LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
dsyevr.f
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1 *> \brief <b> DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY 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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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22 * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
23 * IWORK, LIWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
28 * DOUBLE PRECISION ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER ISUPPZ( * ), IWORK( * )
32 * DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> DSYEVR computes selected eigenvalues and, optionally, eigenvectors
42 *> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
43 *> selected by specifying either a range of values or a range of
44 *> indices for the desired eigenvalues.
45 *>
46 *> DSYEVR first reduces the matrix A to tridiagonal form T with a call
47 *> to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
48 *> the eigenspectrum using Relatively Robust Representations. DSTEMR
49 *> computes eigenvalues by the dqds algorithm, while orthogonal
50 *> eigenvectors are computed from various "good" L D L^T representations
51 *> (also known as Relatively Robust Representations). Gram-Schmidt
52 *> orthogonalization is avoided as far as possible. More specifically,
53 *> the various steps of the algorithm are as follows.
54 *>
55 *> For each unreduced block (submatrix) of T,
56 *> (a) Compute T - sigma I = L D L^T, so that L and D
57 *> define all the wanted eigenvalues to high relative accuracy.
58 *> This means that small relative changes in the entries of D and L
59 *> cause only small relative changes in the eigenvalues and
60 *> eigenvectors. The standard (unfactored) representation of the
61 *> tridiagonal matrix T does not have this property in general.
62 *> (b) Compute the eigenvalues to suitable accuracy.
63 *> If the eigenvectors are desired, the algorithm attains full
64 *> accuracy of the computed eigenvalues only right before
65 *> the corresponding vectors have to be computed, see steps c) and d).
66 *> (c) For each cluster of close eigenvalues, select a new
67 *> shift close to the cluster, find a new factorization, and refine
68 *> the shifted eigenvalues to suitable accuracy.
69 *> (d) For each eigenvalue with a large enough relative separation compute
70 *> the corresponding eigenvector by forming a rank revealing twisted
71 *> factorization. Go back to (c) for any clusters that remain.
72 *>
73 *> The desired accuracy of the output can be specified by the input
74 *> parameter ABSTOL.
75 *>
76 *> For more details, see DSTEMR's documentation and:
77 *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
78 *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
79 *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
80 *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
81 *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
82 *> 2004. Also LAPACK Working Note 154.
83 *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
84 *> tridiagonal eigenvalue/eigenvector problem",
85 *> Computer Science Division Technical Report No. UCB/CSD-97-971,
86 *> UC Berkeley, May 1997.
87 *>
88 *>
89 *> Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
90 *> on machines which conform to the ieee-754 floating point standard.
91 *> DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and
92 *> when partial spectrum requests are made.
93 *>
94 *> Normal execution of DSTEMR may create NaNs and infinities and
95 *> hence may abort due to a floating point exception in environments
96 *> which do not handle NaNs and infinities in the ieee standard default
97 *> manner.
98 *> \endverbatim
99 *
100 * Arguments:
101 * ==========
102 *
103 *> \param[in] JOBZ
104 *> \verbatim
105 *> JOBZ is CHARACTER*1
106 *> = 'N': Compute eigenvalues only;
107 *> = 'V': Compute eigenvalues and eigenvectors.
108 *> \endverbatim
109 *>
110 *> \param[in] RANGE
111 *> \verbatim
112 *> RANGE is CHARACTER*1
113 *> = 'A': all eigenvalues will be found.
114 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
115 *> will be found.
116 *> = 'I': the IL-th through IU-th eigenvalues will be found.
117 *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
118 *> DSTEIN are called
119 *> \endverbatim
120 *>
121 *> \param[in] UPLO
122 *> \verbatim
123 *> UPLO is CHARACTER*1
124 *> = 'U': Upper triangle of A is stored;
125 *> = 'L': Lower triangle of A is stored.
126 *> \endverbatim
127 *>
128 *> \param[in] N
129 *> \verbatim
130 *> N is INTEGER
131 *> The order of the matrix A. N >= 0.
132 *> \endverbatim
133 *>
134 *> \param[in,out] A
135 *> \verbatim
136 *> A is DOUBLE PRECISION array, dimension (LDA, N)
137 *> On entry, the symmetric matrix A. If UPLO = 'U', the
138 *> leading N-by-N upper triangular part of A contains the
139 *> upper triangular part of the matrix A. If UPLO = 'L',
140 *> the leading N-by-N lower triangular part of A contains
141 *> the lower triangular part of the matrix A.
