LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
ssygvx.f
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1 *> \brief \b SSYGVX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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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/ssygvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYGVX( 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 * REAL ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IFAIL( * ), IWORK( * )
32 * REAL A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
33 * $ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> SSYGVX 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 REAL 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 REAL 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 REAL
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 REAL
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 REAL
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*SLAMCH('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 REAL 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 REAL 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 REAL 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 SSYTRD 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: SPOTRF or SSYEVX returned an error code:
269 *> <= N: if INFO = i, SSYEVX 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 *> \date June 2016
287 *
288 *> \ingroup realSYeigen
289 *
290 *> \par Contributors:
291 * ==================
292 *>
293 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
294 *
295 * =====================================================================
296  SUBROUTINE ssygvx( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
297  $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
298  $ LWORK, IWORK, IFAIL, INFO )
299 *
300 * -- LAPACK driver routine (version 3.7.1) --
301 * -- LAPACK is a software package provided by Univ. of Tennessee, --
302 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
303 * June 2016
304 *
305 * .. Scalar Arguments ..
306  CHARACTER JOBZ, RANGE, UPLO
307  INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
308  REAL ABSTOL, VL, VU
309 * ..
310 * .. Array Arguments ..
311  INTEGER IFAIL( * ), IWORK( * )
312  REAL A( lda, * ), B( ldb, * ), W( * ), WORK( * ),
313  $ z( ldz, * )
314 * ..
315 *
316 * =====================================================================
317 *
318 * .. Parameters ..
319  REAL ONE
320  parameter( one = 1.0e+0 )
321 * ..
322 * .. Local Scalars ..
323  LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
324  CHARACTER TRANS
325  INTEGER LWKMIN, LWKOPT, NB
326 * ..
327 * .. External Functions ..
328  LOGICAL LSAME
329  INTEGER ILAENV
330  EXTERNAL ilaenv, lsame
331 * ..
332 * .. External Subroutines ..
333  EXTERNAL spotrf, ssyevx, ssygst, strmm, strsm, xerbla
334 * ..
335 * .. Intrinsic Functions ..
336  INTRINSIC max, min
337 * ..
338 * .. Executable Statements ..
339 *
340 * Test the input parameters.
341 *
342  upper = lsame( uplo, 'U' )
343  wantz = lsame( jobz, 'V' )
344  alleig = lsame( range, 'A' )
345  valeig = lsame( range, 'V' )
346  indeig = lsame( range, 'I' )
347  lquery = ( lwork.EQ.-1 )
348 *
349  info = 0
350  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
351  info = -1
352  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
353  info = -2
354  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
355  info = -3
356  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
357  info = -4
358  ELSE IF( n.LT.0 ) THEN
359  info = -5
360  ELSE IF( lda.LT.max( 1, n ) ) THEN
361  info = -7
362  ELSE IF( ldb.LT.max( 1, n ) ) THEN
363  info = -9
364  ELSE
365  IF( valeig ) THEN
366  IF( n.GT.0 .AND. vu.LE.vl )
367  $ info = -11
368  ELSE IF( indeig ) THEN
369  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
370  info = -12
371  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
372  info = -13
373  END IF
374  END IF
375  END IF
376  IF (info.EQ.0) THEN
377  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
378  info = -18
379  END IF
380  END IF
381 *
382  IF( info.EQ.0 ) THEN
383  lwkmin = max( 1, 8*n )
384  nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
385  lwkopt = max( lwkmin, ( nb + 3 )*n )
386  work( 1 ) = lwkopt
387 *
388  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
389  info = -20
390  END IF
391  END IF
392 *
393  IF( info.NE.0 ) THEN
394  CALL xerbla( 'SSYGVX', -info )
395  RETURN
396  ELSE IF( lquery ) THEN
397  RETURN
398  END IF
399 *
400 * Quick return if possible
401 *
402  m = 0
403  IF( n.EQ.0 ) THEN
404  RETURN
405  END IF
406 *
407 * Form a Cholesky factorization of B.
408 *
409  CALL spotrf( uplo, n, b, ldb, info )
410  IF( info.NE.0 ) THEN
411  info = n + info
412  RETURN
413  END IF
414 *
415 * Transform problem to standard eigenvalue problem and solve.
416 *
417  CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
418  CALL ssyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
419  $ m, w, z, ldz, work, lwork, iwork, ifail, info )
420 *
421  IF( wantz ) THEN
422 *
423 * Backtransform eigenvectors to the original problem.
424 *
425  IF( info.GT.0 )
426  $ m = info - 1
427  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
428 *
429 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
430 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
431 *
432  IF( upper ) THEN
433  trans = 'N'
434  ELSE
435  trans = 'T'
436  END IF
437 *
438  CALL strsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
439  $ ldb, z, ldz )
440 *
441  ELSE IF( itype.EQ.3 ) THEN
442 *
443 * For B*A*x=(lambda)*x;
444 * backtransform eigenvectors: x = L*y or U**T*y
445 *
446  IF( upper ) THEN
447  trans = 'T'
448  ELSE
449  trans = 'N'
450  END IF
451 *
452  CALL strmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
453  $ ldb, z, ldz )
454  END IF
455  END IF
456 *
457 * Set WORK(1) to optimal workspace size.
458 *
459  work( 1 ) = lwkopt
460 *
461  RETURN
462 *
463 * End of SSYGVX
464 *
465  END
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:183
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:109
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:179
subroutine ssygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
SSYGST
Definition: ssygst.f:129
subroutine ssygvx(ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
SSYGVX
Definition: ssygvx.f:299
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine ssyevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ...
Definition: ssyevx.f:255