1 SUBROUTINE DSTEMR( 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 3.2.2) --
7 * -- LAPACK is a software package provided by Univ. of Tennessee, --
8 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
9 * -- June 2010 --
10 *
11 * .. Scalar Arguments ..
12 CHARACTER JOBZ, RANGE
13 LOGICAL TRYRAC
14 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
15 DOUBLE PRECISION VL, VU
16 * ..
17 * .. Array Arguments ..
18 INTEGER ISUPPZ( * ), IWORK( * )
19 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
20 DOUBLE PRECISION Z( LDZ, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * DSTEMR computes selected eigenvalues and, optionally, eigenvectors
27 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
28 * a well defined set of pairwise different real eigenvalues, the corresponding
29 * real eigenvectors are pairwise orthogonal.
30 *
31 * The spectrum may be computed either completely or partially by specifying
32 * either an interval (VL,VU] or a range of indices IL:IU for the desired
33 * eigenvalues.
34 *
35 * Depending on the number of desired eigenvalues, these are computed either
36 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
37 * computed by the use of various suitable L D L^T factorizations near clusters
38 * of close eigenvalues (referred to as RRRs, Relatively Robust
39 * Representations). An informal sketch of the algorithm follows.
40 *
41 * For each unreduced block (submatrix) of T,
42 * (a) Compute T - sigma I = L D L^T, so that L and D
43 * define all the wanted eigenvalues to high relative accuracy.
44 * This means that small relative changes in the entries of D and L
45 * cause only small relative changes in the eigenvalues and
46 * eigenvectors. The standard (unfactored) representation of the
47 * tridiagonal matrix T does not have this property in general.
48 * (b) Compute the eigenvalues to suitable accuracy.
49 * If the eigenvectors are desired, the algorithm attains full
50 * accuracy of the computed eigenvalues only right before
51 * the corresponding vectors have to be computed, see steps c) and d).
52 * (c) For each cluster of close eigenvalues, select a new
53 * shift close to the cluster, find a new factorization, and refine
54 * the shifted eigenvalues to suitable accuracy.
55 * (d) For each eigenvalue with a large enough relative separation compute
56 * the corresponding eigenvector by forming a rank revealing twisted
57 * factorization. Go back to (c) for any clusters that remain.
58 *
59 * For more details, see:
60 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
61 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
62 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
63 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
64 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
65 * 2004. Also LAPACK Working Note 154.
66 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
67 * tridiagonal eigenvalue/eigenvector problem",
68 * Computer Science Division Technical Report No. UCB/CSD-97-971,
69 * UC Berkeley, May 1997.
70 *
71 * Further Details
72 * 1.DSTEMR works only on machines which follow IEEE-754
73 * floating-point standard in their handling of infinities and NaNs.
74 * This permits the use of efficient inner loops avoiding a check for
75 * zero divisors.
76 *
77 * Arguments
78 * =========
79 *
80 * JOBZ (input) CHARACTER*1
81 * = 'N': Compute eigenvalues only;
82 * = 'V': Compute eigenvalues and eigenvectors.
83 *
84 * RANGE (input) CHARACTER*1
85 * = 'A': all eigenvalues will be found.
86 * = 'V': all eigenvalues in the half-open interval (VL,VU]
87 * will be found.
88 * = 'I': the IL-th through IU-th eigenvalues will be found.
89 *
90 * N (input) INTEGER
91 * The order of the matrix. N >= 0.
92 *
93 * D (input/output) DOUBLE PRECISION array, dimension (N)
94 * On entry, the N diagonal elements of the tridiagonal matrix
95 * T. On exit, D is overwritten.
96 *
97 * E (input/output) DOUBLE PRECISION array, dimension (N)
98 * On entry, the (N-1) subdiagonal elements of the tridiagonal
99 * matrix T in elements 1 to N-1 of E. E(N) need not be set on
100 * input, but is used internally as workspace.
101 * On exit, E is overwritten.
102 *
103 * VL (input) DOUBLE PRECISION
104 * VU (input) DOUBLE PRECISION
105 * If RANGE='V', the lower and upper bounds of the interval to
106 * be searched for eigenvalues. VL < VU.
107 * Not referenced if RANGE = 'A' or 'I'.
108 *
109 * IL (input) INTEGER
110 * IU (input) INTEGER
111 * If RANGE='I', the indices (in ascending order) of the
112 * smallest and largest eigenvalues to be returned.
113 * 1 <= IL <= IU <= N, if N > 0.
114 * Not referenced if RANGE = 'A' or 'V'.
115 *
116 * M (output) INTEGER
117 * The total number of eigenvalues found. 0 <= M <= N.
118 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
119 *
120 * W (output) DOUBLE PRECISION array, dimension (N)
121 * The first M elements contain the selected eigenvalues in
122 * ascending order.
123 *
124 * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
125 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
126 * contain the orthonormal eigenvectors of the matrix T
127 * corresponding to the selected eigenvalues, with the i-th
128 * column of Z holding the eigenvector associated with W(i).
129 * If JOBZ = 'N', then Z is not referenced.
130 * Note: the user must ensure that at least max(1,M) columns are
131 * supplied in the array Z; if RANGE = 'V', the exact value of M
132 * is not known in advance and can be computed with a workspace
133 * query by setting NZC = -1, see below.
134 *
135 * LDZ (input) INTEGER
136 * The leading dimension of the array Z. LDZ >= 1, and if
137 * JOBZ = 'V', then LDZ >= max(1,N).
138 *
139 * NZC (input) INTEGER
140 * The number of eigenvectors to be held in the array Z.
141 * If RANGE = 'A', then NZC >= max(1,N).
142 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
143 * If RANGE = 'I', then NZC >= IU-IL+1.
