1: *> \brief \b DSTEMR
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSTEMR + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstemr.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
22: * M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
23: * IWORK, LIWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE
27: * LOGICAL TRYRAC
28: * INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
29: * DOUBLE PRECISION VL, VU
30: * ..
31: * .. Array Arguments ..
32: * INTEGER ISUPPZ( * ), IWORK( * )
33: * DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
34: * DOUBLE PRECISION Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DSTEMR computes selected eigenvalues and, optionally, eigenvectors
44: *> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
45: *> a well defined set of pairwise different real eigenvalues, the corresponding
46: *> real eigenvectors are pairwise orthogonal.
47: *>
48: *> The spectrum may be computed either completely or partially by specifying
49: *> either an interval (VL,VU] or a range of indices IL:IU for the desired
50: *> eigenvalues.
51: *>
52: *> Depending on the number of desired eigenvalues, these are computed either
53: *> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
54: *> computed by the use of various suitable L D L^T factorizations near clusters
55: *> of close eigenvalues (referred to as RRRs, Relatively Robust
56: *> Representations). An informal sketch of the algorithm follows.
57: *>
58: *> For each unreduced block (submatrix) of T,
59: *> (a) Compute T - sigma I = L D L^T, so that L and D
60: *> define all the wanted eigenvalues to high relative accuracy.
61: *> This means that small relative changes in the entries of D and L
62: *> cause only small relative changes in the eigenvalues and
63: *> eigenvectors. The standard (unfactored) representation of the
64: *> tridiagonal matrix T does not have this property in general.
65: *> (b) Compute the eigenvalues to suitable accuracy.
66: *> If the eigenvectors are desired, the algorithm attains full
67: *> accuracy of the computed eigenvalues only right before
68: *> the corresponding vectors have to be computed, see steps c) and d).
69: *> (c) For each cluster of close eigenvalues, select a new
70: *> shift close to the cluster, find a new factorization, and refine
71: *> the shifted eigenvalues to suitable accuracy.
72: *> (d) For each eigenvalue with a large enough relative separation compute
73: *> the corresponding eigenvector by forming a rank revealing twisted
74: *> factorization. Go back to (c) for any clusters that remain.
75: *>
76: *> For more details, see:
77: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
78: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
79: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
80: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
81: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
82: *> 2004. Also LAPACK Working Note 154.
83: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
84: *> tridiagonal eigenvalue/eigenvector problem",
85: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
86: *> UC Berkeley, May 1997.
87: *>
88: *> Further Details
89: *> 1.DSTEMR works only on machines which follow IEEE-754
90: *> floating-point standard in their handling of infinities and NaNs.
91: *> This permits the use of efficient inner loops avoiding a check for
92: *> zero divisors.
93: *> \endverbatim
94: *
95: * Arguments:
96: * ==========
97: *
98: *> \param[in] JOBZ
99: *> \verbatim
100: *> JOBZ is CHARACTER*1
101: *> = 'N': Compute eigenvalues only;
102: *> = 'V': Compute eigenvalues and eigenvectors.
103: *> \endverbatim
104: *>
105: *> \param[in] RANGE
106: *> \verbatim
107: *> RANGE is CHARACTER*1
108: *> = 'A': all eigenvalues will be found.
109: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
110: *> will be found.
111: *> = 'I': the IL-th through IU-th eigenvalues will be found.
112: *> \endverbatim
113: *>
114: *> \param[in] N
115: *> \verbatim
116: *> N is INTEGER
117: *> The order of the matrix. N >= 0.
118: *> \endverbatim
119: *>
120: *> \param[in,out] D
121: *> \verbatim
122: *> D is DOUBLE PRECISION array, dimension (N)
123: *> On entry, the N diagonal elements of the tridiagonal matrix
124: *> T. On exit, D is overwritten.
125: *> \endverbatim
126: *>
127: *> \param[in,out] E
128: *> \verbatim
129: *> E is DOUBLE PRECISION array, dimension (N)
130: *> On entry, the (N-1) subdiagonal elements of the tridiagonal
131: *> matrix T in elements 1 to N-1 of E. E(N) need not be set on
132: *> input, but is used internally as workspace.
