summaryrefslogtreecommitdiff
path: root/SRC/dspgvx.f
blob: 8619ef739fccf8f4924832ee50f502905aa84a8c (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
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
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
*> \brief \b DSPGVX
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPGVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
*                          IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
*                          IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
*       DOUBLE PRECISION   ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
*      $                   Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSPGVX computes selected eigenvalues, and optionally, eigenvectors
*> of a real generalized symmetric-definite eigenproblem, of the form
*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
*> and B are assumed to be symmetric, stored in packed storage, and B
*> is also positive definite.  Eigenvalues and eigenvectors can be
*> selected by specifying either a range of values or a range of indices
*> for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is INTEGER
*>          Specifies the problem type to be solved:
*>          = 1:  A*x = (lambda)*B*x
*>          = 2:  A*B*x = (lambda)*x
*>          = 3:  B*A*x = (lambda)*x
*> \endverbatim
*>
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found.
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found.
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A and B are stored;
*>          = 'L':  Lower triangle of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix pencil (A,B).  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, the contents of AP are destroyed.
*> \endverbatim
*>
*> \param[in,out] BP
*> \verbatim
*>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric matrix
*>          B, packed columnwise in a linear array.  The j-th column of B
*>          is stored in the array BP as follows:
*>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*>
*>          On exit, the triangular factor U or L from the Cholesky
*>          factorization B = U**T*U or B = L*L**T, in the same storage
*>          format as B.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is DOUBLE PRECISION
*>
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is DOUBLE PRECISION
*>
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is DOUBLE PRECISION
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing A to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*DLAMCH('S').
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          On normal exit, the first M elements contain the selected
*>          eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'N', then Z is not referenced.
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          The eigenvectors are normalized as follows:
*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
*>
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and the
*>          index of the eigenvector is returned in IFAIL.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  DPPTRF or DSPEVX returned an error code:
*>             <= N:  if INFO = i, DSPEVX failed to converge;
*>                    i eigenvectors failed to converge.  Their indices
*>                    are stored in array IFAIL.
*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*>                    minor of order i of B is not positive definite.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleOTHEReigen
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
     $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
     $                   IFAIL, INFO )
*
*  -- LAPACK driver 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 ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
* =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
      CHARACTER          TRANS
      INTEGER            J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      UPPER = LSAME( UPLO, 'U' )
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -3
      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL ) THEN
               INFO = -9
            END IF
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 ) THEN
               INFO = -10
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -11
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
            INFO = -16
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSPGVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL DPPTRF( UPLO, N, BP, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
      CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
     $             W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         IF( INFO.GT.0 )
     $      M = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'T'
            END IF
*
            DO 10 J = 1, M
               CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   10       CONTINUE
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**T*y
*
            IF( UPPER ) THEN
               TRANS = 'T'
            ELSE
               TRANS = 'N'
            END IF
*
            DO 20 J = 1, M
               CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
     $                     1 )
   20       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DSPGVX
*
      END