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
|
*> \brief \b CPFTRS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CPFTRS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpftrs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpftrs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpftrs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
* COMPLEX A( 0: * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CPFTRS solves a system of linear equations A*X = B with a Hermitian
*> positive definite matrix A using the Cholesky factorization
*> A = U**H*U or A = L*L**H computed by CPFTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANSR
*> \verbatim
*> TRANSR is CHARACTER*1
*> = 'N': The Normal TRANSR of RFP A is stored;
*> = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of RFP A is stored;
*> = 'L': Lower triangle of RFP A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX array, dimension ( N*(N+1)/2 );
*> The triangular factor U or L from the Cholesky factorization
*> of RFP A = U**H*U or RFP A = L*L**H, as computed by CPFTRF.
*> See note below for more details about RFP A.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complexOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We first consider Standard Packed Format when N is even.
*> We give an example where N = 6.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 05 00
*> 11 12 13 14 15 10 11
*> 22 23 24 25 20 21 22
*> 33 34 35 30 31 32 33
*> 44 45 40 41 42 43 44
*> 55 50 51 52 53 54 55
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*> conjugate-transpose of the first three columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*> conjugate-transpose of the last three columns of AP lower.
*> To denote conjugate we place -- above the element. This covers the
*> case N even and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> -- -- --
*> 03 04 05 33 43 53
*> -- --
*> 13 14 15 00 44 54
*> --
*> 23 24 25 10 11 55
*>
*> 33 34 35 20 21 22
*> --
*> 00 44 45 30 31 32
*> -- --
*> 01 11 55 40 41 42
*> -- -- --
*> 02 12 22 50 51 52
*>
*> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> -- -- -- -- -- -- -- -- -- --
*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
*> -- -- -- -- -- -- -- -- -- --
*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
*> -- -- -- -- -- -- -- -- -- --
*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
*>
*>
*> We next consider Standard Packed Format when N is odd.
*> We give an example where N = 5.
*>
*> AP is Upper AP is Lower
*>
*> 00 01 02 03 04 00
*> 11 12 13 14 10 11
*> 22 23 24 20 21 22
*> 33 34 30 31 32 33
*> 44 40 41 42 43 44
*>
*>
*> Let TRANSR = 'N'. RFP holds AP as follows:
*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*> conjugate-transpose of the first two columns of AP upper.
*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*> conjugate-transpose of the last two columns of AP lower.
*> To denote conjugate we place -- above the element. This covers the
*> case N odd and TRANSR = 'N'.
*>
*> RFP A RFP A
*>
*> -- --
*> 02 03 04 00 33 43
*> --
*> 12 13 14 10 11 44
*>
*> 22 23 24 20 21 22
*> --
*> 00 33 34 30 31 32
*> -- --
*> 01 11 44 40 41 42
*>
*> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
*> transpose of RFP A above. One therefore gets:
*>
*>
*> RFP A RFP A
*>
*> -- -- -- -- -- -- -- -- --
*> 02 12 22 00 01 00 10 20 30 40 50
*> -- -- -- -- -- -- -- -- --
*> 03 13 23 33 11 33 11 21 31 41 51
*> -- -- -- -- -- -- -- -- --
*> 04 14 24 34 44 43 44 22 32 42 52
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
*
* -- LAPACK computational 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 ..
CHARACTER TRANSR, UPLO
INTEGER INFO, LDB, N, NRHS
* ..
* .. Array Arguments ..
COMPLEX A( 0: * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER, NORMALTRANSR
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, CTFSM
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NORMALTRANSR = LSAME( TRANSR, 'N' )
LOWER = LSAME( UPLO, 'L' )
IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
INFO = -1
ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CPFTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
* start execution: there are two triangular solves
*
IF( LOWER ) THEN
CALL CTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, CONE, A, B,
$ LDB )
CALL CTFSM( TRANSR, 'L', UPLO, 'C', 'N', N, NRHS, CONE, A, B,
$ LDB )
ELSE
CALL CTFSM( TRANSR, 'L', UPLO, 'C', 'N', N, NRHS, CONE, A, B,
$ LDB )
CALL CTFSM( TRANSR, 'L', UPLO, 'N', 'N', N, NRHS, CONE, A, B,
$ LDB )
END IF
*
RETURN
*
* End of CPFTRS
*
END
|
