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
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
|
*> \brief \b ZLAGS2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLAGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlags2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlags2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlags2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
* SNV, CSQ, SNQ )
*
* .. Scalar Arguments ..
* LOGICAL UPPER
* DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV
* COMPLEX*16 A2, B2, SNQ, SNU, SNV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
*> that if ( UPPER ) then
*>
*> U**H *A*Q = U**H *( A1 A2 )*Q = ( x 0 )
*> ( 0 A3 ) ( x x )
*> and
*> V**H*B*Q = V**H *( B1 B2 )*Q = ( x 0 )
*> ( 0 B3 ) ( x x )
*>
*> or if ( .NOT.UPPER ) then
*>
*> U**H *A*Q = U**H *( A1 0 )*Q = ( x x )
*> ( A2 A3 ) ( 0 x )
*> and
*> V**H *B*Q = V**H *( B1 0 )*Q = ( x x )
*> ( B2 B3 ) ( 0 x )
*> where
*>
*> U = ( CSU SNU ), V = ( CSV SNV ),
*> ( -SNU**H CSU ) ( -SNV**H CSV )
*>
*> Q = ( CSQ SNQ )
*> ( -SNQ**H CSQ )
*>
*> The rows of the transformed A and B are parallel. Moreover, if the
*> input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
*> of A is not zero. If the input matrices A and B are both not zero,
*> then the transformed (2,2) element of B is not zero, except when the
*> first rows of input A and B are parallel and the second rows are
*> zero.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPPER
*> \verbatim
*> UPPER is LOGICAL
*> = .TRUE.: the input matrices A and B are upper triangular.
*> = .FALSE.: the input matrices A and B are lower triangular.
*> \endverbatim
*>
*> \param[in] A1
*> \verbatim
*> A1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] A2
*> \verbatim
*> A2 is COMPLEX*16
*> \endverbatim
*>
*> \param[in] A3
*> \verbatim
*> A3 is DOUBLE PRECISION
*> On entry, A1, A2 and A3 are elements of the input 2-by-2
*> upper (lower) triangular matrix A.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*> B1 is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] B2
*> \verbatim
*> B2 is COMPLEX*16
*> \endverbatim
*>
*> \param[in] B3
*> \verbatim
*> B3 is DOUBLE PRECISION
*> On entry, B1, B2 and B3 are elements of the input 2-by-2
*> upper (lower) triangular matrix B.
*> \endverbatim
*>
*> \param[out] CSU
*> \verbatim
*> CSU is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNU
*> \verbatim
*> SNU is COMPLEX*16
*> The desired unitary matrix U.
*> \endverbatim
*>
*> \param[out] CSV
*> \verbatim
*> CSV is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNV
*> \verbatim
*> SNV is COMPLEX*16
*> The desired unitary matrix V.
*> \endverbatim
*>
*> \param[out] CSQ
*> \verbatim
*> CSQ is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNQ
*> \verbatim
*> SNQ is COMPLEX*16
*> The desired unitary matrix Q.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16OTHERauxiliary
*
* =====================================================================
SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
$ SNV, CSQ, SNQ )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
LOGICAL UPPER
DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV
COMPLEX*16 A2, B2, SNQ, SNU, SNV
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11,
$ AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2,
$ SNL, SNR, UA11R, UA22R, VB11R, VB22R
COMPLEX*16 B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
$ VB12, VB21, VB22
* ..
* .. External Subroutines ..
EXTERNAL DLASV2, ZLARTG
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG
* ..
* .. Statement Functions ..
DOUBLE PRECISION ABS1
* ..
* .. Statement Function definitions ..
ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
* ..
* .. Executable Statements ..
*
IF( UPPER ) THEN
*
* Input matrices A and B are upper triangular matrices
*
* Form matrix C = A*adj(B) = ( a b )
* ( 0 d )
*
A = A1*B3
D = A3*B1
B = A2*B1 - A1*B2
FB = ABS( B )
*
* Transform complex 2-by-2 matrix C to real matrix by unitary
* diagonal matrix diag(1,D1).
*
D1 = ONE
IF( FB.NE.ZERO )
$ D1 = B / FB
*
* The SVD of real 2 by 2 triangular C
*
* ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 )
* ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T )
*
CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
*
IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
$ THEN
*
* Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
* and (1,2) element of |U|**H *|A| and |V|**H *|B|.
*
UA11R = CSL*A1
UA12 = CSL*A2 + D1*SNL*A3
*
VB11R = CSR*B1
VB12 = CSR*B2 + D1*SNR*B3
*
AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
*
* zero (1,2) elements of U**H *A and V**H *B
*
IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
$ R )
ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
$ R )
ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
$ ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
$ R )
ELSE
CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
$ R )
END IF
*
CSU = CSL
SNU = -D1*SNL
CSV = CSR
SNV = -D1*SNR
*
ELSE
*
* Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
* and (2,2) element of |U|**H *|A| and |V|**H *|B|.
*
UA21 = -DCONJG( D1 )*SNL*A1
UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3
*
VB21 = -DCONJG( D1 )*SNR*B1
VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3
*
AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
*
* zero (2,2) elements of U**H *A and V**H *B, and then swap.
*
IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
$ R )
ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
$ R )
ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
$ ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
$ R )
ELSE
CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
$ R )
END IF
*
CSU = SNL
SNU = D1*CSL
CSV = SNR
SNV = D1*CSR
*
END IF
*
ELSE
*
* Input matrices A and B are lower triangular matrices
*
* Form matrix C = A*adj(B) = ( a 0 )
* ( c d )
*
A = A1*B3
D = A3*B1
C = A2*B3 - A3*B2
FC = ABS( C )
*
* Transform complex 2-by-2 matrix C to real matrix by unitary
* diagonal matrix diag(d1,1).
*
D1 = ONE
IF( FC.NE.ZERO )
$ D1 = C / FC
*
* The SVD of real 2 by 2 triangular C
*
* ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 )
* ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T )
*
CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
*
IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
$ THEN
*
* Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
* and (2,1) element of |U|**H *|A| and |V|**H *|B|.
*
UA21 = -D1*SNR*A1 + CSR*A2
UA22R = CSR*A3
*
VB21 = -D1*SNL*B1 + CSL*B2
VB22R = CSL*B3
*
AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
*
* zero (2,1) elements of U**H *A and V**H *B.
*
IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
$ ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
ELSE
CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
END IF
*
CSU = CSR
SNU = -DCONJG( D1 )*SNR
CSV = CSL
SNV = -DCONJG( D1 )*SNL
*
ELSE
*
* Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
* and (1,1) element of |U|**H *|A| and |V|**H *|B|.
*
UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2
UA12 = DCONJG( D1 )*SNR*A3
*
VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2
VB12 = DCONJG( D1 )*SNL*B3
*
AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
*
* zero (1,1) elements of U**H *A and V**H *B, and then swap.
*
IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
$ ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
ELSE
CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
END IF
*
CSU = SNR
SNU = DCONJG( D1 )*CSR
CSV = SNL
SNV = DCONJG( D1 )*CSL
*
END IF
*
END IF
*
RETURN
*
* End of ZLAGS2
*
END
|
