1 SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
2 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
3 $ IWORK, LIWORK, INFO )
4 IMPLICIT NONE
5 *
6 * -- LAPACK computational routine (version 3.2.1) --
7 *
8 * -- April 2009 --
9 *
10 * -- LAPACK is a software package provided by Univ. of Tennessee, --
11 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12 *
13 * .. Scalar Arguments ..
14 CHARACTER JOBZ, RANGE
15 LOGICAL TRYRAC
16 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
17 DOUBLE PRECISION VL, VU
18 * ..
19 * .. Array Arguments ..
20 INTEGER ISUPPZ( * ), IWORK( * )
21 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
22 COMPLEX*16 Z( LDZ, * )
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
29 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
30 * a well defined set of pairwise different real eigenvalues, the corresponding
31 * real eigenvectors are pairwise orthogonal.
32 *
33 * The spectrum may be computed either completely or partially by specifying
34 * either an interval (VL,VU] or a range of indices IL:IU for the desired
35 * eigenvalues.
36 *
37 * Depending on the number of desired eigenvalues, these are computed either
38 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
39 * computed by the use of various suitable L D L^T factorizations near clusters
40 * of close eigenvalues (referred to as RRRs, Relatively Robust
41 * Representations). An informal sketch of the algorithm follows.
42 *
43 * For each unreduced block (submatrix) of T,
44 * (a) Compute T - sigma I = L D L^T, so that L and D
45 * define all the wanted eigenvalues to high relative accuracy.
46 * This means that small relative changes in the entries of D and L
47 * cause only small relative changes in the eigenvalues and
48 * eigenvectors. The standard (unfactored) representation of the
49 * tridiagonal matrix T does not have this property in general.
50 * (b) Compute the eigenvalues to suitable accuracy.
51 * If the eigenvectors are desired, the algorithm attains full
52 * accuracy of the computed eigenvalues only right before
53 * the corresponding vectors have to be computed, see steps c) and d).
54 * (c) For each cluster of close eigenvalues, select a new
55 * shift close to the cluster, find a new factorization, and refine
56 * the shifted eigenvalues to suitable accuracy.
57 * (d) For each eigenvalue with a large enough relative separation compute
58 * the corresponding eigenvector by forming a rank revealing twisted
59 * factorization. Go back to (c) for any clusters that remain.
60 *
61 * For more details, see:
62 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
63 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
64 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
65 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
66 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
67 * 2004. Also LAPACK Working Note 154.
68 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
69 * tridiagonal eigenvalue/eigenvector problem",
70 * Computer Science Division Technical Report No. UCB/CSD-97-971,
71 * UC Berkeley, May 1997.
72 *
73 * Further Details
74 * 1.ZSTEMR works only on machines which follow IEEE-754
75 * floating-point standard in their handling of infinities and NaNs.
76 * This permits the use of efficient inner loops avoiding a check for
77 * zero divisors.
78 *
79 * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
80 * real symmetric tridiagonal form.
81 *
82 * (Any complex Hermitean tridiagonal matrix has real values on its diagonal
83 * and potentially complex numbers on its off-diagonals. By applying a
84 * similarity transform with an appropriate diagonal matrix
85 * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
86 * matrix can be transformed into a real symmetric matrix and complex
87 * arithmetic can be entirely avoided.)
88 *
89 * While the eigenvectors of the real symmetric tridiagonal matrix are real,
90 * the eigenvectors of original complex Hermitean matrix have complex entries
91 * in general.
92 * Since LAPACK drivers overwrite the matrix data with the eigenvectors,
93 * ZSTEMR accepts complex workspace to facilitate interoperability
94 * with ZUNMTR or ZUPMTR.
95 *
96 * Arguments
97 * =========
98 *
99 * JOBZ (input) CHARACTER*1
100 * = 'N': Compute eigenvalues only;
101 * = 'V': Compute eigenvalues and eigenvectors.
102 *
103 * RANGE (input) CHARACTER*1
104 * = 'A': all eigenvalues will be found.
105 * = 'V': all eigenvalues in the half-open interval (VL,VU]
106 * will be found.
107 * = 'I': the IL-th through IU-th eigenvalues will be found.
108 *
109 * N (input) INTEGER
110 * The order of the matrix. N >= 0.
111 *
112 * D (input/output) DOUBLE PRECISION array, dimension (N)
113 * On entry, the N diagonal elements of the tridiagonal matrix
114 * T. On exit, D is overwritten.
115 *
116 * E (input/output) DOUBLE PRECISION array, dimension (N)
117 * On entry, the (N-1) subdiagonal elements of the tridiagonal
118 * matrix T in elements 1 to N-1 of E. E(N) need not be set on
119 * input, but is used internally as workspace.
120 * On exit, E is overwritten.
121 *
122 * VL (input) DOUBLE PRECISION
123 * VU (input) DOUBLE PRECISION
124 * If RANGE='V', the lower and upper bounds of the interval to
125 * be searched for eigenvalues. VL < VU.
126 * Not referenced if RANGE = 'A' or 'I'.
127 *
128 * IL (input) INTEGER
129 * IU (input) INTEGER
130 * If RANGE='I', the indices (in ascending order) of the
131 * smallest and largest eigenvalues to be returned.
132 * 1 <= IL <= IU <= N, if N > 0.
133 * Not referenced if RANGE = 'A' or 'V'.
134 *
135 * M (output) INTEGER
136 * The total number of eigenvalues found. 0 <= M <= N.
137 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
138 *
139 * W (output) DOUBLE PRECISION array, dimension (N)
140 * The first M elements contain the selected eigenvalues in
141 * ascending order.
142 *
143 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M) )
144 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
145 * contain the orthonormal eigenvectors of the matrix T
146 * corresponding to the selected eigenvalues, with the i-th
147 * column of Z holding the eigenvector associated with W(i).
148 * If JOBZ = 'N', then Z is not referenced.
