LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
ssbevx.f
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1 *> \brief <b> SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
22 * VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
23 * IFAIL, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
28 * REAL ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IFAIL( * ), IWORK( * )
32 * REAL AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
33 * \$ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> SSBEVX computes selected eigenvalues and, optionally, eigenvectors
43 *> of a real symmetric band matrix A. Eigenvalues and eigenvectors can
44 *> be selected by specifying either a range of values or a range of
45 *> indices for the desired eigenvalues.
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] JOBZ
52 *> \verbatim
53 *> JOBZ is CHARACTER*1
54 *> = 'N': Compute eigenvalues only;
55 *> = 'V': Compute eigenvalues and eigenvectors.
56 *> \endverbatim
57 *>
58 *> \param[in] RANGE
59 *> \verbatim
60 *> RANGE is CHARACTER*1
61 *> = 'A': all eigenvalues will be found;
62 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
63 *> will be found;
64 *> = 'I': the IL-th through IU-th eigenvalues will be found.
65 *> \endverbatim
66 *>
67 *> \param[in] UPLO
68 *> \verbatim
69 *> UPLO is CHARACTER*1
70 *> = 'U': Upper triangle of A is stored;
71 *> = 'L': Lower triangle of A is stored.
72 *> \endverbatim
73 *>
74 *> \param[in] N
75 *> \verbatim
76 *> N is INTEGER
77 *> The order of the matrix A. N >= 0.
78 *> \endverbatim
79 *>
80 *> \param[in] KD
81 *> \verbatim
82 *> KD is INTEGER
83 *> The number of superdiagonals of the matrix A if UPLO = 'U',
84 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
85 *> \endverbatim
86 *>
87 *> \param[in,out] AB
88 *> \verbatim
89 *> AB is REAL array, dimension (LDAB, N)
90 *> On entry, the upper or lower triangle of the symmetric band
91 *> matrix A, stored in the first KD+1 rows of the array. The
92 *> j-th column of A is stored in the j-th column of the array AB
93 *> as follows:
94 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
95 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
96 *>
97 *> On exit, AB is overwritten by values generated during the
98 *> reduction to tridiagonal form. If UPLO = 'U', the first
99 *> superdiagonal and the diagonal of the tridiagonal matrix T
100 *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
101 *> the diagonal and first subdiagonal of T are returned in the
102 *> first two rows of AB.
103 *> \endverbatim
104 *>
105 *> \param[in] LDAB
106 *> \verbatim
107 *> LDAB is INTEGER
108 *> The leading dimension of the array AB. LDAB >= KD + 1.
109 *> \endverbatim
110 *>
111 *> \param[out] Q
112 *> \verbatim
113 *> Q is REAL array, dimension (LDQ, N)
114 *> If JOBZ = 'V', the N-by-N orthogonal matrix used in the
115 *> reduction to tridiagonal form.
116 *> If JOBZ = 'N', the array Q is not referenced.
117 *> \endverbatim
118 *>
119 *> \param[in] LDQ
120 *> \verbatim
121 *> LDQ is INTEGER
122 *> The leading dimension of the array Q. If JOBZ = 'V', then
123 *> LDQ >= max(1,N).
124 *> \endverbatim
125 *>
126 *> \param[in] VL
127 *> \verbatim
128 *> VL is REAL
129 *> \endverbatim
130 *>
131 *> \param[in] VU
132 *> \verbatim
133 *> VU is REAL
134 *> If RANGE='V', the lower and upper bounds 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] IL
140 *> \verbatim
141 *> IL is INTEGER
142 *> \endverbatim
143 *>
144 *> \param[in] IU
145 *> \verbatim
146 *> IU is INTEGER
147 *> If RANGE='I', the indices (in ascending order) of the
148 *> smallest and largest eigenvalues to be returned.
149 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
150 *> Not referenced if RANGE = 'A' or 'V'.
151 *> \endverbatim
152 *>
153 *> \param[in] ABSTOL
154 *> \verbatim
155 *> ABSTOL is REAL
156 *> The absolute error tolerance for the eigenvalues.
