dlarfg()

subroutine dlarfg	(	integer	n,
		double precision	alpha,
		double precision, dimension( * )	x,
		integer	incx,
		double precision	tau
	)

DLARFG generates an elementary reflector (Householder matrix).

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Purpose:

 DLARFG 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, 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.

 Otherwise  1 <= tau <= 2.

Parameters

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file dlarfg.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
      DOUBLE PRECISION   ALPHA, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * )
*     ..
*
*  =====================================================================
*
*     .. 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' )
         knt = 0
         IF( abs( beta ).LT.safmin ) THEN
*
*           XNORM, BETA may be inaccurate; scale X and recompute them
*
            rsafmn = one / safmin
   10       CONTINUE
            knt = knt + 1
            CALL dscal( n-1, rsafmn, x, incx )
            beta = beta*rsafmn
            alpha = alpha*rsafmn
            IF( (abs( beta ).LT.safmin) .AND. (knt .LT. 20) )
     $         GO TO 10
*
*           New BETA is at most 1, at least SAFMIN
*
            xnorm = dnrm2( n-1, x, incx )
            beta = -sign( dlapy2( alpha, xnorm ), alpha )
         END IF
         tau = ( beta-alpha ) / beta
         CALL dscal( n-1, one / ( alpha-beta ), 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 DLARFG
*

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