142 *> On exit, the lower triangle (if UPLO='L') or the upper
143 *> triangle (if UPLO='U') of A, including the diagonal, is
144 *> destroyed.
145 *> \endverbatim
146 *>
147 *> \param[in] LDA
148 *> \verbatim
149 *> LDA is INTEGER
150 *> The leading dimension of the array A. LDA >= max(1,N).
151 *> \endverbatim
152 *>
153 *> \param[in] VL
154 *> \verbatim
155 *> VL is DOUBLE PRECISION
156 *> \endverbatim
157 *>
158 *> \param[in] VU
159 *> \verbatim
160 *> VU is DOUBLE PRECISION
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 *> If RANGE='I', the indices (in ascending order) of the
175 *> smallest and largest eigenvalues 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] ABSTOL
181 *> \verbatim
182 *> ABSTOL is DOUBLE PRECISION
183 *> The absolute error tolerance for the eigenvalues.
184 *> An approximate eigenvalue is accepted as converged
185 *> when it is determined to lie in an interval [a,b]
186 *> of width less than or equal to
187 *>
188 *> ABSTOL + EPS * max( |a|,|b| ) ,
189 *>
190 *> where EPS is the machine precision. If ABSTOL is less than
191 *> or equal to zero, then EPS*|T| will be used in its place,
192 *> where |T| is the 1-norm of the tridiagonal matrix obtained
193 *> by reducing A to tridiagonal form.
194 *>
195 *> See "Computing Small Singular Values of Bidiagonal Matrices
196 *> with Guaranteed High Relative Accuracy," by Demmel and
197 *> Kahan, LAPACK Working Note #3.
198 *>
199 *> If high relative accuracy is important, set ABSTOL to
200 *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
201 *> eigenvalues are computed to high relative accuracy when
202 *> possible in future releases. The current code does not
203 *> make any guarantees about high relative accuracy, but
204 *> future releases will. See J. Barlow and J. Demmel,
205 *> "Computing Accurate Eigensystems of Scaled Diagonally
206 *> Dominant Matrices", LAPACK Working Note #7, for a discussion
207 *> of which matrices define their eigenvalues to high relative
208 *> accuracy.
209 *> \endverbatim
210 *>
211 *> \param[out] M
212 *> \verbatim
213 *> M is INTEGER
214 *> The total number of eigenvalues found. 0 <= M <= N.
215 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
216 *> \endverbatim
217 *>
218 *> \param[out] W
219 *> \verbatim
220 *> W is DOUBLE PRECISION array, dimension (N)
221 *> The first M elements contain the selected eigenvalues in
222 *> ascending order.
223 *> \endverbatim
224 *>
225 *> \param[out] Z
226 *> \verbatim
227 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
228 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
229 *> contain the orthonormal eigenvectors of the matrix A
230 *> corresponding to the selected eigenvalues, with the i-th
231 *> column of Z holding the eigenvector associated with W(i).
232 *> If JOBZ = 'N', then Z is not referenced.
233 *> Note: the user must ensure that at least max(1,M) columns are
234 *> supplied in the array Z; if RANGE = 'V', the exact value of M
235 *> is not known in advance and an upper bound must be used.
236 *> Supplying N columns is always safe.
237 *> \endverbatim
238 *>
239 *> \param[in] LDZ
240 *> \verbatim
241 *> LDZ is INTEGER
242 *> The leading dimension of the array Z. LDZ >= 1, and if
243 *> JOBZ = 'V', LDZ >= max(1,N).
244 *> \endverbatim
245 *>
246 *> \param[out] ISUPPZ
247 *> \verbatim
248 *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
249 *> The support of the eigenvectors in Z, i.e., the indices
250 *> indicating the nonzero elements in Z. The i-th eigenvector
251 *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
252 *> ISUPPZ( 2*i ).
253 *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
254 *> \endverbatim
255 *>
256 *> \param[out] WORK
257 *> \verbatim
258 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
259 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
260 *> \endverbatim
261 *>
262 *> \param[in] LWORK
263 *> \verbatim
264 *> LWORK is INTEGER
265 *> The dimension of the array WORK. LWORK >= max(1,26*N).
266 *> For optimal efficiency, LWORK >= (NB+6)*N,
267 *> where NB is the max of the blocksize for DSYTRD and DORMTR
268 *> returned by ILAENV.