144 * If NZC = -1, then a workspace query is assumed; the
145 * routine calculates the number of columns of the array Z that
146 * are needed to hold the eigenvectors.
147 * This value is returned as the first entry of the Z array, and
148 * no error message related to NZC is issued by XERBLA.
149 *
150 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
151 * The support of the eigenvectors in Z, i.e., the indices
152 * indicating the nonzero elements in Z. The i-th computed eigenvector
153 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
154 * ISUPPZ( 2*i ). This is relevant in the case when the matrix
155 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
156 *
157 * TRYRAC (input/output) LOGICAL
158 * If TRYRAC.EQ..TRUE., indicates that the code should check whether
159 * the tridiagonal matrix defines its eigenvalues to high relative
160 * accuracy. If so, the code uses relative-accuracy preserving
161 * algorithms that might be (a bit) slower depending on the matrix.
162 * If the matrix does not define its eigenvalues to high relative
163 * accuracy, the code can uses possibly faster algorithms.
164 * If TRYRAC.EQ..FALSE., the code is not required to guarantee
165 * relatively accurate eigenvalues and can use the fastest possible
166 * techniques.
167 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
168 * does not define its eigenvalues to high relative accuracy.
169 *
170 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
171 * On exit, if INFO = 0, WORK(1) returns the optimal
172 * (and minimal) LWORK.
173 *
174 * LWORK (input) INTEGER
175 * The dimension of the array WORK. LWORK >= max(1,18*N)
176 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
177 * If LWORK = -1, then a workspace query is assumed; the routine
178 * only calculates the optimal size of the WORK array, returns
179 * this value as the first entry of the WORK array, and no error
180 * message related to LWORK is issued by XERBLA.
181 *
182 * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
183 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
184 *
185 * LIWORK (input) INTEGER
186 * The dimension of the array IWORK. LIWORK >= max(1,10*N)
187 * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
188 * if only the eigenvalues are to be computed.
189 * If LIWORK = -1, then a workspace query is assumed; the
190 * routine only calculates the optimal size of the IWORK array,
191 * returns this value as the first entry of the IWORK array, and
192 * no error message related to LIWORK is issued by XERBLA.
193 *
194 * INFO (output) INTEGER
195 * On exit, INFO
196 * = 0: successful exit
197 * < 0: if INFO = -i, the i-th argument had an illegal value
198 * > 0: if INFO = 1X, internal error in DLARRE,
199 * if INFO = 2X, internal error in DLARRV.
200 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
201 * the nonzero error code returned by DLARRE or
202 * DLARRV, respectively.
203 *
204 *
205 * Further Details
206 * ===============
207 *
208 * Based on contributions by
209 * Beresford Parlett, University of California, Berkeley, USA
210 * Jim Demmel, University of California, Berkeley, USA
211 * Inderjit Dhillon, University of Texas, Austin, USA
212 * Osni Marques, LBNL/NERSC, USA
213 * Christof Voemel, University of California, Berkeley, USA
214 *
215 * =====================================================================
216 *
217 * .. Parameters ..
218 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
219 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
220 $ FOUR = 4.0D0,
221 $ MINRGP = 1.0D-3 )
222 * ..
223 * .. Local Scalars ..
224 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
225 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
226 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
227 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
228 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
229 $ NZCMIN, OFFSET, WBEGIN, WEND
230 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
231 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
232 $ THRESH, TMP, TNRM, WL, WU
233 * ..
234 * ..
235 * .. External Functions ..
236 LOGICAL LSAME
237 DOUBLE PRECISION DLAMCH, DLANST
238 EXTERNAL LSAME, DLAMCH, DLANST
239 * ..
240 * .. External Subroutines ..
241 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
242 $ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
243 * ..
244 * .. Intrinsic Functions ..
245 INTRINSIC MAX, MIN, SQRT
246
247
248 * ..
249 * .. Executable Statements ..
250 *
251 * Test the input parameters.
252 *
253 WANTZ = LSAME( JOBZ, 'V' )
254 ALLEIG = LSAME( RANGE, 'A' )
255 VALEIG = LSAME( RANGE, 'V' )
256 INDEIG = LSAME( RANGE, 'I' )
257 *
258 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
259 ZQUERY = ( NZC.EQ.-1 )
260
261 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
262 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
263 * Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
264 IF( WANTZ ) THEN
265 LWMIN = 18*N
266 LIWMIN = 10*N
267 ELSE
268 * need less workspace if only the eigenvalues are wanted
269 LWMIN = 12*N
270 LIWMIN = 8*N
271 ENDIF
272
273 WL = ZERO
274 WU = ZERO
275 IIL = 0
276 IIU = 0
277
278 IF( VALEIG ) THEN
279 * We do not reference VL, VU in the cases RANGE = 'I','A'
280 * The interval (WL, WU] contains all the wanted eigenvalues.
281 * It is either given by the user or computed in DLARRE.
282 WL = VL
283 WU = VU
284 ELSEIF( INDEIG ) THEN
285 * We do not reference IL, IU in the cases RANGE = 'V','A'
286 IIL = IL
287 IIU = IU
288 ENDIF
289 *
290 INFO = 0
291 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
292 INFO = -1
293 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
294 INFO = -2
295 ELSE IF( N.LT.0 ) THEN
296 INFO = -3
297 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
298 INFO = -7
299 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
300 INFO = -8
301 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
302 INFO = -9
303 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
304 INFO = -13
305 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
306 INFO = -17
307 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
308 INFO = -19
309 END IF
310 *
311 * Get machine constants.