133: *> On exit, E is overwritten.
134: *> \endverbatim
135: *>
136: *> \param[in] VL
137: *> \verbatim
138: *> VL is DOUBLE PRECISION
139: *>
140: *> If RANGE='V', the lower bound of the interval to
141: *> be searched for eigenvalues. VL < VU.
142: *> Not referenced if RANGE = 'A' or 'I'.
143: *> \endverbatim
144: *>
145: *> \param[in] VU
146: *> \verbatim
147: *> VU is DOUBLE PRECISION
148: *>
149: *> If RANGE='V', the upper bound of the interval to
150: *> be searched for eigenvalues. VL < VU.
151: *> Not referenced if RANGE = 'A' or 'I'.
152: *> \endverbatim
153: *>
154: *> \param[in] IL
155: *> \verbatim
156: *> IL is INTEGER
157: *>
158: *> If RANGE='I', the index of the
159: *> smallest eigenvalue to be returned.
160: *> 1 <= IL <= IU <= N, if N > 0.
161: *> Not referenced if RANGE = 'A' or 'V'.
162: *> \endverbatim
163: *>
164: *> \param[in] IU
165: *> \verbatim
166: *> IU is INTEGER
167: *>
168: *> If RANGE='I', the index of the
169: *> largest eigenvalue to be returned.
170: *> 1 <= IL <= IU <= N, if N > 0.
171: *> Not referenced if RANGE = 'A' or 'V'.
172: *> \endverbatim
173: *>
174: *> \param[out] M
175: *> \verbatim
176: *> M is INTEGER
177: *> The total number of eigenvalues found. 0 <= M <= N.
178: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
179: *> \endverbatim
180: *>
181: *> \param[out] W
182: *> \verbatim
183: *> W is DOUBLE PRECISION array, dimension (N)
184: *> The first M elements contain the selected eigenvalues in
185: *> ascending order.
186: *> \endverbatim
187: *>
188: *> \param[out] Z
189: *> \verbatim
190: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
191: *> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
192: *> contain the orthonormal eigenvectors of the matrix T
193: *> corresponding to the selected eigenvalues, with the i-th
194: *> column of Z holding the eigenvector associated with W(i).
195: *> If JOBZ = 'N', then Z is not referenced.
196: *> Note: the user must ensure that at least max(1,M) columns are
197: *> supplied in the array Z; if RANGE = 'V', the exact value of M
198: *> is not known in advance and can be computed with a workspace
199: *> query by setting NZC = -1, see below.
200: *> \endverbatim
201: *>
202: *> \param[in] LDZ
203: *> \verbatim
204: *> LDZ is INTEGER
205: *> The leading dimension of the array Z. LDZ >= 1, and if
206: *> JOBZ = 'V', then LDZ >= max(1,N).
207: *> \endverbatim
208: *>
209: *> \param[in] NZC
210: *> \verbatim
211: *> NZC is INTEGER
212: *> The number of eigenvectors to be held in the array Z.
213: *> If RANGE = 'A', then NZC >= max(1,N).
214: *> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
215: *> If RANGE = 'I', then NZC >= IU-IL+1.
216: *> If NZC = -1, then a workspace query is assumed; the
217: *> routine calculates the number of columns of the array Z that
218: *> are needed to hold the eigenvectors.
219: *> This value is returned as the first entry of the Z array, and
220: *> no error message related to NZC is issued by XERBLA.
221: *> \endverbatim
222: *>
223: *> \param[out] ISUPPZ
224: *> \verbatim
225: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
226: *> The support of the eigenvectors in Z, i.e., the indices
227: *> indicating the nonzero elements in Z. The i-th computed eigenvector
228: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
229: *> ISUPPZ( 2*i ). This is relevant in the case when the matrix
230: *> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
231: *> \endverbatim
232: *>
233: *> \param[in,out] TRYRAC
234: *> \verbatim
235: *> TRYRAC is LOGICAL
236: *> If TRYRAC = .TRUE., indicates that the code should check whether
237: *> the tridiagonal matrix defines its eigenvalues to high relative
238: *> accuracy. If so, the code uses relative-accuracy preserving
239: *> algorithms that might be (a bit) slower depending on the matrix.
