LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dsyevr_2stage.f
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1 *> \brief <b> DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
2 *
3 * @precisions fortran d -> s
4 *
5 * =========== DOCUMENTATION ===========
6 *
7 * Online html documentation available at
8 * http://www.netlib.org/lapack/explore-html/
9 *
10 *> \htmlonly
11 *> Download DSYEVR_2STAGE + dependencies
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevr_2stage.f">
13 *> [TGZ]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevr_2stage.f">
15 *> [ZIP]</a>
16 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr_2stage.f">
17 *> [TXT]</a>
18 *> \endhtmlonly
19 *
20 * Definition:
21 * ===========
22 *
23 * SUBROUTINE DSYEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
24 * IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
25 * LWORK, IWORK, LIWORK, INFO )
26 *
27 * IMPLICIT NONE
28 *
29 * .. Scalar Arguments ..
30 * CHARACTER JOBZ, RANGE, UPLO
31 * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
32 * DOUBLE PRECISION ABSTOL, VL, VU
33 * ..
34 * .. Array Arguments ..
35 * INTEGER ISUPPZ( * ), IWORK( * )
36 * DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
37 * ..
38 *
39 *
40 *> \par Purpose:
41 * =============
42 *>
43 *> \verbatim
44 *>
45 *> DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
46 *> of a real symmetric matrix A using the 2stage technique for
47 *> the reduction to tridiagonal. Eigenvalues and eigenvectors can be
48 *> selected by specifying either a range of values or a range of
49 *> indices for the desired eigenvalues.
50 *>
51 *> DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
52 *> to DSYTRD. Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute
53 *> the eigenspectrum using Relatively Robust Representations. DSTEMR
54 *> computes eigenvalues by the dqds algorithm, while orthogonal
55 *> eigenvectors are computed from various "good" L D L^T representations
56 *> (also known as Relatively Robust Representations). Gram-Schmidt
57 *> orthogonalization is avoided as far as possible. More specifically,
58 *> the various steps of the algorithm are as follows.
59 *>
60 *> For each unreduced block (submatrix) of T,
61 *> (a) Compute T - sigma I = L D L^T, so that L and D
62 *> define all the wanted eigenvalues to high relative accuracy.
63 *> This means that small relative changes in the entries of D and L
64 *> cause only small relative changes in the eigenvalues and
65 *> eigenvectors. The standard (unfactored) representation of the
66 *> tridiagonal matrix T does not have this property in general.
67 *> (b) Compute the eigenvalues to suitable accuracy.
68 *> If the eigenvectors are desired, the algorithm attains full
69 *> accuracy of the computed eigenvalues only right before
70 *> the corresponding vectors have to be computed, see steps c) and d).
71 *> (c) For each cluster of close eigenvalues, select a new
72 *> shift close to the cluster, find a new factorization, and refine
73 *> the shifted eigenvalues to suitable accuracy.
74 *> (d) For each eigenvalue with a large enough relative separation compute
75 *> the corresponding eigenvector by forming a rank revealing twisted
76 *> factorization. Go back to (c) for any clusters that remain.
77 *>
78 *> The desired accuracy of the output can be specified by the input
79 *> parameter ABSTOL.
80 *>
81 *> For more details, see DSTEMR's documentation and:
82 *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
83 *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
84 *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
85 *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
86 *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
87 *> 2004. Also LAPACK Working Note 154.
88 *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
89 *> tridiagonal eigenvalue/eigenvector problem",
90 *> Computer Science Division Technical Report No. UCB/CSD-97-971,
91 *> UC Berkeley, May 1997.
92 *>
93 *>
94 *> Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested
95 *> on machines which conform to the ieee-754 floating point standard.
96 *> DSYEVR_2STAGE calls DSTEBZ and SSTEIN on non-ieee machines and
97 *> when partial spectrum requests are made.
98 *>
99 *> Normal execution of DSTEMR may create NaNs and infinities and
100 *> hence may abort due to a floating point exception in environments
101 *> which do not handle NaNs and infinities in the ieee standard default
102 *> manner.
103 *> \endverbatim
104 *
105 * Arguments:
106 * ==========
107 *
108 *> \param[in] JOBZ
109 *> \verbatim
110 *> JOBZ is CHARACTER*1
111 *> = 'N': Compute eigenvalues only;
112 *> = 'V': Compute eigenvalues and eigenvectors.
113 *> Not available in this release.
