summaryrefslogtreecommitdiff
path: root/SRC/clatdf.f
blob: 357f664223df5f66b280fbac58f17b9b8b602fdb (plain)
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
*> \brief \b CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLATDF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clatdf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clatdf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clatdf.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
*                          JPIV )
*
*       .. Scalar Arguments ..
*       INTEGER            IJOB, LDZ, N
*       REAL               RDSCAL, RDSUM
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), JPIV( * )
*       COMPLEX            RHS( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CLATDF computes the contribution to the reciprocal Dif-estimate
*> by solving for x in Z * x = b, where b is chosen such that the norm
*> of x is as large as possible. It is assumed that LU decomposition
*> of Z has been computed by CGETC2. On entry RHS = f holds the
*> contribution from earlier solved sub-systems, and on return RHS = x.
*>
*> The factorization of Z returned by CGETC2 has the form
*> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
*> triangular with unit diagonal elements and U is upper triangular.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          IJOB = 2: First compute an approximative null-vector e
*>              of Z using CGECON, e is normalized and solve for
*>              Zx = +-e - f with the sign giving the greater value of
*>              2-norm(x).  About 5 times as expensive as Default.
*>          IJOB .ne. 2: Local look ahead strategy where
*>              all entries of the r.h.s. b is chosen as either +1 or
*>              -1.  Default.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Z.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is COMPLEX array, dimension (LDZ, N)
*>          On entry, the LU part of the factorization of the n-by-n
*>          matrix Z computed by CGETC2:  Z = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*>          RHS is COMPLEX array, dimension (N).
*>          On entry, RHS contains contributions from other subsystems.
*>          On exit, RHS contains the solution of the subsystem with
*>          entries according to the value of IJOB (see above).
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*>          RDSUM is REAL
*>          On entry, the sum of squares of computed contributions to
*>          the Dif-estimate under computation by CTGSYL, where the
*>          scaling factor RDSCAL (see below) has been factored out.
*>          On exit, the corresponding sum of squares updated with the
*>          contributions from the current sub-system.
*>          If TRANS = 'T' RDSUM is not touched.
*>          NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*>          RDSCAL is REAL
*>          On entry, scaling factor used to prevent overflow in RDSUM.
*>          On exit, RDSCAL is updated w.r.t. the current contributions
*>          in RDSUM.
*>          If TRANS = 'T', RDSCAL is not touched.
*>          NOTE: RDSCAL only makes sense when CTGSY2 is called by
*>          CTGSYL.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= i <= N, row i of the
*>          matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*>          JPIV is INTEGER array, dimension (N).
*>          The pivot indices; for 1 <= j <= N, column j of the
*>          matrix has been interchanged with column JPIV(j).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complexOTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*>  This routine is a further developed implementation of algorithm
*>  BSOLVE in [1] using complete pivoting in the LU factorization.
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*>   [1]   Bo Kagstrom and Lars Westin,
*>         Generalized Schur Methods with Condition Estimators for
*>         Solving the Generalized Sylvester Equation, IEEE Transactions
*>         on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
*>
*>   [2]   Peter Poromaa,
*>         On Efficient and Robust Estimators for the Separation
*>         between two Regular Matrix Pairs with Applications in
*>         Condition Estimation. Report UMINF-95.05, Department of
*>         Computing Science, Umea University, S-901 87 Umea, Sweden,
*>         1995.
*
*  =====================================================================
      SUBROUTINE CLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
     $                   JPIV )
*
*  -- LAPACK auxiliary routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      INTEGER            IJOB, LDZ, N
      REAL               RDSCAL, RDSUM
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), JPIV( * )
      COMPLEX            RHS( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXDIM
      PARAMETER          ( MAXDIM = 2 )
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
      COMPLEX            CONE
      PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, J, K
      REAL               RTEMP, SCALE, SMINU, SPLUS
      COMPLEX            BM, BP, PMONE, TEMP
*     ..
*     .. Local Arrays ..
      REAL               RWORK( MAXDIM )
      COMPLEX            WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
*     ..
