LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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
9*> Download SSYGVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssygvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssygvx.f">
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*> \ingroup realSYeigen
287*
288*> \par Contributors:
289* ==================
290*>
291*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
292*
293* =====================================================================
294 SUBROUTINE ssygvx( 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 REAL ABSTOL, VL, VU
306* ..
307* .. Array Arguments ..
308 INTEGER IFAIL( * ), IWORK( * )
309 REAL A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
310 $ z( ldz, * )
311* ..
312*
313* =====================================================================
314*
315* .. Parameters ..
316 REAL ONE
317 PARAMETER ( ONE = 1.0e+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 ilaenv, lsame
328* ..
329* .. External Subroutines ..
330 EXTERNAL spotrf, ssyevx, ssygst, strmm, strsm, 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, 'SSYTRD', 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( 'SSYGVX', -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 spotrf( 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 ssygst( itype, uplo, n, a, lda, b, ldb, info )
415 CALL ssyevx( 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 strsm( '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 strmm( '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 SSYGVX
461*
462 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:107
subroutine ssygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
SSYGST
Definition: ssygst.f:127
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:297
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:253
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:177
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181