1 SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
2 $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
3 $ RWORK, LRWORK, IWORK, LIWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBZ, RANGE, UPLO
12 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
13 $ M, N
14 DOUBLE PRECISION ABSTOL, VL, VU
15 * ..
16 * .. Array Arguments ..
17 INTEGER ISUPPZ( * ), IWORK( * )
18 DOUBLE PRECISION RWORK( * ), W( * )
19 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHEEVR computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
27 * be selected by specifying either a range of values or a range of
28 * indices for the desired eigenvalues.
29 *
30 * ZHEEVR first reduces the matrix A to tridiagonal form T with a call
31 * to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute
32 * eigenspectrum using Relatively Robust Representations. ZSTEMR
33 * computes eigenvalues by the dqds algorithm, while orthogonal
34 * eigenvectors are computed from various "good" L D L^T representations
35 * (also known as Relatively Robust Representations). Gram-Schmidt
36 * orthogonalization is avoided as far as possible. More specifically,
37 * the various steps of the algorithm are as follows.
38 *
39 * For each unreduced block (submatrix) of T,
40 * (a) Compute T - sigma I = L D L^T, so that L and D
41 * define all the wanted eigenvalues to high relative accuracy.
42 * This means that small relative changes in the entries of D and L
43 * cause only small relative changes in the eigenvalues and
44 * eigenvectors. The standard (unfactored) representation of the
45 * tridiagonal matrix T does not have this property in general.
46 * (b) Compute the eigenvalues to suitable accuracy.
47 * If the eigenvectors are desired, the algorithm attains full
48 * accuracy of the computed eigenvalues only right before
49 * the corresponding vectors have to be computed, see steps c) and d).
50 * (c) For each cluster of close eigenvalues, select a new
51 * shift close to the cluster, find a new factorization, and refine
52 * the shifted eigenvalues to suitable accuracy.
53 * (d) For each eigenvalue with a large enough relative separation compute
54 * the corresponding eigenvector by forming a rank revealing twisted
55 * factorization. Go back to (c) for any clusters that remain.
56 *
57 * The desired accuracy of the output can be specified by the input
58 * parameter ABSTOL.
59 *
60 * For more details, see DSTEMR's documentation and:
61 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
62 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
63 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
64 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
65 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
66 * 2004. Also LAPACK Working Note 154.
67 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
68 * tridiagonal eigenvalue/eigenvector problem",
69 * Computer Science Division Technical Report No. UCB/CSD-97-971,
70 * UC Berkeley, May 1997.
71 *
72 *
73 * Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested
74 * on machines which conform to the ieee-754 floating point standard.
75 * ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
76 * when partial spectrum requests are made.
77 *
78 * Normal execution of ZSTEMR may create NaNs and infinities and
79 * hence may abort due to a floating point exception in environments
80 * which do not handle NaNs and infinities in the ieee standard default
81 * manner.
82 *
83 * Arguments
84 * =========
85 *
86 * JOBZ (input) CHARACTER*1
87 * = 'N': Compute eigenvalues only;
88 * = 'V': Compute eigenvalues and eigenvectors.
89 *
90 * RANGE (input) CHARACTER*1
91 * = 'A': all eigenvalues will be found.
92 * = 'V': all eigenvalues in the half-open interval (VL,VU]
93 * will be found.
94 * = 'I': the IL-th through IU-th eigenvalues will be found.
95 * For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
96 * ZSTEIN are called
97 *
98 * UPLO (input) CHARACTER*1
99 * = 'U': Upper triangle of A is stored;
100 * = 'L': Lower triangle of A is stored.
101 *
102 * N (input) INTEGER
103 * The order of the matrix A. N >= 0.
104 *
105 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
106 * On entry, the Hermitian matrix A. If UPLO = 'U', the
107 * leading N-by-N upper triangular part of A contains the
108 * upper triangular part of the matrix A. If UPLO = 'L',
109 * the leading N-by-N lower triangular part of A contains
110 * the lower triangular part of the matrix A.
111 * On exit, the lower triangle (if UPLO='L') or the upper
112 * triangle (if UPLO='U') of A, including the diagonal, is
113 * destroyed.
114 *
115 * LDA (input) INTEGER
116 * The leading dimension of the array A. LDA >= max(1,N).
117 *
118 * VL (input) DOUBLE PRECISION
119 * VU (input) DOUBLE PRECISION
120 * If RANGE='V', the lower and upper bounds of the interval to
121 * be searched for eigenvalues. VL < VU.
122 * Not referenced if RANGE = 'A' or 'I'.
123 *
124 * IL (input) INTEGER
125 * IU (input) INTEGER
126 * If RANGE='I', the indices (in ascending order) of the
127 * smallest and largest eigenvalues to be returned.
128 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
129 * Not referenced if RANGE = 'A' or 'V'.
