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      1 *> \brief \b SLARFG
      2 *
      3 *  =========== DOCUMENTATION ===========
      4 *
      5 * Online html documentation available at 
      6 *            http://www.netlib.org/lapack/explore-html/ 
      7 *
      8 *> \htmlonly
      9 *> Download SLARFG + dependencies 
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     11 *> [TGZ]</a> 
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     13 *> [ZIP]</a> 
     14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarfg.f"> 
     15 *> [TXT]</a>
     16 *> \endhtmlonly 
     17 *
     18 *  Definition:
     19 *  ===========
     20 *
     21 *       SUBROUTINE SLARFG( N, ALPHA, X, INCX, TAU )
     22 * 
     23 *       .. Scalar Arguments ..
     24 *       INTEGER            INCX, N
     25 *       REAL               ALPHA, TAU
     26 *       ..
     27 *       .. Array Arguments ..
     28 *       REAL               X( * )
     29 *       ..
     30 *  
     31 *
     32 *> \par Purpose:
     33 *  =============
     34 *>
     35 *> \verbatim
     36 *>
     37 *> SLARFG generates a real elementary reflector H of order n, such
     38 *> that
     39 *>
     40 *>       H * ( alpha ) = ( beta ),   H**T * H = I.
     41 *>           (   x   )   (   0  )
     42 *>
     43 *> where alpha and beta are scalars, and x is an (n-1)-element real
     44 *> vector. H is represented in the form
     45 *>
     46 *>       H = I - tau * ( 1 ) * ( 1 v**T ) ,
     47 *>                     ( v )
     48 *>
     49 *> where tau is a real scalar and v is a real (n-1)-element
     50 *> vector.
     51 *>
     52 *> If the elements of x are all zero, then tau = 0 and H is taken to be
     53 *> the unit matrix.
     54 *>
     55 *> Otherwise  1 <= tau <= 2.
     56 *> \endverbatim
     57 *
     58 *  Arguments:
     59 *  ==========
     60 *
     61 *> \param[in] N
     62 *> \verbatim
     63 *>          N is INTEGER
     64 *>          The order of the elementary reflector.
     65 *> \endverbatim
     66 *>
     67 *> \param[in,out] ALPHA
     68 *> \verbatim
     69 *>          ALPHA is REAL
     70 *>          On entry, the value alpha.
     71 *>          On exit, it is overwritten with the value beta.
     72 *> \endverbatim
     73 *>
     74 *> \param[in,out] X
     75 *> \verbatim
     76 *>          X is REAL array, dimension
     77 *>                         (1+(N-2)*abs(INCX))
     78 *>          On entry, the vector x.
     79 *>          On exit, it is overwritten with the vector v.
     80 *> \endverbatim
     81 *>
     82 *> \param[in] INCX
     83 *> \verbatim
     84 *>          INCX is INTEGER
     85 *>          The increment between elements of X. INCX > 0.
     86 *> \endverbatim
     87 *>
     88 *> \param[out] TAU
     89 *> \verbatim
     90 *>          TAU is REAL
     91 *>          The value tau.
     92 *> \endverbatim
     93 *
     94 *  Authors:
     95 *  ========
     96 *
     97 *> \author Univ. of Tennessee 
     98 *> \author Univ. of California Berkeley 
     99 *> \author Univ. of Colorado Denver 
    100 *> \author NAG Ltd. 
    101 *
    102 *> \date November 2011
    103 *
    104 *> \ingroup realOTHERauxiliary
    105 *
    106 *  =====================================================================
    107       SUBROUTINE SLARFG( N, ALPHA, X, INCX, TAU )
    108 *
    109 *  -- LAPACK auxiliary routine (version 3.4.0) --
    110 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    111 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    112 *     November 2011
    113 *
    114 *     .. Scalar Arguments ..
    115       INTEGER            INCX, N
    116       REAL               ALPHA, TAU
    117 *     ..
    118 *     .. Array Arguments ..
    119       REAL               X( * )
    120 *     ..
    121 *
    122 *  =====================================================================
    123 *
    124 *     .. Parameters ..
    125       REAL               ONE, ZERO
    126       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
    127 *     ..
    128 *     .. Local Scalars ..
    129       INTEGER            J, KNT
    130       REAL               BETA, RSAFMN, SAFMIN, XNORM
    131 *     ..
    132 *     .. External Functions ..
    133       REAL               SLAMCH, SLAPY2, SNRM2
    134       EXTERNAL           SLAMCH, SLAPY2, SNRM2
    135 *     ..
    136 *     .. Intrinsic Functions ..
    137       INTRINSIC          ABS, SIGN
    138 *     ..
    139 *     .. External Subroutines ..
    140       EXTERNAL           SSCAL
    141 *     ..
    142 *     .. Executable Statements ..
    143 *
    144       IF( N.LE.1 ) THEN
    145          TAU = ZERO
    146          RETURN
    147       END IF
    148 *
    149       XNORM = SNRM2( N-1, X, INCX )
    150 *
    151       IF( XNORM.EQ.ZERO ) THEN
    152 *
    153 *        H  =  I
    154 *
    155          TAU = ZERO
    156       ELSE
    157 *
    158 *        general case
    159 *
    160          BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA )
    161          SAFMIN = SLAMCH( 'S' ) / SLAMCH( 'E' )
    162          KNT = 0
    163          IF( ABS( BETA ).LT.SAFMIN ) THEN
    164 *
    165 *           XNORM, BETA may be inaccurate; scale X and recompute them
    166 *
    167             RSAFMN = ONE / SAFMIN
    168    10       CONTINUE
    169             KNT = KNT + 1
    170             CALL SSCAL( N-1, RSAFMN, X, INCX )
    171             BETA = BETA*RSAFMN
    172             ALPHA = ALPHA*RSAFMN
    173             IF( ABS( BETA ).LT.SAFMIN )
    174      $         GO TO 10
    175 *
    176 *           New BETA is at most 1, at least SAFMIN
    177 *
    178             XNORM = SNRM2( N-1, X, INCX )
    179             BETA = -SIGN( SLAPY2( ALPHA, XNORM ), ALPHA )
    180          END IF
    181          TAU = ( BETA-ALPHA ) / BETA
    182          CALL SSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
    183 *
    184 *        If ALPHA is subnormal, it may lose relative accuracy
    185 *
    186          DO 20 J = 1, KNT
    187             BETA = BETA*SAFMIN
    188  20      CONTINUE
    189          ALPHA = BETA
    190       END IF
    191 *
    192       RETURN
    193 *
    194 *     End of SLARFG
    195 *
    196       END
    197