157 *> An approximate eigenvalue is accepted as converged
158 *> when it is determined to lie in an interval [a,b]
159 *> of width less than or equal to
160 *>
161 *> ABSTOL + EPS * max( |a|,|b| ) ,
162 *>
163 *> where EPS is the machine precision. If ABSTOL is less than
164 *> or equal to zero, then EPS*|T| will be used in its place,
165 *> where |T| is the 1-norm of the tridiagonal matrix obtained
166 *> by reducing AB to tridiagonal form.
167 *>
168 *> Eigenvalues will be computed most accurately when ABSTOL is
169 *> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
170 *> If this routine returns with INFO>0, indicating that some
171 *> eigenvectors did not converge, try setting ABSTOL to
172 *> 2*SLAMCH('S').
173 *>
174 *> See "Computing Small Singular Values of Bidiagonal Matrices
175 *> with Guaranteed High Relative Accuracy," by Demmel and
176 *> Kahan, LAPACK Working Note #3.
177 *> \endverbatim
178 *>
179 *> \param[out] M
180 *> \verbatim
181 *> M is INTEGER
182 *> The total number of eigenvalues found. 0 <= M <= N.
183 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
184 *> \endverbatim
185 *>
186 *> \param[out] W
187 *> \verbatim
188 *> W is REAL array, dimension (N)
189 *> The first M elements contain the selected eigenvalues in
190 *> ascending order.
191 *> \endverbatim
192 *>
193 *> \param[out] Z
194 *> \verbatim
195 *> Z is REAL array, dimension (LDZ, max(1,M))
196 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
197 *> contain the orthonormal eigenvectors of the matrix A
198 *> corresponding to the selected eigenvalues, with the i-th
199 *> column of Z holding the eigenvector associated with W(i).
200 *> If an eigenvector fails to converge, then that column of Z
201 *> contains the latest approximation to the eigenvector, and the
202 *> index of the eigenvector is returned in IFAIL.
203 *> If JOBZ = 'N', then Z is not referenced.
204 *> Note: the user must ensure that at least max(1,M) columns are
205 *> supplied in the array Z; if RANGE = 'V', the exact value of M
206 *> is not known in advance and an upper bound must be used.
207 *> \endverbatim
208 *>
209 *> \param[in] LDZ
210 *> \verbatim
211 *> LDZ is INTEGER
212 *> The leading dimension of the array Z. LDZ >= 1, and if
213 *> JOBZ = 'V', LDZ >= max(1,N).
214 *> \endverbatim
215 *>
216 *> \param[out] WORK
217 *> \verbatim
218 *> WORK is REAL array, dimension (7*N)
219 *> \endverbatim
220 *>
221 *> \param[out] IWORK
222 *> \verbatim
223 *> IWORK is INTEGER array, dimension (5*N)
224 *> \endverbatim
225 *>
226 *> \param[out] IFAIL
227 *> \verbatim
228 *> IFAIL is INTEGER array, dimension (N)
229 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
230 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
231 *> indices of the eigenvectors that failed to converge.
232 *> If JOBZ = 'N', then IFAIL is not referenced.
233 *> \endverbatim
234 *>
235 *> \param[out] INFO
236 *> \verbatim
237 *> INFO is INTEGER
238 *> = 0: successful exit.
239 *> < 0: if INFO = -i, the i-th argument had an illegal value.
240 *> > 0: if INFO = i, then i eigenvectors failed to converge.
241 *> Their indices are stored in array IFAIL.
242 *> \endverbatim
243 *
244 * Authors:
245 * ========
246 *
247 *> \author Univ. of Tennessee
248 *> \author Univ. of California Berkeley
249 *> \author Univ. of Colorado Denver
250 *> \author NAG Ltd.
251 *
252 *> \date November 2011
253 *
254 *> \ingroup realOTHEReigen
255 *
256 * =====================================================================
257  SUBROUTINE ssbevx( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
258  \$ vu, il, iu, abstol, m, w, z, ldz, work, iwork,
259  \$ ifail, info )
260 *
261 * -- LAPACK driver routine (version 3.4.0) --
262 * -- LAPACK is a software package provided by Univ. of Tennessee, --
263 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264 * November 2011
265 *
266 * .. Scalar Arguments ..
267  CHARACTER jobz, range, uplo
268  INTEGER il, info, iu, kd, ldab, ldq, ldz, m, n
269  REAL abstol, vl, vu
270 * ..
