LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ssbgvx.f
Go to the documentation of this file.
1 *> \brief \b SSBGVX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SSBGVX + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssbgvx.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssbgvx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssbgvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
22 * LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
23 * LDZ, WORK, IWORK, IFAIL, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
28 * $ N
29 * REAL ABSTOL, VL, VU
30 * ..
31 * .. Array Arguments ..
32 * INTEGER IFAIL( * ), IWORK( * )
33 * REAL AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
34 * $ W( * ), WORK( * ), Z( LDZ, * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> SSBGVX computes selected eigenvalues, and optionally, eigenvectors
44 *> of a real generalized symmetric-definite banded eigenproblem, of
45 *> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
46 *> and banded, and B is also positive definite. Eigenvalues and
47 *> eigenvectors can be selected by specifying either all eigenvalues,
48 *> a range of values or a range of indices for the desired eigenvalues.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[in] JOBZ
55 *> \verbatim
56 *> JOBZ is CHARACTER*1
57 *> = 'N': Compute eigenvalues only;
58 *> = 'V': Compute eigenvalues and eigenvectors.
59 *> \endverbatim
60 *>
61 *> \param[in] RANGE
62 *> \verbatim
63 *> RANGE is CHARACTER*1
64 *> = 'A': all eigenvalues will be found.
65 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
66 *> will be found.
67 *> = 'I': the IL-th through IU-th eigenvalues will be found.
68 *> \endverbatim
69 *>
70 *> \param[in] UPLO
71 *> \verbatim
72 *> UPLO is CHARACTER*1
73 *> = 'U': Upper triangles of A and B are stored;
74 *> = 'L': Lower triangles of A and B are stored.
75 *> \endverbatim
76 *>
77 *> \param[in] N
78 *> \verbatim
79 *> N is INTEGER
80 *> The order of the matrices A and B. N >= 0.
81 *> \endverbatim
82 *>
83 *> \param[in] KA
84 *> \verbatim
85 *> KA is INTEGER
86 *> The number of superdiagonals of the matrix A if UPLO = 'U',
87 *> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
88 *> \endverbatim
89 *>
90 *> \param[in] KB
91 *> \verbatim
92 *> KB is INTEGER
93 *> The number of superdiagonals of the matrix B if UPLO = 'U',
94 *> or the number of subdiagonals if UPLO = 'L'. KB >= 0.
95 *> \endverbatim
96 *>
97 *> \param[in,out] AB
98 *> \verbatim
99 *> AB is REAL array, dimension (LDAB, N)
100 *> On entry, the upper or lower triangle of the symmetric band
101 *> matrix A, stored in the first ka+1 rows of the array. The
102 *> j-th column of A is stored in the j-th column of the array AB
103 *> as follows:
104 *> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
105 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
106 *>
107 *> On exit, the contents of AB are destroyed.
108 *> \endverbatim
109 *>
110 *> \param[in] LDAB
111 *> \verbatim
112 *> LDAB is INTEGER
113 *> The leading dimension of the array AB. LDAB >= KA+1.
114 *> \endverbatim
115 *>
116 *> \param[in,out] BB
117 *> \verbatim
118 *> BB is REAL array, dimension (LDBB, N)
119 *> On entry, the upper or lower triangle of the symmetric band
120 *> matrix B, stored in the first kb+1 rows of the array. The
121 *> j-th column of B is stored in the j-th column of the array BB
122 *> as follows:
123 *> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
124 *> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
125 *>
126 *> On exit, the factor S from the split Cholesky factorization
127 *> B = S**T*S, as returned by SPBSTF.
128 *> \endverbatim
129 *>
130 *> \param[in] LDBB
131 *> \verbatim
132 *> LDBB is INTEGER
133 *> The leading dimension of the array BB. LDBB >= KB+1.
134 *> \endverbatim
135 *>
136 *> \param[out] Q
137 *> \verbatim
138 *> Q is REAL array, dimension (LDQ, N)
139 *> If JOBZ = 'V', the n-by-n matrix used in the reduction of
140 *> A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
141 *> and consequently C to tridiagonal form.
142 *> If JOBZ = 'N', the array Q is not referenced.
