xmax = zero
      DO 30 j = 1, n
         xmax = max( xmax, cabs2( x( j ) ) )
   30 continue
      xbnd = xmax
*
      IF( notran ) THEN
*
*        Compute the growth in A * x = b.
*
         IF( upper ) THEN
            jfirst = n
            jlast = 1
            jinc = -1
         ELSE
            jfirst = 1
            jlast = n
            jinc = 1
         END IF
*
         IF( tscal.NE.one ) THEN
            grow = zero
            go to 60
         END IF
*
         IF( nounit ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, G(0) = max{x(i), i=1,...,n}.
*
            grow = half / max( xbnd, smlnum )
            xbnd = grow
            DO 40 j = jfirst, jlast, jinc
*
*              Exit the loop if the growth factor is too small.
*
               IF( grow.LE.smlnum )
     $            go to 60
*
               tjjs = a( j, j )
               tjj = cabs1( tjjs )
*
               IF( tjj.GE.smlnum ) THEN
*
*                 M(j) = G(j-1) / abs(A(j,j))
*
                  xbnd = min( xbnd, min( one, tjj )*grow )
               ELSE
*
*                 M(j) could overflow, set XBND to 0.
*
                  xbnd = zero
               END IF
*
               IF( tjj+cnorm( j ).GE.smlnum ) THEN
*
*                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
                  grow = grow*( tjj / ( tjj+cnorm( j ) ) )
               ELSE
*
*                 G(j) could overflow, set GROW to 0.
*
                  grow = zero
               END IF
   40       continue
            grow = xbnd
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            grow = min( one, half / max( xbnd, smlnum ) )
            DO 50 j = jfirst, jlast, jinc
*
*              Exit the loop if the growth factor is too small.
*
               IF( grow.LE.smlnum )
     $            go to 60
*
*              G(j) = G(j-1)*( 1 + CNORM(j) )
*
               grow = grow*( one / ( one+cnorm( j ) ) )
   50       continue
         END IF
   60    continue
*
      ELSE
*
*        Compute the growth in A**T * x = b  or  A**H * x = b.
*
         IF( upper ) THEN
            jfirst = 1
            jlast = n
            jinc = 1
         ELSE
            jfirst = n
            jlast = 1
            jinc = -1
         END IF
*
         IF( tscal.NE.one ) THEN
            grow = zero
            go to 90
         END IF
*
         IF( nounit ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, M(0) = max{x(i), i=1,...,n}.
*
            grow = half / max( xbnd, smlnum )
            xbnd = grow
            DO 70 j = jfirst, jlast, jinc
*
*              Exit the loop if the growth factor is too small.
*
               IF( grow.LE.smlnum )
     $            go to 90
*
*              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
               xj = one + cnorm( j )
               grow = min( grow, xbnd / xj )
*
               tjjs = a( j, j )
               tjj = cabs1( tjjs )
*
               IF( tjj.GE.smlnum ) THEN
*
*                 M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
                  IF( xj.GT.tjj )
     $               xbnd = xbnd*( tjj / xj )
               ELSE
*
*                 M(j) could overflow, set XBND to 0.
*
                  xbnd = zero
               END IF
   70       continue
            grow = min( grow, xbnd )
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            grow = min( one, half / max( xbnd, smlnum ) )
            DO 80 j = jfirst, jlast, jinc
*
*              Exit the loop if the growth factor is too small.
*
               IF( grow.LE.smlnum )
     $            go to 90
*
*              G(j) = ( 1 + CNORM(j) )*G(j-1)
*
               xj = one + cnorm( j )
               grow = grow / xj
   80       continue
         END IF
   90    continue
      END IF
*
      IF( ( grow*tscal ).GT.smlnum ) THEN
*
*        Use the Level 2 BLAS solve if the reciprocal of the bound on
*        elements of X is not too small.
*
         CALL ctrsv( uplo, trans, diag, n, a, lda, x, 1 )
      ELSE
*
*        Use a Level 1 BLAS solve, scaling intermediate results.