130 *
131 * ABSTOL (input) DOUBLE PRECISION
132 * The absolute error tolerance for the eigenvalues.
133 * An approximate eigenvalue is accepted as converged
134 * when it is determined to lie in an interval [a,b]
135 * of width less than or equal to
136 *
137 * ABSTOL + EPS * max( |a|,|b| ) ,
138 *
139 * where EPS is the machine precision. If ABSTOL is less than
140 * or equal to zero, then EPS*|T| will be used in its place,
141 * where |T| is the 1-norm of the tridiagonal matrix obtained
142 * by reducing A to tridiagonal form.
143 *
144 * See "Computing Small Singular Values of Bidiagonal Matrices
145 * with Guaranteed High Relative Accuracy," by Demmel and
146 * Kahan, LAPACK Working Note #3.
147 *
148 * If high relative accuracy is important, set ABSTOL to
149 * DLAMCH( 'Safe minimum' ). Doing so will guarantee that
150 * eigenvalues are computed to high relative accuracy when
151 * possible in future releases. The current code does not
152 * make any guarantees about high relative accuracy, but
153 * furutre releases will. See J. Barlow and J. Demmel,
154 * "Computing Accurate Eigensystems of Scaled Diagonally
155 * Dominant Matrices", LAPACK Working Note #7, for a discussion
156 * of which matrices define their eigenvalues to high relative
157 * accuracy.
158 *
159 * M (output) INTEGER
160 * The total number of eigenvalues found. 0 <= M <= N.
161 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
162 *
163 * W (output) DOUBLE PRECISION array, dimension (N)
164 * The first M elements contain the selected eigenvalues in
165 * ascending order.
166 *
167 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
168 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
169 * contain the orthonormal eigenvectors of the matrix A
170 * corresponding to the selected eigenvalues, with the i-th
171 * column of Z holding the eigenvector associated with W(i).
172 * If JOBZ = 'N', then Z is not referenced.
173 * Note: the user must ensure that at least max(1,M) columns are
174 * supplied in the array Z; if RANGE = 'V', the exact value of M
175 * is not known in advance and an upper bound must be used.
176 *
177 * LDZ (input) INTEGER
178 * The leading dimension of the array Z. LDZ >= 1, and if
179 * JOBZ = 'V', LDZ >= max(1,N).
180 *
181 * ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
182 * The support of the eigenvectors in Z, i.e., the indices
183 * indicating the nonzero elements in Z. The i-th eigenvector
184 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
185 * ISUPPZ( 2*i ).
186 * Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
187 *
188 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
189 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
190 *
191 * LWORK (input) INTEGER
192 * The length of the array WORK. LWORK >= max(1,2*N).
193 * For optimal efficiency, LWORK >= (NB+1)*N,
194 * where NB is the max of the blocksize for ZHETRD and for
195 * ZUNMTR as returned by ILAENV.
196 *
197 * If LWORK = -1, then a workspace query is assumed; the routine
198 * only calculates the optimal sizes of the WORK, RWORK and
199 * IWORK arrays, returns these values as the first entries of
200 * the WORK, RWORK and IWORK arrays, and no error message
201 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
202 *
203 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
204 * On exit, if INFO = 0, RWORK(1) returns the optimal
205 * (and minimal) LRWORK.
206 *
207 * LRWORK (input) INTEGER
208 * The length of the array RWORK. LRWORK >= max(1,24*N).
209 *
210 * If LRWORK = -1, then a workspace query is assumed; the
211 * routine only calculates the optimal sizes of the WORK, RWORK
212 * and IWORK arrays, returns these values as the first entries
213 * of the WORK, RWORK and IWORK arrays, and no error message
214 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
215 *
216 * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
217 * On exit, if INFO = 0, IWORK(1) returns the optimal
218 * (and minimal) LIWORK.
219 *
220 * LIWORK (input) INTEGER
221 * The dimension of the array IWORK. LIWORK >= max(1,10*N).
222 *
223 * If LIWORK = -1, then a workspace query is assumed; the
224 * routine only calculates the optimal sizes of the WORK, RWORK
225 * and IWORK arrays, returns these values as the first entries
226 * of the WORK, RWORK and IWORK arrays, and no error message
227 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
228 *
229 * INFO (output) INTEGER
230 * = 0: successful exit
231 * < 0: if INFO = -i, the i-th argument had an illegal value
232 * > 0: Internal error
233 *
234 * Further Details
235 * ===============
236 *
237 * Based on contributions by
238 * Inderjit Dhillon, IBM Almaden, USA
239 * Osni Marques, LBNL/NERSC, USA
240 * Ken Stanley, Computer Science Division, University of
241 * California at Berkeley, USA
242 * Jason Riedy, Computer Science Division, University of
243 * California at Berkeley, USA
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248 DOUBLE PRECISION ZERO, ONE, TWO
249 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
250 * ..
251 * .. Local Scalars ..