149 * Note: the user must ensure that at least max(1,M) columns are
150 * supplied in the array Z; if RANGE = 'V', the exact value of M
151 * is not known in advance and can be computed with a workspace
152 * query by setting NZC = -1, see below.
153 *
154 * LDZ (input) INTEGER
155 * The leading dimension of the array Z. LDZ >= 1, and if
156 * JOBZ = 'V', then LDZ >= max(1,N).
157 *
158 * NZC (input) INTEGER
159 * The number of eigenvectors to be held in the array Z.
160 * If RANGE = 'A', then NZC >= max(1,N).
161 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
162 * If RANGE = 'I', then NZC >= IU-IL+1.
163 * If NZC = -1, then a workspace query is assumed; the
164 * routine calculates the number of columns of the array Z that
165 * are needed to hold the eigenvectors.
166 * This value is returned as the first entry of the Z array, and
167 * no error message related to NZC is issued by XERBLA.
168 *
169 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
170 * The support of the eigenvectors in Z, i.e., the indices
171 * indicating the nonzero elements in Z. The i-th computed eigenvector
172 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
173 * ISUPPZ( 2*i ). This is relevant in the case when the matrix
174 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
175 *
176 * TRYRAC (input/output) LOGICAL
177 * If TRYRAC.EQ..TRUE., indicates that the code should check whether
178 * the tridiagonal matrix defines its eigenvalues to high relative
179 * accuracy. If so, the code uses relative-accuracy preserving
180 * algorithms that might be (a bit) slower depending on the matrix.
181 * If the matrix does not define its eigenvalues to high relative
182 * accuracy, the code can uses possibly faster algorithms.
183 * If TRYRAC.EQ..FALSE., the code is not required to guarantee
184 * relatively accurate eigenvalues and can use the fastest possible
185 * techniques.
186 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
187 * does not define its eigenvalues to high relative accuracy.
188 *
189 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
190 * On exit, if INFO = 0, WORK(1) returns the optimal
191 * (and minimal) LWORK.
192 *
193 * LWORK (input) INTEGER
194 * The dimension of the array WORK. LWORK >= max(1,18*N)
195 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
196 * If LWORK = -1, then a workspace query is assumed; the routine
197 * only calculates the optimal size of the WORK array, returns
198 * this value as the first entry of the WORK array, and no error
199 * message related to LWORK is issued by XERBLA.
200 *
201 * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
202 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
203 *
204 * LIWORK (input) INTEGER
205 * The dimension of the array IWORK. LIWORK >= max(1,10*N)
206 * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
207 * if only the eigenvalues are to be computed.
208 * If LIWORK = -1, then a workspace query is assumed; the
209 * routine only calculates the optimal size of the IWORK array,
210 * returns this value as the first entry of the IWORK array, and
211 * no error message related to LIWORK is issued by XERBLA.
212 *
213 * INFO (output) INTEGER
214 * On exit, INFO
215 * = 0: successful exit
216 * < 0: if INFO = -i, the i-th argument had an illegal value
217 * > 0: if INFO = 1X, internal error in DLARRE,
218 * if INFO = 2X, internal error in ZLARRV.
219 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
220 * the nonzero error code returned by DLARRE or
221 * ZLARRV, respectively.
222 *
223 *
224 * Further Details
225 * ===============
226 *
227 * Based on contributions by
228 * Beresford Parlett, University of California, Berkeley, USA
229 * Jim Demmel, University of California, Berkeley, USA
230 * Inderjit Dhillon, University of Texas, Austin, USA
231 * Osni Marques, LBNL/NERSC, USA
232 * Christof Voemel, University of California, Berkeley, USA
233 *
234 * =====================================================================
235 *
236 * .. Parameters ..
237 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
238 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
239 $ FOUR = 4.0D0,
240 $ MINRGP = 1.0D-3 )
241 * ..
242 * .. Local Scalars ..
243 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
244 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
245 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
246 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
247 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
248 $ NZCMIN, OFFSET, WBEGIN, WEND
249 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
250 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
251 $ THRESH, TMP, TNRM, WL, WU
252 * ..
253 * ..
254 * .. External Functions ..
255 LOGICAL LSAME
256 DOUBLE PRECISION DLAMCH, DLANST
257 EXTERNAL LSAME, DLAMCH, DLANST
258 * ..
259 * .. External Subroutines ..
260 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
261 $ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
262 * ..
263 * .. Intrinsic Functions ..
264 INTRINSIC MAX, MIN, SQRT
265
266
267 * ..
268 * .. Executable Statements ..
269 *
270 * Test the input parameters.
271 *
272 WANTZ = LSAME( JOBZ, 'V' )
273 ALLEIG = LSAME( RANGE, 'A' )
274 VALEIG = LSAME( RANGE, 'V' )
275 INDEIG = LSAME( RANGE, 'I' )
276 *
277 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
278 ZQUERY = ( NZC.EQ.-1 )
279
280 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
281 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
282 * Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
283 IF( WANTZ ) THEN
284 LWMIN = 18*N
285 LIWMIN = 10*N
286 ELSE
287 * need less workspace if only the eigenvalues are wanted
288 LWMIN = 12*N
289 LIWMIN = 8*N
290 ENDIF
291
292 WL = ZERO
293 WU = ZERO
294 IIL = 0
295 IIU = 0
296
297 IF( VALEIG ) THEN
298 * We do not reference VL, VU in the cases RANGE = 'I','A'
299 * The interval (WL, WU] contains all the wanted eigenvalues.
300 * It is either given by the user or computed in DLARRE.
301 WL = VL
302 WU = VU
303 ELSEIF( INDEIG ) THEN
304 * We do not reference IL, IU in the cases RANGE = 'V','A'
305 IIL = IL
306 IIU = IU
307 ENDIF
308 *
309 INFO = 0
310 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
311 INFO = -1
312 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
313 INFO = -2
314 ELSE IF( N.LT.0 ) THEN
315 INFO = -3
316 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
317 INFO = -7
318 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
319 INFO = -8
320 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
321 INFO = -9
322 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
323 INFO = -13
324 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
325 INFO = -17
326 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
327 INFO = -19
328 END IF
329 *
330 * Get machine constants.
