LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dsygvx.f
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1 *> \brief \b DSYGST
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
22 * VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
23 * LWORK, IWORK, IFAIL, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
28 * DOUBLE PRECISION ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IFAIL( * ), IWORK( * )
32 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
33 * \$ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> DSYGVX computes selected eigenvalues, and optionally, eigenvectors
43 *> of a real generalized symmetric-definite eigenproblem, of the form
44 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
45 *> and B are assumed to be symmetric and B is also positive definite.
46 *> Eigenvalues and eigenvectors can be selected by specifying either a
47 *> range of values or a range of indices for the desired eigenvalues.
48 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] ITYPE
54 *> \verbatim
55 *> ITYPE is INTEGER
56 *> Specifies the problem type to be solved:
57 *> = 1: A*x = (lambda)*B*x
58 *> = 2: A*B*x = (lambda)*x
59 *> = 3: B*A*x = (lambda)*x
60 *> \endverbatim
61 *>
62 *> \param[in] JOBZ
63 *> \verbatim
64 *> JOBZ is CHARACTER*1
65 *> = 'N': Compute eigenvalues only;
66 *> = 'V': Compute eigenvalues and eigenvectors.
67 *> \endverbatim
68 *>
69 *> \param[in] RANGE
70 *> \verbatim
71 *> RANGE is CHARACTER*1
72 *> = 'A': all eigenvalues will be found.
73 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
74 *> will be found.
75 *> = 'I': the IL-th through IU-th eigenvalues will be found.
76 *> \endverbatim
77 *>
78 *> \param[in] UPLO
79 *> \verbatim
80 *> UPLO is CHARACTER*1
81 *> = 'U': Upper triangle of A and B are stored;
82 *> = 'L': Lower triangle of A and B are stored.
83 *> \endverbatim
84 *>
85 *> \param[in] N
86 *> \verbatim
87 *> N is INTEGER
88 *> The order of the matrix pencil (A,B). N >= 0.
89 *> \endverbatim
90 *>
91 *> \param[in,out] A
92 *> \verbatim
93 *> A is DOUBLE PRECISION array, dimension (LDA, N)
94 *> On entry, the symmetric matrix A. If UPLO = 'U', the
95 *> leading N-by-N upper triangular part of A contains the
96 *> upper triangular part of the matrix A. If UPLO = 'L',
97 *> the leading N-by-N lower triangular part of A contains
98 *> the lower triangular part of the matrix A.
99 *>
100 *> On exit, the lower triangle (if UPLO='L') or the upper
101 *> triangle (if UPLO='U') of A, including the diagonal, is
102 *> destroyed.
103 *> \endverbatim
104 *>
105 *> \param[in] LDA
106 *> \verbatim
107 *> LDA is INTEGER
108 *> The leading dimension of the array A. LDA >= max(1,N).
109 *> \endverbatim
110 *>
111 *> \param[in,out] B
112 *> \verbatim
113 *> B is DOUBLE PRECISION array, dimension (LDB, N)
114 *> On entry, the symmetric matrix B. If UPLO = 'U', the
115 *> leading N-by-N upper triangular part of B contains the
116 *> upper triangular part of the matrix B. If UPLO = 'L',
117 *> the leading N-by-N lower triangular part of B contains
118 *> the lower triangular part of the matrix B.
119 *>
120 *> On exit, if INFO <= N, the part of B containing the matrix is
121 *> overwritten by the triangular factor U or L from the Cholesky
122 *> factorization B = U**T*U or B = L*L**T.
123 *> \endverbatim
124 *>
125 *> \param[in] LDB
126 *> \verbatim
127 *> LDB is INTEGER
128 *> The leading dimension of the array B. LDB >= max(1,N).
129 *> \endverbatim
130 *>
131 *> \param[in] VL
132 *> \verbatim
133 *> VL is DOUBLE PRECISION
134 *> \endverbatim
135 *>
136 *> \param[in] VU
137 *> \verbatim
138 *> VU is DOUBLE PRECISION
139 *> If RANGE='V', the lower and upper bounds of the interval to
140 *> be searched for eigenvalues. VL < VU.
141 *> Not referenced if RANGE = 'A' or 'I'.
142 *> \endverbatim
143 *>
144 *> \param[in] IL
145 *> \verbatim
146 *> IL is INTEGER
147 *> \endverbatim
148 *>
149 *> \param[in] IU
150 *> \verbatim
151 *> IU is INTEGER
152 *> If RANGE='I', the indices (in ascending order) of the
153 *> smallest and largest eigenvalues to be returned.
154 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
155 *> Not referenced if RANGE = 'A' or 'V'.
156 *> \endverbatim
157 *>
158 *> \param[in] ABSTOL
159 *> \verbatim
160 *> ABSTOL is DOUBLE PRECISION
161 *> The absolute error tolerance for the eigenvalues.