312 *
313 SAFMIN = DLAMCH( 'Safe minimum' )
314 EPS = DLAMCH( 'Precision' )
315 SMLNUM = SAFMIN / EPS
316 BIGNUM = ONE / SMLNUM
317 RMIN = SQRT( SMLNUM )
318 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
319 *
320 IF( INFO.EQ.0 ) THEN
321 WORK( 1 ) = LWMIN
322 IWORK( 1 ) = LIWMIN
323 *
324 IF( WANTZ .AND. ALLEIG ) THEN
325 NZCMIN = N
326 ELSE IF( WANTZ .AND. VALEIG ) THEN
327 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
328 $ NZCMIN, ITMP, ITMP2, INFO )
329 ELSE IF( WANTZ .AND. INDEIG ) THEN
330 NZCMIN = IIU-IIL+1
331 ELSE
332 * WANTZ .EQ. FALSE.
333 NZCMIN = 0
334 ENDIF
335 IF( ZQUERY .AND. INFO.EQ.0 ) THEN
336 Z( 1,1 ) = NZCMIN
337 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
338 INFO = -14
339 END IF
340 END IF
341
342 IF( INFO.NE.0 ) THEN
343 *
344 CALL XERBLA( 'DSTEMR', -INFO )
345 *
346 RETURN
347 ELSE IF( LQUERY .OR. ZQUERY ) THEN
348 RETURN
349 END IF
350 *
351 * Handle N = 0, 1, and 2 cases immediately
352 *
353 M = 0
354 IF( N.EQ.0 )
355 $ RETURN
356 *
357 IF( N.EQ.1 ) THEN
358 IF( ALLEIG .OR. INDEIG ) THEN
359 M = 1
360 W( 1 ) = D( 1 )
361 ELSE
362 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
363 M = 1
364 W( 1 ) = D( 1 )
365 END IF
366 END IF
367 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
368 Z( 1, 1 ) = ONE
369 ISUPPZ(1) = 1
370 ISUPPZ(2) = 1
371 END IF
372 RETURN
373 END IF
374 *
375 IF( N.EQ.2 ) THEN
376 IF( .NOT.WANTZ ) THEN
377 CALL DLAE2( D(1), E(1), D(2), R1, R2 )
378 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
379 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
380 END IF
381 IF( ALLEIG.OR.
382 $ (VALEIG.AND.(R2.GT.WL).AND.
383 $ (R2.LE.WU)).OR.
384 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
385 M = M+1
386 W( M ) = R2
387 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
388 Z( 1, M ) = -SN
389 Z( 2, M ) = CS
390 * Note: At most one of SN and CS can be zero.
391 IF (SN.NE.ZERO) THEN
392 IF (CS.NE.ZERO) THEN
393 ISUPPZ(2*M-1) = 1
394 ISUPPZ(2*M) = 2
395 ELSE
396 ISUPPZ(2*M-1) = 1
397 ISUPPZ(2*M) = 1
398 END IF
399 ELSE
400 ISUPPZ(2*M-1) = 2
401 ISUPPZ(2*M) = 2
402 END IF
403 ENDIF
404 ENDIF
405 IF( ALLEIG.OR.
406 $ (VALEIG.AND.(R1.GT.WL).AND.
407 $ (R1.LE.WU)).OR.
408 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
409 M = M+1
410 W( M ) = R1
411 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
412 Z( 1, M ) = CS
413 Z( 2, M ) = SN
414 * Note: At most one of SN and CS can be zero.
415 IF (SN.NE.ZERO) THEN
416 IF (CS.NE.ZERO) THEN
417 ISUPPZ(2*M-1) = 1
418 ISUPPZ(2*M) = 2
419 ELSE
420 ISUPPZ(2*M-1) = 1
421 ISUPPZ(2*M) = 1
422 END IF
423 ELSE
424 ISUPPZ(2*M-1) = 2
425 ISUPPZ(2*M) = 2
426 END IF
427 ENDIF
428 ENDIF
429 RETURN
430 END IF
431
432 * Continue with general N
433
434 INDGRS = 1
435 INDERR = 2*N + 1
436 INDGP = 3*N + 1
437 INDD = 4*N + 1
438 INDE2 = 5*N + 1
439 INDWRK = 6*N + 1
440 *
441 IINSPL = 1
442 IINDBL = N + 1
443 IINDW = 2*N + 1
444 IINDWK = 3*N + 1
445 *
446 * Scale matrix to allowable range, if necessary.
447 * The allowable range is related to the PIVMIN parameter; see the
448 * comments in DLARRD. The preference for scaling small values
449 * up is heuristic; we expect users' matrices not to be close to the
450 * RMAX threshold.
451 *
452 SCALE = ONE
453 TNRM = DLANST( 'M', N, D, E )
454 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
455 SCALE = RMIN / TNRM
456 ELSE IF( TNRM.GT.RMAX ) THEN
457 SCALE = RMAX / TNRM
458 END IF
459 IF( SCALE.NE.ONE ) THEN
460 CALL DSCAL( N, SCALE, D, 1 )
461 CALL DSCAL( N-1, SCALE, E, 1 )
462 TNRM = TNRM*SCALE
463 IF( VALEIG ) THEN
464 * If eigenvalues in interval have to be found,
465 * scale (WL, WU] accordingly
466 WL = WL*SCALE
467 WU = WU*SCALE
468 ENDIF
469 END IF
470 *
471 * Compute the desired eigenvalues of the tridiagonal after splitting
472 * into smaller subblocks if the corresponding off-diagonal elements
473 * are small
474 * THRESH is the splitting parameter for DLARRE
475 * A negative THRESH forces the old splitting criterion based on the
476 * size of the off-diagonal. A positive THRESH switches to splitting
477 * which preserves relative accuracy.
478 *
479 IF( TRYRAC ) THEN
480 * Test whether the matrix warrants the more expensive relative approach.
