LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dsygvx.f
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1 *> \brief \b DSYGVX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DSYGVX + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
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 *> If RANGE='V', the lower bound of the interval to
135 *> be searched for eigenvalues. VL < VU.
136 *> Not referenced if RANGE = 'A' or 'I'.
137 *> \endverbatim
138 *>
139 *> \param[in] VU
140 *> \verbatim
141 *> VU is DOUBLE PRECISION
142 *> If RANGE='V', the upper bound of the interval to
143 *> be searched for eigenvalues. VL < VU.
144 *> Not referenced if RANGE = 'A' or 'I'.
145 *> \endverbatim
146 *>
147 *> \param[in] IL
148 *> \verbatim
149 *> IL is INTEGER
150 *> If RANGE='I', the index of the
151 *> smallest eigenvalue to be returned.
152 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
153 *> Not referenced if RANGE = 'A' or 'V'.
154 *> \endverbatim
155 *>
156 *> \param[in] IU
157 *> \verbatim
158 *> IU is INTEGER
159 *> If RANGE='I', the index of the
160 *> largest eigenvalue to be returned.
161 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
162 *> Not referenced if RANGE = 'A' or 'V'.
163 *> \endverbatim
164 *>
165 *> \param[in] ABSTOL
166 *> \verbatim
167 *> ABSTOL is DOUBLE PRECISION
168 *> The absolute error tolerance for the eigenvalues.
169 *> An approximate eigenvalue is accepted as converged
170 *> when it is determined to lie in an interval [a,b]
171 *> of width less than or equal to
172 *>
173 *> ABSTOL + EPS * max( |a|,|b| ) ,
174 *>
175 *> where EPS is the machine precision. If ABSTOL is less than
176 *> or equal to zero, then EPS*|T| will be used in its place,
177 *> where |T| is the 1-norm of the tridiagonal matrix obtained
178 *> by reducing C to tridiagonal form, where C is the symmetric
179 *> matrix of the standard symmetric problem to which the
180 *> generalized problem is transformed.
181 *>
182 *> Eigenvalues will be computed most accurately when ABSTOL is
183 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
184 *> If this routine returns with INFO>0, indicating that some
185 *> eigenvectors did not converge, try setting ABSTOL to
186 *> 2*DLAMCH('S').
187 *> \endverbatim
188 *>
189 *> \param[out] M
190 *> \verbatim
191 *> M is INTEGER
192 *> The total number of eigenvalues found. 0 <= M <= N.
193 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
194 *> \endverbatim
195 *>
196 *> \param[out] W
197 *> \verbatim
198 *> W is DOUBLE PRECISION array, dimension (N)
199 *> On normal exit, the first M elements contain the selected
200 *> eigenvalues in ascending order.
201 *> \endverbatim
202 *>
203 *> \param[out] Z
204 *> \verbatim
205 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
206 *> If JOBZ = 'N', then Z is not referenced.
207 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
208 *> contain the orthonormal eigenvectors of the matrix A
209 *> corresponding to the selected eigenvalues, with the i-th
210 *> column of Z holding the eigenvector associated with W(i).
211 *> The eigenvectors are normalized as follows:
212 *> if ITYPE = 1 or 2, Z**T*B*Z = I;
213 *> if ITYPE = 3, Z**T*inv(B)*Z = I.
214 *>
215 *> If an eigenvector fails to converge, then that column of Z
216 *> contains the latest approximation to the eigenvector, and the
217 *> index of the eigenvector is returned in IFAIL.
218 *> Note: the user must ensure that at least max(1,M) columns are
219 *> supplied in the array Z; if RANGE = 'V', the exact value of M
220 *> is not known in advance and an upper bound must be used.
221 *> \endverbatim
222 *>
223 *> \param[in] LDZ
224 *> \verbatim
225 *> LDZ is INTEGER
226 *> The leading dimension of the array Z. LDZ >= 1, and if
227 *> JOBZ = 'V', LDZ >= max(1,N).
228 *> \endverbatim
229 *>
230 *> \param[out] WORK
231 *> \verbatim
232 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
233 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
234 *> \endverbatim
235 *>
236 *> \param[in] LWORK
237 *> \verbatim
238 *> LWORK is INTEGER
239 *> The length of the array WORK. LWORK >= max(1,8*N).
240 *> For optimal efficiency, LWORK >= (NB+3)*N,
241 *> where NB is the blocksize for DSYTRD returned by ILAENV.
242 *>
243 *> If LWORK = -1, then a workspace query is assumed; the routine
244 *> only calculates the optimal size of the WORK array, returns
245 *> this value as the first entry of the WORK array, and no error
246 *> message related to LWORK is issued by XERBLA.
247 *> \endverbatim
248 *>
249 *> \param[out] IWORK
250 *> \verbatim
251 *> IWORK is INTEGER array, dimension (5*N)
252 *> \endverbatim
253 *>
254 *> \param[out] IFAIL
255 *> \verbatim
256 *> IFAIL is INTEGER array, dimension (N)
257 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
258 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
259 *> indices of the eigenvectors that failed to converge.
260 *> If JOBZ = 'N', then IFAIL is not referenced.
261 *> \endverbatim
262 *>
263 *> \param[out] INFO
264 *> \verbatim
265 *> INFO is INTEGER
266 *> = 0: successful exit
267 *> < 0: if INFO = -i, the i-th argument had an illegal value
268 *> > 0: DPOTRF or DSYEVX returned an error code:
269 *> <= N: if INFO = i, DSYEVX failed to converge;
270 *> i eigenvectors failed to converge. Their indices
271 *> are stored in array IFAIL.
272 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
273 *> minor of order i of B is not positive definite.