252 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
253 $ WANTZ, TRYRAC
254 CHARACTER ORDER
255 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
256 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
257 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
258 $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
259 $ LWKOPT, LWMIN, NB, NSPLIT
260 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
261 $ SIGMA, SMLNUM, TMP1, VLL, VUU
262 * ..
263 * .. External Functions ..
264 LOGICAL LSAME
265 INTEGER ILAENV
266 DOUBLE PRECISION DLAMCH, ZLANSY
267 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
268 * ..
269 * .. External Subroutines ..
270 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
271 $ ZHETRD, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
272 * ..
273 * .. Intrinsic Functions ..
274 INTRINSIC DBLE, MAX, MIN, SQRT
275 * ..
276 * .. Executable Statements ..
277 *
278 * Test the input parameters.
279 *
280 IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
281 *
282 LOWER = LSAME( UPLO, 'L' )
283 WANTZ = LSAME( JOBZ, 'V' )
284 ALLEIG = LSAME( RANGE, 'A' )
285 VALEIG = LSAME( RANGE, 'V' )
286 INDEIG = LSAME( RANGE, 'I' )
287 *
288 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
289 $ ( LIWORK.EQ.-1 ) )
290 *
291 LRWMIN = MAX( 1, 24*N )
292 LIWMIN = MAX( 1, 10*N )
293 LWMIN = MAX( 1, 2*N )
294 *
295 INFO = 0
296 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
297 INFO = -1
298 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
299 INFO = -2
300 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
301 INFO = -3
302 ELSE IF( N.LT.0 ) THEN
303 INFO = -4
304 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
305 INFO = -6
306 ELSE
307 IF( VALEIG ) THEN
308 IF( N.GT.0 .AND. VU.LE.VL )
309 $ INFO = -8
310 ELSE IF( INDEIG ) THEN
311 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
312 INFO = -9
313 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
314 INFO = -10
315 END IF
316 END IF
317 END IF
318 IF( INFO.EQ.0 ) THEN
319 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
320 INFO = -15
321 END IF
322 END IF
323 *
324 IF( INFO.EQ.0 ) THEN
325 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
326 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
327 LWKOPT = MAX( ( NB+1 )*N, LWMIN )
328 WORK( 1 ) = LWKOPT
329 RWORK( 1 ) = LRWMIN
330 IWORK( 1 ) = LIWMIN
331 *
332 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
333 INFO = -18
334 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
335 INFO = -20
336 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
337 INFO = -22
338 END IF
339 END IF
340 *
341 IF( INFO.NE.0 ) THEN
342 CALL XERBLA( 'ZHEEVR', -INFO )
343 RETURN
344 ELSE IF( LQUERY ) THEN
345 RETURN
346 END IF
347 *
348 * Quick return if possible
349 *
350 M = 0
351 IF( N.EQ.0 ) THEN
352 WORK( 1 ) = 1
353 RETURN
354 END IF
355 *
356 IF( N.EQ.1 ) THEN
357 WORK( 1 ) = 2
358 IF( ALLEIG .OR. INDEIG ) THEN
359 M = 1
360 W( 1 ) = DBLE( A( 1, 1 ) )
361 ELSE
362 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
363 $ THEN
364 M = 1
365 W( 1 ) = DBLE( A( 1, 1 ) )
366 END IF
367 END IF
368 IF( WANTZ ) THEN
369 Z( 1, 1 ) = ONE
370 ISUPPZ( 1 ) = 1
371 ISUPPZ( 2 ) = 1
372 END IF
373 RETURN
374 END IF
375 *
376 * Get machine constants.
377 *
378 SAFMIN = DLAMCH( 'Safe minimum' )
379 EPS = DLAMCH( '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 END IF
393 ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
394 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
395 ISCALE = 1
396 SIGMA = RMIN / ANRM
397 ELSE IF( ANRM.GT.RMAX ) THEN
398 ISCALE = 1
399 SIGMA = RMAX / ANRM
400 END IF
401 IF( ISCALE.EQ.1 ) THEN
402 IF( LOWER ) THEN
403 DO 10 J = 1, N
404 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
405 10 CONTINUE
406 ELSE
407 DO 20 J = 1, N
408 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
409 20 CONTINUE
410 END IF
411 IF( ABSTOL.GT.0 )
412 $ ABSTLL = ABSTOL*SIGMA
413 IF( VALEIG ) THEN
414 VLL = VL*SIGMA
415 VUU = VU*SIGMA
416 END IF
417 END IF
418
419 * Initialize indices into workspaces. Note: The IWORK indices are
420 * used only if DSTERF or ZSTEMR fail.
421
422 * WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
423 * elementary reflectors used in ZHETRD.
424 INDTAU = 1
425 * INDWK is the starting offset of the remaining complex workspace,
426 * and LLWORK is the remaining complex workspace size.
427 INDWK = INDTAU + N
428 LLWORK = LWORK - INDWK + 1
429
430 * RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
431 * entries.