331 *
332 SAFMIN = DLAMCH( 'Safe minimum' )
333 EPS = DLAMCH( 'Precision' )
334 SMLNUM = SAFMIN / EPS
335 BIGNUM = ONE / SMLNUM
336 RMIN = SQRT( SMLNUM )
337 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
338 *
339 IF( INFO.EQ.0 ) THEN
340 WORK( 1 ) = LWMIN
341 IWORK( 1 ) = LIWMIN
342 *
343 IF( WANTZ .AND. ALLEIG ) THEN
344 NZCMIN = N
345 ELSE IF( WANTZ .AND. VALEIG ) THEN
346 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
347 $ NZCMIN, ITMP, ITMP2, INFO )
348 ELSE IF( WANTZ .AND. INDEIG ) THEN
349 NZCMIN = IIU-IIL+1
350 ELSE
351 * WANTZ .EQ. FALSE.
352 NZCMIN = 0
353 ENDIF
354 IF( ZQUERY .AND. INFO.EQ.0 ) THEN
355 Z( 1,1 ) = NZCMIN
356 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
357 INFO = -14
358 END IF
359 END IF
360
361 IF( INFO.NE.0 ) THEN
362 *
363 CALL XERBLA( 'ZSTEMR', -INFO )
364 *
365 RETURN
366 ELSE IF( LQUERY .OR. ZQUERY ) THEN
367 RETURN
368 END IF
369 *
370 * Handle N = 0, 1, and 2 cases immediately
371 *
372 M = 0
373 IF( N.EQ.0 )
374 $ RETURN
375 *
376 IF( N.EQ.1 ) THEN
377 IF( ALLEIG .OR. INDEIG ) THEN
378 M = 1
379 W( 1 ) = D( 1 )
380 ELSE
381 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
382 M = 1
383 W( 1 ) = D( 1 )
384 END IF
385 END IF
386 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
387 Z( 1, 1 ) = ONE
388 ISUPPZ(1) = 1
389 ISUPPZ(2) = 1
390 END IF
391 RETURN
392 END IF
393 *
394 IF( N.EQ.2 ) THEN
395 IF( .NOT.WANTZ ) THEN
396 CALL DLAE2( D(1), E(1), D(2), R1, R2 )
397 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
398 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
399 END IF
400 IF( ALLEIG.OR.
401 $ (VALEIG.AND.(R2.GT.WL).AND.
402 $ (R2.LE.WU)).OR.
403 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
404 M = M+1
405 W( M ) = R2
406 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
407 Z( 1, M ) = -SN
408 Z( 2, M ) = CS
409 * Note: At most one of SN and CS can be zero.
410 IF (SN.NE.ZERO) THEN
411 IF (CS.NE.ZERO) THEN
412 ISUPPZ(2*M-1) = 1
413 ISUPPZ(2*M-1) = 2
414 ELSE
415 ISUPPZ(2*M-1) = 1
416 ISUPPZ(2*M-1) = 1
417 END IF
418 ELSE
419 ISUPPZ(2*M-1) = 2
420 ISUPPZ(2*M) = 2
421 END IF
422 ENDIF
423 ENDIF
424 IF( ALLEIG.OR.
425 $ (VALEIG.AND.(R1.GT.WL).AND.
426 $ (R1.LE.WU)).OR.
427 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
428 M = M+1
429 W( M ) = R1
430 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
431 Z( 1, M ) = CS
432 Z( 2, M ) = SN
433 * Note: At most one of SN and CS can be zero.
434 IF (SN.NE.ZERO) THEN
435 IF (CS.NE.ZERO) THEN
436 ISUPPZ(2*M-1) = 1
437 ISUPPZ(2*M-1) = 2
438 ELSE
439 ISUPPZ(2*M-1) = 1
440 ISUPPZ(2*M-1) = 1
441 END IF
442 ELSE
443 ISUPPZ(2*M-1) = 2
444 ISUPPZ(2*M) = 2
445 END IF
446 ENDIF
447 ENDIF
448 RETURN
449 END IF
450
451 * Continue with general N
452
453 INDGRS = 1
454 INDERR = 2*N + 1
455 INDGP = 3*N + 1
456 INDD = 4*N + 1
457 INDE2 = 5*N + 1
458 INDWRK = 6*N + 1
459 *
460 IINSPL = 1
461 IINDBL = N + 1
462 IINDW = 2*N + 1
463 IINDWK = 3*N + 1
464 *
465 * Scale matrix to allowable range, if necessary.
466 * The allowable range is related to the PIVMIN parameter; see the
467 * comments in DLARRD. The preference for scaling small values
468 * up is heuristic; we expect users' matrices not to be close to the
469 * RMAX threshold.
470 *
471 SCALE = ONE
472 TNRM = DLANST( 'M', N, D, E )
473 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
474 SCALE = RMIN / TNRM
475 ELSE IF( TNRM.GT.RMAX ) THEN
476 SCALE = RMAX / TNRM
477 END IF
478 IF( SCALE.NE.ONE ) THEN
479 CALL DSCAL( N, SCALE, D, 1 )
480 CALL DSCAL( N-1, SCALE, E, 1 )
481 TNRM = TNRM*SCALE
482 IF( VALEIG ) THEN
483 * If eigenvalues in interval have to be found,
484 * scale (WL, WU] accordingly
485 WL = WL*SCALE
486 WU = WU*SCALE
487 ENDIF
488 END IF
489 *
490 * Compute the desired eigenvalues of the tridiagonal after splitting
491 * into smaller subblocks if the corresponding off-diagonal elements
492 * are small
493 * THRESH is the splitting parameter for DLARRE
494 * A negative THRESH forces the old splitting criterion based on the
495 * size of the off-diagonal. A positive THRESH switches to splitting
496 * which preserves relative accuracy.