271 * .. Array Arguments ..
272  INTEGER ifail( * ), iwork( * )
273  REAL ab( ldab, * ), q( ldq, * ), w( * ), work( * ),
274  \$ z( ldz, * )
275 * ..
276 *
277 * =====================================================================
278 *
279 * .. Parameters ..
280  REAL zero, one
281  parameter( zero = 0.0e0, one = 1.0e0 )
282 * ..
283 * .. Local Scalars ..
284  LOGICAL alleig, indeig, lower, test, valeig, wantz
285  CHARACTER order
286  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
287  \$ indisp, indiwo, indwrk, iscale, itmp1, j, jj,
288  \$ nsplit
289  REAL abstll, anrm, bignum, eps, rmax, rmin, safmin,
290  \$ sigma, smlnum, tmp1, vll, vuu
291 * ..
292 * .. External Functions ..
293  LOGICAL lsame
294  REAL slamch, slansb
295  EXTERNAL lsame, slamch, slansb
296 * ..
297 * .. External Subroutines ..
298  EXTERNAL scopy, sgemv, slacpy, slascl, ssbtrd, sscal,
300 * ..
301 * .. Intrinsic Functions ..
302  INTRINSIC max, min, sqrt
303 * ..
304 * .. Executable Statements ..
305 *
306 * Test the input parameters.
307 *
308  wantz = lsame( jobz, 'V' )
309  alleig = lsame( range, 'A' )
310  valeig = lsame( range, 'V' )
311  indeig = lsame( range, 'I' )
312  lower = lsame( uplo, 'L' )
313 *
314  info = 0
315  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
316  info = -1
317  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
318  info = -2
319  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
320  info = -3
321  ELSE IF( n.LT.0 ) THEN
322  info = -4
323  ELSE IF( kd.LT.0 ) THEN
324  info = -5
325  ELSE IF( ldab.LT.kd+1 ) THEN
326  info = -7
327  ELSE IF( wantz .AND. ldq.LT.max( 1, n ) ) THEN
328  info = -9
329  ELSE
330  IF( valeig ) THEN
331  IF( n.GT.0 .AND. vu.LE.vl )
332  \$ info = -11
333  ELSE IF( indeig ) THEN
334  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
335  info = -12
336  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
337  info = -13
338  END IF
339  END IF
340  END IF
341  IF( info.EQ.0 ) THEN
342  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
343  \$ info = -18
344  END IF
345 *
346  IF( info.NE.0 ) THEN
347  CALL xerbla( 'SSBEVX', -info )
348  return
349  END IF
350 *
351 * Quick return if possible
352 *
353  m = 0
354  IF( n.EQ.0 )
355  \$ return
356 *
357  IF( n.EQ.1 ) THEN
358  m = 1
359  IF( lower ) THEN
360  tmp1 = ab( 1, 1 )
361  ELSE
362  tmp1 = ab( kd+1, 1 )
363  END IF
364  IF( valeig ) THEN
365  IF( .NOT.( vl.LT.tmp1 .AND. vu.GE.tmp1 ) )
366  \$ m = 0
367  END IF
368  IF( m.EQ.1 ) THEN
369  w( 1 ) = tmp1
370  IF( wantz )
371  \$ z( 1, 1 ) = one
372  END IF
373  return
374  END IF
375 *
376 * Get machine constants.
377 *
378  safmin = slamch( 'Safe minimum' )
379  eps = slamch( 'Precision' )
380  smlnum = safmin / eps
381  bignum = one / smlnum
382  rmin = sqrt( smlnum )
383  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
384 *
385 * Scale matrix to allowable range, if necessary.
386 *
387  iscale = 0
388  abstll = abstol
389  IF ( valeig ) THEN
390  vll = vl
391  vuu = vu
392  ELSE
393  vll = zero
394  vuu = zero
395  ENDIF
396  anrm = slansb( 'M', uplo, n, kd, ab, ldab, work )
397  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
398  iscale = 1
399  sigma = rmin / anrm
400  ELSE IF( anrm.GT.rmax ) THEN
401  iscale = 1
402  sigma = rmax / anrm
403  END IF
404  IF( iscale.EQ.1 ) THEN
405  IF( lower ) THEN
406  CALL slascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
407  ELSE
408  CALL slascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
409  END IF
410  IF( abstol.GT.0 )
411  \$ abstll = abstol*sigma
412  IF( valeig ) THEN
413  vll = vl*sigma
414  vuu = vu*sigma
415  END IF
416  END IF
417 *
418 * Call SSBTRD to reduce symmetric band matrix to tridiagonal form.