432 INDRD = 1
433 * RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
434 * tridiagonal matrix from ZHETRD.
435 INDRE = INDRD + N
436 * RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
437 * -written by ZSTEMR (the DSTERF path copies the diagonal to W).
438 INDRDD = INDRE + N
439 * RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
440 * -written while computing the eigenvalues in DSTERF and ZSTEMR.
441 INDREE = INDRDD + N
442 * INDRWK is the starting offset of the left-over real workspace, and
443 * LLRWORK is the remaining workspace size.
444 INDRWK = INDREE + N
445 LLRWORK = LRWORK - INDRWK + 1
446
447 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
448 * stores the block indices of each of the M<=N eigenvalues.
449 INDIBL = 1
450 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
451 * stores the starting and finishing indices of each block.
452 INDISP = INDIBL + N
453 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
454 * that corresponding to eigenvectors that fail to converge in
455 * DSTEIN. This information is discarded; if any fail, the driver
456 * returns INFO > 0.
457 INDIFL = INDISP + N
458 * INDIWO is the offset of the remaining integer workspace.
459 INDIWO = INDISP + N
460
461 *
462 * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
463 *
464 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ),
465 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
466 *
467 * If all eigenvalues are desired
468 * then call DSTERF or ZSTEMR and ZUNMTR.
469 *
470 TEST = .FALSE.
471 IF( INDEIG ) THEN
472 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
473 TEST = .TRUE.
474 END IF
475 END IF
476 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
477 IF( .NOT.WANTZ ) THEN
478 CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
479 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
480 CALL DSTERF( N, W, RWORK( INDREE ), INFO )
481 ELSE
482 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
483 CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
484 *
485 IF (ABSTOL .LE. TWO*N*EPS) THEN
486 TRYRAC = .TRUE.
487 ELSE
488 TRYRAC = .FALSE.
489 END IF
490 CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
491 $ RWORK( INDREE ), VL, VU, IL, IU, M, W,
492 $ Z, LDZ, N, ISUPPZ, TRYRAC,
493 $ RWORK( INDRWK ), LLRWORK,
494 $ IWORK, LIWORK, INFO )
495 *
496 * Apply unitary matrix used in reduction to tridiagonal
497 * form to eigenvectors returned by ZSTEIN.
498 *
499 IF( WANTZ .AND. INFO.EQ.0 ) THEN
500 INDWKN = INDWK
501 LLWRKN = LWORK - INDWKN + 1
502 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
503 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
504 $ LLWRKN, IINFO )
505 END IF
506 END IF
507 *
508 *
509 IF( INFO.EQ.0 ) THEN
510 M = N
511 GO TO 30
512 END IF
513 INFO = 0
514 END IF
515 *
516 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
517 * Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
518 *
519 IF( WANTZ ) THEN
520 ORDER = 'B'
521 ELSE
522 ORDER = 'E'
523 END IF
524
525 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
526 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
527 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
528 $ IWORK( INDIWO ), INFO )
529 *
530 IF( WANTZ ) THEN
531 CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
532 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
533 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
534 $ INFO )
535 *
536 * Apply unitary matrix used in reduction to tridiagonal
537 * form to eigenvectors returned by ZSTEIN.
538 *
539 INDWKN = INDWK
540 LLWRKN = LWORK - INDWKN + 1
541 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
542 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
543 END IF
544 *
545 * If matrix was scaled, then rescale eigenvalues appropriately.
546 *
547 30 CONTINUE
548 IF( ISCALE.EQ.1 ) THEN
549 IF( INFO.EQ.0 ) THEN
550 IMAX = M
551 ELSE
552 IMAX = INFO - 1
553 END IF
554 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
555 END IF
556 *
557 * If eigenvalues are not in order, then sort them, along with
558 * eigenvectors.
559 *
560 IF( WANTZ ) THEN
561 DO 50 J = 1, M - 1
562 I = 0
563 TMP1 = W( J )
564 DO 40 JJ = J + 1, M
565 IF( W( JJ ).LT.TMP1 ) THEN
566 I = JJ
567 TMP1 = W( JJ )
568 END IF
569 40 CONTINUE
570 *
571 IF( I.NE.0 ) THEN
572 ITMP1 = IWORK( INDIBL+I-1 )
573 W( I ) = W( J )
574 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
575 W( J ) = TMP1
576 IWORK( INDIBL+J-1 ) = ITMP1
577 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
578 END IF
579 50 CONTINUE
580 END IF
581 *
582 * Set WORK(1) to optimal workspace size.
583 *
584 WORK( 1 ) = LWKOPT
585 RWORK( 1 ) = LRWMIN
586 IWORK( 1 ) = LIWMIN
587 *
588 RETURN
589 *
590 * End of ZHEEVR
591 *
592 END
2 $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
3 $ RWORK, LRWORK, IWORK, LIWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBZ, RANGE, UPLO
12 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
13 $ M, N
14 DOUBLE PRECISION ABSTOL, VL, VU
15 * ..