497 *
498 IF( TRYRAC ) THEN
499 * Test whether the matrix warrants the more expensive relative approach.
500 CALL DLARRR( N, D, E, IINFO )
501 ELSE
502 * The user does not care about relative accurately eigenvalues
503 IINFO = -1
504 ENDIF
505 * Set the splitting criterion
506 IF (IINFO.EQ.0) THEN
507 THRESH = EPS
508 ELSE
509 THRESH = -EPS
510 * relative accuracy is desired but T does not guarantee it
511 TRYRAC = .FALSE.
512 ENDIF
513 *
514 IF( TRYRAC ) THEN
515 * Copy original diagonal, needed to guarantee relative accuracy
516 CALL DCOPY(N,D,1,WORK(INDD),1)
517 ENDIF
518 * Store the squares of the offdiagonal values of T
519 DO 5 J = 1, N-1
520 WORK( INDE2+J-1 ) = E(J)**2
521 5 CONTINUE
522
523 * Set the tolerance parameters for bisection
524 IF( .NOT.WANTZ ) THEN
525 * DLARRE computes the eigenvalues to full precision.
526 RTOL1 = FOUR * EPS
527 RTOL2 = FOUR * EPS
528 ELSE
529 * DLARRE computes the eigenvalues to less than full precision.
530 * ZLARRV will refine the eigenvalue approximations, and we only
531 * need less accurate initial bisection in DLARRE.
532 * Note: these settings do only affect the subset case and DLARRE
533 RTOL1 = SQRT(EPS)
534 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
535 ENDIF
536 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
537 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
538 $ IWORK( IINSPL ), M, W, WORK( INDERR ),
539 $ WORK( INDGP ), IWORK( IINDBL ),
540 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
541 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
542 IF( IINFO.NE.0 ) THEN
543 INFO = 10 + ABS( IINFO )
544 RETURN
545 END IF
546 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
547 * part of the spectrum. All desired eigenvalues are contained in
548 * (WL,WU]
549
550
551 IF( WANTZ ) THEN
552 *
553 * Compute the desired eigenvectors corresponding to the computed
554 * eigenvalues
555 *
556 CALL ZLARRV( N, WL, WU, D, E,
557 $ PIVMIN, IWORK( IINSPL ), M,
558 $ 1, M, MINRGP, RTOL1, RTOL2,
559 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
560 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
561 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
562 IF( IINFO.NE.0 ) THEN
563 INFO = 20 + ABS( IINFO )
564 RETURN
565 END IF
566 ELSE
567 * DLARRE computes eigenvalues of the (shifted) root representation
568 * ZLARRV returns the eigenvalues of the unshifted matrix.
569 * However, if the eigenvectors are not desired by the user, we need
570 * to apply the corresponding shifts from DLARRE to obtain the
571 * eigenvalues of the original matrix.
572 DO 20 J = 1, M
573 ITMP = IWORK( IINDBL+J-1 )
574 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
575 20 CONTINUE
576 END IF
577 *
578
579 IF ( TRYRAC ) THEN
580 * Refine computed eigenvalues so that they are relatively accurate
581 * with respect to the original matrix T.
582 IBEGIN = 1
583 WBEGIN = 1
584 DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
585 IEND = IWORK( IINSPL+JBLK-1 )
586 IN = IEND - IBEGIN + 1
587 WEND = WBEGIN - 1
588 * check if any eigenvalues have to be refined in this block
589 36 CONTINUE
590 IF( WEND.LT.M ) THEN
591 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
592 WEND = WEND + 1
593 GO TO 36
594 END IF
595 END IF
596 IF( WEND.LT.WBEGIN ) THEN
597 IBEGIN = IEND + 1
598 GO TO 39
599 END IF
600
601 OFFSET = IWORK(IINDW+WBEGIN-1)-1
602 IFIRST = IWORK(IINDW+WBEGIN-1)
603 ILAST = IWORK(IINDW+WEND-1)
604 RTOL2 = FOUR * EPS
605 CALL DLARRJ( IN,
606 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
607 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
608 $ WORK( INDERR+WBEGIN-1 ),
609 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
610 $ TNRM, IINFO )
611 IBEGIN = IEND + 1
612 WBEGIN = WEND + 1
613 39 CONTINUE
614 ENDIF
615 *
616 * If matrix was scaled, then rescale eigenvalues appropriately.
617 *
618 IF( SCALE.NE.ONE ) THEN
619 CALL DSCAL( M, ONE / SCALE, W, 1 )
620 END IF
621 *
622 * If eigenvalues are not in increasing order, then sort them,
623 * possibly along with eigenvectors.
624 *
625 IF( NSPLIT.GT.1 ) THEN
626 IF( .NOT. WANTZ ) THEN
627 CALL DLASRT( 'I', M, W, IINFO )
628 IF( IINFO.NE.0 ) THEN
629 INFO = 3
630 RETURN
631 END IF
632 ELSE
633 DO 60 J = 1, M - 1
634 I = 0
635 TMP = W( J )
636 DO 50 JJ = J + 1, M
637 IF( W( JJ ).LT.TMP ) THEN
638 I = JJ
639 TMP = W( JJ )
640 END IF
641 50 CONTINUE
642 IF( I.NE.0 ) THEN
643 W( I ) = W( J )
644 W( J ) = TMP
645 IF( WANTZ ) THEN
646 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
647 ITMP = ISUPPZ( 2*I-1 )
648 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
649 ISUPPZ( 2*J-1 ) = ITMP
650 ITMP = ISUPPZ( 2*I )
651 ISUPPZ( 2*I ) = ISUPPZ( 2*J )
652 ISUPPZ( 2*J ) = ITMP
653 END IF
654 END IF
655 60 CONTINUE
656 END IF
657 ENDIF
658 *
659 *
660 WORK( 1 ) = LWMIN
661 IWORK( 1 ) = LIWMIN
662 RETURN
663 *
664 * End of ZSTEMR
665 *
666 END
2 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
3 $ IWORK, LIWORK, INFO )
4 IMPLICIT NONE
5 *
6 * -- LAPACK computational routine (version 3.2.1) --
7 *
8 * -- April 2009 --
9 *
10 * -- LAPACK is a software package provided by Univ. of Tennessee, --
11 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12 *
13 * .. Scalar Arguments ..