419 *
420  indd = 1
421  inde = indd + n
422  indwrk = inde + n
423  CALL ssbtrd( jobz, uplo, n, kd, ab, ldab, work( indd ),
424  \$ work( inde ), q, ldq, work( indwrk ), iinfo )
425 *
426 * If all eigenvalues are desired and ABSTOL is less than or equal
427 * to zero, then call SSTERF or SSTEQR. If this fails for some
428 * eigenvalue, then try SSTEBZ.
429 *
430  test = .false.
431  IF (indeig) THEN
432  IF (il.EQ.1 .AND. iu.EQ.n) THEN
433  test = .true.
434  END IF
435  END IF
436  IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
437  CALL scopy( n, work( indd ), 1, w, 1 )
438  indee = indwrk + 2*n
439  IF( .NOT.wantz ) THEN
440  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
441  CALL ssterf( n, w, work( indee ), info )
442  ELSE
443  CALL slacpy( 'A', n, n, q, ldq, z, ldz )
444  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
445  CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
446  \$ work( indwrk ), info )
447  IF( info.EQ.0 ) THEN
448  DO 10 i = 1, n
449  ifail( i ) = 0
450  10 continue
451  END IF
452  END IF
453  IF( info.EQ.0 ) THEN
454  m = n
455  go to 30
456  END IF
457  info = 0
458  END IF
459 *
460 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
461 *
462  IF( wantz ) THEN
463  order = 'B'
464  ELSE
465  order = 'E'
466  END IF
467  indibl = 1
468  indisp = indibl + n
469  indiwo = indisp + n
470  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
471  \$ work( indd ), work( inde ), m, nsplit, w,
472  \$ iwork( indibl ), iwork( indisp ), work( indwrk ),
473  \$ iwork( indiwo ), info )
474 *
475  IF( wantz ) THEN
476  CALL sstein( n, work( indd ), work( inde ), m, w,
477  \$ iwork( indibl ), iwork( indisp ), z, ldz,
478  \$ work( indwrk ), iwork( indiwo ), ifail, info )
479 *
480 * Apply orthogonal matrix used in reduction to tridiagonal
481 * form to eigenvectors returned by SSTEIN.
482 *
483  DO 20 j = 1, m
484  CALL scopy( n, z( 1, j ), 1, work( 1 ), 1 )
485  CALL sgemv( 'N', n, n, one, q, ldq, work, 1, zero,
486  \$ z( 1, j ), 1 )
487  20 continue
488  END IF
489 *
490 * If matrix was scaled, then rescale eigenvalues appropriately.
491 *
492  30 continue
493  IF( iscale.EQ.1 ) THEN
494  IF( info.EQ.0 ) THEN
495  imax = m
496  ELSE
497  imax = info - 1
498  END IF
499  CALL sscal( imax, one / sigma, w, 1 )
500  END IF
501 *
502 * If eigenvalues are not in order, then sort them, along with
503 * eigenvectors.
504 *
505  IF( wantz ) THEN
506  DO 50 j = 1, m - 1
507  i = 0
508  tmp1 = w( j )
509  DO 40 jj = j + 1, m
510  IF( w( jj ).LT.tmp1 ) THEN
511  i = jj
512  tmp1 = w( jj )
513  END IF
514  40 continue
515 *
516  IF( i.NE.0 ) THEN
517  itmp1 = iwork( indibl+i-1 )
518  w( i ) = w( j )
519  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
520  w( j ) = tmp1
521  iwork( indibl+j-1 ) = itmp1
522  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
523  IF( info.NE.0 ) THEN
524  itmp1 = ifail( i )
525  ifail( i ) = ifail( j )
526  ifail( j ) = itmp1
527  END IF
528  END IF
529  50 continue
530  END IF
531 *
532  return
533 *
534 * End of SSBEVX
535 *
536  END