*
         IF( xmax.GT.bignum*half ) THEN
*
*           Scale X so that its components are less than or equal to
*           BIGNUM in absolute value.
*
            scale = ( bignum*half ) / xmax
            CALL csscal( n, scale, x, 1 )
            xmax = bignum
         ELSE
            xmax = xmax*two
         END IF
*
         IF( notran ) THEN
*
*           Solve A * x = b
*
            DO 110 j = jfirst, jlast, jinc
*
*              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
               xj = cabs1( x( j ) )
               IF( nounit ) THEN
                  tjjs = a( j, j )*tscal
               ELSE
                  tjjs = tscal
                  IF( tscal.EQ.one )
     $               go to 105
               END IF
                  tjj = cabs1( tjjs )
                  IF( tjj.GT.smlnum ) THEN
*
*                    abs(A(j,j)) > SMLNUM:
*
                     IF( tjj.LT.one ) THEN
                        IF( xj.GT.tjj*bignum ) THEN
*
*                          Scale x by 1/b(j).
*
                           rec = one / xj
                           CALL csscal( n, rec, x, 1 )
                           scale = scale*rec
                           xmax = xmax*rec
                        END IF
                     END IF
                     x( j ) = cladiv( x( j ), tjjs )
                     xj = cabs1( x( j ) )
                  ELSE IF( tjj.GT.zero ) THEN
*
*                    0 < abs(A(j,j)) <= SMLNUM:
*
                     IF( xj.GT.tjj*bignum ) THEN
*
*                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
*                       to avoid overflow when dividing by A(j,j).
*
                        rec = ( tjj*bignum ) / xj
                        IF( cnorm( j ).GT.one ) THEN
*
*                          Scale by 1/CNORM(j) to avoid overflow when
*                          multiplying x(j) times column j.
*
                           rec = rec / cnorm( j )
                        END IF
                        CALL csscal( n, rec, x, 1 )
                        scale = scale*rec
                        xmax = xmax*rec
                     END IF
                     x( j ) = cladiv( x( j ), tjjs )
                     xj = cabs1( x( j ) )
                  ELSE
*
*                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                    scale = 0, and compute a solution to A*x = 0.
*
                     DO 100 i = 1, n
                        x( i ) = zero
  100                continue
                     x( j ) = one
                     xj = one
                     scale = zero
                     xmax = zero
                  END IF
  105          continue
*
*              Scale x if necessary to avoid overflow when adding a
*              multiple of column j of A.
*
               IF( xj.GT.one ) THEN
                  rec = one / xj
                  IF( cnorm( j ).GT.( bignum-xmax )*rec ) THEN
*
*                    Scale x by 1/(2*abs(x(j))).
*
                     rec = rec*half
                     CALL csscal( n, rec, x, 1 )
                     scale = scale*rec
                  END IF
               ELSE IF( xj*cnorm( j ).GT.( bignum-xmax ) ) THEN
*
*                 Scale x by 1/2.
*
                  CALL csscal( n, half, x, 1 )
                  scale = scale*half
               END IF
*
               IF( upper ) THEN
                  IF( j.GT.1 ) THEN
*
*                    Compute the update
*                       x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
                     CALL caxpy( j-1, -x( j )*tscal, a( 1, j ), 1, x,
     $                           1 )
                     i = icamax( j-1, x, 1 )
                     xmax = cabs1( x( i ) )
                  END IF
               ELSE
                  IF( j.LT.n ) THEN
*
*                    Compute the update
*                       x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
                     CALL caxpy( n-j, -x( j )*tscal, a( j+1, j ), 1,
     $                           x( j+1 ), 1 )
                     i = j + icamax( n-j, x( j+1 ), 1 )
                     xmax = cabs1( x( i ) )
                  END IF
               END IF
  110       continue
*
         ELSE IF( lsame( trans, 'T' ) ) THEN
*
*           Solve A**T * x = b
*
            DO 150 j = jfirst, jlast, jinc
*
*              Compute x(j) = b(j) - sum A(k,j)*x(k).