16 * .. Array Arguments ..
17 INTEGER ISUPPZ( * ), IWORK( * )
18 DOUBLE PRECISION RWORK( * ), W( * )
19 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHEEVR computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
27 * be selected by specifying either a range of values or a range of
28 * indices for the desired eigenvalues.
29 *
30 * ZHEEVR first reduces the matrix A to tridiagonal form T with a call
31 * to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute
32 * eigenspectrum using Relatively Robust Representations. ZSTEMR
33 * computes eigenvalues by the dqds algorithm, while orthogonal
34 * eigenvectors are computed from various "good" L D L^T representations
35 * (also known as Relatively Robust Representations). Gram-Schmidt
36 * orthogonalization is avoided as far as possible. More specifically,
37 * the various steps of the algorithm are as follows.
38 *
39 * For each unreduced block (submatrix) of T,
40 * (a) Compute T - sigma I = L D L^T, so that L and D
41 * define all the wanted eigenvalues to high relative accuracy.
42 * This means that small relative changes in the entries of D and L
43 * cause only small relative changes in the eigenvalues and
44 * eigenvectors. The standard (unfactored) representation of the
45 * tridiagonal matrix T does not have this property in general.
46 * (b) Compute the eigenvalues to suitable accuracy.
47 * If the eigenvectors are desired, the algorithm attains full
48 * accuracy of the computed eigenvalues only right before
49 * the corresponding vectors have to be computed, see steps c) and d).
50 * (c) For each cluster of close eigenvalues, select a new
51 * shift close to the cluster, find a new factorization, and refine
52 * the shifted eigenvalues to suitable accuracy.
53 * (d) For each eigenvalue with a large enough relative separation compute
54 * the corresponding eigenvector by forming a rank revealing twisted
55 * factorization. Go back to (c) for any clusters that remain.
56 *
57 * The desired accuracy of the output can be specified by the input
58 * parameter ABSTOL.
59 *
60 * For more details, see DSTEMR's documentation and:
61 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
62 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
63 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
64 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
65 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
66 * 2004. Also LAPACK Working Note 154.
67 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
68 * tridiagonal eigenvalue/eigenvector problem",
69 * Computer Science Division Technical Report No. UCB/CSD-97-971,
70 * UC Berkeley, May 1997.
71 *
72 *
73 * Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested
74 * on machines which conform to the ieee-754 floating point standard.
75 * ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
76 * when partial spectrum requests are made.
77 *
78 * Normal execution of ZSTEMR may create NaNs and infinities and
79 * hence may abort due to a floating point exception in environments
80 * which do not handle NaNs and infinities in the ieee standard default
81 * manner.
82 *
83 * Arguments
84 * =========
85 *
86 * JOBZ (input) CHARACTER*1
87 * = 'N': Compute eigenvalues only;
88 * = 'V': Compute eigenvalues and eigenvectors.
89 *
90 * RANGE (input) CHARACTER*1
91 * = 'A': all eigenvalues will be found.
92 * = 'V': all eigenvalues in the half-open interval (VL,VU]
93 * will be found.
94 * = 'I': the IL-th through IU-th eigenvalues will be found.
95 * For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
96 * ZSTEIN are called
97 *
98 * UPLO (input) CHARACTER*1
99 * = 'U': Upper triangle of A is stored;
100 * = 'L': Lower triangle of A is stored.
101 *
102 * N (input) INTEGER
103 * The order of the matrix A. N >= 0.
104 *
105 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
106 * On entry, the Hermitian matrix A. If UPLO = 'U', the
107 * leading N-by-N upper triangular part of A contains the
108 * upper triangular part of the matrix A. If UPLO = 'L',
109 * the leading N-by-N lower triangular part of A contains
110 * the lower triangular part of the matrix A.
111 * On exit, the lower triangle (if UPLO='L') or the upper
112 * triangle (if UPLO='U') of A, including the diagonal, is
113 * destroyed.
114 *
115 * LDA (input) INTEGER
116 * The leading dimension of the array A. LDA >= max(1,N).
117 *
118 * VL (input) DOUBLE PRECISION
119 * VU (input) DOUBLE PRECISION
120 * If RANGE='V', the lower and upper bounds of the interval to
121 * be searched for eigenvalues. VL < VU.
122 * Not referenced if RANGE = 'A' or 'I'.
123 *
124 * IL (input) INTEGER
125 * IU (input) INTEGER
126 * If RANGE='I', the indices (in ascending order) of the
127 * smallest and largest eigenvalues to be returned.
128 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
129 * Not referenced if RANGE = 'A' or 'V'.
130 *
131 * ABSTOL (input) DOUBLE PRECISION
132 * The absolute error tolerance for the eigenvalues.