14 CHARACTER JOBZ, RANGE
15 LOGICAL TRYRAC
16 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
17 DOUBLE PRECISION VL, VU
18 * ..
19 * .. Array Arguments ..
20 INTEGER ISUPPZ( * ), IWORK( * )
21 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
22 COMPLEX*16 Z( LDZ, * )
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
29 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
30 * a well defined set of pairwise different real eigenvalues, the corresponding
31 * real eigenvectors are pairwise orthogonal.
32 *
33 * The spectrum may be computed either completely or partially by specifying
34 * either an interval (VL,VU] or a range of indices IL:IU for the desired
35 * eigenvalues.
36 *
37 * Depending on the number of desired eigenvalues, these are computed either
38 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
39 * computed by the use of various suitable L D L^T factorizations near clusters
40 * of close eigenvalues (referred to as RRRs, Relatively Robust
41 * Representations). An informal sketch of the algorithm follows.
42 *
43 * For each unreduced block (submatrix) of T,
44 * (a) Compute T - sigma I = L D L^T, so that L and D
45 * define all the wanted eigenvalues to high relative accuracy.
46 * This means that small relative changes in the entries of D and L
47 * cause only small relative changes in the eigenvalues and
48 * eigenvectors. The standard (unfactored) representation of the
49 * tridiagonal matrix T does not have this property in general.
50 * (b) Compute the eigenvalues to suitable accuracy.
51 * If the eigenvectors are desired, the algorithm attains full
52 * accuracy of the computed eigenvalues only right before
53 * the corresponding vectors have to be computed, see steps c) and d).
54 * (c) For each cluster of close eigenvalues, select a new
55 * shift close to the cluster, find a new factorization, and refine
56 * the shifted eigenvalues to suitable accuracy.
57 * (d) For each eigenvalue with a large enough relative separation compute
58 * the corresponding eigenvector by forming a rank revealing twisted
59 * factorization. Go back to (c) for any clusters that remain.
60 *
61 * For more details, see:
62 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
63 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
64 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
65 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
66 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
67 * 2004. Also LAPACK Working Note 154.
68 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
69 * tridiagonal eigenvalue/eigenvector problem",
70 * Computer Science Division Technical Report No. UCB/CSD-97-971,
71 * UC Berkeley, May 1997.
72 *
73 * Further Details
74 * 1.ZSTEMR works only on machines which follow IEEE-754
75 * floating-point standard in their handling of infinities and NaNs.
76 * This permits the use of efficient inner loops avoiding a check for
77 * zero divisors.
78 *
79 * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
80 * real symmetric tridiagonal form.
81 *
82 * (Any complex Hermitean tridiagonal matrix has real values on its diagonal
83 * and potentially complex numbers on its off-diagonals. By applying a
84 * similarity transform with an appropriate diagonal matrix
85 * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
86 * matrix can be transformed into a real symmetric matrix and complex
87 * arithmetic can be entirely avoided.)
88 *
89 * While the eigenvectors of the real symmetric tridiagonal matrix are real,
90 * the eigenvectors of original complex Hermitean matrix have complex entries
91 * in general.
92 * Since LAPACK drivers overwrite the matrix data with the eigenvectors,
93 * ZSTEMR accepts complex workspace to facilitate interoperability
94 * with ZUNMTR or ZUPMTR.
95 *
96 * Arguments
97 * =========
98 *
99 * JOBZ (input) CHARACTER*1
100 * = 'N': Compute eigenvalues only;
101 * = 'V': Compute eigenvalues and eigenvectors.
102 *
103 * RANGE (input) CHARACTER*1
104 * = 'A': all eigenvalues will be found.
105 * = 'V': all eigenvalues in the half-open interval (VL,VU]
106 * will be found.
107 * = 'I': the IL-th through IU-th eigenvalues will be found.
108 *
109 * N (input) INTEGER
110 * The order of the matrix. N >= 0.
111 *
112 * D (input/output) DOUBLE PRECISION array, dimension (N)
113 * On entry, the N diagonal elements of the tridiagonal matrix
114 * T. On exit, D is overwritten.
115 *
116 * E (input/output) DOUBLE PRECISION array, dimension (N)
117 * On entry, the (N-1) subdiagonal elements of the tridiagonal
118 * matrix T in elements 1 to N-1 of E. E(N) need not be set on
119 * input, but is used internally as workspace.
120 * On exit, E is overwritten.
121 *
122 * VL (input) DOUBLE PRECISION
123 * VU (input) DOUBLE PRECISION
124 * If RANGE='V', the lower and upper bounds of the interval to
125 * be searched for eigenvalues. VL < VU.
126 * Not referenced if RANGE = 'A' or 'I'.
127 *
128 * IL (input) INTEGER
129 * IU (input) INTEGER
130 * If RANGE='I', the indices (in ascending order) of the
131 * smallest and largest eigenvalues to be returned.
132 * 1 <= IL <= IU <= N, if N > 0.
133 * Not referenced if RANGE = 'A' or 'V'.
134 *
135 * M (output) INTEGER
136 * The total number of eigenvalues found. 0 <= M <= N.
137 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
138 *
139 * W (output) DOUBLE PRECISION array, dimension (N)
140 * The first M elements contain the selected eigenvalues in
141 * ascending order.
142 *
143 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M) )
144 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
145 * contain the orthonormal eigenvectors of the matrix T
146 * corresponding to the selected eigenvalues, with the i-th
147 * column of Z holding the eigenvector associated with W(i).
148 * If JOBZ = 'N', then Z is not referenced.