143 *> \endverbatim
144 *>
145 *> \param[in] LDQ
146 *> \verbatim
147 *> LDQ is INTEGER
148 *> The leading dimension of the array Q. If JOBZ = 'N',
149 *> LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
150 *> \endverbatim
151 *>
152 *> \param[in] VL
153 *> \verbatim
154 *> VL is REAL
155 *>
156 *> If RANGE='V', the lower bound of the interval to
157 *> be searched for eigenvalues. VL < VU.
158 *> Not referenced if RANGE = 'A' or 'I'.
159 *> \endverbatim
160 *>
161 *> \param[in] VU
162 *> \verbatim
163 *> VU is REAL
164 *>
165 *> If RANGE='V', the upper bound of the interval to
166 *> be searched for eigenvalues. VL < VU.
167 *> Not referenced if RANGE = 'A' or 'I'.
168 *> \endverbatim
169 *>
170 *> \param[in] IL
171 *> \verbatim
172 *> IL is INTEGER
173 *>
174 *> If RANGE='I', the index of the
175 *> smallest eigenvalue to be returned.
176 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
177 *> Not referenced if RANGE = 'A' or 'V'.
178 *> \endverbatim
179 *>
180 *> \param[in] IU
181 *> \verbatim
182 *> IU is INTEGER
183 *>
184 *> If RANGE='I', the index of the
185 *> largest eigenvalue to be returned.
186 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
187 *> Not referenced if RANGE = 'A' or 'V'.
188 *> \endverbatim
189 *>
190 *> \param[in] ABSTOL
191 *> \verbatim
192 *> ABSTOL is REAL
193 *> The absolute error tolerance for the eigenvalues.
194 *> An approximate eigenvalue is accepted as converged
195 *> when it is determined to lie in an interval [a,b]
196 *> of width less than or equal to
197 *>
198 *> ABSTOL + EPS * max( |a|,|b| ) ,
199 *>
200 *> where EPS is the machine precision. If ABSTOL is less than
201 *> or equal to zero, then EPS*|T| will be used in its place,
202 *> where |T| is the 1-norm of the tridiagonal matrix obtained
203 *> by reducing A to tridiagonal form.
204 *>
205 *> Eigenvalues will be computed most accurately when ABSTOL is
206 *> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
207 *> If this routine returns with INFO>0, indicating that some
208 *> eigenvectors did not converge, try setting ABSTOL to
209 *> 2*SLAMCH('S').
210 *> \endverbatim
211 *>
212 *> \param[out] M
213 *> \verbatim
214 *> M is INTEGER
215 *> The total number of eigenvalues found. 0 <= M <= N.
216 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
217 *> \endverbatim
218 *>
219 *> \param[out] W
220 *> \verbatim
221 *> W is REAL array, dimension (N)
222 *> If INFO = 0, the eigenvalues in ascending order.
223 *> \endverbatim
224 *>
225 *> \param[out] Z
226 *> \verbatim
227 *> Z is REAL array, dimension (LDZ, N)
228 *> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
229 *> eigenvectors, with the i-th column of Z holding the
230 *> eigenvector associated with W(i). The eigenvectors are
231 *> normalized so Z**T*B*Z = I.
232 *> If JOBZ = 'N', then Z is not referenced.
233 *> \endverbatim
234 *>
235 *> \param[in] LDZ
236 *> \verbatim
237 *> LDZ is INTEGER
238 *> The leading dimension of the array Z. LDZ >= 1, and if
239 *> JOBZ = 'V', LDZ >= max(1,N).
240 *> \endverbatim
241 *>
242 *> \param[out] WORK
243 *> \verbatim
244 *> WORK is REAL array, dimension (7*N)
245 *> \endverbatim
246 *>
247 *> \param[out] IWORK
248 *> \verbatim
249 *> IWORK is INTEGER array, dimension (5*N)
250 *> \endverbatim
251 *>
252 *> \param[out] IFAIL
253 *> \verbatim
254 *> IFAIL is INTEGER array, dimension (M)
255 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
256 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
257 *> indices of the eigenvalues that failed to converge.
258 *> If JOBZ = 'N', then IFAIL is not referenced.