240: *> If the matrix does not define its eigenvalues to high relative
241: *> accuracy, the code can uses possibly faster algorithms.
242: *> If TRYRAC = .FALSE., the code is not required to guarantee
243: *> relatively accurate eigenvalues and can use the fastest possible
244: *> techniques.
245: *> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
246: *> does not define its eigenvalues to high relative accuracy.
247: *> \endverbatim
248: *>
249: *> \param[out] WORK
250: *> \verbatim
251: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
252: *> On exit, if INFO = 0, WORK(1) returns the optimal
253: *> (and minimal) LWORK.
254: *> \endverbatim
255: *>
256: *> \param[in] LWORK
257: *> \verbatim
258: *> LWORK is INTEGER
259: *> The dimension of the array WORK. LWORK >= max(1,18*N)
260: *> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
261: *> If LWORK = -1, then a workspace query is assumed; the routine
262: *> only calculates the optimal size of the WORK array, returns
263: *> this value as the first entry of the WORK array, and no error
264: *> message related to LWORK is issued by XERBLA.
265: *> \endverbatim
266: *>
267: *> \param[out] IWORK
268: *> \verbatim
269: *> IWORK is INTEGER array, dimension (LIWORK)
270: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
271: *> \endverbatim
272: *>
273: *> \param[in] LIWORK
274: *> \verbatim
275: *> LIWORK is INTEGER
276: *> The dimension of the array IWORK. LIWORK >= max(1,10*N)
277: *> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
278: *> if only the eigenvalues are to be computed.
279: *> If LIWORK = -1, then a workspace query is assumed; the
280: *> routine only calculates the optimal size of the IWORK array,
281: *> returns this value as the first entry of the IWORK array, and
282: *> no error message related to LIWORK is issued by XERBLA.
283: *> \endverbatim
284: *>
285: *> \param[out] INFO
286: *> \verbatim
287: *> INFO is INTEGER
288: *> On exit, INFO
289: *> = 0: successful exit
290: *> < 0: if INFO = -i, the i-th argument had an illegal value
291: *> > 0: if INFO = 1X, internal error in DLARRE,
292: *> if INFO = 2X, internal error in DLARRV.
293: *> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
294: *> the nonzero error code returned by DLARRE or
295: *> DLARRV, respectively.
296: *> \endverbatim
297: *
298: * Authors:
299: * ========
300: *
301: *> \author Univ. of Tennessee
302: *> \author Univ. of California Berkeley
303: *> \author Univ. of Colorado Denver
304: *> \author NAG Ltd.
305: *
306: *> \ingroup doubleOTHERcomputational
307: *
308: *> \par Contributors:
309: * ==================
310: *>
311: *> Beresford Parlett, University of California, Berkeley, USA \n
312: *> Jim Demmel, University of California, Berkeley, USA \n
313: *> Inderjit Dhillon, University of Texas, Austin, USA \n
314: *> Osni Marques, LBNL/NERSC, USA \n
315: *> Christof Voemel, University of California, Berkeley, USA
316: *
317: * =====================================================================
318: SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
319: $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
320: $ IWORK, LIWORK, INFO )
321: *
322: * -- LAPACK computational routine --
323: * -- LAPACK is a software package provided by Univ. of Tennessee, --
324: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
325: *
326: * .. Scalar Arguments ..
327: CHARACTER JOBZ, RANGE
328: LOGICAL TRYRAC
329: INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
330: DOUBLE PRECISION VL, VU
331: * ..
332: * .. Array Arguments ..
333: INTEGER ISUPPZ( * ), IWORK( * )
334: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
335: DOUBLE PRECISION Z( LDZ, * )
336: * ..
337: *
338: * =====================================================================
339: *
340: * .. Parameters ..