114 *> \endverbatim
115 *>
116 *> \param[in] RANGE
117 *> \verbatim
118 *> RANGE is CHARACTER*1
119 *> = 'A': all eigenvalues will be found.
120 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
121 *> will be found.
122 *> = 'I': the IL-th through IU-th eigenvalues will be found.
123 *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
124 *> DSTEIN are called
125 *> \endverbatim
126 *>
127 *> \param[in] UPLO
128 *> \verbatim
129 *> UPLO is CHARACTER*1
130 *> = 'U': Upper triangle of A is stored;
131 *> = 'L': Lower triangle of A is stored.
132 *> \endverbatim
133 *>
134 *> \param[in] N
135 *> \verbatim
136 *> N is INTEGER
137 *> The order of the matrix A. N >= 0.
138 *> \endverbatim
139 *>
140 *> \param[in,out] A
141 *> \verbatim
142 *> A is DOUBLE PRECISION array, dimension (LDA, N)
143 *> On entry, the symmetric matrix A. If UPLO = 'U', the
144 *> leading N-by-N upper triangular part of A contains the
145 *> upper triangular part of the matrix A. If UPLO = 'L',
146 *> the leading N-by-N lower triangular part of A contains
147 *> the lower triangular part of the matrix A.
148 *> On exit, the lower triangle (if UPLO='L') or the upper
149 *> triangle (if UPLO='U') of A, including the diagonal, is
150 *> destroyed.
151 *> \endverbatim
152 *>
153 *> \param[in] LDA
154 *> \verbatim
155 *> LDA is INTEGER
156 *> The leading dimension of the array A. LDA >= max(1,N).
157 *> \endverbatim
158 *>
159 *> \param[in] VL
160 *> \verbatim
161 *> VL is DOUBLE PRECISION
162 *> If RANGE='V', the lower bound of the interval to
163 *> be searched for eigenvalues. VL < VU.
164 *> Not referenced if RANGE = 'A' or 'I'.
165 *> \endverbatim
166 *>
167 *> \param[in] VU
168 *> \verbatim
169 *> VU is DOUBLE PRECISION
170 *> If RANGE='V', the upper bound of the interval to
171 *> be searched for eigenvalues. VL < VU.
172 *> Not referenced if RANGE = 'A' or 'I'.
173 *> \endverbatim
174 *>
175 *> \param[in] IL
176 *> \verbatim
177 *> IL is INTEGER
178 *> If RANGE='I', the index of the
179 *> smallest eigenvalue to be returned.
180 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
181 *> Not referenced if RANGE = 'A' or 'V'.
182 *> \endverbatim
183 *>
184 *> \param[in] IU
185 *> \verbatim
186 *> IU is INTEGER
187 *> If RANGE='I', the index of the
188 *> largest eigenvalue to be returned.
189 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
190 *> Not referenced if RANGE = 'A' or 'V'.
191 *> \endverbatim
192 *>
193 *> \param[in] ABSTOL
194 *> \verbatim
195 *> ABSTOL is DOUBLE PRECISION
196 *> The absolute error tolerance for the eigenvalues.
197 *> An approximate eigenvalue is accepted as converged
198 *> when it is determined to lie in an interval [a,b]
199 *> of width less than or equal to
200 *>
201 *> ABSTOL + EPS * max( |a|,|b| ) ,
202 *>
203 *> where EPS is the machine precision. If ABSTOL is less than
204 *> or equal to zero, then EPS*|T| will be used in its place,
205 *> where |T| is the 1-norm of the tridiagonal matrix obtained
206 *> by reducing A to tridiagonal form.
207 *>
208 *> See "Computing Small Singular Values of Bidiagonal Matrices
209 *> with Guaranteed High Relative Accuracy," by Demmel and
210 *> Kahan, LAPACK Working Note #3.
211 *>
212 *> If high relative accuracy is important, set ABSTOL to
213 *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
214 *> eigenvalues are computed to high relative accuracy when
215 *> possible in future releases. The current code does not
216 *> make any guarantees about high relative accuracy, but
217 *> future releases will. See J. Barlow and J. Demmel,
218 *> "Computing Accurate Eigensystems of Scaled Diagonally
219 *> Dominant Matrices", LAPACK Working Note #7, for a discussion
220 *> of which matrices define their eigenvalues to high relative
221 *> accuracy.
222 *> \endverbatim
223 *>
224 *> \param[out] M
225 *> \verbatim
226 *> M is INTEGER
227 *> The total number of eigenvalues found. 0 <= M <= N.
