LAPACK 3.12.0
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*> principal minor of order i of B is not positive.
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 hegvx
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 REAL SROUNDUP_LWORK
328 EXTERNAL ilaenv, lsame, sroundup_lwork
329* ..
330* .. External Subroutines ..
331 EXTERNAL spotrf, ssyevx, ssygst, strmm, strsm, xerbla
332* ..
333* .. Intrinsic Functions ..
334 INTRINSIC max, min
335* ..
336* .. Executable Statements ..
337*
338* Test the input parameters.
339*
340 upper = lsame( uplo, 'U' )
341 wantz = lsame( jobz, 'V' )
342 alleig = lsame( range, 'A' )
343 valeig = lsame( range, 'V' )
344 indeig = lsame( range, 'I' )
345 lquery = ( lwork.EQ.-1 )
346*
347 info = 0
348 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
349 info = -1
350 ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
351 info = -2
352 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
353 info = -3
354 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
355 info = -4
356 ELSE IF( n.LT.0 ) THEN
357 info = -5
358 ELSE IF( lda.LT.max( 1, n ) ) THEN
359 info = -7
360 ELSE IF( ldb.LT.max( 1, n ) ) THEN
361 info = -9
362 ELSE
363 IF( valeig ) THEN
364 IF( n.GT.0 .AND. vu.LE.vl )
365 $ info = -11
366 ELSE IF( indeig ) THEN
367 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
368 info = -12
369 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
370 info = -13
371 END IF
372 END IF
373 END IF
374 IF (info.EQ.0) THEN
375 IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
376 info = -18
377 END IF
378 END IF
379*
380 IF( info.EQ.0 ) THEN
381 lwkmin = max( 1, 8*n )
382 nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
383 lwkopt = max( lwkmin, ( nb + 3 )*n )
384 work( 1 ) = sroundup_lwork(lwkopt)
385*
386 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
387 info = -20
388 END IF
389 END IF
390*
391 IF( info.NE.0 ) THEN
392 CALL xerbla( 'SSYGVX', -info )
393 RETURN
394 ELSE IF( lquery ) THEN
395 RETURN
396 END IF
397*
398* Quick return if possible
399*
400 m = 0
401 IF( n.EQ.0 ) THEN
402 RETURN
403 END IF
404*
405* Form a Cholesky factorization of B.
406*
407 CALL spotrf( uplo, n, b, ldb, info )
408 IF( info.NE.0 ) THEN
409 info = n + info
410 RETURN
411 END IF
412*
413* Transform problem to standard eigenvalue problem and solve.
414*
415 CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
416 CALL ssyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
417 $ m, w, z, ldz, work, lwork, iwork, ifail, info )
418*
419 IF( wantz ) THEN
420*
421* Backtransform eigenvectors to the original problem.
422*
423 IF( info.GT.0 )
424 $ m = info - 1
425 IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
426*
427* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
428* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
429*
430 IF( upper ) THEN
431 trans = 'N'
432 ELSE
433 trans = 'T'
434 END IF
435*
436 CALL strsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
437 $ ldb, z, ldz )
438*
439 ELSE IF( itype.EQ.3 ) THEN
440*
441* For B*A*x=(lambda)*x;
442* backtransform eigenvectors: x = L*y or U**T*y
443*
444 IF( upper ) THEN
445 trans = 'T'
446 ELSE
447 trans = 'N'
448 END IF
449*
450 CALL strmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
451 $ ldb, z, ldz )
452 END IF
453 END IF
454*
455* Set WORK(1) to optimal workspace size.
456*
457 work( 1 ) = sroundup_lwork(lwkopt)
458*
459 RETURN
460*
461* End of SSYGVX
462*
463 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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 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 spotrf(uplo, n, a, lda, info)
SPOTRF
Definition spotrf.f:107
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