133 * An approximate eigenvalue is accepted as converged
134 * when it is determined to lie in an interval [a,b]
135 * of width less than or equal to
136 *
137 * ABSTOL + EPS * max( |a|,|b| ) ,
138 *
139 * where EPS is the machine precision. If ABSTOL is less than
140 * or equal to zero, then EPS*|T| will be used in its place,
141 * where |T| is the 1-norm of the tridiagonal matrix obtained
142 * by reducing A to tridiagonal form.
143 *
144 * See "Computing Small Singular Values of Bidiagonal Matrices
145 * with Guaranteed High Relative Accuracy," by Demmel and
146 * Kahan, LAPACK Working Note #3.
147 *
148 * If high relative accuracy is important, set ABSTOL to
149 * DLAMCH( 'Safe minimum' ). Doing so will guarantee that
150 * eigenvalues are computed to high relative accuracy when
151 * possible in future releases. The current code does not
152 * make any guarantees about high relative accuracy, but
153 * furutre releases will. See J. Barlow and J. Demmel,
154 * "Computing Accurate Eigensystems of Scaled Diagonally
155 * Dominant Matrices", LAPACK Working Note #7, for a discussion
156 * of which matrices define their eigenvalues to high relative
157 * accuracy.
158 *
159 * M (output) INTEGER
160 * The total number of eigenvalues found. 0 <= M <= N.
161 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
162 *
163 * W (output) DOUBLE PRECISION array, dimension (N)
164 * The first M elements contain the selected eigenvalues in
165 * ascending order.
166 *
167 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
168 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
169 * contain the orthonormal eigenvectors of the matrix A
170 * corresponding to the selected eigenvalues, with the i-th
171 * column of Z holding the eigenvector associated with W(i).
172 * If JOBZ = 'N', then Z is not referenced.
173 * Note: the user must ensure that at least max(1,M) columns are
174 * supplied in the array Z; if RANGE = 'V', the exact value of M
175 * is not known in advance and an upper bound must be used.
176 *
177 * LDZ (input) INTEGER
178 * The leading dimension of the array Z. LDZ >= 1, and if
179 * JOBZ = 'V', LDZ >= max(1,N).
180 *
181 * ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
182 * The support of the eigenvectors in Z, i.e., the indices
183 * indicating the nonzero elements in Z. The i-th eigenvector
184 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
185 * ISUPPZ( 2*i ).
186 * Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
187 *
188 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
189 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
190 *
191 * LWORK (input) INTEGER
192 * The length of the array WORK. LWORK >= max(1,2*N).
193 * For optimal efficiency, LWORK >= (NB+1)*N,
194 * where NB is the max of the blocksize for ZHETRD and for
195 * ZUNMTR as returned by ILAENV.
196 *
197 * If LWORK = -1, then a workspace query is assumed; the routine
198 * only calculates the optimal sizes of the WORK, RWORK and
199 * IWORK arrays, returns these values as the first entries of
200 * the WORK, RWORK and IWORK arrays, and no error message
201 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
202 *
203 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
204 * On exit, if INFO = 0, RWORK(1) returns the optimal
205 * (and minimal) LRWORK.
206 *
207 * LRWORK (input) INTEGER
208 * The length of the array RWORK. LRWORK >= max(1,24*N).
209 *
210 * If LRWORK = -1, then a workspace query is assumed; the
211 * routine only calculates the optimal sizes of the WORK, RWORK
212 * and IWORK arrays, returns these values as the first entries
213 * of the WORK, RWORK and IWORK arrays, and no error message
214 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
215 *
216 * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
217 * On exit, if INFO = 0, IWORK(1) returns the optimal
218 * (and minimal) LIWORK.
219 *
220 * LIWORK (input) INTEGER
221 * The dimension of the array IWORK. LIWORK >= max(1,10*N).
222 *
223 * If LIWORK = -1, then a workspace query is assumed; the
224 * routine only calculates the optimal sizes of the WORK, RWORK
225 * and IWORK arrays, returns these values as the first entries
226 * of the WORK, RWORK and IWORK arrays, and no error message
227 * related to LWORK or LRWORK or LIWORK is issued by XERBLA.
228 *
229 * INFO (output) INTEGER
230 * = 0: successful exit
231 * < 0: if INFO = -i, the i-th argument had an illegal value
232 * > 0: Internal error
233 *
234 * Further Details
235 * ===============
236 *
237 * Based on contributions by
238 * Inderjit Dhillon, IBM Almaden, USA
239 * Osni Marques, LBNL/NERSC, USA
240 * Ken Stanley, Computer Science Division, University of
241 * California at Berkeley, USA
242 * Jason Riedy, Computer Science Division, University of
243 * California at Berkeley, USA
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248 DOUBLE PRECISION ZERO, ONE, TWO
249 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
250 * ..
251 * .. Local Scalars ..
