LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
dsygvd.f
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1 *> \brief \b DSYGVD
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
22 * LWORK, IWORK, LIWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER JOBZ, UPLO
26 * INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IWORK( * )
30 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
40 *> of a real generalized symmetric-definite eigenproblem, of the form
41 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
42 *> B are assumed to be symmetric and B is also positive definite.
43 *> If eigenvectors are desired, it uses a divide and conquer algorithm.
44 *>
45 *> The divide and conquer algorithm makes very mild assumptions about
46 *> floating point arithmetic. It will work on machines with a guard
47 *> digit in add/subtract, or on those binary machines without guard
48 *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
49 *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
50 *> without guard digits, but we know of none.
51 *> \endverbatim
52 *
53 * Arguments:
54 * ==========
55 *
56 *> \param[in] ITYPE
57 *> \verbatim
58 *> ITYPE is INTEGER
59 *> Specifies the problem type to be solved:
60 *> = 1: A*x = (lambda)*B*x
61 *> = 2: A*B*x = (lambda)*x
62 *> = 3: B*A*x = (lambda)*x
63 *> \endverbatim
64 *>
65 *> \param[in] JOBZ
66 *> \verbatim
67 *> JOBZ is CHARACTER*1
68 *> = 'N': Compute eigenvalues only;
69 *> = 'V': Compute eigenvalues and eigenvectors.
70 *> \endverbatim
71 *>
72 *> \param[in] UPLO
73 *> \verbatim
74 *> UPLO is CHARACTER*1
75 *> = 'U': Upper triangles of A and B are stored;
76 *> = 'L': Lower triangles of A and B are stored.
77 *> \endverbatim
78 *>
79 *> \param[in] N
80 *> \verbatim
81 *> N is INTEGER
82 *> The order of the matrices A and B. N >= 0.
83 *> \endverbatim
84 *>
85 *> \param[in,out] A
86 *> \verbatim
87 *> A is DOUBLE PRECISION array, dimension (LDA, N)
88 *> On entry, the symmetric matrix A. If UPLO = 'U', the
89 *> leading N-by-N upper triangular part of A contains the
90 *> upper triangular part of the matrix A. If UPLO = 'L',
91 *> the leading N-by-N lower triangular part of A contains
92 *> the lower triangular part of the matrix A.
93 *>
94 *> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
95 *> matrix Z of eigenvectors. The eigenvectors are normalized
96 *> as follows:
97 *> if ITYPE = 1 or 2, Z**T*B*Z = I;
98 *> if ITYPE = 3, Z**T*inv(B)*Z = I.
99 *> If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
100 *> or the lower triangle (if UPLO='L') of A, including the
101 *> diagonal, is destroyed.
102 *> \endverbatim
103 *>
104 *> \param[in] LDA
105 *> \verbatim
106 *> LDA is INTEGER
107 *> The leading dimension of the array A. LDA >= max(1,N).
108 *> \endverbatim
109 *>
110 *> \param[in,out] B
111 *> \verbatim
112 *> B is DOUBLE PRECISION array, dimension (LDB, N)
113 *> On entry, the symmetric matrix B. If UPLO = 'U', the
114 *> leading N-by-N upper triangular part of B contains the
115 *> upper triangular part of the matrix B. If UPLO = 'L',
116 *> the leading N-by-N lower triangular part of B contains
117 *> the lower triangular part of the matrix B.
118 *>
119 *> On exit, if INFO <= N, the part of B containing the matrix is
120 *> overwritten by the triangular factor U or L from the Cholesky
121 *> factorization B = U**T*U or B = L*L**T.
122 *> \endverbatim
123 *>
124 *> \param[in] LDB
125 *> \verbatim
126 *> LDB is INTEGER
127 *> The leading dimension of the array B. LDB >= max(1,N).
128 *> \endverbatim
129 *>
130 *> \param[out] W
131 *> \verbatim
132 *> W is DOUBLE PRECISION array, dimension (N)
133 *> If INFO = 0, the eigenvalues in ascending order.