149 * Note: the user must ensure that at least max(1,M) columns are
150 * supplied in the array Z; if RANGE = 'V', the exact value of M
151 * is not known in advance and can be computed with a workspace
152 * query by setting NZC = -1, see below.
153 *
154 * LDZ (input) INTEGER
155 * The leading dimension of the array Z. LDZ >= 1, and if
156 * JOBZ = 'V', then LDZ >= max(1,N).
157 *
158 * NZC (input) INTEGER
159 * The number of eigenvectors to be held in the array Z.
160 * If RANGE = 'A', then NZC >= max(1,N).
161 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
162 * If RANGE = 'I', then NZC >= IU-IL+1.
163 * If NZC = -1, then a workspace query is assumed; the
164 * routine calculates the number of columns of the array Z that
165 * are needed to hold the eigenvectors.
166 * This value is returned as the first entry of the Z array, and
167 * no error message related to NZC is issued by XERBLA.
168 *
169 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
170 * The support of the eigenvectors in Z, i.e., the indices
171 * indicating the nonzero elements in Z. The i-th computed eigenvector
172 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
173 * ISUPPZ( 2*i ). This is relevant in the case when the matrix
174 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
175 *
176 * TRYRAC (input/output) LOGICAL
177 * If TRYRAC.EQ..TRUE., indicates that the code should check whether
178 * the tridiagonal matrix defines its eigenvalues to high relative
179 * accuracy. If so, the code uses relative-accuracy preserving
180 * algorithms that might be (a bit) slower depending on the matrix.
181 * If the matrix does not define its eigenvalues to high relative
182 * accuracy, the code can uses possibly faster algorithms.
183 * If TRYRAC.EQ..FALSE., the code is not required to guarantee
184 * relatively accurate eigenvalues and can use the fastest possible
185 * techniques.
186 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
187 * does not define its eigenvalues to high relative accuracy.
188 *
189 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
190 * On exit, if INFO = 0, WORK(1) returns the optimal
191 * (and minimal) LWORK.
192 *
193 * LWORK (input) INTEGER
194 * The dimension of the array WORK. LWORK >= max(1,18*N)
195 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
196 * If LWORK = -1, then a workspace query is assumed; the routine
197 * only calculates the optimal size of the WORK array, returns
198 * this value as the first entry of the WORK array, and no error
199 * message related to LWORK is issued by XERBLA.
200 *
201 * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
202 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
203 *
204 * LIWORK (input) INTEGER
205 * The dimension of the array IWORK. LIWORK >= max(1,10*N)
206 * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
207 * if only the eigenvalues are to be computed.
208 * If LIWORK = -1, then a workspace query is assumed; the
209 * routine only calculates the optimal size of the IWORK array,
210 * returns this value as the first entry of the IWORK array, and
211 * no error message related to LIWORK is issued by XERBLA.
212 *
213 * INFO (output) INTEGER
214 * On exit, INFO
215 * = 0: successful exit
216 * < 0: if INFO = -i, the i-th argument had an illegal value
217 * > 0: if INFO = 1X, internal error in DLARRE,
218 * if INFO = 2X, internal error in ZLARRV.
219 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
220 * the nonzero error code returned by DLARRE or
221 * ZLARRV, respectively.
222 *
223 *
224 * Further Details
225 * ===============
226 *
227 * Based on contributions by
228 * Beresford Parlett, University of California, Berkeley, USA
229 * Jim Demmel, University of California, Berkeley, USA
230 * Inderjit Dhillon, University of Texas, Austin, USA
231 * Osni Marques, LBNL/NERSC, USA
232 * Christof Voemel, University of California, Berkeley, USA
233 *
234 * =====================================================================
235 *
236 * .. Parameters ..
237 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
238 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
239 $ FOUR = 4.0D0,
240 $ MINRGP = 1.0D-3 )
241 * ..
242 * .. Local Scalars ..
243 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
244 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
245 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
246 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
247 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
248 $ NZCMIN, OFFSET, WBEGIN, WEND
249 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
250 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
251 $ THRESH, TMP, TNRM, WL, WU
252 * ..
253 * ..
254 * .. External Functions ..
255 LOGICAL LSAME
256 DOUBLE PRECISION DLAMCH, DLANST
257 EXTERNAL LSAME, DLAMCH, DLANST
258 * ..
259 * .. External Subroutines ..
260 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
261 $ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
262 * ..
263 * .. Intrinsic Functions ..
264 INTRINSIC MAX, MIN, SQRT
265
266
267 * ..
268 * .. Executable Statements ..
269 *
270 * Test the input parameters.
271 *
272 WANTZ = LSAME( JOBZ, 'V' )
273 ALLEIG = LSAME( RANGE, 'A' )
274 VALEIG = LSAME( RANGE, 'V' )
275 INDEIG = LSAME( RANGE, 'I' )
276 *
277 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
278 ZQUERY = ( NZC.EQ.-1 )
279
280 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
281 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
282 * Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
283 IF( WANTZ ) THEN
284 LWMIN = 18*N
285 LIWMIN = 10*N
286 ELSE
287 * need less workspace if only the eigenvalues are wanted
288 LWMIN = 12*N
289 LIWMIN = 8*N
290 ENDIF
291
292 WL = ZERO
293 WU = ZERO
294 IIL = 0
295 IIU = 0
296
297 IF( VALEIG ) THEN
298 * We do not reference VL, VU in the cases RANGE = 'I','A'
299 * The interval (WL, WU] contains all the wanted eigenvalues.
300 * It is either given by the user or computed in DLARRE.
301 WL = VL
302 WU = VU
303 ELSEIF( INDEIG ) THEN
304 * We do not reference IL, IU in the cases RANGE = 'V','A'
305 IIL = IL
306 IIU = IU
307 ENDIF
308 *
309 INFO = 0
310 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
311 INFO = -1
312 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
313 INFO = -2
314 ELSE IF( N.LT.0 ) THEN
315 INFO = -3
316 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
317 INFO = -7
318 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
319 INFO = -8
320 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
321 INFO = -9
322 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
323 INFO = -13
324 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
325 INFO = -17
326 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
327 INFO = -19
328 END IF
329 *
330 * Get machine constants.