252 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
253 $ WANTZ, TRYRAC
254 CHARACTER ORDER
255 INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
256 $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
257 $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
258 $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
259 $ LWKOPT, LWMIN, NB, NSPLIT
260 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
261 $ SIGMA, SMLNUM, TMP1, VLL, VUU
262 * ..
263 * .. External Functions ..
264 LOGICAL LSAME
265 INTEGER ILAENV
266 DOUBLE PRECISION DLAMCH, ZLANSY
267 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
268 * ..
269 * .. External Subroutines ..
270 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
271 $ ZHETRD, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
272 * ..
273 * .. Intrinsic Functions ..
274 INTRINSIC DBLE, MAX, MIN, SQRT
275 * ..
276 * .. Executable Statements ..
277 *
278 * Test the input parameters.
279 *
280 IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
281 *
282 LOWER = LSAME( UPLO, 'L' )
283 WANTZ = LSAME( JOBZ, 'V' )
284 ALLEIG = LSAME( RANGE, 'A' )
285 VALEIG = LSAME( RANGE, 'V' )
286 INDEIG = LSAME( RANGE, 'I' )
287 *
288 LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
289 $ ( LIWORK.EQ.-1 ) )
290 *
291 LRWMIN = MAX( 1, 24*N )
292 LIWMIN = MAX( 1, 10*N )
293 LWMIN = MAX( 1, 2*N )
294 *
295 INFO = 0
296 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
297 INFO = -1
298 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
299 INFO = -2
300 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
301 INFO = -3
302 ELSE IF( N.LT.0 ) THEN
303 INFO = -4
304 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
305 INFO = -6
306 ELSE
307 IF( VALEIG ) THEN
308 IF( N.GT.0 .AND. VU.LE.VL )
309 $ INFO = -8
310 ELSE IF( INDEIG ) THEN
311 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
312 INFO = -9
313 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
314 INFO = -10
315 END IF
316 END IF
317 END IF
318 IF( INFO.EQ.0 ) THEN
319 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
320 INFO = -15
321 END IF
322 END IF
323 *
324 IF( INFO.EQ.0 ) THEN
325 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
326 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
327 LWKOPT = MAX( ( NB+1 )*N, LWMIN )
328 WORK( 1 ) = LWKOPT
329 RWORK( 1 ) = LRWMIN
330 IWORK( 1 ) = LIWMIN
331 *
332 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
333 INFO = -18
334 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
335 INFO = -20
336 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
337 INFO = -22
338 END IF
339 END IF
340 *
341 IF( INFO.NE.0 ) THEN
342 CALL XERBLA( 'ZHEEVR', -INFO )
343 RETURN
344 ELSE IF( LQUERY ) THEN
345 RETURN
346 END IF
347 *
348 * Quick return if possible
349 *
350 M = 0
351 IF( N.EQ.0 ) THEN
352 WORK( 1 ) = 1
353 RETURN
354 END IF
355 *
356 IF( N.EQ.1 ) THEN
357 WORK( 1 ) = 2
358 IF( ALLEIG .OR. INDEIG ) THEN
359 M = 1
360 W( 1 ) = DBLE( A( 1, 1 ) )
361 ELSE
362 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
363 $ THEN
364 M = 1
365 W( 1 ) = DBLE( A( 1, 1 ) )
366 END IF
367 END IF
368 IF( WANTZ ) THEN
369 Z( 1, 1 ) = ONE
370 ISUPPZ( 1 ) = 1
371 ISUPPZ( 2 ) = 1
372 END IF
373 RETURN
374 END IF
375 *
376 * Get machine constants.
377 *
378 SAFMIN = DLAMCH( 'Safe minimum' )
379 EPS = DLAMCH( '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 END IF
393 ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
394 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
395 ISCALE = 1
396 SIGMA = RMIN / ANRM
397 ELSE IF( ANRM.GT.RMAX ) THEN
398 ISCALE = 1
399 SIGMA = RMAX / ANRM
400 END IF
401 IF( ISCALE.EQ.1 ) THEN
402 IF( LOWER ) THEN
403 DO 10 J = 1, N
404 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
405 10 CONTINUE
406 ELSE
407 DO 20 J = 1, N
408 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
409 20 CONTINUE
410 END IF
411 IF( ABSTOL.GT.0 )
412 $ ABSTLL = ABSTOL*SIGMA
413 IF( VALEIG ) THEN
414 VLL = VL*SIGMA
415 VUU = VU*SIGMA
416 END IF
417 END IF
418
419 * Initialize indices into workspaces. Note: The IWORK indices are
420 * used only if DSTERF or ZSTEMR fail.
421
422 * WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
423 * elementary reflectors used in ZHETRD.
424 INDTAU = 1
425 * INDWK is the starting offset of the remaining complex workspace,
426 * and LLWORK is the remaining complex workspace size.