274 *> The factorization of B could not be completed and
275 *> no eigenvalues or eigenvectors were computed.
276 *> \endverbatim
277 *
278 * Authors:
279 * ========
280 *
281 *> \author Univ. of Tennessee
282 *> \author Univ. of California Berkeley
283 *> \author Univ. of Colorado Denver
284 *> \author NAG Ltd.
285 *
286 *> \ingroup doubleSYeigen
287 *
288 *> \par Contributors:
289 * ==================
290 *>
291 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
292 *
293 * =====================================================================
294  SUBROUTINE dsygvx( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
295  $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
296  $ LWORK, IWORK, IFAIL, INFO )
297 *
298 * -- LAPACK driver routine --
299 * -- LAPACK is a software package provided by Univ. of Tennessee, --
300 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
301 *
302 * .. Scalar Arguments ..
303  CHARACTER JOBZ, RANGE, UPLO
304  INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
305  DOUBLE PRECISION ABSTOL, VL, VU
306 * ..
307 * .. Array Arguments ..
308  INTEGER IFAIL( * ), IWORK( * )
309  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
310  $ z( ldz, * )
311 * ..
312 *
313 * =====================================================================
314 *
315 * .. Parameters ..
316  DOUBLE PRECISION ONE
317  PARAMETER ( ONE = 1.0d+0 )
318 * ..
319 * .. Local Scalars ..
320  LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
321  CHARACTER TRANS
322  INTEGER LWKMIN, LWKOPT, NB
323 * ..
324 * .. External Functions ..
325  LOGICAL LSAME
326  INTEGER ILAENV
327  EXTERNAL lsame, ilaenv
328 * ..
329 * .. External Subroutines ..
330  EXTERNAL dpotrf, dsyevx, dsygst, dtrmm, dtrsm, xerbla
331 * ..
332 * .. Intrinsic Functions ..
333  INTRINSIC max, min
334 * ..
335 * .. Executable Statements ..
336 *
337 * Test the input parameters.
338 *
339  upper = lsame( uplo, 'U' )
340  wantz = lsame( jobz, 'V' )
341  alleig = lsame( range, 'A' )
342  valeig = lsame( range, 'V' )
343  indeig = lsame( range, 'I' )
344  lquery = ( lwork.EQ.-1 )
345 *
346  info = 0
347  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
348  info = -1
349  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
350  info = -2
351  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
352  info = -3
353  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
354  info = -4
355  ELSE IF( n.LT.0 ) THEN
356  info = -5
357  ELSE IF( lda.LT.max( 1, n ) ) THEN
358  info = -7
359  ELSE IF( ldb.LT.max( 1, n ) ) THEN
360  info = -9
361  ELSE
362  IF( valeig ) THEN
363  IF( n.GT.0 .AND. vu.LE.vl )
364  $ info = -11
365  ELSE IF( indeig ) THEN
366  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
367  info = -12
368  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
369  info = -13
370  END IF
371  END IF
372  END IF
373  IF (info.EQ.0) THEN
374  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
375  info = -18
376  END IF
377  END IF
378 *
379  IF( info.EQ.0 ) THEN
380  lwkmin = max( 1, 8*n )
381  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
382  lwkopt = max( lwkmin, ( nb + 3 )*n )
383  work( 1 ) = lwkopt
384 *
385  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
386  info = -20
387  END IF
388  END IF
389 *
390  IF( info.NE.0 ) THEN
391  CALL xerbla( 'DSYGVX', -info )
392  RETURN
393  ELSE IF( lquery ) THEN
394  RETURN
395  END IF
396 *
397 * Quick return if possible
398 *
399  m = 0
400  IF( n.EQ.0 ) THEN
401  RETURN
402  END IF
403 *
404 * Form a Cholesky factorization of B.
405 *
406  CALL dpotrf( uplo, n, b, ldb, info )
407  IF( info.NE.0 ) THEN
408  info = n + info
409  RETURN
410  END IF
411 *
412 * Transform problem to standard eigenvalue problem and solve.
413 *
414  CALL dsygst( itype, uplo, n, a, lda, b, ldb, info )
415  CALL dsyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
416  $ m, w, z, ldz, work, lwork, iwork, ifail, info )
417 *
418  IF( wantz ) THEN
419 *
420 * Backtransform eigenvectors to the original problem.
421 *
422  IF( info.GT.0 )
423  $ m = info - 1
424  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
425 *
426 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
427 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
428 *
429  IF( upper ) THEN
430  trans = 'N'
431  ELSE
432  trans = 'T'
433  END IF
434 *
435  CALL dtrsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
436  $ ldb, z, ldz )
437 *
438  ELSE IF( itype.EQ.3 ) THEN
439 *
440 * For B*A*x=(lambda)*x;
441 * backtransform eigenvectors: x = L*y or U**T*y
442 *
443  IF( upper ) THEN
444  trans = 'T'
445  ELSE
446  trans = 'N'
447  END IF
448 *
449  CALL dtrmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
450  $ ldb, z, ldz )
451  END IF
452  END IF
453 *
454 * Set WORK(1) to optimal workspace size.
455 *
456  work( 1 ) = lwkopt
457 *
458  RETURN
459 *
460 * End of DSYGVX
461 *
462  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dtrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRSM
Definition: dtrsm.f:181
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:177
subroutine dpotrf(UPLO, N, A, LDA, INFO)
DPOTRF
Definition: dpotrf.f:107
subroutine dsygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
DSYGST
Definition: dsygst.f:127
subroutine dsygvx(ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYGVX
Definition: dsygvx.f:297
subroutine dsyevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
Definition: dsyevx.f:253