162 *> An approximate eigenvalue is accepted as converged
163 *> when it is determined to lie in an interval [a,b]
164 *> of width less than or equal to
165 *>
166 *> ABSTOL + EPS * max( |a|,|b| ) ,
167 *>
168 *> where EPS is the machine precision. If ABSTOL is less than
169 *> or equal to zero, then EPS*|T| will be used in its place,
170 *> where |T| is the 1-norm of the tridiagonal matrix obtained
171 *> by reducing C to tridiagonal form, where C is the symmetric
172 *> matrix of the standard symmetric problem to which the
173 *> generalized problem is transformed.
174 *>
175 *> Eigenvalues will be computed most accurately when ABSTOL is
176 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
177 *> If this routine returns with INFO>0, indicating that some
178 *> eigenvectors did not converge, try setting ABSTOL to
179 *> 2*DLAMCH('S').
180 *> \endverbatim
181 *>
182 *> \param[out] M
183 *> \verbatim
184 *> M is INTEGER
185 *> The total number of eigenvalues found. 0 <= M <= N.
186 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
187 *> \endverbatim
188 *>
189 *> \param[out] W
190 *> \verbatim
191 *> W is DOUBLE PRECISION array, dimension (N)
192 *> On normal exit, the first M elements contain the selected
193 *> eigenvalues in ascending order.
194 *> \endverbatim
195 *>
196 *> \param[out] Z
197 *> \verbatim
198 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
199 *> If JOBZ = 'N', then Z is not referenced.
200 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
201 *> contain the orthonormal eigenvectors of the matrix A
202 *> corresponding to the selected eigenvalues, with the i-th
203 *> column of Z holding the eigenvector associated with W(i).
204 *> The eigenvectors are normalized as follows:
205 *> if ITYPE = 1 or 2, Z**T*B*Z = I;
206 *> if ITYPE = 3, Z**T*inv(B)*Z = I.
207 *>
208 *> If an eigenvector fails to converge, then that column of Z
209 *> contains the latest approximation to the eigenvector, and the
210 *> index of the eigenvector is returned in IFAIL.
211 *> Note: the user must ensure that at least max(1,M) columns are
212 *> supplied in the array Z; if RANGE = 'V', the exact value of M
213 *> is not known in advance and an upper bound must be used.
214 *> \endverbatim
215 *>
216 *> \param[in] LDZ
217 *> \verbatim
218 *> LDZ is INTEGER
219 *> The leading dimension of the array Z. LDZ >= 1, and if
220 *> JOBZ = 'V', LDZ >= max(1,N).
221 *> \endverbatim
222 *>
223 *> \param[out] WORK
224 *> \verbatim
225 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
226 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
227 *> \endverbatim
228 *>
229 *> \param[in] LWORK
230 *> \verbatim
231 *> LWORK is INTEGER
232 *> The length of the array WORK. LWORK >= max(1,8*N).
233 *> For optimal efficiency, LWORK >= (NB+3)*N,
234 *> where NB is the blocksize for DSYTRD returned by ILAENV.
235 *>
236 *> If LWORK = -1, then a workspace query is assumed; the routine
237 *> only calculates the optimal size of the WORK array, returns
238 *> this value as the first entry of the WORK array, and no error
239 *> message related to LWORK is issued by XERBLA.
240 *> \endverbatim
241 *>
242 *> \param[out] IWORK
243 *> \verbatim
244 *> IWORK is INTEGER array, dimension (5*N)
245 *> \endverbatim
246 *>
247 *> \param[out] IFAIL
248 *> \verbatim
249 *> IFAIL is INTEGER array, dimension (N)
250 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
251 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
252 *> indices of the eigenvectors that failed to converge.
253 *> If JOBZ = 'N', then IFAIL is not referenced.
254 *> \endverbatim
255 *>
256 *> \param[out] INFO
257 *> \verbatim
258 *> INFO is INTEGER
259 *> = 0: successful exit
260 *> < 0: if INFO = -i, the i-th argument had an illegal value
261 *> > 0: DPOTRF or DSYEVX returned an error code:
262 *> <= N: if INFO = i, DSYEVX failed to converge;
263 *> i eigenvectors failed to converge. Their indices
264 *> are stored in array IFAIL.
265 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
266 *> minor of order i of B is not positive definite.
267 *> The factorization of B could not be completed and
268 *> no eigenvalues or eigenvectors were computed.
269 *> \endverbatim
270 *
271 * Authors:
272 * ========
273 *
274 *> \author Univ. of Tennessee
275 *> \author Univ. of California Berkeley
276 *> \author Univ. of Colorado Denver
277 *> \author NAG Ltd.