481 CALL DLARRR( N, D, E, IINFO )
482 ELSE
483 * The user does not care about relative accurately eigenvalues
484 IINFO = -1
485 ENDIF
486 * Set the splitting criterion
487 IF (IINFO.EQ.0) THEN
488 THRESH = EPS
489 ELSE
490 THRESH = -EPS
491 * relative accuracy is desired but T does not guarantee it
492 TRYRAC = .FALSE.
493 ENDIF
494 *
495 IF( TRYRAC ) THEN
496 * Copy original diagonal, needed to guarantee relative accuracy
497 CALL DCOPY(N,D,1,WORK(INDD),1)
498 ENDIF
499 * Store the squares of the offdiagonal values of T
500 DO 5 J = 1, N-1
501 WORK( INDE2+J-1 ) = E(J)**2
502 5 CONTINUE
503
504 * Set the tolerance parameters for bisection
505 IF( .NOT.WANTZ ) THEN
506 * DLARRE computes the eigenvalues to full precision.
507 RTOL1 = FOUR * EPS
508 RTOL2 = FOUR * EPS
509 ELSE
510 * DLARRE computes the eigenvalues to less than full precision.
511 * DLARRV will refine the eigenvalue approximations, and we can
512 * need less accurate initial bisection in DLARRE.
513 * Note: these settings do only affect the subset case and DLARRE
514 RTOL1 = SQRT(EPS)
515 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
516 ENDIF
517 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
518 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
519 $ IWORK( IINSPL ), M, W, WORK( INDERR ),
520 $ WORK( INDGP ), IWORK( IINDBL ),
521 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
522 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
523 IF( IINFO.NE.0 ) THEN
524 INFO = 10 + ABS( IINFO )
525 RETURN
526 END IF
527 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
528 * part of the spectrum. All desired eigenvalues are contained in
529 * (WL,WU]
530
531
532 IF( WANTZ ) THEN
533 *
534 * Compute the desired eigenvectors corresponding to the computed
535 * eigenvalues
536 *
537 CALL DLARRV( N, WL, WU, D, E,
538 $ PIVMIN, IWORK( IINSPL ), M,
539 $ 1, M, MINRGP, RTOL1, RTOL2,
540 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
541 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
542 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
543 IF( IINFO.NE.0 ) THEN
544 INFO = 20 + ABS( IINFO )
545 RETURN
546 END IF
547 ELSE
548 * DLARRE computes eigenvalues of the (shifted) root representation
549 * DLARRV returns the eigenvalues of the unshifted matrix.
550 * However, if the eigenvectors are not desired by the user, we need
551 * to apply the corresponding shifts from DLARRE to obtain the
552 * eigenvalues of the original matrix.
553 DO 20 J = 1, M
554 ITMP = IWORK( IINDBL+J-1 )
555 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
556 20 CONTINUE
557 END IF
558 *
559
560 IF ( TRYRAC ) THEN
561 * Refine computed eigenvalues so that they are relatively accurate
562 * with respect to the original matrix T.
563 IBEGIN = 1
564 WBEGIN = 1
565 DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
566 IEND = IWORK( IINSPL+JBLK-1 )
567 IN = IEND - IBEGIN + 1
568 WEND = WBEGIN - 1
569 * check if any eigenvalues have to be refined in this block
570 36 CONTINUE
571 IF( WEND.LT.M ) THEN
572 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
573 WEND = WEND + 1
574 GO TO 36
575 END IF
576 END IF
577 IF( WEND.LT.WBEGIN ) THEN
578 IBEGIN = IEND + 1
579 GO TO 39
580 END IF
581
582 OFFSET = IWORK(IINDW+WBEGIN-1)-1
583 IFIRST = IWORK(IINDW+WBEGIN-1)
584 ILAST = IWORK(IINDW+WEND-1)
585 RTOL2 = FOUR * EPS
586 CALL DLARRJ( IN,
587 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
588 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
589 $ WORK( INDERR+WBEGIN-1 ),
590 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
591 $ TNRM, IINFO )
592 IBEGIN = IEND + 1
593 WBEGIN = WEND + 1
594 39 CONTINUE
595 ENDIF
596 *
597 * If matrix was scaled, then rescale eigenvalues appropriately.
598 *
599 IF( SCALE.NE.ONE ) THEN
600 CALL DSCAL( M, ONE / SCALE, W, 1 )
601 END IF
602 *
603 * If eigenvalues are not in increasing order, then sort them,
604 * possibly along with eigenvectors.
605 *
606 IF( NSPLIT.GT.1 ) THEN
607 IF( .NOT. WANTZ ) THEN
608 CALL DLASRT( 'I', M, W, IINFO )
609 IF( IINFO.NE.0 ) THEN
610 INFO = 3
611 RETURN
612 END IF
613 ELSE
614 DO 60 J = 1, M - 1
615 I = 0
616 TMP = W( J )
617 DO 50 JJ = J + 1, M
618 IF( W( JJ ).LT.TMP ) THEN
619 I = JJ
620 TMP = W( JJ )
621 END IF
622 50 CONTINUE
623 IF( I.NE.0 ) THEN
624 W( I ) = W( J )
625 W( J ) = TMP
626 IF( WANTZ ) THEN
627 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
628 ITMP = ISUPPZ( 2*I-1 )
629 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
630 ISUPPZ( 2*J-1 ) = ITMP
631 ITMP = ISUPPZ( 2*I )
632 ISUPPZ( 2*I ) = ISUPPZ( 2*J )
633 ISUPPZ( 2*J ) = ITMP
634 END IF
635 END IF
636 60 CONTINUE
637 END IF
638 ENDIF
639 *
640 *
641 WORK( 1 ) = LWMIN
642 IWORK( 1 ) = LIWMIN
643 RETURN
644 *
645 * End of DSTEMR
646 *
647 END
2 $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
3 $ IWORK, LIWORK, INFO )
4 IMPLICIT NONE
5 *
6 * -- LAPACK computational routine (version 3.2.2) --
7 * -- LAPACK is a software package provided by Univ. of Tennessee, --
8 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
9 * -- June 2010 --
10 *
11 * .. Scalar Arguments ..
