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
|
*> \brief \b DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlags2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlags2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlags2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
* SNV, CSQ, SNQ )
*
* .. Scalar Arguments ..
* LOGICAL UPPER
* DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
* $ SNU, SNV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
*> that if ( UPPER ) then
*>
*> U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 )
*> ( 0 A3 ) ( x x )
*> and
*> V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 )
*> ( 0 B3 ) ( x x )
*>
*> or if ( .NOT.UPPER ) then
*>
*> U**T *A*Q = U**T *( A1 0 )*Q = ( x x )
*> ( A2 A3 ) ( 0 x )
*> and
*> V**T*B*Q = V**T*( B1 0 )*Q = ( x x )
*> ( B2 B3 ) ( 0 x )
*>
*> The rows of the transformed A and B are parallel, where
*>
*> U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ )
*> ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ )
*>
*> Z**T denotes the transpose of Z.
*>
*> \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 DOUBLE PRECISION
*> \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 DOUBLE PRECISION
*> \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 DOUBLE PRECISION
*> The desired orthogonal matrix U.
*> \endverbatim
*>
*> \param[out] CSV
*> \verbatim
*> CSV is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNV
*> \verbatim
*> SNV is DOUBLE PRECISION
*> The desired orthogonal matrix V.
*> \endverbatim
*>
*> \param[out] CSQ
*> \verbatim
*> CSQ is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SNQ
*> \verbatim
*> SNQ is DOUBLE PRECISION
*> The desired orthogonal matrix Q.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
$ SNV, CSQ, SNQ )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
LOGICAL UPPER
DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
$ SNU, SNV
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
$ AVB21, AVB22, B, C, CSL, CSR, D, R, S1, S2,
$ SNL, SNR, UA11, UA11R, UA12, UA21, UA22, UA22R,
$ VB11, VB11R, VB12, VB21, VB22, VB22R
* ..
* .. External Subroutines ..
EXTERNAL DLARTG, DLASV2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. 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
*
* 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, B, 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**T *A and V**T *B,
* and (1,2) element of |U|**T *|A| and |V|**T *|B|.
*
UA11R = CSL*A1
UA12 = CSL*A2 + SNL*A3
*
VB11R = CSR*B1
VB12 = CSR*B2 + SNR*B3
*
AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 )
AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 )
*
* zero (1,2) elements of U**T *A and V**T *B
*
IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN
IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 /
$ ( ABS( VB11R )+ABS( VB12 ) ) ) THEN
CALL DLARTG( -UA11R, UA12, CSQ, SNQ, R )
ELSE
CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R )
END IF
*
CSU = CSL
SNU = -SNL
CSV = CSR
SNV = -SNR
*
ELSE
*
* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
* and (2,2) element of |U|**T *|A| and |V|**T *|B|.
*
UA21 = -SNL*A1
UA22 = -SNL*A2 + CSL*A3
*
VB21 = -SNR*B1
VB22 = -SNR*B2 + CSR*B3
*
AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 )
AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 )
*
* zero (2,2) elements of U**T*A and V**T*B, and then swap.
*
IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN
IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 /
$ ( ABS( VB21 )+ABS( VB22 ) ) ) THEN
CALL DLARTG( -UA21, UA22, CSQ, SNQ, R )
ELSE
CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( -VB21, VB22, CSQ, SNQ, R )
END IF
*
CSU = SNL
SNU = CSL
CSV = SNR
SNV = 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
*
* 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, C, 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**T *A and V**T *B,
* and (2,1) element of |U|**T *|A| and |V|**T *|B|.
*
UA21 = -SNR*A1 + CSR*A2
UA22R = CSR*A3
*
VB21 = -SNL*B1 + CSL*B2
VB22R = CSL*B3
*
AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 )
AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 )
*
* zero (2,1) elements of U**T *A and V**T *B.
*
IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN
IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 /
$ ( ABS( VB21 )+ABS( VB22R ) ) ) THEN
CALL DLARTG( UA22R, UA21, CSQ, SNQ, R )
ELSE
CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( VB22R, VB21, CSQ, SNQ, R )
END IF
*
CSU = CSR
SNU = -SNR
CSV = CSL
SNV = -SNL
*
ELSE
*
* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
* and (1,1) element of |U|**T *|A| and |V|**T *|B|.
*
UA11 = CSR*A1 + SNR*A2
UA12 = SNR*A3
*
VB11 = CSL*B1 + SNL*B2
VB12 = SNL*B3
*
AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 )
AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 )
*
* zero (1,1) elements of U**T*A and V**T*B, and then swap.
*
IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN
IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 /
$ ( ABS( VB11 )+ABS( VB12 ) ) ) THEN
CALL DLARTG( UA12, UA11, CSQ, SNQ, R )
ELSE
CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
END IF
ELSE
CALL DLARTG( VB12, VB11, CSQ, SNQ, R )
END IF
*
CSU = SNR
SNU = CSR
CSV = SNL
SNV = CSL
*
END IF
*
END IF
*
RETURN
*
* End of DLAGS2
*
END
|
