SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU ) 4,6
!
!  -- LAPACK auxiliary routine (version 3.1) --
!     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
!     November 2006
!
!     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   ALPHA, TAU
!     ..
!     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
!     ..
!
!  Purpose
!  =======
!
!  DLARFG generates a real elementary reflector H of order n, such
!  that
!
!        H * ( alpha ) = ( beta ),   H' * H = I.
!            (   x   )   (   0  )
!
!  where alpha and beta are scalars, and x is an (n-1)-element real
!  vector. H is represented in the form
!
!        H = I - tau * ( 1 ) * ( 1 v' ) ,
!                      ( v )
!
!  where tau is a real scalar and v is a real (n-1)-element
!  vector.
!
!  If the elements of x are all zero, then tau = 0 and H is taken to be
!  the unit matrix.
!
!  Otherwise  1 <= tau <= 2.
!
!  Arguments
!  =========
!
!  N       (input) INTEGER
!          The order of the elementary reflector.
!
!  ALPHA   (input/output) DOUBLE PRECISION
!          On entry, the value alpha.
!          On exit, it is overwritten with the value beta.
!
!  X       (input/output) DOUBLE PRECISION array, dimension
!                         (1+(N-2)*abs(INCX))
!          On entry, the vector x.
!          On exit, it is overwritten with the vector v.
!
!  INCX    (input) INTEGER
!          The increment between elements of X. INCX > 0.
!
!  TAU     (output) DOUBLE PRECISION
!          The value tau.
!
!  =====================================================================
!
!     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
!     ..
!     .. Local Scalars ..
      INTEGER            J, KNT
      DOUBLE PRECISION   BETA, RSAFMN, SAFMIN, XNORM
!     ..
!     .. External Functions ..
!     DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
!     EXTERNAL           DLAMCH, DLAPY2, DNRM2
!     ..
!     .. Intrinsic Functions ..
      INTRINSIC          ABS, SIGN
!     ..
!     .. External Subroutines ..
!     EXTERNAL           DSCAL
!     ..
!     .. Executable Statements ..
!
      IF( N.LE.1 ) THEN
         TAU = ZERO
         RETURN
      END IF
!
      XNORM = DNRM2( N-1, X, INCX )
!
      IF( XNORM.EQ.ZERO ) THEN
!
!        H  =  I
!
         TAU = ZERO
      ELSE
!
!        general case
!
         BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
         IF( ABS( BETA ).LT.SAFMIN ) THEN
!
!           XNORM, BETA may be inaccurate; scale X and recompute them
!
            RSAFMN = ONE / SAFMIN
            KNT = 0
   10       CONTINUE
            KNT = KNT + 1
            CALL DSCAL( N-1, RSAFMN, X, INCX )
            BETA = BETA*RSAFMN
            ALPHA = ALPHA*RSAFMN
            IF( ABS( BETA ).LT.SAFMIN ) &
               GO TO 10
!
!           New BETA is at most 1, at least SAFMIN
!
            XNORM = DNRM2( N-1, X, INCX )
            BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
            TAU = ( BETA-ALPHA ) / BETA
            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
!
!           If ALPHA is subnormal, it may lose relative accuracy
!
            ALPHA = BETA
            DO 20 J = 1, KNT
               ALPHA = ALPHA*SAFMIN
   20       CONTINUE
         ELSE
            TAU = ( BETA-ALPHA ) / BETA
            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
            ALPHA = BETA
         END IF
      END IF
!
      RETURN
!
!     End of DLARFG
!
      END SUBROUTINE DLARFG