*     .. External Subroutines ..
      EXTERNAL           CAXPY, CCOPY, CGECON, CGESC2, CLASSQ, CLASWP,
     $                   CSCAL
*     ..
*     .. External Functions ..
      REAL               SCASUM
      COMPLEX            CDOTC
      EXTERNAL           SCASUM, CDOTC
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, REAL, SQRT
*     ..
*     .. Executable Statements ..
*
      IF( IJOB.NE.2 ) THEN
*
*        Apply permutations IPIV to RHS
*
         CALL CLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
*
*        Solve for L-part choosing RHS either to +1 or -1.
*
         PMONE = -CONE
         DO 10 J = 1, N - 1
            BP = RHS( J ) + CONE
            BM = RHS( J ) - CONE
            SPLUS = ONE
*
*           Lockahead for L- part RHS(1:N-1) = +-1
*           SPLUS and SMIN computed more efficiently than in BSOLVE[1].
*
            SPLUS = SPLUS + REAL( CDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
     $              J ), 1 ) )
            SMINU = REAL( CDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
            SPLUS = SPLUS*REAL( RHS( J ) )
            IF( SPLUS.GT.SMINU ) THEN
               RHS( J ) = BP
            ELSE IF( SMINU.GT.SPLUS ) THEN
               RHS( J ) = BM
            ELSE
*
*              In this case the updating sums are equal and we can
*              choose RHS(J) +1 or -1. The first time this happens we
*              choose -1, thereafter +1. This is a simple way to get
*              good estimates of matrices like Byers well-known example
*              (see [1]). (Not done in BSOLVE.)
*
               RHS( J ) = RHS( J ) + PMONE
               PMONE = CONE
            END IF
*
*           Compute the remaining r.h.s.
*
            TEMP = -RHS( J )
            CALL CAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
   10    CONTINUE
*
*        Solve for U- part, lockahead for RHS(N) = +-1. This is not done
*        In BSOLVE and will hopefully give us a better estimate because
*        any ill-conditioning of the original matrix is transfered to U
*        and not to L. U(N, N) is an approximation to sigma_min(LU).
*
         CALL CCOPY( N-1, RHS, 1, WORK, 1 )
         WORK( N ) = RHS( N ) + CONE
         RHS( N ) = RHS( N ) - CONE
         SPLUS = ZERO
         SMINU = ZERO
         DO 30 I = N, 1, -1
            TEMP = CONE / Z( I, I )
            WORK( I ) = WORK( I )*TEMP
            RHS( I ) = RHS( I )*TEMP
            DO 20 K = I + 1, N
               WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
               RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
   20       CONTINUE
            SPLUS = SPLUS + ABS( WORK( I ) )
            SMINU = SMINU + ABS( RHS( I ) )
   30    CONTINUE
         IF( SPLUS.GT.SMINU )
     $      CALL CCOPY( N, WORK, 1, RHS, 1 )
*
*        Apply the permutations JPIV to the computed solution (RHS)
*
         CALL CLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
*        Compute the sum of squares
*
         CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM )
         RETURN
      END IF
*
*     ENTRY IJOB = 2
*
*     Compute approximate nullvector XM of Z
*
      CALL CGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
      CALL CCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
*     Compute RHS
*
      CALL CLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
      TEMP = CONE / SQRT( CDOTC( N, XM, 1, XM, 1 ) )
      CALL CSCAL( N, TEMP, XM, 1 )
      CALL CCOPY( N, XM, 1, XP, 1 )
      CALL CAXPY( N, CONE, RHS, 1, XP, 1 )
      CALL CAXPY( N, -CONE, XM, 1, RHS, 1 )
      CALL CGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
      CALL CGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
      IF( SCASUM( N, XP, 1 ).GT.SCASUM( N, RHS, 1 ) )
     $   CALL CCOPY( N, XP, 1, RHS, 1 )
*
*     Compute the sum of squares
*
      CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM )
      RETURN
*
*     End of CLATDF
*
      END