269 *>
270 *> If LWORK = -1, then a workspace query is assumed; the routine
271 *> only calculates the optimal size of the WORK array, returns
272 *> this value as the first entry of the WORK array, and no error
273 *> message related to LWORK is issued by XERBLA.
274 *> \endverbatim
275 *>
276 *> \param[out] IWORK
277 *> \verbatim
278 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
279 *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
280 *> \endverbatim
281 *>
282 *> \param[in] LIWORK
283 *> \verbatim
284 *> LIWORK is INTEGER
285 *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
286 *>
287 *> If LIWORK = -1, then a workspace query is assumed; the
288 *> routine only calculates the optimal size of the IWORK array,
289 *> returns this value as the first entry of the IWORK array, and
290 *> no error message related to LIWORK is issued by XERBLA.
291 *> \endverbatim
292 *>
293 *> \param[out] INFO
294 *> \verbatim
295 *> INFO is INTEGER
296 *> = 0: successful exit
297 *> < 0: if INFO = -i, the i-th argument had an illegal value
298 *> > 0: Internal error
299 *> \endverbatim
300 *
301 * Authors:
302 * ========
303 *
304 *> \author Univ. of Tennessee
305 *> \author Univ. of California Berkeley
306 *> \author Univ. of Colorado Denver
307 *> \author NAG Ltd.
308 *
309 *> \date September 2012
310 *
311 *> \ingroup doubleSYeigen
312 *
313 *> \par Contributors:
314 * ==================
315 *>
316 *> Inderjit Dhillon, IBM Almaden, USA \n
317 *> Osni Marques, LBNL/NERSC, USA \n
318 *> Ken Stanley, Computer Science Division, University of
319 *> California at Berkeley, USA \n
320 *> Jason Riedy, Computer Science Division, University of
321 *> California at Berkeley, USA \n
322 *>
323 * =====================================================================
324  SUBROUTINE dsyevr( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
325  $ abstol, m, w, z, ldz, isuppz, work, lwork,
326  $ iwork, liwork, info )
327 *
328 * -- LAPACK driver routine (version 3.4.2) --
329 * -- LAPACK is a software package provided by Univ. of Tennessee, --
330 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
331 * September 2012
332 *
333 * .. Scalar Arguments ..
334  CHARACTER JOBZ, RANGE, UPLO
335  INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
336  DOUBLE PRECISION ABSTOL, VL, VU
337 * ..
338 * .. Array Arguments ..
339  INTEGER ISUPPZ( * ), IWORK( * )
340  DOUBLE PRECISION A( lda, * ), W( * ), WORK( * ), Z( ldz, * )
341 * ..
342 *
343 * =====================================================================
344 *
345 * .. Parameters ..
346  DOUBLE PRECISION ZERO, ONE, TWO
347  parameter( zero = 0.0d+0, one = 1.0d+0, two = 2.0d+0 )
348 * ..
349 * .. Local Scalars ..
350  LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
351  $ tryrac
352  CHARACTER ORDER
353  INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
354  $ indee, indibl, indifl, indisp, indiwo, indtau,
355  $ indwk, indwkn, iscale, j, jj, liwmin,
356  $ llwork, llwrkn, lwkopt, lwmin, nb, nsplit
357  DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
358  $ sigma, smlnum, tmp1, vll, vuu
359 * ..
360 * .. External Functions ..
361  LOGICAL LSAME
362  INTEGER ILAENV
363  DOUBLE PRECISION DLAMCH, DLANSY
364  EXTERNAL lsame, ilaenv, dlamch, dlansy
365 * ..
366 * .. External Subroutines ..
367  EXTERNAL dcopy, dormtr, dscal, dstebz, dstemr, dstein,
369 * ..
370 * .. Intrinsic Functions ..
371  INTRINSIC max, min, sqrt
372 * ..
373 * .. Executable Statements ..
374 *
375 * Test the input parameters.