259 *> \endverbatim
260 *>
261 *> \param[out] INFO
262 *> \verbatim
263 *> INFO is INTEGER
264 *> = 0: successful exit
265 *> < 0: if INFO = -i, the i-th argument had an illegal value
266 *> <= N: if INFO = i, then i eigenvectors failed to converge.
267 *> Their indices are stored in IFAIL.
268 *> > N: SPBSTF returned an error code; i.e.,
269 *> if INFO = N + i, for 1 <= i <= N, then the leading
270 *> minor of order i of B is not positive definite.
271 *> The factorization of B could not be completed and
272 *> no eigenvalues or eigenvectors were computed.
273 *> \endverbatim
274 *
275 * Authors:
276 * ========
277 *
278 *> \author Univ. of Tennessee
279 *> \author Univ. of California Berkeley
280 *> \author Univ. of Colorado Denver
281 *> \author NAG Ltd.
282 *
283 *> \ingroup realOTHEReigen
284 *
285 *> \par Contributors:
286 * ==================
287 *>
288 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
289 *
290 * =====================================================================
291  SUBROUTINE ssbgvx( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
292  $ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
293  $ LDZ, WORK, IWORK, IFAIL, INFO )
294 *
295 * -- LAPACK driver routine --
296 * -- LAPACK is a software package provided by Univ. of Tennessee, --
297 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
298 *
299 * .. Scalar Arguments ..
300  CHARACTER JOBZ, RANGE, UPLO
301  INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
302  $ n
303  REAL ABSTOL, VL, VU
304 * ..
305 * .. Array Arguments ..
306  INTEGER IFAIL( * ), IWORK( * )
307  REAL AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
308  $ w( * ), work( * ), z( ldz, * )
309 * ..
310 *
311 * =====================================================================
312 *
313 * .. Parameters ..
314  REAL ZERO, ONE
315  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
316 * ..
317 * .. Local Scalars ..
318  LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
319  CHARACTER ORDER, VECT
320  INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
321  $ indiwo, indwrk, itmp1, j, jj, nsplit
322  REAL TMP1
323 * ..
324 * .. External Functions ..
325  LOGICAL LSAME
326  EXTERNAL LSAME
327 * ..
328 * .. External Subroutines ..
329  EXTERNAL scopy, sgemv, slacpy, spbstf, ssbgst, ssbtrd,
331 * ..
332 * .. Intrinsic Functions ..
333  INTRINSIC min
334 * ..
335 * .. Executable Statements ..
336 *
337 * Test the input parameters.
338 *
339  wantz = lsame( jobz, 'V' )
340  upper = lsame( uplo, 'U' )
341  alleig = lsame( range, 'A' )
342  valeig = lsame( range, 'V' )
343  indeig = lsame( range, 'I' )
344 *
345  info = 0
346  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
347  info = -1
348  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
349  info = -2
350  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
351  info = -3
352  ELSE IF( n.LT.0 ) THEN
353  info = -4
354  ELSE IF( ka.LT.0 ) THEN
355  info = -5
356  ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
357  info = -6
358  ELSE IF( ldab.LT.ka+1 ) THEN
359  info = -8
360  ELSE IF( ldbb.LT.kb+1 ) THEN
361  info = -10
362  ELSE IF( ldq.LT.1 .OR. ( wantz .AND. ldq.LT.n ) ) THEN
363  info = -12
364  ELSE
365  IF( valeig ) THEN
366  IF( n.GT.0 .AND. vu.LE.vl )
367  $ info = -14
368  ELSE IF( indeig ) THEN
369  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
370  info = -15
371  ELSE IF ( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
372  info = -16
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 = -21
379  END IF
380  END IF
381 *
382  IF( info.NE.0 ) THEN
383  CALL xerbla( 'SSBGVX', -info )
384  RETURN
385  END IF
386 *
387 * Quick return if possible
388 *
389  m = 0
390  IF( n.EQ.0 )
391  $ RETURN
392 *
393 * Form a split Cholesky factorization of B.
394 *
395  CALL spbstf( uplo, n, kb, bb, ldbb, info )
396  IF( info.NE.0 ) THEN
397  info = n + info
398  RETURN
399  END IF
400 *
401 * Transform problem to standard eigenvalue problem.