*                                    k<>j
*
               xj = cabs1( x( j ) )
               uscal = tscal
               rec = one / max( xmax, one )
               IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
*
*                 If x(j) could overflow, scale x by 1/(2*XMAX).
*
                  rec = rec*half
                  IF( nounit ) THEN
                     tjjs = a( j, j )*tscal
                  ELSE
                     tjjs = tscal
                  END IF
                     tjj = cabs1( tjjs )
                     IF( tjj.GT.one ) THEN
*
*                       Divide by A(j,j) when scaling x if A(j,j) > 1.
*
                        rec = min( one, rec*tjj )
                        uscal = cladiv( uscal, tjjs )
                     END IF
                  IF( rec.LT.one ) THEN
                     CALL csscal( n, rec, x, 1 )
                     scale = scale*rec
                     xmax = xmax*rec
                  END IF
               END IF
*
               csumj = zero
               IF( uscal.EQ.cmplx( one ) ) THEN
*
*                 If the scaling needed for A in the dot product is 1,
*                 call CDOTU to perform the dot product.
*
                  IF( upper ) THEN
                     csumj = cdotu( j-1, a( 1, j ), 1, x, 1 )
                  ELSE IF( j.LT.n ) THEN
                     csumj = cdotu( n-j, a( j+1, j ), 1, x( j+1 ), 1 )
                  END IF
               ELSE
*
*                 Otherwise, use in-line code for the dot product.
*
                  IF( upper ) THEN
                     DO 120 i = 1, j - 1
                        csumj = csumj + ( a( i, j )*uscal )*x( i )
  120                continue
                  ELSE IF( j.LT.n ) THEN
                     DO 130 i = j + 1, n
                        csumj = csumj + ( a( i, j )*uscal )*x( i )
  130                continue
                  END IF
               END IF
*
               IF( uscal.EQ.cmplx( tscal ) ) THEN
*
*                 Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
*                 was not used to scale the dotproduct.
*
                  x( j ) = x( j ) - csumj
                  xj = cabs1( x( j ) )
                  IF( nounit ) THEN
                     tjjs = a( j, j )*tscal
                  ELSE
                     tjjs = tscal
                     IF( tscal.EQ.one )
     $                  go to 145
                  END IF
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                     tjj = cabs1( tjjs )
                     IF( tjj.GT.smlnum ) THEN
*
*                       abs(A(j,j)) > SMLNUM:
*
                        IF( tjj.LT.one ) THEN
                           IF( xj.GT.tjj*bignum ) THEN
*
*                             Scale X by 1/abs(x(j)).
*
                              rec = one / xj
                              CALL csscal( n, rec, x, 1 )
                              scale = scale*rec
                              xmax = xmax*rec
                           END IF
                        END IF
                        x( j ) = cladiv( x( j ), tjjs )
                     ELSE IF( tjj.GT.zero ) THEN
*
*                       0 < abs(A(j,j)) <= SMLNUM:
*
                        IF( xj.GT.tjj*bignum ) THEN
*
*                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
                           rec = ( tjj*bignum ) / xj
                           CALL csscal( n, rec, x, 1 )
                           scale = scale*rec
                           xmax = xmax*rec
                        END IF
                        x( j ) = cladiv( x( j ), tjjs )
                     ELSE
*
*                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                       scale = 0 and compute a solution to A**T *x = 0.
*
                        DO 140 i = 1, n
                           x( i ) = zero
  140                   continue
                        x( j ) = one
                        scale = zero
                        xmax = zero
                     END IF
  145             continue
               ELSE
*
*                 Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
*                 product has already been divided by 1/A(j,j).
*
                  x( j ) = cladiv( x( j ), tjjs ) - csumj
               END IF
               xmax = max( xmax, cabs1( x( j ) ) )
  150       continue
*
         ELSE
*
*           Solve A**H * x = b
*
            DO 190 j = jfirst, jlast, jinc
*
*              Compute x(j) = b(j) - sum A(k,j)*x(k).
*                                    k<>j
*
               xj = cabs1( x( j ) )
               uscal = tscal
               rec = one / max( xmax, one )
               IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
*
*                 If x(j) could overflow, scale x by 1/(2*XMAX).
