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*> \brief \b DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFGP + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfgp.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfgp.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
*
* .. Scalar Arguments ..
* INTEGER INCX, N
* DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the elementary reflector.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> On entry, the value alpha.
*> On exit, it is overwritten with the value beta.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*> INCX is INTEGER
*> The increment between elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The value tau.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup larfgp
*
* =====================================================================
SUBROUTINE DLARFGP( 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
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
*
END
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