402 *
403  CALL ssbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq,
404  $ work, iinfo )
405 *
406 * Reduce symmetric band matrix to tridiagonal form.
407 *
408  indd = 1
409  inde = indd + n
410  indwrk = inde + n
411  IF( wantz ) THEN
412  vect = 'U'
413  ELSE
414  vect = 'N'
415  END IF
416  CALL ssbtrd( vect, uplo, n, ka, ab, ldab, work( indd ),
417  $ work( inde ), q, ldq, work( indwrk ), iinfo )
418 *
419 * If all eigenvalues are desired and ABSTOL is less than or equal
420 * to zero, then call SSTERF or SSTEQR. If this fails for some
421 * eigenvalue, then try SSTEBZ.
422 *
423  test = .false.
424  IF( indeig ) THEN
425  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
426  test = .true.
427  END IF
428  END IF
429  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
430  CALL scopy( n, work( indd ), 1, w, 1 )
431  indee = indwrk + 2*n
432  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
433  IF( .NOT.wantz ) THEN
434  CALL ssterf( n, w, work( indee ), info )
435  ELSE
436  CALL slacpy( 'A', n, n, q, ldq, z, ldz )
437  CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
438  $ work( indwrk ), info )
439  IF( info.EQ.0 ) THEN
440  DO 10 i = 1, n
441  ifail( i ) = 0
442  10 CONTINUE
443  END IF
444  END IF
445  IF( info.EQ.0 ) THEN
446  m = n
447  GO TO 30
448  END IF
449  info = 0
450  END IF
451 *
452 * Otherwise, call SSTEBZ and, if eigenvectors are desired,
453 * call SSTEIN.
454 *
455  IF( wantz ) THEN
456  order = 'B'
457  ELSE
458  order = 'E'
459  END IF
460  indibl = 1
461  indisp = indibl + n
462  indiwo = indisp + n
463  CALL sstebz( range, order, n, vl, vu, il, iu, abstol,
464  $ work( indd ), work( inde ), m, nsplit, w,
465  $ iwork( indibl ), iwork( indisp ), work( indwrk ),
466  $ iwork( indiwo ), info )
467 *
468  IF( wantz ) THEN
469  CALL sstein( n, work( indd ), work( inde ), m, w,
470  $ iwork( indibl ), iwork( indisp ), z, ldz,
471  $ work( indwrk ), iwork( indiwo ), ifail, info )
472 *
473 * Apply transformation matrix used in reduction to tridiagonal
474 * form to eigenvectors returned by SSTEIN.
475 *
476  DO 20 j = 1, m
477  CALL scopy( n, z( 1, j ), 1, work( 1 ), 1 )
478  CALL sgemv( 'N', n, n, one, q, ldq, work, 1, zero,
479  $ z( 1, j ), 1 )
480  20 CONTINUE
481  END IF
482 *
483  30 CONTINUE
484 *
485 * If eigenvalues are not in order, then sort them, along with
486 * eigenvectors.
487 *
488  IF( wantz ) THEN
489  DO 50 j = 1, m - 1
490  i = 0
491  tmp1 = w( j )
492  DO 40 jj = j + 1, m
493  IF( w( jj ).LT.tmp1 ) THEN
494  i = jj
495  tmp1 = w( jj )
496  END IF
497  40 CONTINUE
498 *
499  IF( i.NE.0 ) THEN
500  itmp1 = iwork( indibl+i-1 )
501  w( i ) = w( j )
502  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
503  w( j ) = tmp1
504  iwork( indibl+j-1 ) = itmp1
505  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
506  IF( info.NE.0 ) THEN
507  itmp1 = ifail( i )
508  ifail( i ) = ifail( j )
509  ifail( j ) = itmp1
510  END IF
511  END IF
512  50 CONTINUE
513  END IF
514 *
515  RETURN
516 *
517 * End of SSBGVX
518 *
519  END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:131
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:273
subroutine spbstf(UPLO, N, KD, AB, LDAB, INFO)
SPBSTF
Definition: spbstf.f:152
subroutine ssbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
SSBTRD
Definition: ssbtrd.f:163
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:174
subroutine ssbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
SSBGST
Definition: ssbgst.f:159
subroutine ssbgvx(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSBGVX
Definition: ssbgvx.f:294
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156