341: DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
342: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
343: $ FOUR = 4.0D0,
344: $ MINRGP = 1.0D-3 )
345: * ..
346: * .. Local Scalars ..
347: LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
348: INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
349: $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
350: $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
351: $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
352: $ NZCMIN, OFFSET, WBEGIN, WEND
353: DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
354: $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
355: $ THRESH, TMP, TNRM, WL, WU
356: * ..
357: * ..
358: * .. External Functions ..
359: LOGICAL LSAME
360: DOUBLE PRECISION DLAMCH, DLANST
361: EXTERNAL LSAME, DLAMCH, DLANST
362: * ..
363: * .. External Subroutines ..
364: EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
365: $ DLARRR, DLARRV, DLASRT, DSCAL, DSWAP, XERBLA
366: * ..
367: * .. Intrinsic Functions ..
368: INTRINSIC MAX, MIN, SQRT
369:
370:
371: * ..
372: * .. Executable Statements ..
373: *
374: * Test the input parameters.
375: *
376: WANTZ = LSAME( JOBZ, 'V' )
377: ALLEIG = LSAME( RANGE, 'A' )
378: VALEIG = LSAME( RANGE, 'V' )
379: INDEIG = LSAME( RANGE, 'I' )
380: *
381: LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
382: ZQUERY = ( NZC.EQ.-1 )
383:
384: * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
385: * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
386: * Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N.
387: IF( WANTZ ) THEN
388: LWMIN = 18*N
389: LIWMIN = 10*N
390: ELSE
391: * need less workspace if only the eigenvalues are wanted
392: LWMIN = 12*N
393: LIWMIN = 8*N
394: ENDIF
395:
396: WL = ZERO
397: WU = ZERO
398: IIL = 0
399: IIU = 0
400: NSPLIT = 0
401:
402: IF( VALEIG ) THEN
403: * We do not reference VL, VU in the cases RANGE = 'I','A'
404: * The interval (WL, WU] contains all the wanted eigenvalues.
405: * It is either given by the user or computed in DLARRE.
406: WL = VL
407: WU = VU
408: ELSEIF( INDEIG ) THEN
409: * We do not reference IL, IU in the cases RANGE = 'V','A'
410: IIL = IL
411: IIU = IU
412: ENDIF
413: *
414: INFO = 0
415: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
416: INFO = -1
417: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
418: INFO = -2
419: ELSE IF( N.LT.0 ) THEN
420: INFO = -3
421: ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
422: INFO = -7
423: ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
424: INFO = -8
425: ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
426: INFO = -9
427: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
428: INFO = -13
429: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
430: INFO = -17
431: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
432: INFO = -19
433: END IF
434: *
435: * Get machine constants.
436: *
437: SAFMIN = DLAMCH( 'Safe minimum' )
438: EPS = DLAMCH( 'Precision' )
439: SMLNUM = SAFMIN / EPS
440: BIGNUM = ONE / SMLNUM
441: RMIN = SQRT( SMLNUM )
442: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
443: *
444: IF( INFO.EQ.0 ) THEN
445: WORK( 1 ) = LWMIN
446: IWORK( 1 ) = LIWMIN
447: *
448: IF( WANTZ .AND. ALLEIG ) THEN
449: NZCMIN = N
450: ELSE IF( WANTZ .AND. VALEIG ) THEN
451: CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
452: $ NZCMIN, ITMP, ITMP2, INFO )
453: ELSE IF( WANTZ .AND. INDEIG ) THEN
454: NZCMIN = IIU-IIL+1
455: ELSE
456: * WANTZ .EQ. FALSE.
