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ssbgvx.f
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1 *> \brief \b SSBGST
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 *> \endverbatim
156 *>
157 *> \param[in] VU
158 *> \verbatim
159 *> VU is REAL
160 *>
161 *> If RANGE='V', the lower and upper bounds of the interval to
162 *> be searched for eigenvalues. VL < VU.
163 *> Not referenced if RANGE = 'A' or 'I'.
164 *> \endverbatim
165 *>
166 *> \param[in] IL
167 *> \verbatim
168 *> IL is INTEGER
169 *> \endverbatim
170 *>
171 *> \param[in] IU
172 *> \verbatim
173 *> IU is INTEGER
174 *>
175 *> If RANGE='I', the indices (in ascending order) of the
176 *> smallest and largest eigenvalues to be returned.
177 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
178 *> Not referenced if RANGE = 'A' or 'V'.
179 *> \endverbatim
180 *>
181 *> \param[in] ABSTOL
182 *> \verbatim
183 *> ABSTOL is REAL
184 *> The absolute error tolerance for the eigenvalues.
185 *> An approximate eigenvalue is accepted as converged
186 *> when it is determined to lie in an interval [a,b]
187 *> of width less than or equal to
188 *>
189 *> ABSTOL + EPS * max( |a|,|b| ) ,
190 *>
191 *> where EPS is the machine precision. If ABSTOL is less than
192 *> or equal to zero, then EPS*|T| will be used in its place,
193 *> where |T| is the 1-norm of the tridiagonal matrix obtained
194 *> by reducing A to tridiagonal form.
195 *>
196 *> Eigenvalues will be computed most accurately when ABSTOL is
197 *> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
198 *> If this routine returns with INFO>0, indicating that some
199 *> eigenvectors did not converge, try setting ABSTOL to
200 *> 2*SLAMCH('S').
201 *> \endverbatim
202 *>
203 *> \param[out] M
204 *> \verbatim
205 *> M is INTEGER
206 *> The total number of eigenvalues found. 0 <= M <= N.
207 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
208 *> \endverbatim
209 *>
210 *> \param[out] W
211 *> \verbatim
212 *> W is REAL array, dimension (N)
213 *> If INFO = 0, the eigenvalues in ascending order.
214 *> \endverbatim
215 *>
216 *> \param[out] Z
217 *> \verbatim
218 *> Z is REAL array, dimension (LDZ, N)
219 *> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
220 *> eigenvectors, with the i-th column of Z holding the
221 *> eigenvector associated with W(i). The eigenvectors are
222 *> normalized so Z**T*B*Z = I.
223 *> If JOBZ = 'N', then Z is not referenced.
224 *> \endverbatim
225 *>
226 *> \param[in] LDZ
227 *> \verbatim
228 *> LDZ is INTEGER
229 *> The leading dimension of the array Z. LDZ >= 1, and if
230 *> JOBZ = 'V', LDZ >= max(1,N).
231 *> \endverbatim
232 *>
233 *> \param[out] WORK
234 *> \verbatim
235 *> WORK is REAL array, dimension (7N)
236 *> \endverbatim
237 *>
238 *> \param[out] IWORK
239 *> \verbatim
240 *> IWORK is INTEGER array, dimension (5N)
241 *> \endverbatim
242 *>
243 *> \param[out] IFAIL
244 *> \verbatim
245 *> IFAIL is INTEGER array, dimension (M)
246 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
247 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
248 *> indices of the eigenvalues that failed to converge.
249 *> If JOBZ = 'N', then IFAIL is not referenced.
250 *> \endverbatim
251 *>
252 *> \param[out] INFO
253 *> \verbatim
254 *> INFO is INTEGER
255 *> = 0 : successful exit
256 *> < 0 : if INFO = -i, the i-th argument had an illegal value
257 *> <= N: if INFO = i, then i eigenvectors failed to converge.
258 *> Their indices are stored in IFAIL.
259 *> > N : SPBSTF returned an error code; i.e.,
260 *> if INFO = N + i, for 1 <= i <= N, then the leading
261 *> minor of order i of B is not positive definite.