228 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
229 *> \endverbatim
230 *>
231 *> \param[out] W
232 *> \verbatim
233 *> W is DOUBLE PRECISION array, dimension (N)
234 *> The first M elements contain the selected eigenvalues in
235 *> ascending order.
236 *> \endverbatim
237 *>
238 *> \param[out] Z
239 *> \verbatim
240 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
241 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
242 *> contain the orthonormal eigenvectors of the matrix A
243 *> corresponding to the selected eigenvalues, with the i-th
244 *> column of Z holding the eigenvector associated with W(i).
245 *> If JOBZ = 'N', then Z is not referenced.
246 *> Note: the user must ensure that at least max(1,M) columns are
247 *> supplied in the array Z; if RANGE = 'V', the exact value of M
248 *> is not known in advance and an upper bound must be used.
249 *> Supplying N columns is always safe.
250 *> \endverbatim
251 *>
252 *> \param[in] LDZ
253 *> \verbatim
254 *> LDZ is INTEGER
255 *> The leading dimension of the array Z. LDZ >= 1, and if
256 *> JOBZ = 'V', LDZ >= max(1,N).
257 *> \endverbatim
258 *>
259 *> \param[out] ISUPPZ
260 *> \verbatim
261 *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
262 *> The support of the eigenvectors in Z, i.e., the indices
263 *> indicating the nonzero elements in Z. The i-th eigenvector
264 *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
265 *> ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
266 *> matrix). The support of the eigenvectors of A is typically
267 *> 1:N because of the orthogonal transformations applied by DORMTR.
268 *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
269 *> \endverbatim
270 *>
271 *> \param[out] WORK
272 *> \verbatim
273 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
274 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
275 *> \endverbatim
276 *>
277 *> \param[in] LWORK
278 *> \verbatim
279 *> LWORK is INTEGER
280 *> The dimension of the array WORK.
281 *> If JOBZ = 'N' and N > 1, LWORK must be queried.
282 *> LWORK = MAX(1, 26*N, dimension) where
283 *> dimension = max(stage1,stage2) + (KD+1)*N + 5*N
284 *> = N*KD + N*max(KD+1,FACTOPTNB)
285 *> + max(2*KD*KD, KD*NTHREADS)
286 *> + (KD+1)*N + 5*N
287 *> where KD is the blocking size of the reduction,
288 *> FACTOPTNB is the blocking used by the QR or LQ
289 *> algorithm, usually FACTOPTNB=128 is a good choice
290 *> NTHREADS is the number of threads used when
291 *> openMP compilation is enabled, otherwise =1.
292 *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
293 *>
294 *> If LWORK = -1, then a workspace query is assumed; the routine
295 *> only calculates the optimal size of the WORK array, returns
296 *> this value as the first entry of the WORK array, and no error
297 *> message related to LWORK is issued by XERBLA.
298 *> \endverbatim
299 *>
300 *> \param[out] IWORK
301 *> \verbatim
302 *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
303 *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
304 *> \endverbatim
305 *>
306 *> \param[in] LIWORK
307 *> \verbatim
308 *> LIWORK is INTEGER
309 *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
310 *>
311 *> If LIWORK = -1, then a workspace query is assumed; the
312 *> routine only calculates the optimal size of the IWORK array,
313 *> returns this value as the first entry of the IWORK array, and
314 *> no error message related to LIWORK is issued by XERBLA.
315 *> \endverbatim
316 *>
317 *> \param[out] INFO
318 *> \verbatim
319 *> INFO is INTEGER
320 *> = 0: successful exit
321 *> < 0: if INFO = -i, the i-th argument had an illegal value
322 *> > 0: Internal error
323 *> \endverbatim
324 *
325 * Authors:
326 * ========
327 *
328 *> \author Univ. of Tennessee
329 *> \author Univ. of California Berkeley
330 *> \author Univ. of Colorado Denver
331 *> \author NAG Ltd.
332 *
333 *> \ingroup doubleSYeigen
334 *
335 *> \par Contributors:
336 * ==================
337 *>
338 *> Inderjit Dhillon, IBM Almaden, USA \n
339 *> Osni Marques, LBNL/NERSC, USA \n
340 *> Ken Stanley, Computer Science Division, University of
341 *> California at Berkeley, USA \n
342 *> Jason Riedy, Computer Science Division, University of
343 *> California at Berkeley, USA \n
344 *>
345 *> \par Further Details:
346 * =====================
347 *>
348 *> \verbatim
349 *>
350 *> All details about the 2stage techniques are available in:
351 *>
352 *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
353 *> Parallel reduction to condensed forms for symmetric eigenvalue problems
354 *> using aggregated fine-grained and memory-aware kernels. In Proceedings
355 *> of 2011 International Conference for High Performance Computing,
356 *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
357 *> Article 8 , 11 pages.