331 *
332 SAFMIN = DLAMCH( 'Safe minimum' )
333 EPS = DLAMCH( 'Precision' )
334 SMLNUM = SAFMIN / EPS
335 BIGNUM = ONE / SMLNUM
336 RMIN = SQRT( SMLNUM )
337 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
338 *
339 IF( INFO.EQ.0 ) THEN
340 WORK( 1 ) = LWMIN
341 IWORK( 1 ) = LIWMIN
342 *
343 IF( WANTZ .AND. ALLEIG ) THEN
344 NZCMIN = N
345 ELSE IF( WANTZ .AND. VALEIG ) THEN
346 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
347 $ NZCMIN, ITMP, ITMP2, INFO )
348 ELSE IF( WANTZ .AND. INDEIG ) THEN
349 NZCMIN = IIU-IIL+1
350 ELSE
351 * WANTZ .EQ. FALSE.
352 NZCMIN = 0
353 ENDIF
354 IF( ZQUERY .AND. INFO.EQ.0 ) THEN
355 Z( 1,1 ) = NZCMIN
356 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
357 INFO = -14
358 END IF
359 END IF
360
361 IF( INFO.NE.0 ) THEN
362 *
363 CALL XERBLA( 'ZSTEMR', -INFO )
364 *
365 RETURN
366 ELSE IF( LQUERY .OR. ZQUERY ) THEN
367 RETURN
368 END IF
369 *
370 * Handle N = 0, 1, and 2 cases immediately
371 *
372 M = 0
373 IF( N.EQ.0 )
374 $ RETURN
375 *
376 IF( N.EQ.1 ) THEN
377 IF( ALLEIG .OR. INDEIG ) THEN
378 M = 1
379 W( 1 ) = D( 1 )
380 ELSE
381 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
382 M = 1
383 W( 1 ) = D( 1 )
384 END IF
385 END IF
386 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
387 Z( 1, 1 ) = ONE
388 ISUPPZ(1) = 1
389 ISUPPZ(2) = 1
390 END IF
391 RETURN
392 END IF
393 *
394 IF( N.EQ.2 ) THEN
395 IF( .NOT.WANTZ ) THEN
396 CALL DLAE2( D(1), E(1), D(2), R1, R2 )
397 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
398 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
399 END IF
400 IF( ALLEIG.OR.
401 $ (VALEIG.AND.(R2.GT.WL).AND.
402 $ (R2.LE.WU)).OR.
403 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
404 M = M+1
405 W( M ) = R2
406 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
407 Z( 1, M ) = -SN
408 Z( 2, M ) = CS
409 * Note: At most one of SN and CS can be zero.
410 IF (SN.NE.ZERO) THEN
411 IF (CS.NE.ZERO) THEN
412 ISUPPZ(2*M-1) = 1
413 ISUPPZ(2*M-1) = 2
414 ELSE
415 ISUPPZ(2*M-1) = 1
416 ISUPPZ(2*M-1) = 1
417 END IF
418 ELSE
419 ISUPPZ(2*M-1) = 2
420 ISUPPZ(2*M) = 2
421 END IF
422 ENDIF
423 ENDIF
424 IF( ALLEIG.OR.
425 $ (VALEIG.AND.(R1.GT.WL).AND.
426 $ (R1.LE.WU)).OR.
427 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
428 M = M+1
429 W( M ) = R1
430 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
431 Z( 1, M ) = CS
432 Z( 2, M ) = SN
433 * Note: At most one of SN and CS can be zero.
434 IF (SN.NE.ZERO) THEN
435 IF (CS.NE.ZERO) THEN
436 ISUPPZ(2*M-1) = 1
437 ISUPPZ(2*M-1) = 2
438 ELSE
439 ISUPPZ(2*M-1) = 1
440 ISUPPZ(2*M-1) = 1
441 END IF
442 ELSE
443 ISUPPZ(2*M-1) = 2
444 ISUPPZ(2*M) = 2
445 END IF
446 ENDIF
447 ENDIF
448 RETURN
449 END IF
450
451 * Continue with general N
452
453 INDGRS = 1
454 INDERR = 2*N + 1
455 INDGP = 3*N + 1
456 INDD = 4*N + 1
457 INDE2 = 5*N + 1
458 INDWRK = 6*N + 1
459 *
460 IINSPL = 1
461 IINDBL = N + 1
462 IINDW = 2*N + 1
463 IINDWK = 3*N + 1
464 *
465 * Scale matrix to allowable range, if necessary.
466 * The allowable range is related to the PIVMIN parameter; see the
467 * comments in DLARRD. The preference for scaling small values
468 * up is heuristic; we expect users' matrices not to be close to the
469 * RMAX threshold.
470 *
471 SCALE = ONE
472 TNRM = DLANST( 'M', N, D, E )
473 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
474 SCALE = RMIN / TNRM
475 ELSE IF( TNRM.GT.RMAX ) THEN
476 SCALE = RMAX / TNRM
477 END IF
478 IF( SCALE.NE.ONE ) THEN
479 CALL DSCAL( N, SCALE, D, 1 )
480 CALL DSCAL( N-1, SCALE, E, 1 )
481 TNRM = TNRM*SCALE
482 IF( VALEIG ) THEN
483 * If eigenvalues in interval have to be found,
484 * scale (WL, WU] accordingly
485 WL = WL*SCALE
486 WU = WU*SCALE
487 ENDIF
488 END IF
489 *
490 * Compute the desired eigenvalues of the tridiagonal after splitting
491 * into smaller subblocks if the corresponding off-diagonal elements
492 * are small
493 * THRESH is the splitting parameter for DLARRE
494 * A negative THRESH forces the old splitting criterion based on the
495 * size of the off-diagonal. A positive THRESH switches to splitting
496 * which preserves relative accuracy.