12 CHARACTER JOBZ, RANGE
13 LOGICAL TRYRAC
14 INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
15 DOUBLE PRECISION VL, VU
16 * ..
17 * .. Array Arguments ..
18 INTEGER ISUPPZ( * ), IWORK( * )
19 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
20 DOUBLE PRECISION Z( LDZ, * )
21 * ..
22 *
23 * Purpose
24 * =======
25 *
26 * DSTEMR computes selected eigenvalues and, optionally, eigenvectors
27 * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
28 * a well defined set of pairwise different real eigenvalues, the corresponding
29 * real eigenvectors are pairwise orthogonal.
30 *
31 * The spectrum may be computed either completely or partially by specifying
32 * either an interval (VL,VU] or a range of indices IL:IU for the desired
33 * eigenvalues.
34 *
35 * Depending on the number of desired eigenvalues, these are computed either
36 * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
37 * computed by the use of various suitable L D L^T factorizations near clusters
38 * of close eigenvalues (referred to as RRRs, Relatively Robust
39 * Representations). An informal sketch of the algorithm follows.
40 *
41 * For each unreduced block (submatrix) of T,
42 * (a) Compute T - sigma I = L D L^T, so that L and D
43 * define all the wanted eigenvalues to high relative accuracy.
44 * This means that small relative changes in the entries of D and L
45 * cause only small relative changes in the eigenvalues and
46 * eigenvectors. The standard (unfactored) representation of the
47 * tridiagonal matrix T does not have this property in general.
48 * (b) Compute the eigenvalues to suitable accuracy.
49 * If the eigenvectors are desired, the algorithm attains full
50 * accuracy of the computed eigenvalues only right before
51 * the corresponding vectors have to be computed, see steps c) and d).
52 * (c) For each cluster of close eigenvalues, select a new
53 * shift close to the cluster, find a new factorization, and refine
54 * the shifted eigenvalues to suitable accuracy.
55 * (d) For each eigenvalue with a large enough relative separation compute
56 * the corresponding eigenvector by forming a rank revealing twisted
57 * factorization. Go back to (c) for any clusters that remain.
58 *
59 * For more details, see:
60 * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
61 * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
62 * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
63 * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
64 * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
65 * 2004. Also LAPACK Working Note 154.
66 * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
67 * tridiagonal eigenvalue/eigenvector problem",
68 * Computer Science Division Technical Report No. UCB/CSD-97-971,
69 * UC Berkeley, May 1997.
70 *
71 * Further Details
72 * 1.DSTEMR works only on machines which follow IEEE-754
73 * floating-point standard in their handling of infinities and NaNs.
74 * This permits the use of efficient inner loops avoiding a check for
75 * zero divisors.
76 *
77 * Arguments
78 * =========
79 *
80 * JOBZ (input) CHARACTER*1
81 * = 'N': Compute eigenvalues only;
82 * = 'V': Compute eigenvalues and eigenvectors.
83 *
84 * RANGE (input) CHARACTER*1
85 * = 'A': all eigenvalues will be found.
86 * = 'V': all eigenvalues in the half-open interval (VL,VU]
87 * will be found.
88 * = 'I': the IL-th through IU-th eigenvalues will be found.
89 *
90 * N (input) INTEGER
91 * The order of the matrix. N >= 0.
92 *
93 * D (input/output) DOUBLE PRECISION array, dimension (N)
94 * On entry, the N diagonal elements of the tridiagonal matrix
95 * T. On exit, D is overwritten.
96 *
97 * E (input/output) DOUBLE PRECISION array, dimension (N)
98 * On entry, the (N-1) subdiagonal elements of the tridiagonal
99 * matrix T in elements 1 to N-1 of E. E(N) need not be set on
100 * input, but is used internally as workspace.
101 * On exit, E is overwritten.
102 *
103 * VL (input) DOUBLE PRECISION
104 * VU (input) DOUBLE PRECISION
105 * If RANGE='V', the lower and upper bounds of the interval to
106 * be searched for eigenvalues. VL < VU.
107 * Not referenced if RANGE = 'A' or 'I'.
108 *
109 * IL (input) INTEGER
110 * IU (input) INTEGER
111 * If RANGE='I', the indices (in ascending order) of the
112 * smallest and largest eigenvalues to be returned.
113 * 1 <= IL <= IU <= N, if N > 0.
114 * Not referenced if RANGE = 'A' or 'V'.
115 *
116 * M (output) INTEGER
117 * The total number of eigenvalues found. 0 <= M <= N.
118 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
119 *
120 * W (output) DOUBLE PRECISION array, dimension (N)
121 * The first M elements contain the selected eigenvalues in
122 * ascending order.
123 *
124 * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
125 * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
126 * contain the orthonormal eigenvectors of the matrix T
127 * corresponding to the selected eigenvalues, with the i-th
128 * column of Z holding the eigenvector associated with W(i).
129 * If JOBZ = 'N', then Z is not referenced.
130 * Note: the user must ensure that at least max(1,M) columns are
131 * supplied in the array Z; if RANGE = 'V', the exact value of M
132 * is not known in advance and can be computed with a workspace
133 * query by setting NZC = -1, see below.
134 *
135 * LDZ (input) INTEGER
136 * The leading dimension of the array Z. LDZ >= 1, and if
137 * JOBZ = 'V', then LDZ >= max(1,N).
138 *
139 * NZC (input) INTEGER
140 * The number of eigenvectors to be held in the array Z.
141 * If RANGE = 'A', then NZC >= max(1,N).
142 * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
143 * If RANGE = 'I', then NZC >= IU-IL+1.