262 *> The factorization of B could not be completed and
263 *> no eigenvalues or eigenvectors were computed.
264 *> \endverbatim
265 *
266 * Authors:
267 * ========
268 *
269 *> \author Univ. of Tennessee
270 *> \author Univ. of California Berkeley
271 *> \author Univ. of Colorado Denver
272 *> \author NAG Ltd.
273 *
274 *> \date November 2011
275 *
276 *> \ingroup realOTHEReigen
277 *
278 *> \par Contributors:
279 * ==================
280 *>
281 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
282 *
283 * =====================================================================
284  SUBROUTINE ssbgvx( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
285  $ ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z,
286  $ ldz, work, iwork, ifail, info )
287 *
288 * -- LAPACK driver routine (version 3.4.0) --
289 * -- LAPACK is a software package provided by Univ. of Tennessee, --
290 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
291 * November 2011
292 *
293 * .. Scalar Arguments ..
294  CHARACTER jobz, range, uplo
295  INTEGER il, info, iu, ka, kb, ldab, ldbb, ldq, ldz, m,
296  $ n
297  REAL abstol, vl, vu
298 * ..
299 * .. Array Arguments ..
300  INTEGER ifail( * ), iwork( * )
301  REAL ab( ldab, * ), bb( ldbb, * ), q( ldq, * ),
302  $ w( * ), work( * ), z( ldz, * )
303 * ..
304 *
305 * =====================================================================
306 *
307 * .. Parameters ..
308  REAL zero, one
309  parameter( zero = 0.0e+0, one = 1.0e+0 )
310 * ..
311 * .. Local Scalars ..
312  LOGICAL alleig, indeig, test, upper, valeig, wantz
313  CHARACTER order, vect
314  INTEGER i, iinfo, indd, inde, indee, indibl, indisp,
315  $ indiwo, indwrk, itmp1, j, jj, nsplit
316  REAL tmp1
317 * ..
318 * .. External Functions ..
319  LOGICAL lsame
320  EXTERNAL lsame
321 * ..
322 * .. External Subroutines ..
323  EXTERNAL scopy, sgemv, slacpy, spbstf, ssbgst, ssbtrd,
325 * ..
326 * .. Intrinsic Functions ..
327  INTRINSIC min
328 * ..
329 * .. Executable Statements ..
330 *
331 * Test the input parameters.
332 *
333  wantz = lsame( jobz, 'V' )
334  upper = lsame( uplo, 'U' )
335  alleig = lsame( range, 'A' )
336  valeig = lsame( range, 'V' )
337  indeig = lsame( range, 'I' )
338 *
339  info = 0
340  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
341  info = -1
342  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
343  info = -2
344  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
345  info = -3
346  ELSE IF( n.LT.0 ) THEN
347  info = -4
348  ELSE IF( ka.LT.0 ) THEN
349  info = -5
350  ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
351  info = -6
352  ELSE IF( ldab.LT.ka+1 ) THEN
353  info = -8
354  ELSE IF( ldbb.LT.kb+1 ) THEN
355  info = -10
356  ELSE IF( ldq.LT.1 .OR. ( wantz .AND. ldq.LT.n ) ) THEN
357  info = -12
358  ELSE
359  IF( valeig ) THEN
360  IF( n.GT.0 .AND. vu.LE.vl )
361  $ info = -14
362  ELSE IF( indeig ) THEN
363  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
364  info = -15
365  ELSE IF ( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
366  info = -16
367  END IF
368  END IF
369  END IF
370  IF( info.EQ.0) THEN
371  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
372  info = -21
373  END IF
374  END IF
375 *
376  IF( info.NE.0 ) THEN
377  CALL xerbla( 'SSBGVX', -info )
378  return
379  END IF
380 *
381 * Quick return if possible
382 *
383  m = 0
384  IF( n.EQ.0 )
385  $ return
386 *
387 * Form a split Cholesky factorization of B.