497 *
498 IF( TRYRAC ) THEN
499 * Test whether the matrix warrants the more expensive relative approach.
500 CALL DLARRR( N, D, E, IINFO )
501 ELSE
502 * The user does not care about relative accurately eigenvalues
503 IINFO = -1
504 ENDIF
505 * Set the splitting criterion
506 IF (IINFO.EQ.0) THEN
507 THRESH = EPS
508 ELSE
509 THRESH = -EPS
510 * relative accuracy is desired but T does not guarantee it
511 TRYRAC = .FALSE.
512 ENDIF
513 *
514 IF( TRYRAC ) THEN
515 * Copy original diagonal, needed to guarantee relative accuracy
516 CALL DCOPY(N,D,1,WORK(INDD),1)
517 ENDIF
518 * Store the squares of the offdiagonal values of T
519 DO 5 J = 1, N-1
520 WORK( INDE2+J-1 ) = E(J)**2
521 5 CONTINUE
522
523 * Set the tolerance parameters for bisection
524 IF( .NOT.WANTZ ) THEN
525 * DLARRE computes the eigenvalues to full precision.
526 RTOL1 = FOUR * EPS
527 RTOL2 = FOUR * EPS
528 ELSE
529 * DLARRE computes the eigenvalues to less than full precision.
530 * ZLARRV will refine the eigenvalue approximations, and we only
531 * need less accurate initial bisection in DLARRE.
532 * Note: these settings do only affect the subset case and DLARRE
533 RTOL1 = SQRT(EPS)
534 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
535 ENDIF
536 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
537 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
538 $ IWORK( IINSPL ), M, W, WORK( INDERR ),
539 $ WORK( INDGP ), IWORK( IINDBL ),
540 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
541 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
542 IF( IINFO.NE.0 ) THEN
543 INFO = 10 + ABS( IINFO )
544 RETURN
545 END IF
546 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
547 * part of the spectrum. All desired eigenvalues are contained in
548 * (WL,WU]
549
550
551 IF( WANTZ ) THEN
552 *
553 * Compute the desired eigenvectors corresponding to the computed
554 * eigenvalues
555 *
556 CALL ZLARRV( N, WL, WU, D, E,
557 $ PIVMIN, IWORK( IINSPL ), M,
558 $ 1, M, MINRGP, RTOL1, RTOL2,
559 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
560 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
561 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
562 IF( IINFO.NE.0 ) THEN
563 INFO = 20 + ABS( IINFO )
564 RETURN
565 END IF
566 ELSE
567 * DLARRE computes eigenvalues of the (shifted) root representation
568 * ZLARRV returns the eigenvalues of the unshifted matrix.
569 * However, if the eigenvectors are not desired by the user, we need
570 * to apply the corresponding shifts from DLARRE to obtain the
571 * eigenvalues of the original matrix.
572 DO 20 J = 1, M
573 ITMP = IWORK( IINDBL+J-1 )
574 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
575 20 CONTINUE
576 END IF
577 *
578
579 IF ( TRYRAC ) THEN
580 * Refine computed eigenvalues so that they are relatively accurate
581 * with respect to the original matrix T.
582 IBEGIN = 1
583 WBEGIN = 1
584 DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
585 IEND = IWORK( IINSPL+JBLK-1 )
586 IN = IEND - IBEGIN + 1
587 WEND = WBEGIN - 1
588 * check if any eigenvalues have to be refined in this block
589 36 CONTINUE
590 IF( WEND.LT.M ) THEN
591 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
592 WEND = WEND + 1
593 GO TO 36
594 END IF
595 END IF
596 IF( WEND.LT.WBEGIN ) THEN
597 IBEGIN = IEND + 1
598 GO TO 39
599 END IF
600
601 OFFSET = IWORK(IINDW+WBEGIN-1)-1
602 IFIRST = IWORK(IINDW+WBEGIN-1)
603 ILAST = IWORK(IINDW+WEND-1)
604 RTOL2 = FOUR * EPS
605 CALL DLARRJ( IN,
606 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
607 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
608 $ WORK( INDERR+WBEGIN-1 ),
609 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
610 $ TNRM, IINFO )
611 IBEGIN = IEND + 1
612 WBEGIN = WEND + 1
613 39 CONTINUE
614 ENDIF
615 *
616 * If matrix was scaled, then rescale eigenvalues appropriately.
617 *
618 IF( SCALE.NE.ONE ) THEN
619 CALL DSCAL( M, ONE / SCALE, W, 1 )
620 END IF
621 *
622 * If eigenvalues are not in increasing order, then sort them,
623 * possibly along with eigenvectors.
624 *
625 IF( NSPLIT.GT.1 ) THEN
626 IF( .NOT. WANTZ ) THEN
627 CALL DLASRT( 'I', M, W, IINFO )
628 IF( IINFO.NE.0 ) THEN
629 INFO = 3
630 RETURN
631 END IF
632 ELSE
633 DO 60 J = 1, M - 1
634 I = 0
635 TMP = W( J )
636 DO 50 JJ = J + 1, M
637 IF( W( JJ ).LT.TMP ) THEN
638 I = JJ
639 TMP = W( JJ )
640 END IF
641 50 CONTINUE
642 IF( I.NE.0 ) THEN
643 W( I ) = W( J )
644 W( J ) = TMP
645 IF( WANTZ ) THEN
646 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
647 ITMP = ISUPPZ( 2*I-1 )
648 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
649 ISUPPZ( 2*J-1 ) = ITMP
650 ITMP = ISUPPZ( 2*I )
651 ISUPPZ( 2*I ) = ISUPPZ( 2*J )
652 ISUPPZ( 2*J ) = ITMP
653 END IF
654 END IF
655 60 CONTINUE
656 END IF
657 ENDIF
658 *
659 *
660 WORK( 1 ) = LWMIN
661 IWORK( 1 ) = LIWMIN
662 RETURN
663 *
664 * End of ZSTEMR
665 *
666 END