zlarfg()

subroutine zlarfg	(	integer	n,
		complex*16	alpha,
		complex*16, dimension( * )	x,
		integer	incx,
		complex*16	tau
	)

ZLARFG generates an elementary reflector (Householder matrix).

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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 .

Parameters

[in]	N	N is INTEGER The order of the elementary reflector.
[in,out]	ALPHA	ALPHA is COMPLEX*16 On entry, the value alpha. On exit, it is overwritten with the value beta.
[in,out]	X	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.
[in]	INCX	INCX is INTEGER The increment between elements of X. INCX > 0.
[out]	TAU	TAU is COMPLEX*16 The value tau.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file zlarfg.f.

*
*  -- 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   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) .AND. (knt .LT. 20) )
     $         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
*

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