427 INDWK = INDTAU + N
428 LLWORK = LWORK - INDWK + 1
429
430 * RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
431 * entries.
432 INDRD = 1
433 * RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
434 * tridiagonal matrix from ZHETRD.
435 INDRE = INDRD + N
436 * RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
437 * -written by ZSTEMR (the DSTERF path copies the diagonal to W).
438 INDRDD = INDRE + N
439 * RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
440 * -written while computing the eigenvalues in DSTERF and ZSTEMR.
441 INDREE = INDRDD + N
442 * INDRWK is the starting offset of the left-over real workspace, and
443 * LLRWORK is the remaining workspace size.
444 INDRWK = INDREE + N
445 LLRWORK = LRWORK - INDRWK + 1
446
447 * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
448 * stores the block indices of each of the M<=N eigenvalues.
449 INDIBL = 1
450 * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
451 * stores the starting and finishing indices of each block.
452 INDISP = INDIBL + N
453 * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
454 * that corresponding to eigenvectors that fail to converge in
455 * DSTEIN. This information is discarded; if any fail, the driver
456 * returns INFO > 0.
457 INDIFL = INDISP + N
458 * INDIWO is the offset of the remaining integer workspace.
459 INDIWO = INDISP + N
460
461 *
462 * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
463 *
464 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ),
465 $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
466 *
467 * If all eigenvalues are desired
468 * then call DSTERF or ZSTEMR and ZUNMTR.
469 *
470 TEST = .FALSE.
471 IF( INDEIG ) THEN
472 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
473 TEST = .TRUE.
474 END IF
475 END IF
476 IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
477 IF( .NOT.WANTZ ) THEN
478 CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
479 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
480 CALL DSTERF( N, W, RWORK( INDREE ), INFO )
481 ELSE
482 CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
483 CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
484 *
485 IF (ABSTOL .LE. TWO*N*EPS) THEN
486 TRYRAC = .TRUE.
487 ELSE
488 TRYRAC = .FALSE.
489 END IF
490 CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
491 $ RWORK( INDREE ), VL, VU, IL, IU, M, W,
492 $ Z, LDZ, N, ISUPPZ, TRYRAC,
493 $ RWORK( INDRWK ), LLRWORK,
494 $ IWORK, LIWORK, INFO )
495 *
496 * Apply unitary matrix used in reduction to tridiagonal
497 * form to eigenvectors returned by ZSTEIN.
498 *
499 IF( WANTZ .AND. INFO.EQ.0 ) THEN
500 INDWKN = INDWK
501 LLWRKN = LWORK - INDWKN + 1
502 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
503 $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
504 $ LLWRKN, IINFO )
505 END IF
506 END IF
507 *
508 *
509 IF( INFO.EQ.0 ) THEN
510 M = N
511 GO TO 30
512 END IF
513 INFO = 0
514 END IF
515 *
516 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
517 * Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
518 *
519 IF( WANTZ ) THEN
520 ORDER = 'B'
521 ELSE
522 ORDER = 'E'
523 END IF
524
525 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
526 $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
527 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
528 $ IWORK( INDIWO ), INFO )
529 *
530 IF( WANTZ ) THEN
531 CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
532 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
533 $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
534 $ INFO )
535 *
536 * Apply unitary matrix used in reduction to tridiagonal
537 * form to eigenvectors returned by ZSTEIN.
538 *
539 INDWKN = INDWK
540 LLWRKN = LWORK - INDWKN + 1
541 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
542 $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
543 END IF
544 *
545 * If matrix was scaled, then rescale eigenvalues appropriately.
546 *
547 30 CONTINUE
548 IF( ISCALE.EQ.1 ) THEN
549 IF( INFO.EQ.0 ) THEN
550 IMAX = M
551 ELSE
552 IMAX = INFO - 1
553 END IF
554 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
555 END IF
556 *
557 * If eigenvalues are not in order, then sort them, along with
558 * eigenvectors.
559 *
560 IF( WANTZ ) THEN
561 DO 50 J = 1, M - 1
562 I = 0
563 TMP1 = W( J )
564 DO 40 JJ = J + 1, M
565 IF( W( JJ ).LT.TMP1 ) THEN
566 I = JJ
567 TMP1 = W( JJ )
568 END IF
569 40 CONTINUE
570 *
571 IF( I.NE.0 ) THEN
572 ITMP1 = IWORK( INDIBL+I-1 )
573 W( I ) = W( J )
574 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
575 W( J ) = TMP1
576 IWORK( INDIBL+J-1 ) = ITMP1
577 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
578 END IF
579 50 CONTINUE
580 END IF
581 *
582 * Set WORK(1) to optimal workspace size.
583 *
584 WORK( 1 ) = LWKOPT
585 RWORK( 1 ) = LRWMIN
586 IWORK( 1 ) = LIWMIN
587 *
588 RETURN
589 *
590 * End of ZHEEVR
591 *
592 END