Annotation of rpl/lapack/lapack/dlarfg.f, revision 1.9

1.9     ! bertrand    1: *> \brief \b DLARFG
        !             2: *
        !             3: *  =========== DOCUMENTATION ===========
        !             4: *
        !             5: * Online html documentation available at 
        !             6: *            http://www.netlib.org/lapack/explore-html/ 
        !             7: *
        !             8: *> \htmlonly
        !             9: *> Download DLARFG + dependencies 
        !            10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfg.f"> 
        !            11: *> [TGZ]</a> 
        !            12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfg.f"> 
        !            13: *> [ZIP]</a> 
        !            14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfg.f"> 
        !            15: *> [TXT]</a>
        !            16: *> \endhtmlonly 
        !            17: *
        !            18: *  Definition:
        !            19: *  ===========
        !            20: *
        !            21: *       SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
        !            22: * 
        !            23: *       .. Scalar Arguments ..
        !            24: *       INTEGER            INCX, N
        !            25: *       DOUBLE PRECISION   ALPHA, TAU
        !            26: *       ..
        !            27: *       .. Array Arguments ..
        !            28: *       DOUBLE PRECISION   X( * )
        !            29: *       ..
        !            30: *  
        !            31: *
        !            32: *> \par Purpose:
        !            33: *  =============
        !            34: *>
        !            35: *> \verbatim
        !            36: *>
        !            37: *> DLARFG 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 DOUBLE PRECISION
        !            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 DOUBLE PRECISION 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 DOUBLE PRECISION
        !            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 doubleOTHERauxiliary
        !           105: *
        !           106: *  =====================================================================
1.1       bertrand  107:       SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
                    108: *
1.9     ! bertrand  109: *  -- LAPACK auxiliary routine (version 3.4.0) --
1.1       bertrand  110: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    111: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9     ! bertrand  112: *     November 2011
1.1       bertrand  113: *
                    114: *     .. Scalar Arguments ..
                    115:       INTEGER            INCX, N
                    116:       DOUBLE PRECISION   ALPHA, TAU
                    117: *     ..
                    118: *     .. Array Arguments ..
                    119:       DOUBLE PRECISION   X( * )
                    120: *     ..
                    121: *
                    122: *  =====================================================================
                    123: *
                    124: *     .. Parameters ..
                    125:       DOUBLE PRECISION   ONE, ZERO
                    126:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
                    127: *     ..
                    128: *     .. Local Scalars ..
                    129:       INTEGER            J, KNT
                    130:       DOUBLE PRECISION   BETA, RSAFMN, SAFMIN, XNORM
                    131: *     ..
                    132: *     .. External Functions ..
                    133:       DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
                    134:       EXTERNAL           DLAMCH, DLAPY2, DNRM2
                    135: *     ..
                    136: *     .. Intrinsic Functions ..
                    137:       INTRINSIC          ABS, SIGN
                    138: *     ..
                    139: *     .. External Subroutines ..
                    140:       EXTERNAL           DSCAL
                    141: *     ..
                    142: *     .. Executable Statements ..
                    143: *
                    144:       IF( N.LE.1 ) THEN
                    145:          TAU = ZERO
                    146:          RETURN
                    147:       END IF
                    148: *
                    149:       XNORM = DNRM2( 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( DLAPY2( ALPHA, XNORM ), ALPHA )
                    161:          SAFMIN = DLAMCH( 'S' ) / DLAMCH( '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 DSCAL( 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 = DNRM2( N-1, X, INCX )
                    179:             BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
                    180:          END IF
                    181:          TAU = ( BETA-ALPHA ) / BETA
                    182:          CALL DSCAL( 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 DLARFG
                    195: *
                    196:       END

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