144 * If NZC = -1, then a workspace query is assumed; the
145 * routine calculates the number of columns of the array Z that
146 * are needed to hold the eigenvectors.
147 * This value is returned as the first entry of the Z array, and
148 * no error message related to NZC is issued by XERBLA.
149 *
150 * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
151 * The support of the eigenvectors in Z, i.e., the indices
152 * indicating the nonzero elements in Z. The i-th computed eigenvector
153 * is nonzero only in elements ISUPPZ( 2*i-1 ) through
154 * ISUPPZ( 2*i ). This is relevant in the case when the matrix
155 * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
156 *
157 * TRYRAC (input/output) LOGICAL
158 * If TRYRAC.EQ..TRUE., indicates that the code should check whether
159 * the tridiagonal matrix defines its eigenvalues to high relative
160 * accuracy. If so, the code uses relative-accuracy preserving
161 * algorithms that might be (a bit) slower depending on the matrix.
162 * If the matrix does not define its eigenvalues to high relative
163 * accuracy, the code can uses possibly faster algorithms.
164 * If TRYRAC.EQ..FALSE., the code is not required to guarantee
165 * relatively accurate eigenvalues and can use the fastest possible
166 * techniques.
167 * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
168 * does not define its eigenvalues to high relative accuracy.
169 *
170 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
171 * On exit, if INFO = 0, WORK(1) returns the optimal
172 * (and minimal) LWORK.
173 *
174 * LWORK (input) INTEGER
175 * The dimension of the array WORK. LWORK >= max(1,18*N)
176 * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
177 * If LWORK = -1, then a workspace query is assumed; the routine
178 * only calculates the optimal size of the WORK array, returns
179 * this value as the first entry of the WORK array, and no error
180 * message related to LWORK is issued by XERBLA.
181 *
182 * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
183 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
184 *
185 * LIWORK (input) INTEGER
186 * The dimension of the array IWORK. LIWORK >= max(1,10*N)
187 * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
188 * if only the eigenvalues are to be computed.
189 * If LIWORK = -1, then a workspace query is assumed; the
190 * routine only calculates the optimal size of the IWORK array,
191 * returns this value as the first entry of the IWORK array, and
192 * no error message related to LIWORK is issued by XERBLA.
193 *
194 * INFO (output) INTEGER
195 * On exit, INFO
196 * = 0: successful exit
197 * < 0: if INFO = -i, the i-th argument had an illegal value
198 * > 0: if INFO = 1X, internal error in DLARRE,
199 * if INFO = 2X, internal error in DLARRV.
200 * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
201 * the nonzero error code returned by DLARRE or
202 * DLARRV, respectively.
203 *
204 *
205 * Further Details
206 * ===============
207 *
208 * Based on contributions by
209 * Beresford Parlett, University of California, Berkeley, USA
210 * Jim Demmel, University of California, Berkeley, USA
211 * Inderjit Dhillon, University of Texas, Austin, USA
212 * Osni Marques, LBNL/NERSC, USA
213 * Christof Voemel, University of California, Berkeley, USA
214 *
215 * =====================================================================
216 *
217 * .. Parameters ..
218 DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
219 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
220 $ FOUR = 4.0D0,
221 $ MINRGP = 1.0D-3 )
222 * ..
223 * .. Local Scalars ..
224 LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
225 INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
226 $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
227 $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
228 $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
229 $ NZCMIN, OFFSET, WBEGIN, WEND
230 DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
231 $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
232 $ THRESH, TMP, TNRM, WL, WU
233 * ..
234 * ..
235 * .. External Functions ..
236 LOGICAL LSAME
237 DOUBLE PRECISION DLAMCH, DLANST
238 EXTERNAL LSAME, DLAMCH, DLANST
239 * ..
240 * .. External Subroutines ..
241 EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
242 $ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
243 * ..
244 * .. Intrinsic Functions ..
245 INTRINSIC MAX, MIN, SQRT
246
247
248 * ..
249 * .. Executable Statements ..
250 *
251 * Test the input parameters.
252 *
253 WANTZ = LSAME( JOBZ, 'V' )
254 ALLEIG = LSAME( RANGE, 'A' )
255 VALEIG = LSAME( RANGE, 'V' )
256 INDEIG = LSAME( RANGE, 'I' )
257 *
258 LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
259 ZQUERY = ( NZC.EQ.-1 )
260
261 * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
262 * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
263 * Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
264 IF( WANTZ ) THEN
265 LWMIN = 18*N
266 LIWMIN = 10*N
267 ELSE
268 * need less workspace if only the eigenvalues are wanted
269 LWMIN = 12*N
270 LIWMIN = 8*N
271 ENDIF
272
273 WL = ZERO
274 WU = ZERO
275 IIL = 0
276 IIU = 0
277
278 IF( VALEIG ) THEN
279 * We do not reference VL, VU in the cases RANGE = 'I','A'
280 * The interval (WL, WU] contains all the wanted eigenvalues.
281 * It is either given by the user or computed in DLARRE.
282 WL = VL
283 WU = VU
284 ELSEIF( INDEIG ) THEN
285 * We do not reference IL, IU in the cases RANGE = 'V','A'
286 IIL = IL
287 IIU = IU
288 ENDIF
289 *
290 INFO = 0
291 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
292 INFO = -1
293 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
294 INFO = -2
295 ELSE IF( N.LT.0 ) THEN
296 INFO = -3
297 ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
298 INFO = -7
299 ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
300 INFO = -8
301 ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
302 INFO = -9
303 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
304 INFO = -13
305 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
306 INFO = -17
307 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
308 INFO = -19
309 END IF
310 *
311 * Get machine constants.
