zlarfg.f

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      SUBROUTINE ZLARFG( N, ALPHA, X, INCX, TAU )
*
*  -- LAPACK auxiliary routine (version 3.3.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*  -- April 2011                                                      --
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      COMPLEX*16         ALPHA, TAU
*     ..
*     .. Array Arguments ..
      COMPLEX*16         X( * )
*     ..
*
*  Purpose
*  =======
*
*  ZLARFG 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, with beta real, 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.
*
*  Otherwise  1 <= real(tau) <= 2  and  abs(tau-1) <= 1 .
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The order of the elementary reflector.
*
*  ALPHA   (input/output) COMPLEX*16
*          On entry, the value alpha.
*          On exit, it is overwritten with the value beta.
*
*  X       (input/output) COMPLEX*16 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) COMPLEX*16
*          The value tau.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, KNT
      DOUBLE PRECISION   ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLAPY3, DZNRM2
      COMPLEX*16         ZLADIV
      EXTERNAL           DLAMCH, DLAPY3, 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 .AND. ALPHI.EQ.ZERO ) THEN
*
*        H  =  I
*
         TAU = ZERO
      ELSE
*
*        general case
*
         BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
         SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
         RSAFMN = ONE / SAFMIN
*
         KNT = 0
         IF( ABS( BETA ).LT.SAFMIN ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
   10       CONTINUE
            KNT = KNT + 1
            CALL ZDSCAL( N-1, RSAFMN, X, INCX )
            BETA = BETA*RSAFMN
            ALPHI = ALPHI*RSAFMN
            ALPHR = ALPHR*RSAFMN
            IF( ABS( BETA ).LT.SAFMIN )
     $         GO TO 10
*
*           New BETA is at most 1, at least SAFMIN
*
            XNORM = DZNRM2( N-1, X, INCX )
            ALPHA = DCMPLX( ALPHR, ALPHI )
            BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
         END IF
         TAU = DCMPLX( ( BETA-ALPHR ) / BETA, -ALPHI / BETA )
         ALPHA = ZLADIV( DCMPLX( ONE ), ALPHA-BETA )
         CALL ZSCAL( N-1, ALPHA, X, INCX )
*
*        If ALPHA is subnormal, it may lose relative accuracy
*
         DO 20 J = 1, KNT
            BETA = BETA*SAFMIN
 20      CONTINUE
         ALPHA = BETA
      END IF
*
      RETURN
*
*     End of ZLARFG
*
      END