1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
|
SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INCX, N
DOUBLE PRECISION ALPHA, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION X( * )
* ..
*
* Purpose
* =======
*
* DLARFG generates a real elementary reflector H of order n, such
* that
*
* H * ( alpha ) = ( beta ), H' * H = I.
* ( x ) ( 0 )
*
* where alpha and beta are scalars, and x is an (n-1)-element real
* vector. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v' ) ,
* ( 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.
*
* Otherwise 1 <= tau <= 2.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the elementary reflector.
*
* ALPHA (input/output) DOUBLE PRECISION
* On entry, the value alpha.
* On exit, it is overwritten with the value beta.
*
* X (input/output) DOUBLE PRECISION array, dimension
* (1+(N-2)*abs(INCX))
* On entry, the vector x.
* On exit, it is overwritten with the vector v.
*
* INCX (input) INTEGER
* The increment between elements of X. INCX > 0.
*
* TAU (output) DOUBLE PRECISION
* The value tau.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
DOUBLE PRECISION BETA, RSAFMN, SAFMIN, 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.1 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = DNRM2( N-1, X, INCX )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = I
*
TAU = ZERO
ELSE
*
* general case
*
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
RSAFMN = ONE / SAFMIN
KNT = 0
10 CONTINUE
KNT = KNT + 1
CALL DSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHA = ALPHA*RSAFMN
IF( ABS( BETA ).LT.SAFMIN )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = DNRM2( N-1, X, INCX )
BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
TAU = ( BETA-ALPHA ) / BETA
CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
*
* If ALPHA is subnormal, it may lose relative accuracy
*
ALPHA = BETA
DO 20 J = 1, KNT
ALPHA = ALPHA*SAFMIN
20 CONTINUE
ELSE
TAU = ( BETA-ALPHA ) / BETA
CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
ALPHA = BETA
END IF
END IF
*
RETURN
*
* End of DLARFG
*
END
|
