zlarfgp.f

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*> \brief \b ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLARFGP( N, ALPHA, X, INCX, TAU )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, N
*       COMPLEX*16         ALPHA, TAU
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         X( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZLARFGP generates a complex elementary reflector H of order n, such
*> that
*>
*>       H**H * ( alpha ) = ( beta ),   H**H * H = I.
*>              (   x   )   (   0  )
*>
*> where alpha and beta are scalars, beta is real and non-negative, and
*> x is an (n-1)-element complex vector.  H is represented in the form
*>
*>       H = I - tau * ( 1 ) * ( 1 v**H ) ,
*>                     ( v )
*>
*> where tau is a complex scalar and v is a complex (n-1)-element
*> vector. Note that H is not hermitian.
*>
*> If the elements of x are all zero and alpha is real, 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 COMPLEX*16
*>          On entry, the value alpha.
*>          On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is COMPLEX*16 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 COMPLEX*16
*>          The value tau.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*
*  =====================================================================
      SUBROUTINE zlarfgp( 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
      COMPLEX*16         ALPHA, TAU
*     ..
*     .. Array Arguments ..
      COMPLEX*16         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   ALPHI, ALPHR, BETA, BIGNUM, SMLNUM, XNORM
      COMPLEX*16         SAVEALPHA
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY3, DLAPY2, DZNRM2
      COMPLEX*16         ZLADIV
      EXTERNAL           dlamch, dlapy3, dlapy2, dznrm2, zladiv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dcmplx, dimag, sign
*     ..
*     .. External Subroutines ..
      EXTERNAL           zdscal, zscal
*     ..
*     .. Executable Statements ..
*
      IF( n.LE.0 ) THEN
         tau = zero
         RETURN
      END IF
*
      xnorm = dznrm2( n-1, x, incx )
      alphr = dble( alpha )
      alphi = dimag( alpha )
*
      IF( xnorm.EQ.zero ) THEN
*
*        H  =  [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0.
*
         IF( alphi.EQ.zero ) THEN
            IF( alphr.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 ) = zero
               END DO
               alpha = -alpha
            END IF
         ELSE
*           Only "reflecting" the diagonal entry to be real and non-negative.
            xnorm = dlapy2( alphr, alphi )
            tau = dcmplx( one - alphr / xnorm, -alphi / xnorm )
            DO j = 1, n-1
               x( 1 + (j-1)*incx ) = zero
            END DO
            alpha = xnorm
         END IF
      ELSE
*
*        general case
*
         beta = sign( dlapy3( alphr, alphi, xnorm ), alphr )
         smlnum = dlamch( 'S' ) / dlamch( 'E' )
         bignum = one / smlnum
*
         knt = 0
         IF( abs( beta ).LT.smlnum ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
   10       CONTINUE
            knt = knt + 1
            CALL zdscal( n-1, bignum, x, incx )
            beta = beta*bignum
            alphi = alphi*bignum
            alphr = alphr*bignum
            IF( (abs( beta ).LT.smlnum) .AND. (knt .LT. 20) )
     $         GO TO 10
*
*           New BETA is at most 1, at least SMLNUM
*
            xnorm = dznrm2( n-1, x, incx )
            alpha = dcmplx( alphr, alphi )
            beta = sign( dlapy3( alphr, alphi, xnorm ), alphr )
         END IF
         savealpha = alpha
         alpha = alpha + beta
         IF( beta.LT.zero ) THEN
            beta = -beta
            tau = -alpha / beta
         ELSE
            alphr = alphi * (alphi/dble( alpha ))
            alphr = alphr + xnorm * (xnorm/dble( alpha ))
            tau = dcmplx( alphr/beta, -alphi/beta )
            alpha = dcmplx( -alphr, alphi )
         END IF
         alpha = zladiv( dcmplx( one ), alpha )
*
         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 (or TWO or whatever makes a nonnegative real number for BETA).
*
*           (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
*           (Thanks Pat. Thanks MathWorks.)
*
            alphr = dble( savealpha )
            alphi = dimag( savealpha )
            IF( alphi.EQ.zero ) THEN
               IF( alphr.GE.zero ) THEN
                  tau = zero
               ELSE
                  tau = two
                  DO j = 1, n-1
                     x( 1 + (j-1)*incx ) = zero
                  END DO
                  beta = dble( -savealpha )
               END IF
            ELSE
               xnorm = dlapy2( alphr, alphi )
               tau = dcmplx( one - alphr / xnorm, -alphi / xnorm )
               DO j = 1, n-1
                  x( 1 + (j-1)*incx ) = zero
               END DO
               beta = xnorm
            END IF
*
         ELSE
*
*           This is the general case.
*
            CALL zscal( n-1, 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 ZLARFGP
*
      END