376 *
377  ieeeok = ilaenv( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
378 *
379  lower = lsame( uplo, 'L' )
380  wantz = lsame( jobz, 'V' )
381  alleig = lsame( range, 'A' )
382  valeig = lsame( range, 'V' )
383  indeig = lsame( range, 'I' )
384 *
385  lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
386 *
387  lwmin = max( 1, 26*n )
388  liwmin = max( 1, 10*n )
389 *
390  info = 0
391  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
392  info = -1
393  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
394  info = -2
395  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
396  info = -3
397  ELSE IF( n.LT.0 ) THEN
398  info = -4
399  ELSE IF( lda.LT.max( 1, n ) ) THEN
400  info = -6
401  ELSE
402  IF( valeig ) THEN
403  IF( n.GT.0 .AND. vu.LE.vl )
404  $ info = -8
405  ELSE IF( indeig ) THEN
406  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
407  info = -9
408  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
409  info = -10
410  END IF
411  END IF
412  END IF
413  IF( info.EQ.0 ) THEN
414  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
415  info = -15
416  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
417  info = -18
418  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
419  info = -20
420  END IF
421  END IF
422 *
423  IF( info.EQ.0 ) THEN
424  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
425  nb = max( nb, ilaenv( 1, 'DORMTR', uplo, n, -1, -1, -1 ) )
426  lwkopt = max( ( nb+1 )*n, lwmin )
427  work( 1 ) = lwkopt
428  iwork( 1 ) = liwmin
429  END IF
430 *
431  IF( info.NE.0 ) THEN
432  CALL xerbla( 'DSYEVR', -info )
433  RETURN
434  ELSE IF( lquery ) THEN
435  RETURN
436  END IF
437 *
438 * Quick return if possible
439 *
440  m = 0
441  IF( n.EQ.0 ) THEN
442  work( 1 ) = 1
443  RETURN
444  END IF
445 *
446  IF( n.EQ.1 ) THEN
447  work( 1 ) = 7
448  IF( alleig .OR. indeig ) THEN
449  m = 1
450  w( 1 ) = a( 1, 1 )
451  ELSE
452  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
453  m = 1
454  w( 1 ) = a( 1, 1 )
455  END IF
456  END IF
457  IF( wantz ) THEN
458  z( 1, 1 ) = one
459  isuppz( 1 ) = 1
460  isuppz( 2 ) = 1
461  END IF
462  RETURN
463  END IF
464 *
465 * Get machine constants.
466 *
467  safmin = dlamch( 'Safe minimum' )
468  eps = dlamch( 'Precision' )
469  smlnum = safmin / eps
470  bignum = one / smlnum
471  rmin = sqrt( smlnum )
472  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
473 *
474 * Scale matrix to allowable range, if necessary.
475 *
476  iscale = 0
477  abstll = abstol
478  IF (valeig) THEN
479  vll = vl
480  vuu = vu
481  END IF
482  anrm = dlansy( 'M', uplo, n, a, lda, work )
483  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
484  iscale = 1
485  sigma = rmin / anrm
486  ELSE IF( anrm.GT.rmax ) THEN
487  iscale = 1
488  sigma = rmax / anrm
489  END IF
490  IF( iscale.EQ.1 ) THEN
491  IF( lower ) THEN
492  DO 10 j = 1, n
493  CALL dscal( n-j+1, sigma, a( j, j ), 1 )
494  10 CONTINUE
495  ELSE
496  DO 20 j = 1, n
497  CALL dscal( j, sigma, a( 1, j ), 1 )
498  20 CONTINUE
499  END IF
500  IF( abstol.GT.0 )
501  $ abstll = abstol*sigma
502  IF( valeig ) THEN
503  vll = vl*sigma
504  vuu = vu*sigma
505  END IF
506  END IF
507 
508 * Initialize indices into workspaces. Note: The IWORK indices are
509 * used only if DSTERF or DSTEMR fail.
510 
511 * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
512 * elementary reflectors used in DSYTRD.
513  indtau = 1
514 * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
515  indd = indtau + n
516 * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
517 * tridiagonal matrix from DSYTRD.
518  inde = indd + n
519 * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
520 * -written by DSTEMR (the DSTERF path copies the diagonal to W).
521  inddd = inde + n
522 * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
523 * -written while computing the eigenvalues in DSTERF and DSTEMR.
524  indee = inddd + n
525 * INDWK is the starting offset of the left-over workspace, and
526 * LLWORK is the remaining workspace size.
527  indwk = indee + n
528  llwork = lwork - indwk + 1
529 
530 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
531 * stores the block indices of each of the M<=N eigenvalues.
532  indibl = 1
533 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
534 * stores the starting and finishing indices of each block.
535  indisp = indibl + n
536 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
537 * that corresponding to eigenvectors that fail to converge in
538 * DSTEIN. This information is discarded; if any fail, the driver
539 * returns INFO > 0.
540  indifl = indisp + n
541 * INDIWO is the offset of the remaining integer workspace.
