dlarfgp()

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

DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

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Purpose:

 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.

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 103 of file dlarfgp.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   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
*

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