358 *> http://doi.acm.org/10.1145/2063384.2063394
359 *>
360 *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
361 *> An improved parallel singular value algorithm and its implementation
362 *> for multicore hardware, In Proceedings of 2013 International Conference
363 *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
364 *> Denver, Colorado, USA, 2013.
365 *> Article 90, 12 pages.
366 *> http://doi.acm.org/10.1145/2503210.2503292
367 *>
368 *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
369 *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
370 *> calculations based on fine-grained memory aware tasks.
371 *> International Journal of High Performance Computing Applications.
372 *> Volume 28 Issue 2, Pages 196-209, May 2014.
373 *> http://hpc.sagepub.com/content/28/2/196
374 *>
375 *> \endverbatim
376 *
377 * =====================================================================
378  SUBROUTINE dsyevr_2stage( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
379  $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
380  $ LWORK, IWORK, LIWORK, INFO )
381 *
382  IMPLICIT NONE
383 *
384 * -- LAPACK driver routine --
385 * -- LAPACK is a software package provided by Univ. of Tennessee, --
386 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
387 *
388 * .. Scalar Arguments ..
389  CHARACTER JOBZ, RANGE, UPLO
390  INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
391  DOUBLE PRECISION ABSTOL, VL, VU
392 * ..
393 * .. Array Arguments ..
394  INTEGER ISUPPZ( * ), IWORK( * )
395  DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
396 * ..
397 *
398 * =====================================================================
399 *
400 * .. Parameters ..
401  DOUBLE PRECISION ZERO, ONE, TWO
402  PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0, two = 2.0d+0 )
403 * ..
404 * .. Local Scalars ..
405  LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
406  $ TRYRAC
407  CHARACTER ORDER
408  INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
409  $ indee, indibl, indifl, indisp, indiwo, indtau,
410  $ indwk, indwkn, iscale, j, jj, liwmin,
411  $ llwork, llwrkn, lwmin, nsplit,
412  $ lhtrd, lwtrd, kd, ib, indhous
413  DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
414  $ SIGMA, SMLNUM, TMP1, VLL, VUU
415 * ..
416 * .. External Functions ..
417  LOGICAL LSAME
418  INTEGER ILAENV, ILAENV2STAGE
419  DOUBLE PRECISION DLAMCH, DLANSY
420  EXTERNAL lsame, dlamch, dlansy, ilaenv, ilaenv2stage
421 * ..
422 * .. External Subroutines ..
423  EXTERNAL dcopy, dormtr, dscal, dstebz, dstemr, dstein,
425 * ..
426 * .. Intrinsic Functions ..
427  INTRINSIC max, min, sqrt
428 * ..
429 * .. Executable Statements ..
430 *
431 * Test the input parameters.