388 *
389  CALL spbstf( uplo, n, kb, bb, ldbb, info )
390  IF( info.NE.0 ) THEN
391  info = n + info
392  return
393  END IF
394 *
395 * Transform problem to standard eigenvalue problem.
396 *
397  CALL ssbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq,
398  $ work, iinfo )
399 *
400 * Reduce symmetric band matrix to tridiagonal form.
401 *
402  indd = 1
403  inde = indd + n
404  indwrk = inde + n
405  IF( wantz ) THEN
406  vect = 'U'
407  ELSE
408  vect = 'N'
409  END IF
410  CALL ssbtrd( vect, uplo, n, ka, ab, ldab, work( indd ),
411  $ work( inde ), q, ldq, work( indwrk ), iinfo )
412 *
413 * If all eigenvalues are desired and ABSTOL is less than or equal
414 * to zero, then call SSTERF or SSTEQR. If this fails for some
415 * eigenvalue, then try SSTEBZ.
416 *
417  test = .false.
418  IF( indeig ) THEN
419  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
420  test = .true.
421  END IF
422  END IF
423  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
424  CALL scopy( n, work( indd ), 1, w, 1 )
425  indee = indwrk + 2*n
426  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
427  IF( .NOT.wantz ) THEN
428  CALL ssterf( n, w, work( indee ), info )
429  ELSE
430  CALL slacpy( 'A', n, n, q, ldq, z, ldz )
431  CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
432  $ work( indwrk ), info )
433  IF( info.EQ.0 ) THEN
434  DO 10 i = 1, n
435  ifail( i ) = 0
436  10 continue
437  END IF
438  END IF
439  IF( info.EQ.0 ) THEN
440  m = n
441  go to 30
442  END IF
443  info = 0
444  END IF
445 *
446 * Otherwise, call SSTEBZ and, if eigenvectors are desired,
447 * call SSTEIN.
448 *
449  IF( wantz ) THEN
450  order = 'B'
451  ELSE
452  order = 'E'
453  END IF
454  indibl = 1
455  indisp = indibl + n
456  indiwo = indisp + n
457  CALL sstebz( range, order, n, vl, vu, il, iu, abstol,
458  $ work( indd ), work( inde ), m, nsplit, w,
459  $ iwork( indibl ), iwork( indisp ), work( indwrk ),
460  $ iwork( indiwo ), info )
461 *
462  IF( wantz ) THEN
463  CALL sstein( n, work( indd ), work( inde ), m, w,
464  $ iwork( indibl ), iwork( indisp ), z, ldz,
465  $ work( indwrk ), iwork( indiwo ), ifail, info )
466 *
467 * Apply transformation matrix used in reduction to tridiagonal
468 * form to eigenvectors returned by SSTEIN.
469 *
470  DO 20 j = 1, m
471  CALL scopy( n, z( 1, j ), 1, work( 1 ), 1 )
472  CALL sgemv( 'N', n, n, one, q, ldq, work, 1, zero,
473  $ z( 1, j ), 1 )
474  20 continue
475  END IF
476 *
477  30 continue
478 *
479 * If eigenvalues are not in order, then sort them, along with
480 * eigenvectors.
481 *
482  IF( wantz ) THEN
483  DO 50 j = 1, m - 1
484  i = 0
485  tmp1 = w( j )
486  DO 40 jj = j + 1, m
487  IF( w( jj ).LT.tmp1 ) THEN
488  i = jj
489  tmp1 = w( jj )
490  END IF
491  40 continue
492 *
493  IF( i.NE.0 ) THEN
494  itmp1 = iwork( indibl+i-1 )
495  w( i ) = w( j )
496  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
497  w( j ) = tmp1
498  iwork( indibl+j-1 ) = itmp1
499  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
500  IF( info.NE.0 ) THEN
501  itmp1 = ifail( i )
502  ifail( i ) = ifail( j )
503  ifail( j ) = itmp1
504  END IF
505  END IF
506  50 continue
507  END IF
508 *
509  return
510 *
511 * End of SSBGVX
512 *
513  END