312 *
313 SAFMIN = DLAMCH( 'Safe minimum' )
314 EPS = DLAMCH( 'Precision' )
315 SMLNUM = SAFMIN / EPS
316 BIGNUM = ONE / SMLNUM
317 RMIN = SQRT( SMLNUM )
318 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
319 *
320 IF( INFO.EQ.0 ) THEN
321 WORK( 1 ) = LWMIN
322 IWORK( 1 ) = LIWMIN
323 *
324 IF( WANTZ .AND. ALLEIG ) THEN
325 NZCMIN = N
326 ELSE IF( WANTZ .AND. VALEIG ) THEN
327 CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
328 $ NZCMIN, ITMP, ITMP2, INFO )
329 ELSE IF( WANTZ .AND. INDEIG ) THEN
330 NZCMIN = IIU-IIL+1
331 ELSE
332 * WANTZ .EQ. FALSE.
333 NZCMIN = 0
334 ENDIF
335 IF( ZQUERY .AND. INFO.EQ.0 ) THEN
336 Z( 1,1 ) = NZCMIN
337 ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
338 INFO = -14
339 END IF
340 END IF
341
342 IF( INFO.NE.0 ) THEN
343 *
344 CALL XERBLA( 'DSTEMR', -INFO )
345 *
346 RETURN
347 ELSE IF( LQUERY .OR. ZQUERY ) THEN
348 RETURN
349 END IF
350 *
351 * Handle N = 0, 1, and 2 cases immediately
352 *
353 M = 0
354 IF( N.EQ.0 )
355 $ RETURN
356 *
357 IF( N.EQ.1 ) THEN
358 IF( ALLEIG .OR. INDEIG ) THEN
359 M = 1
360 W( 1 ) = D( 1 )
361 ELSE
362 IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
363 M = 1
364 W( 1 ) = D( 1 )
365 END IF
366 END IF
367 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
368 Z( 1, 1 ) = ONE
369 ISUPPZ(1) = 1
370 ISUPPZ(2) = 1
371 END IF
372 RETURN
373 END IF
374 *
375 IF( N.EQ.2 ) THEN
376 IF( .NOT.WANTZ ) THEN
377 CALL DLAE2( D(1), E(1), D(2), R1, R2 )
378 ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
379 CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
380 END IF
381 IF( ALLEIG.OR.
382 $ (VALEIG.AND.(R2.GT.WL).AND.
383 $ (R2.LE.WU)).OR.
384 $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
385 M = M+1
386 W( M ) = R2
387 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
388 Z( 1, M ) = -SN
389 Z( 2, M ) = CS
390 * Note: At most one of SN and CS can be zero.
391 IF (SN.NE.ZERO) THEN
392 IF (CS.NE.ZERO) THEN
393 ISUPPZ(2*M-1) = 1
394 ISUPPZ(2*M) = 2
395 ELSE
396 ISUPPZ(2*M-1) = 1
397 ISUPPZ(2*M) = 1
398 END IF
399 ELSE
400 ISUPPZ(2*M-1) = 2
401 ISUPPZ(2*M) = 2
402 END IF
403 ENDIF
404 ENDIF
405 IF( ALLEIG.OR.
406 $ (VALEIG.AND.(R1.GT.WL).AND.
407 $ (R1.LE.WU)).OR.
408 $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
409 M = M+1
410 W( M ) = R1
411 IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
412 Z( 1, M ) = CS
413 Z( 2, M ) = SN
414 * Note: At most one of SN and CS can be zero.
415 IF (SN.NE.ZERO) THEN
416 IF (CS.NE.ZERO) THEN
417 ISUPPZ(2*M-1) = 1
418 ISUPPZ(2*M) = 2
419 ELSE
420 ISUPPZ(2*M-1) = 1
421 ISUPPZ(2*M) = 1
422 END IF
423 ELSE
424 ISUPPZ(2*M-1) = 2
425 ISUPPZ(2*M) = 2
426 END IF
427 ENDIF
428 ENDIF
429 RETURN
430 END IF
431
432 * Continue with general N
433
434 INDGRS = 1
435 INDERR = 2*N + 1
436 INDGP = 3*N + 1
437 INDD = 4*N + 1
438 INDE2 = 5*N + 1
439 INDWRK = 6*N + 1
440 *
441 IINSPL = 1
442 IINDBL = N + 1
443 IINDW = 2*N + 1
444 IINDWK = 3*N + 1
445 *
446 * Scale matrix to allowable range, if necessary.
447 * The allowable range is related to the PIVMIN parameter; see the
448 * comments in DLARRD. The preference for scaling small values
449 * up is heuristic; we expect users' matrices not to be close to the
450 * RMAX threshold.
451 *
452 SCALE = ONE
453 TNRM = DLANST( 'M', N, D, E )
454 IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
455 SCALE = RMIN / TNRM
456 ELSE IF( TNRM.GT.RMAX ) THEN
457 SCALE = RMAX / TNRM
458 END IF
459 IF( SCALE.NE.ONE ) THEN
460 CALL DSCAL( N, SCALE, D, 1 )
461 CALL DSCAL( N-1, SCALE, E, 1 )
462 TNRM = TNRM*SCALE
463 IF( VALEIG ) THEN
464 * If eigenvalues in interval have to be found,
465 * scale (WL, WU] accordingly
466 WL = WL*SCALE
467 WU = WU*SCALE
468 ENDIF
469 END IF
470 *
471 * Compute the desired eigenvalues of the tridiagonal after splitting
472 * into smaller subblocks if the corresponding off-diagonal elements
473 * are small
474 * THRESH is the splitting parameter for DLARRE
475 * A negative THRESH forces the old splitting criterion based on the
476 * size of the off-diagonal. A positive THRESH switches to splitting
477 * which preserves relative accuracy.
478 *
479 IF( TRYRAC ) THEN
480 * Test whether the matrix warrants the more expensive relative approach.
