1      SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
2*
3*  -- LAPACK auxiliary routine (version 3.0) --
4*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
5*     Courant Institute, Argonne National Lab, and Rice University
6*     September 30, 1994
7*
8*     .. Scalar Arguments ..
9      INTEGER            INCX, N
10      DOUBLE PRECISION   ALPHA, TAU
11*     ..
12*     .. Array Arguments ..
13      DOUBLE PRECISION   X( * )
14*     ..
15*
16*  Purpose
17*  =======
18*
19*  DLARFG generates a real elementary reflector H of order n, such
20*  that
21*
22*        H * ( alpha ) = ( beta ),   H' * H = I.
23*            (   x   )   (   0  )
24*
25*  where alpha and beta are scalars, and x is an (n-1)-element real
26*  vector. H is represented in the form
27*
28*        H = I - tau * ( 1 ) * ( 1 v' ) ,
29*                      ( v )
30*
31*  where tau is a real scalar and v is a real (n-1)-element
32*  vector.
33*
34*  If the elements of x are all zero, then tau = 0 and H is taken to be
35*  the unit matrix.
36*
37*  Otherwise  1 <= tau <= 2.
38*
39*  Arguments
40*  =========
41*
42*  N       (input) INTEGER
43*          The order of the elementary reflector.
44*
45*  ALPHA   (input/output) DOUBLE PRECISION
46*          On entry, the value alpha.
47*          On exit, it is overwritten with the value beta.
48*
49*  X       (input/output) DOUBLE PRECISION array, dimension
50*                         (1+(N-2)*abs(INCX))
51*          On entry, the vector x.
52*          On exit, it is overwritten with the vector v.
53*
54*  INCX    (input) INTEGER
55*          The increment between elements of X. INCX > 0.
56*
57*  TAU     (output) DOUBLE PRECISION
58*          The value tau.
59*
60*  =====================================================================
61*
62*     .. Parameters ..
63      DOUBLE PRECISION   ONE, ZERO
64      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
65*     ..
66*     .. Local Scalars ..
67      INTEGER            J, KNT
68      DOUBLE PRECISION   BETA, RSAFMN, SAFMIN, XNORM
69*     ..
70*     .. External Functions ..
71      DOUBLE PRECISION   DLAMCH, DLAPY2, DNRM2
72      EXTERNAL           DLAMCH, DLAPY2, DNRM2
73*     ..
74*     .. Intrinsic Functions ..
75      INTRINSIC          ABS, SIGN
76*     ..
77*     .. External Subroutines ..
78      EXTERNAL           DSCAL
79*     ..
80*     .. Executable Statements ..
81*
82      IF( N.LE.1 ) THEN
83         TAU = ZERO
84         RETURN
85      END IF
86*
87      XNORM = DNRM2( N-1, X, INCX )
88*
89      IF( XNORM.EQ.ZERO ) THEN
90*
91*        H  =  I
92*
93         TAU = ZERO
94      ELSE
95*
96*        general case
97*
98         BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
99         SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
100         IF( ABS( BETA ).LT.SAFMIN ) THEN
101*
102*           XNORM, BETA may be inaccurate; scale X and recompute them
103*
104            RSAFMN = ONE / SAFMIN
105            KNT = 0
106   10       CONTINUE
107            KNT = KNT + 1
108            CALL DSCAL( N-1, RSAFMN, X, INCX )
109            BETA = BETA*RSAFMN
110            ALPHA = ALPHA*RSAFMN
111            IF( ABS( BETA ).LT.SAFMIN )
112     $         GO TO 10
113*
114*           New BETA is at most 1, at least SAFMIN
115*
116            XNORM = DNRM2( N-1, X, INCX )
117            BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
118            TAU = ( BETA-ALPHA ) / BETA
119            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
120*
121*           If ALPHA is subnormal, it may lose relative accuracy
122*
123            ALPHA = BETA
124            DO 20 J = 1, KNT
125               ALPHA = ALPHA*SAFMIN
126   20       CONTINUE
127         ELSE
128            TAU = ( BETA-ALPHA ) / BETA
129            CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
130            ALPHA = BETA
131         END IF
132      END IF
133*
134      RETURN
135*
136*     End of DLARFG
137*
138      END
139