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
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
|
SUBROUTINE CLARFP( N, ALPHA, X, INCX, TAU )
*
* -- LAPACK auxiliary routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INCX, N
COMPLEX ALPHA, TAU
* ..
* .. Array Arguments ..
COMPLEX X( * )
* ..
*
* Purpose
* =======
*
* CLARFP generates a complex elementary reflector H of order n, such
* that
*
* H' * ( alpha ) = ( beta ), H' * H = I.
* ( x ) ( 0 )
*
* where alpha and beta are scalars, beta is real and non-negative, and
* x is an (n-1)-element complex vector. H is represented in the form
*
* H = I - tau * ( 1 ) * ( 1 v' ) ,
* ( v )
*
* where tau is a complex scalar and v is a complex (n-1)-element
* vector. Note that H is not hermitian.
*
* If the elements of x are all zero and alpha is real, then tau = 0
* and H is taken to be the unit matrix.
*
* Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the elementary reflector.
*
* ALPHA (input/output) COMPLEX
* On entry, the value alpha.
* On exit, it is overwritten with the value beta.
*
* X (input/output) COMPLEX 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) COMPLEX
* The value tau.
*
* =====================================================================
*
* .. Parameters ..
REAL TWO, ONE, ZERO
PARAMETER ( TWO = 2.0E+0, ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER J, KNT
REAL ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
* ..
* .. External Functions ..
REAL SCNRM2, SLAMCH, SLAPY3, SLAPY2
COMPLEX CLADIV
EXTERNAL SCNRM2, SLAMCH, SLAPY3, SLAPY2, CLADIV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, REAL, SIGN
* ..
* .. External Subroutines ..
EXTERNAL CSCAL, CSSCAL
* ..
* .. Executable Statements ..
*
IF( N.LE.0 ) THEN
TAU = ZERO
RETURN
END IF
*
XNORM = SCNRM2( N-1, X, INCX )
ALPHR = REAL( ALPHA )
ALPHI = AIMAG( ALPHA )
*
IF( XNORM.EQ.ZERO ) THEN
*
* H = [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0.
*
IF( ALPHI.EQ.ZERO ) THEN
IF( ALPHR.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 ) = ZERO
END DO
ALPHA = -ALPHA
END IF
ELSE
! Only "reflecting" the diagonal entry to be real and non-negative.
XNORM = SLAPY2( ALPHR, ALPHI )
TAU = CMPLX( ONE - ALPHR / XNORM, -ALPHI / XNORM )
DO J = 1, N-1
X( 1 + (J-1)*INCX ) = ZERO
END DO
ALPHA = XNORM
END IF
ELSE
*
* general case
*
BETA = SIGN( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
SAFMIN = SLAMCH( 'S' ) / SLAMCH( 'E' )
RSAFMN = ONE / SAFMIN
*
KNT = 0
IF( ABS( BETA ).LT.SAFMIN ) THEN
*
* XNORM, BETA may be inaccurate; scale X and recompute them
*
10 CONTINUE
KNT = KNT + 1
CALL CSSCAL( N-1, RSAFMN, X, INCX )
BETA = BETA*RSAFMN
ALPHI = ALPHI*RSAFMN
ALPHR = ALPHR*RSAFMN
IF( ABS( BETA ).LT.SAFMIN )
$ GO TO 10
*
* New BETA is at most 1, at least SAFMIN
*
XNORM = SCNRM2( N-1, X, INCX )
ALPHA = CMPLX( ALPHR, ALPHI )
BETA = SIGN( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
END IF
ALPHA = ALPHA + BETA
IF( BETA.LT.ZERO ) THEN
BETA = -BETA
TAU = -ALPHA / BETA
ELSE
ALPHR = ALPHI * (ALPHI/REAL( ALPHA ))
ALPHR = ALPHR + XNORM * (XNORM/REAL( ALPHA ))
TAU = CMPLX( ALPHR/BETA, -ALPHI/BETA )
ALPHA = CMPLX( -ALPHR, ALPHI )
END IF
ALPHA = CLADIV( CMPLX( ONE ), ALPHA )
CALL CSCAL( N-1, ALPHA, X, INCX )
*
* If BETA is subnormal, it may lose relative accuracy
*
DO 20 J = 1, KNT
BETA = BETA*SAFMIN
20 CONTINUE
ALPHA = BETA
END IF
*
RETURN
*
* End of CLARFP
*
END
|