481 CALL DLARRR( N, D, E, IINFO )
482 ELSE
483 * The user does not care about relative accurately eigenvalues
484 IINFO = -1
485 ENDIF
486 * Set the splitting criterion
487 IF (IINFO.EQ.0) THEN
488 THRESH = EPS
489 ELSE
490 THRESH = -EPS
491 * relative accuracy is desired but T does not guarantee it
492 TRYRAC = .FALSE.
493 ENDIF
494 *
495 IF( TRYRAC ) THEN
496 * Copy original diagonal, needed to guarantee relative accuracy
497 CALL DCOPY(N,D,1,WORK(INDD),1)
498 ENDIF
499 * Store the squares of the offdiagonal values of T
500 DO 5 J = 1, N-1
501 WORK( INDE2+J-1 ) = E(J)**2
502 5 CONTINUE
503
504 * Set the tolerance parameters for bisection
505 IF( .NOT.WANTZ ) THEN
506 * DLARRE computes the eigenvalues to full precision.
507 RTOL1 = FOUR * EPS
508 RTOL2 = FOUR * EPS
509 ELSE
510 * DLARRE computes the eigenvalues to less than full precision.
511 * DLARRV will refine the eigenvalue approximations, and we can
512 * need less accurate initial bisection in DLARRE.
513 * Note: these settings do only affect the subset case and DLARRE
514 RTOL1 = SQRT(EPS)
515 RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
516 ENDIF
517 CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
518 $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
519 $ IWORK( IINSPL ), M, W, WORK( INDERR ),
520 $ WORK( INDGP ), IWORK( IINDBL ),
521 $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
522 $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
523 IF( IINFO.NE.0 ) THEN
524 INFO = 10 + ABS( IINFO )
525 RETURN
526 END IF
527 * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
528 * part of the spectrum. All desired eigenvalues are contained in
529 * (WL,WU]
530
531
532 IF( WANTZ ) THEN
533 *
534 * Compute the desired eigenvectors corresponding to the computed
535 * eigenvalues
536 *
537 CALL DLARRV( N, WL, WU, D, E,
538 $ PIVMIN, IWORK( IINSPL ), M,
539 $ 1, M, MINRGP, RTOL1, RTOL2,
540 $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
541 $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
542 $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
543 IF( IINFO.NE.0 ) THEN
544 INFO = 20 + ABS( IINFO )
545 RETURN
546 END IF
547 ELSE
548 * DLARRE computes eigenvalues of the (shifted) root representation
549 * DLARRV returns the eigenvalues of the unshifted matrix.
550 * However, if the eigenvectors are not desired by the user, we need
551 * to apply the corresponding shifts from DLARRE to obtain the
552 * eigenvalues of the original matrix.
553 DO 20 J = 1, M
554 ITMP = IWORK( IINDBL+J-1 )
555 W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
556 20 CONTINUE
557 END IF
558 *
559
560 IF ( TRYRAC ) THEN
561 * Refine computed eigenvalues so that they are relatively accurate
562 * with respect to the original matrix T.
563 IBEGIN = 1
564 WBEGIN = 1
565 DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
566 IEND = IWORK( IINSPL+JBLK-1 )
567 IN = IEND - IBEGIN + 1
568 WEND = WBEGIN - 1
569 * check if any eigenvalues have to be refined in this block
570 36 CONTINUE
571 IF( WEND.LT.M ) THEN
572 IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
573 WEND = WEND + 1
574 GO TO 36
575 END IF
576 END IF
577 IF( WEND.LT.WBEGIN ) THEN
578 IBEGIN = IEND + 1
579 GO TO 39
580 END IF
581
582 OFFSET = IWORK(IINDW+WBEGIN-1)-1
583 IFIRST = IWORK(IINDW+WBEGIN-1)
584 ILAST = IWORK(IINDW+WEND-1)
585 RTOL2 = FOUR * EPS
586 CALL DLARRJ( IN,
587 $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
588 $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
589 $ WORK( INDERR+WBEGIN-1 ),
590 $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
591 $ TNRM, IINFO )
592 IBEGIN = IEND + 1
593 WBEGIN = WEND + 1
594 39 CONTINUE
595 ENDIF
596 *
597 * If matrix was scaled, then rescale eigenvalues appropriately.
598 *
599 IF( SCALE.NE.ONE ) THEN
600 CALL DSCAL( M, ONE / SCALE, W, 1 )
601 END IF
602 *
603 * If eigenvalues are not in increasing order, then sort them,
604 * possibly along with eigenvectors.
605 *
606 IF( NSPLIT.GT.1 ) THEN
607 IF( .NOT. WANTZ ) THEN
608 CALL DLASRT( 'I', M, W, IINFO )
609 IF( IINFO.NE.0 ) THEN
610 INFO = 3
611 RETURN
612 END IF
613 ELSE
614 DO 60 J = 1, M - 1
615 I = 0
616 TMP = W( J )
617 DO 50 JJ = J + 1, M
618 IF( W( JJ ).LT.TMP ) THEN
619 I = JJ
620 TMP = W( JJ )
621 END IF
622 50 CONTINUE
623 IF( I.NE.0 ) THEN
624 W( I ) = W( J )
625 W( J ) = TMP
626 IF( WANTZ ) THEN
627 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
628 ITMP = ISUPPZ( 2*I-1 )
629 ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
630 ISUPPZ( 2*J-1 ) = ITMP
631 ITMP = ISUPPZ( 2*I )
632 ISUPPZ( 2*I ) = ISUPPZ( 2*J )
633 ISUPPZ( 2*J ) = ITMP
634 END IF
635 END IF
636 60 CONTINUE
637 END IF
638 ENDIF
639 *
640 *
641 WORK( 1 ) = LWMIN
642 IWORK( 1 ) = LIWMIN
643 RETURN
644 *
645 * End of DSTEMR
646 *
647 END