*> \brief \b DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, N
*       DOUBLE PRECISION   ALPHA, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLARFGP generates a real elementary reflector H of order n, such
*> that
*>
*>       H * ( alpha ) = ( beta ),   H**T * H = I.
*>           (   x   )   (   0  )
*>
*> where alpha and beta are scalars, beta is non-negative, and x is
*> an (n-1)-element real vector.  H is represented in the form
*>
*>       H = I - tau * ( 1 ) * ( 1 v**T ) ,
*>                     ( 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>          On entry, the value alpha.
*>          On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension
*>                         (1+(N-2)*abs(INCX))
*>          On entry, the vector x.
*>          On exit, it is overwritten with the vector v.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>          The value tau.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfgp
*
*  =====================================================================
      SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      DOUBLE PRECISION   ALPHA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   TWO, ONE, ZERO
      PARAMETER          ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, KNT
      DOUBLE PRECISION   BETA, BIGNUM, EPS, SAVEALPHA, SMLNUM, 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.0 ) THEN
         TAU = ZERO
         RETURN
      END IF
*
      EPS = DLAMCH( 'Precision' )
      XNORM = DNRM2( N-1, X, INCX )
*
      IF( XNORM.LE.EPS*ABS(ALPHA) ) THEN
*
*        H  =  [+/-1, 0; I], sign chosen so ALPHA >= 0.
*
         IF( ALPHA.GE.ZERO ) THEN
*           When TAU.eq.ZERO, the vector is special-cased to be
*           all zeros in the application routines.  We do not need
*           to clear it.
            TAU = ZERO
         ELSE
*           However, the application routines rely on explicit
*           zero checks when TAU.ne.ZERO, and we must clear X.
            TAU = TWO
            DO J = 1, N-1
               X( 1 + (J-1)*INCX ) = 0
            END DO
            ALPHA = -ALPHA
         END IF
      ELSE
*
*        general case
*
         BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
         KNT = 0
         IF( ABS( BETA ).LT.SMLNUM ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
            BIGNUM = ONE / SMLNUM
   10       CONTINUE
            KNT = KNT + 1
            CALL DSCAL( N-1, BIGNUM, X, INCX )
            BETA = BETA*BIGNUM
            ALPHA = ALPHA*BIGNUM
            IF( (ABS( BETA ).LT.SMLNUM) .AND. (KNT .LT. 20) )
     $         GO TO 10
*
*           New BETA is at most 1, at least SMLNUM
*
            XNORM = DNRM2( N-1, X, INCX )
            BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
         END IF
         SAVEALPHA = ALPHA
         ALPHA = ALPHA + BETA
         IF( BETA.LT.ZERO ) THEN
            BETA = -BETA
            TAU = -ALPHA / BETA
         ELSE
            ALPHA = XNORM * (XNORM/ALPHA)
            TAU = ALPHA / BETA
            ALPHA = -ALPHA
         END IF
*
         IF ( ABS(TAU).LE.SMLNUM ) THEN
*
*           In the case where the computed TAU ends up being a denormalized number,
*           it loses relative accuracy. This is a BIG problem. Solution: flush TAU
*           to ZERO. This explains the next IF statement.
*
*           (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
*           (Thanks Pat. Thanks MathWorks.)
*
            IF( SAVEALPHA.GE.ZERO ) THEN
               TAU = ZERO
            ELSE
               TAU = TWO
               DO J = 1, N-1
                  X( 1 + (J-1)*INCX ) = 0
               END DO
               BETA = -SAVEALPHA
            END IF
*
         ELSE
*
*           This is the general case.
*
            CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
*
         END IF
*
*        If BETA is subnormal, it may lose relative accuracy
*
         DO 20 J = 1, KNT
            BETA = BETA*SMLNUM
 20      CONTINUE
         ALPHA = BETA
      END IF
*
      RETURN
*
*     End of DLARFGP
*
      END