134 *> \endverbatim
135 *>
136 *> \param[out] WORK
137 *> \verbatim
138 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
139 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
140 *> \endverbatim
141 *>
142 *> \param[in] LWORK
143 *> \verbatim
144 *> LWORK is INTEGER
145 *> The dimension of the array WORK.
146 *> If N <= 1, LWORK >= 1.
147 *> If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
148 *> If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
149 *>
150 *> If LWORK = -1, then a workspace query is assumed; the routine
151 *> only calculates the optimal sizes of the WORK and IWORK
152 *> arrays, returns these values as the first entries of the WORK
153 *> and IWORK arrays, and no error message related to LWORK or
154 *> LIWORK is issued by XERBLA.
155 *> \endverbatim
156 *>
157 *> \param[out] IWORK
158 *> \verbatim
159 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
160 *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
161 *> \endverbatim
162 *>
163 *> \param[in] LIWORK
164 *> \verbatim
165 *> LIWORK is INTEGER
166 *> The dimension of the array IWORK.
167 *> If N <= 1, LIWORK >= 1.
168 *> If JOBZ = 'N' and N > 1, LIWORK >= 1.
169 *> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
170 *>
171 *> If LIWORK = -1, then a workspace query is assumed; the
172 *> routine only calculates the optimal sizes of the WORK and
173 *> IWORK arrays, returns these values as the first entries of
174 *> the WORK and IWORK arrays, and no error message related to
175 *> LWORK or LIWORK is issued by XERBLA.
176 *> \endverbatim
177 *>
178 *> \param[out] INFO
179 *> \verbatim
180 *> INFO is INTEGER
181 *> = 0: successful exit
182 *> < 0: if INFO = -i, the i-th argument had an illegal value
183 *> > 0: DPOTRF or DSYEVD returned an error code:
184 *> <= N: if INFO = i and JOBZ = 'N', then the algorithm
185 *> failed to converge; i off-diagonal elements of an
186 *> intermediate tridiagonal form did not converge to
187 *> zero;
188 *> if INFO = i and JOBZ = 'V', then the algorithm
189 *> failed to compute an eigenvalue while working on
190 *> the submatrix lying in rows and columns INFO/(N+1)
191 *> through mod(INFO,N+1);
192 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
193 *> minor of order i of B is not positive definite.
194 *> The factorization of B could not be completed and
195 *> no eigenvalues or eigenvectors were computed.
196 *> \endverbatim
197 *
198 * Authors:
199 * ========
200 *
201 *> \author Univ. of Tennessee
202 *> \author Univ. of California Berkeley
203 *> \author Univ. of Colorado Denver
204 *> \author NAG Ltd.
205 *
206 *> \date November 2015
207 *
208 *> \ingroup doubleSYeigen
209 *
210 *> \par Further Details:
211 * =====================
212 *>
213 *> \verbatim
214 *>
215 *> Modified so that no backsubstitution is performed if DSYEVD fails to
216 *> converge (NEIG in old code could be greater than N causing out of
217 *> bounds reference to A - reported by Ralf Meyer). Also corrected the
218 *> description of INFO and the test on ITYPE. Sven, 16 Feb 05.
219 *> \endverbatim
220 *
221 *> \par Contributors:
222 * ==================
223 *>
224 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
225 *>
226 * =====================================================================
227  SUBROUTINE dsygvd( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
228  \$ lwork, iwork, liwork, info )
229 *
230 * -- LAPACK driver routine (version 3.6.0) --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 * November 2015
234 *
235 * .. Scalar Arguments ..
236  CHARACTER JOBZ, UPLO
237  INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
238 * ..
239 * .. Array Arguments ..
240  INTEGER IWORK( * )
241  DOUBLE PRECISION A( lda, * ), B( ldb, * ), W( * ), WORK( * )
242 * ..
243 *
244 * =====================================================================
245 *
246 * .. Parameters ..
247  DOUBLE PRECISION ONE
248  parameter ( one = 1.0d+0 )
249 * ..
250 * .. Local Scalars ..
251  LOGICAL LQUERY, UPPER, WANTZ
252  CHARACTER TRANS
253  INTEGER LIOPT, LIWMIN, LOPT, LWMIN
254 * ..