278 *
279 *> \date November 2011
280 *
281 *> \ingroup doubleSYeigen
282 *
283 *> \par Contributors:
284 * ==================
285 *>
286 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
287 *
288 * =====================================================================
289  SUBROUTINE dsygvx( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
290  \$ vl, vu, il, iu, abstol, m, w, z, ldz, work,
291  \$ lwork, iwork, ifail, info )
292 *
293 * -- LAPACK driver routine (version 3.4.0) --
294 * -- LAPACK is a software package provided by Univ. of Tennessee, --
295 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
296 * November 2011
297 *
298 * .. Scalar Arguments ..
299  CHARACTER jobz, range, uplo
300  INTEGER il, info, itype, iu, lda, ldb, ldz, lwork, m, n
301  DOUBLE PRECISION abstol, vl, vu
302 * ..
303 * .. Array Arguments ..
304  INTEGER ifail( * ), iwork( * )
305  DOUBLE PRECISION a( lda, * ), b( ldb, * ), w( * ), work( * ),
306  \$ z( ldz, * )
307 * ..
308 *
309 * =====================================================================
310 *
311 * .. Parameters ..
312  DOUBLE PRECISION one
313  parameter( one = 1.0d+0 )
314 * ..
315 * .. Local Scalars ..
316  LOGICAL alleig, indeig, lquery, upper, valeig, wantz
317  CHARACTER trans
318  INTEGER lwkmin, lwkopt, nb
319 * ..
320 * .. External Functions ..
321  LOGICAL lsame
322  INTEGER ilaenv
323  EXTERNAL lsame, ilaenv
324 * ..
325 * .. External Subroutines ..
326  EXTERNAL dpotrf, dsyevx, dsygst, dtrmm, dtrsm, xerbla
327 * ..
328 * .. Intrinsic Functions ..
329  INTRINSIC max, min
330 * ..
331 * .. Executable Statements ..
332 *
333 * Test the input parameters.
334 *
335  upper = lsame( uplo, 'U' )
336  wantz = lsame( jobz, 'V' )
337  alleig = lsame( range, 'A' )
338  valeig = lsame( range, 'V' )
339  indeig = lsame( range, 'I' )
340  lquery = ( lwork.EQ.-1 )
341 *
342  info = 0
343  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
344  info = -1
345  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
346  info = -2
347  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
348  info = -3
349  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
350  info = -4
351  ELSE IF( n.LT.0 ) THEN
352  info = -5
353  ELSE IF( lda.LT.max( 1, n ) ) THEN
354  info = -7
355  ELSE IF( ldb.LT.max( 1, n ) ) THEN
356  info = -9
357  ELSE
358  IF( valeig ) THEN
359  IF( n.GT.0 .AND. vu.LE.vl )
360  \$ info = -11
361  ELSE IF( indeig ) THEN
362  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
363  info = -12
364  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
365  info = -13
366  END IF
367  END IF
368  END IF
369  IF (info.EQ.0) THEN
370  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
371  info = -18
372  END IF
373  END IF
374 *
375  IF( info.EQ.0 ) THEN
376  lwkmin = max( 1, 8*n )
377  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
378  lwkopt = max( lwkmin, ( nb + 3 )*n )
379  work( 1 ) = lwkopt
380 *
381  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
382  info = -20
383  END IF
384  END IF
385 *
386  IF( info.NE.0 ) THEN
387  CALL xerbla( 'DSYGVX', -info )
388  return
389  ELSE IF( lquery ) THEN
390  return
391  END IF
392 *
393 * Quick return if possible
394 *
395  m = 0
396  IF( n.EQ.0 ) THEN
397  return
398  END IF
399 *
400 * Form a Cholesky factorization of B.
401 *
402  CALL dpotrf( uplo, n, b, ldb, info )
403  IF( info.NE.0 ) THEN
404  info = n + info
405  return
406  END IF
407 *
408 * Transform problem to standard eigenvalue problem and solve.
409 *
410  CALL dsygst( itype, uplo, n, a, lda, b, ldb, info )
411  CALL dsyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
412  \$ m, w, z, ldz, work, lwork, iwork, ifail, info )
413 *
414  IF( wantz ) THEN
415 *
416 * Backtransform eigenvectors to the original problem.
417 *
418  IF( info.GT.0 )
419  \$ m = info - 1
420  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
421 *
422 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
423 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
424 *
425  IF( upper ) THEN
426  trans = 'N'
427  ELSE
428  trans = 'T'
429  END IF
430 *
431  CALL dtrsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
432  \$ ldb, z, ldz )
433 *
434  ELSE IF( itype.EQ.3 ) THEN
435 *
436 * For B*A*x=(lambda)*x;
437 * backtransform eigenvectors: x = L*y or U**T*y
438 *
439  IF( upper ) THEN
440  trans = 'T'
441  ELSE
442  trans = 'N'
443  END IF
444 *
445  CALL dtrmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
446  \$ ldb, z, ldz )
447  END IF
448  END IF
449 *
450 * Set WORK(1) to optimal workspace size.
451 *
452  work( 1 ) = lwkopt
453 *
454  return
455 *
456 * End of DSYGVX
457 *
458  END