*
                  rec = rec*half
                  IF( nounit ) THEN
                     tjjs = conjg( a( j, j ) )*tscal
                  ELSE
                     tjjs = tscal
                  END IF
                     tjj = cabs1( tjjs )
                     IF( tjj.GT.one ) THEN
*
*                       Divide by A(j,j) when scaling x if A(j,j) > 1.
*
                        rec = min( one, rec*tjj )
                        uscal = cladiv( uscal, tjjs )
                     END IF
                  IF( rec.LT.one ) THEN
                     CALL csscal( n, rec, x, 1 )
                     scale = scale*rec
                     xmax = xmax*rec
                  END IF
               END IF
*
               csumj = zero
               IF( uscal.EQ.cmplx( one ) ) THEN
*
*                 If the scaling needed for A in the dot product is 1,
*                 call CDOTC to perform the dot product.
*
                  IF( upper ) THEN
                     csumj = cdotc( j-1, a( 1, j ), 1, x, 1 )
                  ELSE IF( j.LT.n ) THEN
                     csumj = cdotc( n-j, a( j+1, j ), 1, x( j+1 ), 1 )
                  END IF
               ELSE
*
*                 Otherwise, use in-line code for the dot product.
*
                  IF( upper ) THEN
                     DO 160 i = 1, j - 1
                        csumj = csumj + ( conjg( a( i, j ) )*uscal )*
     $                          x( i )
  160                continue
                  ELSE IF( j.LT.n ) THEN
                     DO 170 i = j + 1, n
                        csumj = csumj + ( conjg( a( i, j ) )*uscal )*
     $                          x( i )
  170                continue
                  END IF
               END IF
*
               IF( uscal.EQ.cmplx( tscal ) ) THEN
*
*                 Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
*                 was not used to scale the dotproduct.
*
                  x( j ) = x( j ) - csumj
                  xj = cabs1( x( j ) )
                  IF( nounit ) THEN
                     tjjs = conjg( a( j, j ) )*tscal
                  ELSE
                     tjjs = tscal
                     IF( tscal.EQ.one )
     $                  go to 185
                  END IF
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                     tjj = cabs1( tjjs )
                     IF( tjj.GT.smlnum ) THEN
*
*                       abs(A(j,j)) > SMLNUM:
*
                        IF( tjj.LT.one ) THEN
                           IF( xj.GT.tjj*bignum ) THEN
*
*                             Scale X by 1/abs(x(j)).
*
                              rec = one / xj
                              CALL csscal( n, rec, x, 1 )
                              scale = scale*rec
                              xmax = xmax*rec
                           END IF
                        END IF
                        x( j ) = cladiv( x( j ), tjjs )
                     ELSE IF( tjj.GT.zero ) THEN
*
*                       0 < abs(A(j,j)) <= SMLNUM:
*
                        IF( xj.GT.tjj*bignum ) THEN
*
*                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
                           rec = ( tjj*bignum ) / xj
                           CALL csscal( n, rec, x, 1 )
                           scale = scale*rec
                           xmax = xmax*rec
                        END IF
                        x( j ) = cladiv( x( j ), tjjs )
                     ELSE
*
*                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
*                       scale = 0 and compute a solution to A**H *x = 0.
*
                        DO 180 i = 1, n
                           x( i ) = zero
  180                   continue
                        x( j ) = one
                        scale = zero
                        xmax = zero
                     END IF
  185             continue
               ELSE
*
*                 Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
*                 product has already been divided by 1/A(j,j).
*
                  x( j ) = cladiv( x( j ), tjjs ) - csumj
               END IF
               xmax = max( xmax, cabs1( x( j ) ) )
  190       continue
         END IF
         scale = scale / tscal
      END IF
*
*     Scale the column norms by 1/TSCAL for return.
*
      IF( tscal.NE.one ) THEN
         CALL sscal( n, one / tscal, cnorm, 1 )
      END IF
*
      return
*
*     End of CLATRS
*
      END