432 *
433  ieeeok = ilaenv( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
434 *
435  lower = lsame( uplo, 'L' )
436  wantz = lsame( jobz, 'V' )
437  alleig = lsame( range, 'A' )
438  valeig = lsame( range, 'V' )
439  indeig = lsame( range, 'I' )
440 *
441  lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
442 *
443  kd = ilaenv2stage( 1, 'DSYTRD_2STAGE', jobz, n, -1, -1, -1 )
444  ib = ilaenv2stage( 2, 'DSYTRD_2STAGE', jobz, n, kd, -1, -1 )
445  lhtrd = ilaenv2stage( 3, 'DSYTRD_2STAGE', jobz, n, kd, ib, -1 )
446  lwtrd = ilaenv2stage( 4, 'DSYTRD_2STAGE', jobz, n, kd, ib, -1 )
447  lwmin = max( 26*n, 5*n + lhtrd + lwtrd )
448  liwmin = max( 1, 10*n )
449 *
450  info = 0
451  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
452  info = -1
453  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
454  info = -2
455  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
456  info = -3
457  ELSE IF( n.LT.0 ) THEN
458  info = -4
459  ELSE IF( lda.LT.max( 1, n ) ) THEN
460  info = -6
461  ELSE
462  IF( valeig ) THEN
463  IF( n.GT.0 .AND. vu.LE.vl )
464  $ info = -8
465  ELSE IF( indeig ) THEN
466  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
467  info = -9
468  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
469  info = -10
470  END IF
471  END IF
472  END IF
473  IF( info.EQ.0 ) THEN
474  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
475  info = -15
476  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
477  info = -18
478  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
479  info = -20
480  END IF
481  END IF
482 *
483  IF( info.EQ.0 ) THEN
484 * NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
485 * NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
486 * LWKOPT = MAX( ( NB+1 )*N, LWMIN )
487  work( 1 ) = lwmin
488  iwork( 1 ) = liwmin
489  END IF
490 *
491  IF( info.NE.0 ) THEN
492  CALL xerbla( 'DSYEVR_2STAGE', -info )
493  RETURN
494  ELSE IF( lquery ) THEN
495  RETURN
496  END IF
497 *
498 * Quick return if possible
499 *
500  m = 0
501  IF( n.EQ.0 ) THEN
502  work( 1 ) = 1
503  RETURN
504  END IF
505 *
506  IF( n.EQ.1 ) THEN
507  work( 1 ) = 7
508  IF( alleig .OR. indeig ) THEN
509  m = 1
510  w( 1 ) = a( 1, 1 )
511  ELSE
512  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
513  m = 1
514  w( 1 ) = a( 1, 1 )
515  END IF
516  END IF
517  IF( wantz ) THEN
518  z( 1, 1 ) = one
519  isuppz( 1 ) = 1
520  isuppz( 2 ) = 1
521  END IF
522  RETURN
523  END IF
524 *
525 * Get machine constants.
526 *
527  safmin = dlamch( 'Safe minimum' )
528  eps = dlamch( 'Precision' )
529  smlnum = safmin / eps
530  bignum = one / smlnum
531  rmin = sqrt( smlnum )
532  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
533 *
534 * Scale matrix to allowable range, if necessary.
535 *
536  iscale = 0
537  abstll = abstol
538  IF (valeig) THEN
539  vll = vl
540  vuu = vu
541  END IF
542  anrm = dlansy( 'M', uplo, n, a, lda, work )
543  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
544  iscale = 1
545  sigma = rmin / anrm
546  ELSE IF( anrm.GT.rmax ) THEN
547  iscale = 1
548  sigma = rmax / anrm
549  END IF
550  IF( iscale.EQ.1 ) THEN
551  IF( lower ) THEN
552  DO 10 j = 1, n
553  CALL dscal( n-j+1, sigma, a( j, j ), 1 )
554  10 CONTINUE
555  ELSE
556  DO 20 j = 1, n
557  CALL dscal( j, sigma, a( 1, j ), 1 )
558  20 CONTINUE
559  END IF
560  IF( abstol.GT.0 )
561  $ abstll = abstol*sigma
562  IF( valeig ) THEN
563  vll = vl*sigma
564  vuu = vu*sigma
565  END IF
566  END IF
567 
568 * Initialize indices into workspaces. Note: The IWORK indices are
569 * used only if DSTERF or DSTEMR fail.
570 
571 * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
572 * elementary reflectors used in DSYTRD.
573  indtau = 1
574 * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
575  indd = indtau + n
576 * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
577 * tridiagonal matrix from DSYTRD.
578  inde = indd + n
579 * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
580 * -written by DSTEMR (the DSTERF path copies the diagonal to W).
581  inddd = inde + n
582 * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
583 * -written while computing the eigenvalues in DSTERF and DSTEMR.
584  indee = inddd + n
585 * INDHOUS is the starting offset Householder storage of stage 2
586  indhous = indee + n
587 * INDWK is the starting offset of the left-over workspace, and
588 * LLWORK is the remaining workspace size.
589  indwk = indhous + lhtrd
590  llwork = lwork - indwk + 1
591 
592 
593 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
594 * stores the block indices of each of the M<=N eigenvalues.
595  indibl = 1
596 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
597 * stores the starting and finishing indices of each block.
598  indisp = indibl + n
599 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
600 * that corresponding to eigenvectors that fail to converge in
601 * DSTEIN. This information is discarded; if any fail, the driver
602 * returns INFO > 0.
603  indifl = indisp + n
604 * INDIWO is the offset of the remaining integer workspace.