457: NZCMIN = 0
458: ENDIF
459: IF( ZQUERY .AND. INFO.EQ.0 ) THEN
460: Z( 1,1 ) = NZCMIN
461: ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
462: INFO = -14
463: END IF
464: END IF
465:
466: IF( INFO.NE.0 ) THEN
467: *
468: CALL XERBLA( 'DSTEMR', -INFO )
469: *
470: RETURN
471: ELSE IF( LQUERY .OR. ZQUERY ) THEN
472: RETURN
473: END IF
474: *
475: * Handle N = 0, 1, and 2 cases immediately
476: *
477: M = 0
478: IF( N.EQ.0 )
479: $ RETURN
480: *
481: IF( N.EQ.1 ) THEN
482: IF( ALLEIG .OR. INDEIG ) THEN
483: M = 1
484: W( 1 ) = D( 1 )
485: ELSE
486: IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
487: M = 1
488: W( 1 ) = D( 1 )
489: END IF
490: END IF
491: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
492: Z( 1, 1 ) = ONE
493: ISUPPZ(1) = 1
494: ISUPPZ(2) = 1
495: END IF
496: RETURN
497: END IF
498: *
499: IF( N.EQ.2 ) THEN
500: IF( .NOT.WANTZ ) THEN
501: CALL DLAE2( D(1), E(1), D(2), R1, R2 )
502: ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
503: CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
504: END IF
505: IF( ALLEIG.OR.
506: $ (VALEIG.AND.(R2.GT.WL).AND.
507: $ (R2.LE.WU)).OR.
508: $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
509: M = M+1
510: W( M ) = R2
511: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
512: Z( 1, M ) = -SN
513: Z( 2, M ) = CS
514: * Note: At most one of SN and CS can be zero.
515: IF (SN.NE.ZERO) THEN
516: IF (CS.NE.ZERO) THEN
517: ISUPPZ(2*M-1) = 1
518: ISUPPZ(2*M) = 2
519: ELSE
520: ISUPPZ(2*M-1) = 1
521: ISUPPZ(2*M) = 1
522: END IF
523: ELSE
524: ISUPPZ(2*M-1) = 2
525: ISUPPZ(2*M) = 2
526: END IF
527: ENDIF
528: ENDIF
529: IF( ALLEIG.OR.
530: $ (VALEIG.AND.(R1.GT.WL).AND.
531: $ (R1.LE.WU)).OR.
532: $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
533: M = M+1
534: W( M ) = R1
535: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
536: Z( 1, M ) = CS
537: Z( 2, M ) = SN
538: * Note: At most one of SN and CS can be zero.
539: IF (SN.NE.ZERO) THEN
540: IF (CS.NE.ZERO) THEN
541: ISUPPZ(2*M-1) = 1
542: ISUPPZ(2*M) = 2
543: ELSE
544: ISUPPZ(2*M-1) = 1
545: ISUPPZ(2*M) = 1
546: END IF
547: ELSE
548: ISUPPZ(2*M-1) = 2
549: ISUPPZ(2*M) = 2
550: END IF
551: ENDIF
552: ENDIF
553:
554: ELSE
555:
556: * Continue with general N
557:
558: INDGRS = 1
559: INDERR = 2*N + 1
560: INDGP = 3*N + 1
561: INDD = 4*N + 1
562: INDE2 = 5*N + 1
563: INDWRK = 6*N + 1
564: *
565: IINSPL = 1
566: IINDBL = N + 1
567: IINDW = 2*N + 1
568: IINDWK = 3*N + 1
569: *
570: * Scale matrix to allowable range, if necessary.
571: * The allowable range is related to the PIVMIN parameter; see the
572: * comments in DLARRD. The preference for scaling small values
573: * up is heuristic; we expect users' matrices not to be close to the
574: * RMAX threshold.
575: *
576: SCALE = ONE
577: TNRM = DLANST( 'M', N, D, E )
578: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
579: SCALE = RMIN / TNRM
580: ELSE IF( TNRM.GT.RMAX ) THEN
581: SCALE = RMAX / TNRM
582: END IF
583: IF( SCALE.NE.ONE ) THEN
584: CALL DSCAL( N, SCALE, D, 1 )
585: CALL DSCAL( N-1, SCALE, E, 1 )
586: TNRM = TNRM*SCALE
587: IF( VALEIG ) THEN
588: * If eigenvalues in interval have to be found,
589: * scale (WL, WU] accordingly
590: WL = WL*SCALE
591: WU = WU*SCALE
592: ENDIF
593: END IF
594: *
595: * Compute the desired eigenvalues of the tridiagonal after splitting
596: * into smaller subblocks if the corresponding off-diagonal elements
597: * are small
598: * THRESH is the splitting parameter for DLARRE
599: * A negative THRESH forces the old splitting criterion based on the
600: * size of the off-diagonal. A positive THRESH switches to splitting
601: * which preserves relative accuracy.