255 * .. External Functions ..
256  LOGICAL LSAME
257  EXTERNAL lsame
258 * ..
259 * .. External Subroutines ..
260  EXTERNAL dpotrf, dsyevd, dsygst, dtrmm, dtrsm, xerbla
261 * ..
262 * .. Intrinsic Functions ..
263  INTRINSIC dble, max
264 * ..
265 * .. Executable Statements ..
266 *
267 * Test the input parameters.
268 *
269  wantz = lsame( jobz, 'V' )
270  upper = lsame( uplo, 'U' )
271  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
272 *
273  info = 0
274  IF( n.LE.1 ) THEN
275  liwmin = 1
276  lwmin = 1
277  ELSE IF( wantz ) THEN
278  liwmin = 3 + 5*n
279  lwmin = 1 + 6*n + 2*n**2
280  ELSE
281  liwmin = 1
282  lwmin = 2*n + 1
283  END IF
284  lopt = lwmin
285  liopt = liwmin
286  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
287  info = -1
288  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
289  info = -2
290  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
291  info = -3
292  ELSE IF( n.LT.0 ) THEN
293  info = -4
294  ELSE IF( lda.LT.max( 1, n ) ) THEN
295  info = -6
296  ELSE IF( ldb.LT.max( 1, n ) ) THEN
297  info = -8
298  END IF
299 *
300  IF( info.EQ.0 ) THEN
301  work( 1 ) = lopt
302  iwork( 1 ) = liopt
303 *
304  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
305  info = -11
306  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
307  info = -13
308  END IF
309  END IF
310 *
311  IF( info.NE.0 ) THEN
312  CALL xerbla( 'DSYGVD', -info )
313  RETURN
314  ELSE IF( lquery ) THEN
315  RETURN
316  END IF
317 *
318 * Quick return if possible
319 *
320  IF( n.EQ.0 )
321  \$ RETURN
322 *
323 * Form a Cholesky factorization of B.
324 *
325  CALL dpotrf( uplo, n, b, ldb, info )
326  IF( info.NE.0 ) THEN
327  info = n + info
328  RETURN
329  END IF
330 *
331 * Transform problem to standard eigenvalue problem and solve.
332 *
333  CALL dsygst( itype, uplo, n, a, lda, b, ldb, info )
334  CALL dsyevd( jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork,
335  \$ info )
336  lopt = max( dble( lopt ), dble( work( 1 ) ) )
337  liopt = max( dble( liopt ), dble( iwork( 1 ) ) )
338 *
339  IF( wantz .AND. info.EQ.0 ) THEN
340 *
341 * Backtransform eigenvectors to the original problem.
342 *
343  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
344 *
345 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
346 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
347 *
348  IF( upper ) THEN
349  trans = 'N'
350  ELSE
351  trans = 'T'
352  END IF
353 *
354  CALL dtrsm( 'Left', uplo, trans, 'Non-unit', n, n, one,
355  \$ b, ldb, a, lda )
356 *
357  ELSE IF( itype.EQ.3 ) THEN
358 *
359 * For B*A*x=(lambda)*x;
360 * backtransform eigenvectors: x = L*y or U**T*y
361 *
362  IF( upper ) THEN
363  trans = 'T'
364  ELSE
365  trans = 'N'
366  END IF
367 *
368  CALL dtrmm( 'Left', uplo, trans, 'Non-unit', n, n, one,
369  \$ b, ldb, a, lda )
370  END IF
371  END IF
372 *
373  work( 1 ) = lopt
374  iwork( 1 ) = liopt
375 *
376  RETURN
377 *
378 * End of DSYGVD
379 *
380  END
subroutine dtrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRSM
Definition: dtrsm.f:183
subroutine dpotrf(UPLO, N, A, LDA, INFO)
DPOTRF
Definition: dpotrf.f:109
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:179
subroutine dsygvd(ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, IWORK, LIWORK, INFO)
DSYGVD
Definition: dsygvd.f:229
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dsyevd(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ...
Definition: dsyevd.f:187
subroutine dsygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
DSYGST
Definition: dsygst.f:129