605  indiwo = indifl + n
606 
607 *
608 * Call DSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
609 *
610 *
611  CALL dsytrd_2stage( jobz, uplo, n, a, lda, work( indd ),
612  $ work( inde ), work( indtau ), work( indhous ),
613  $ lhtrd, work( indwk ), llwork, iinfo )
614 *
615 * If all eigenvalues are desired
616 * then call DSTERF or DSTEMR and DORMTR.
617 *
618  IF( ( alleig .OR. ( indeig .AND. il.EQ.1 .AND. iu.EQ.n ) ) .AND.
619  $ ieeeok.EQ.1 ) THEN
620  IF( .NOT.wantz ) THEN
621  CALL dcopy( n, work( indd ), 1, w, 1 )
622  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
623  CALL dsterf( n, w, work( indee ), info )
624  ELSE
625  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
626  CALL dcopy( n, work( indd ), 1, work( inddd ), 1 )
627 *
628  IF (abstol .LE. two*n*eps) THEN
629  tryrac = .true.
630  ELSE
631  tryrac = .false.
632  END IF
633  CALL dstemr( jobz, 'A', n, work( inddd ), work( indee ),
634  $ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
635  $ tryrac, work( indwk ), lwork, iwork, liwork,
636  $ info )
637 *
638 *
639 *
640 * Apply orthogonal matrix used in reduction to tridiagonal
641 * form to eigenvectors returned by DSTEMR.
642 *
643  IF( wantz .AND. info.EQ.0 ) THEN
644  indwkn = inde
645  llwrkn = lwork - indwkn + 1
646  CALL dormtr( 'L', uplo, 'N', n, m, a, lda,
647  $ work( indtau ), z, ldz, work( indwkn ),
648  $ llwrkn, iinfo )
649  END IF
650  END IF
651 *
652 *
653  IF( info.EQ.0 ) THEN
654 * Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
655 * undefined.
656  m = n
657  GO TO 30
658  END IF
659  info = 0
660  END IF
661 *
662 * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
663 * Also call DSTEBZ and DSTEIN if DSTEMR fails.
664 *
665  IF( wantz ) THEN
666  order = 'B'
667  ELSE
668  order = 'E'
669  END IF
670 
671  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
672  $ work( indd ), work( inde ), m, nsplit, w,
673  $ iwork( indibl ), iwork( indisp ), work( indwk ),
674  $ iwork( indiwo ), info )
675 *
676  IF( wantz ) THEN
677  CALL dstein( n, work( indd ), work( inde ), m, w,
678  $ iwork( indibl ), iwork( indisp ), z, ldz,
679  $ work( indwk ), iwork( indiwo ), iwork( indifl ),
680  $ info )
681 *
682 * Apply orthogonal matrix used in reduction to tridiagonal
683 * form to eigenvectors returned by DSTEIN.
684 *
685  indwkn = inde
686  llwrkn = lwork - indwkn + 1
687  CALL dormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
688  $ ldz, work( indwkn ), llwrkn, iinfo )
689  END IF
690 *
691 * If matrix was scaled, then rescale eigenvalues appropriately.
692 *
693 * Jump here if DSTEMR/DSTEIN succeeded.
694  30 CONTINUE
695  IF( iscale.EQ.1 ) THEN
696  IF( info.EQ.0 ) THEN
697  imax = m
698  ELSE
699  imax = info - 1
700  END IF
701  CALL dscal( imax, one / sigma, w, 1 )
702  END IF
703 *
704 * If eigenvalues are not in order, then sort them, along with
705 * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
706 * It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
707 * not return this detailed information to the user.
708 *
709  IF( wantz ) THEN
710  DO 50 j = 1, m - 1
711  i = 0
712  tmp1 = w( j )
713  DO 40 jj = j + 1, m
714  IF( w( jj ).LT.tmp1 ) THEN
715  i = jj
716  tmp1 = w( jj )
717  END IF
718  40 CONTINUE
719 *
720  IF( i.NE.0 ) THEN
721  w( i ) = w( j )
722  w( j ) = tmp1
723  CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
724  END IF
725  50 CONTINUE
726  END IF
727 *
728 * Set WORK(1) to optimal workspace size.
729 *
730  work( 1 ) = lwmin
731  iwork( 1 ) = liwmin
732 *
733  RETURN
734 *
735 * End of DSYEVR_2STAGE
736 *
737  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ
Definition: dstebz.f:273
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
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:321
subroutine dstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
Definition: dstein.f:174
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:171
subroutine dsytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
DSYTRD_2STAGE
subroutine dsyevr_2stage(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY mat...