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
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
|
SUBROUTINE CLARGV( N, X, INCX, Y, INCY, C, INCC )
*
* -- LAPACK auxiliary routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INCC, INCX, INCY, N
* ..
* .. Array Arguments ..
REAL C( * )
COMPLEX X( * ), Y( * )
* ..
*
* Purpose
* =======
*
* CLARGV generates a vector of complex plane rotations with real
* cosines, determined by elements of the complex vectors x and y.
* For i = 1,2,...,n
*
* ( c(i) s(i) ) ( x(i) ) = ( r(i) )
* ( -conjg(s(i)) c(i) ) ( y(i) ) = ( 0 )
*
* where c(i)**2 + ABS(s(i))**2 = 1
*
* The following conventions are used (these are the same as in CLARTG,
* but differ from the BLAS1 routine CROTG):
* If y(i)=0, then c(i)=1 and s(i)=0.
* If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.
*
* Arguments
* =========
*
* N (input) INTEGER
* The number of plane rotations to be generated.
*
* X (input/output) COMPLEX array, dimension (1+(N-1)*INCX)
* On entry, the vector x.
* On exit, x(i) is overwritten by r(i), for i = 1,...,n.
*
* INCX (input) INTEGER
* The increment between elements of X. INCX > 0.
*
* Y (input/output) COMPLEX array, dimension (1+(N-1)*INCY)
* On entry, the vector y.
* On exit, the sines of the plane rotations.
*
* INCY (input) INTEGER
* The increment between elements of Y. INCY > 0.
*
* C (output) REAL array, dimension (1+(N-1)*INCC)
* The cosines of the plane rotations.
*
* INCC (input) INTEGER
* The increment between elements of C. INCC > 0.
*
* Further Details
* ======= =======
*
* 6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel
*
* =====================================================================
*
* .. Parameters ..
REAL TWO, ONE, ZERO
PARAMETER ( TWO = 2.0E+0, ONE = 1.0E+0, ZERO = 0.0E+0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL FIRST
INTEGER COUNT, I, IC, IX, IY, J
REAL CS, D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN,
$ SAFMN2, SAFMX2, SCALE
COMPLEX F, FF, FS, G, GS, R, SN
* ..
* .. External Functions ..
REAL SLAMCH, SLAPY2
EXTERNAL SLAMCH, SLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, CONJG, INT, LOG, MAX, REAL,
$ SQRT
* ..
* .. Statement Functions ..
REAL ABS1, ABSSQ
* ..
* .. Save statement ..
SAVE FIRST, SAFMX2, SAFMIN, SAFMN2
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Statement Function definitions ..
ABS1( FF ) = MAX( ABS( REAL( FF ) ), ABS( AIMAG( FF ) ) )
ABSSQ( FF ) = REAL( FF )**2 + AIMAG( FF )**2
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
SAFMIN = SLAMCH( 'S' )
EPS = SLAMCH( 'E' )
SAFMN2 = SLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( SLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
END IF
IX = 1
IY = 1
IC = 1
DO 60 I = 1, N
F = X( IX )
G = Y( IY )
*
* Use identical algorithm as in CLARTG
*
SCALE = MAX( ABS1( F ), ABS1( G ) )
FS = F
GS = G
COUNT = 0
IF( SCALE.GE.SAFMX2 ) THEN
10 CONTINUE
COUNT = COUNT + 1
FS = FS*SAFMN2
GS = GS*SAFMN2
SCALE = SCALE*SAFMN2
IF( SCALE.GE.SAFMX2 )
$ GO TO 10
ELSE IF( SCALE.LE.SAFMN2 ) THEN
IF( G.EQ.CZERO ) THEN
CS = ONE
SN = CZERO
R = F
GO TO 50
END IF
20 CONTINUE
COUNT = COUNT - 1
FS = FS*SAFMX2
GS = GS*SAFMX2
SCALE = SCALE*SAFMX2
IF( SCALE.LE.SAFMN2 )
$ GO TO 20
END IF
F2 = ABSSQ( FS )
G2 = ABSSQ( GS )
IF( F2.LE.MAX( G2, ONE )*SAFMIN ) THEN
*
* This is a rare case: F is very small.
*
IF( F.EQ.CZERO ) THEN
CS = ZERO
R = SLAPY2( REAL( G ), AIMAG( G ) )
* Do complex/real division explicitly with two real
* divisions
D = SLAPY2( REAL( GS ), AIMAG( GS ) )
SN = CMPLX( REAL( GS ) / D, -AIMAG( GS ) / D )
GO TO 50
END IF
F2S = SLAPY2( REAL( FS ), AIMAG( FS ) )
* G2 and G2S are accurate
* G2 is at least SAFMIN, and G2S is at least SAFMN2
G2S = SQRT( G2 )
* Error in CS from underflow in F2S is at most
* UNFL / SAFMN2 .lt. sqrt(UNFL*EPS) .lt. EPS
* If MAX(G2,ONE)=G2, then F2 .lt. G2*SAFMIN,
* and so CS .lt. sqrt(SAFMIN)
* If MAX(G2,ONE)=ONE, then F2 .lt. SAFMIN
* and so CS .lt. sqrt(SAFMIN)/SAFMN2 = sqrt(EPS)
* Therefore, CS = F2S/G2S / sqrt( 1 + (F2S/G2S)**2 ) = F2S/G2S
CS = F2S / G2S
* Make sure abs(FF) = 1
* Do complex/real division explicitly with 2 real divisions
IF( ABS1( F ).GT.ONE ) THEN
D = SLAPY2( REAL( F ), AIMAG( F ) )
FF = CMPLX( REAL( F ) / D, AIMAG( F ) / D )
ELSE
DR = SAFMX2*REAL( F )
DI = SAFMX2*AIMAG( F )
D = SLAPY2( DR, DI )
FF = CMPLX( DR / D, DI / D )
END IF
SN = FF*CMPLX( REAL( GS ) / G2S, -AIMAG( GS ) / G2S )
R = CS*F + SN*G
ELSE
*
* This is the most common case.
* Neither F2 nor F2/G2 are less than SAFMIN
* F2S cannot overflow, and it is accurate
*
F2S = SQRT( ONE+G2 / F2 )
* Do the F2S(real)*FS(complex) multiply with two real
* multiplies
R = CMPLX( F2S*REAL( FS ), F2S*AIMAG( FS ) )
CS = ONE / F2S
D = F2 + G2
* Do complex/real division explicitly with two real divisions
SN = CMPLX( REAL( R ) / D, AIMAG( R ) / D )
SN = SN*CONJG( GS )
IF( COUNT.NE.0 ) THEN
IF( COUNT.GT.0 ) THEN
DO 30 J = 1, COUNT
R = R*SAFMX2
30 CONTINUE
ELSE
DO 40 J = 1, -COUNT
R = R*SAFMN2
40 CONTINUE
END IF
END IF
END IF
50 CONTINUE
C( IC ) = CS
Y( IY ) = SN
X( IX ) = R
IC = IC + INCC
IY = IY + INCY
IX = IX + INCX
60 CONTINUE
RETURN
*
* End of CLARGV
*
END
|