542  indiwo = indifl + n
543 
544 *
545 * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
546 *
547  CALL dsytrd( uplo, n, a, lda, work( indd ), work( inde ),
548  $ work( indtau ), work( indwk ), llwork, iinfo )
549 *
550 * If all eigenvalues are desired
551 * then call DSTERF or DSTEMR and DORMTR.
552 *
553  IF( ( alleig .OR. ( indeig .AND. il.EQ.1 .AND. iu.EQ.n ) ) .AND.
554  $ ieeeok.EQ.1 ) THEN
555  IF( .NOT.wantz ) THEN
556  CALL dcopy( n, work( indd ), 1, w, 1 )
557  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
558  CALL dsterf( n, w, work( indee ), info )
559  ELSE
560  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
561  CALL dcopy( n, work( indd ), 1, work( inddd ), 1 )
562 *
563  IF (abstol .LE. two*n*eps) THEN
564  tryrac = .true.
565  ELSE
566  tryrac = .false.
567  END IF
568  CALL dstemr( jobz, 'A', n, work( inddd ), work( indee ),
569  $ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
570  $ tryrac, work( indwk ), lwork, iwork, liwork,
571  $ info )
572 *
573 *
574 *
575 * Apply orthogonal matrix used in reduction to tridiagonal
576 * form to eigenvectors returned by DSTEIN.
577 *
578  IF( wantz .AND. info.EQ.0 ) THEN
579  indwkn = inde
580  llwrkn = lwork - indwkn + 1
581  CALL dormtr( 'L', uplo, 'N', n, m, a, lda,
582  $ work( indtau ), z, ldz, work( indwkn ),
583  $ llwrkn, iinfo )
584  END IF
585  END IF
586 *
587 *
588  IF( info.EQ.0 ) THEN
589 * Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
590 * undefined.
591  m = n
592  GO TO 30
593  END IF
594  info = 0
595  END IF
596 *
597 * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
598 * Also call DSTEBZ and DSTEIN if DSTEMR fails.
599 *
600  IF( wantz ) THEN
601  order = 'B'
602  ELSE
603  order = 'E'
604  END IF
605 
606  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
607  $ work( indd ), work( inde ), m, nsplit, w,
608  $ iwork( indibl ), iwork( indisp ), work( indwk ),
609  $ iwork( indiwo ), info )
610 *
611  IF( wantz ) THEN
612  CALL dstein( n, work( indd ), work( inde ), m, w,
613  $ iwork( indibl ), iwork( indisp ), z, ldz,
614  $ work( indwk ), iwork( indiwo ), iwork( indifl ),
615  $ info )
616 *
617 * Apply orthogonal matrix used in reduction to tridiagonal
618 * form to eigenvectors returned by DSTEIN.
619 *
620  indwkn = inde
621  llwrkn = lwork - indwkn + 1
622  CALL dormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
623  $ ldz, work( indwkn ), llwrkn, iinfo )
624  END IF
625 *
626 * If matrix was scaled, then rescale eigenvalues appropriately.
627 *
628 * Jump here if DSTEMR/DSTEIN succeeded.
629  30 CONTINUE
630  IF( iscale.EQ.1 ) THEN
631  IF( info.EQ.0 ) THEN
632  imax = m
633  ELSE
634  imax = info - 1
635  END IF
636  CALL dscal( imax, one / sigma, w, 1 )
637  END IF
638 *
639 * If eigenvalues are not in order, then sort them, along with
640 * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
641 * It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
642 * not return this detailed information to the user.
643 *
644  IF( wantz ) THEN
645  DO 50 j = 1, m - 1
646  i = 0
647  tmp1 = w( j )
648  DO 40 jj = j + 1, m
649  IF( w( jj ).LT.tmp1 ) THEN
650  i = jj
651  tmp1 = w( jj )
652  END IF
653  40 CONTINUE
654 *
655  IF( i.NE.0 ) THEN
656  w( i ) = w( j )
657  w( j ) = tmp1
658  CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
659  END IF
660  50 CONTINUE
661  END IF
662 *
663 * Set WORK(1) to optimal workspace size.
664 *
665  work( 1 ) = lwkopt
666  iwork( 1 ) = liwmin
667 *
668  RETURN
669 *
670 * End of DSYEVR
671 *
672  END
subroutine dstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
Definition: dstein.f:176
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:173
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:88
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:53
subroutine dstemr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEMR
Definition: dstemr.f:314
subroutine dsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD
Definition: dsytrd.f:194
subroutine dsyevr(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ...
Definition: dsyevr.f:327
subroutine dstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ
Definition: dstebz.f:265
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53