602: *
603: IF( TRYRAC ) THEN
604: * Test whether the matrix warrants the more expensive relative approach.
605: CALL DLARRR( N, D, E, IINFO )
606: ELSE
607: * The user does not care about relative accurately eigenvalues
608: IINFO = -1
609: ENDIF
610: * Set the splitting criterion
611: IF (IINFO.EQ.0) THEN
612: THRESH = EPS
613: ELSE
614: THRESH = -EPS
615: * relative accuracy is desired but T does not guarantee it
616: TRYRAC = .FALSE.
617: ENDIF
618: *
619: IF( TRYRAC ) THEN
620: * Copy original diagonal, needed to guarantee relative accuracy
621: CALL DCOPY(N,D,1,WORK(INDD),1)
622: ENDIF
623: * Store the squares of the offdiagonal values of T
624: DO 5 J = 1, N-1
625: WORK( INDE2+J-1 ) = E(J)**2
626: 5 CONTINUE
627:
628: * Set the tolerance parameters for bisection
629: IF( .NOT.WANTZ ) THEN
630: * DLARRE computes the eigenvalues to full precision.
631: RTOL1 = FOUR * EPS
632: RTOL2 = FOUR * EPS
633: ELSE
634: * DLARRE computes the eigenvalues to less than full precision.
635: * DLARRV will refine the eigenvalue approximations, and we can
636: * need less accurate initial bisection in DLARRE.
637: * Note: these settings do only affect the subset case and DLARRE
638: RTOL1 = SQRT(EPS)
639: RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
640: ENDIF
641: CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
642: $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
643: $ IWORK( IINSPL ), M, W, WORK( INDERR ),
644: $ WORK( INDGP ), IWORK( IINDBL ),
645: $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
646: $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
647: IF( IINFO.NE.0 ) THEN
648: INFO = 10 + ABS( IINFO )
649: RETURN
650: END IF
651: * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
652: * part of the spectrum. All desired eigenvalues are contained in
653: * (WL,WU]
654:
655:
656: IF( WANTZ ) THEN
657: *
658: * Compute the desired eigenvectors corresponding to the computed
659: * eigenvalues
660: *
661: CALL DLARRV( N, WL, WU, D, E,
662: $ PIVMIN, IWORK( IINSPL ), M,
663: $ 1, M, MINRGP, RTOL1, RTOL2,
664: $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
665: $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
666: $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
667: IF( IINFO.NE.0 ) THEN
668: INFO = 20 + ABS( IINFO )
669: RETURN
670: END IF
671: ELSE
672: * DLARRE computes eigenvalues of the (shifted) root representation
673: * DLARRV returns the eigenvalues of the unshifted matrix.
674: * However, if the eigenvectors are not desired by the user, we need
675: * to apply the corresponding shifts from DLARRE to obtain the
676: * eigenvalues of the original matrix.
677: DO 20 J = 1, M
678: ITMP = IWORK( IINDBL+J-1 )
679: W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
680: 20 CONTINUE
681: END IF
682: *
683:
684: IF ( TRYRAC ) THEN
685: * Refine computed eigenvalues so that they are relatively accurate
686: * with respect to the original matrix T.
687: IBEGIN = 1
688: WBEGIN = 1
689: DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
690: IEND = IWORK( IINSPL+JBLK-1 )
691: IN = IEND - IBEGIN + 1
692: WEND = WBEGIN - 1
693: * check if any eigenvalues have to be refined in this block
694: 36 CONTINUE
695: IF( WEND.LT.M ) THEN
696: IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
697: WEND = WEND + 1
698: GO TO 36
699: END IF
700: END IF
701: IF( WEND.LT.WBEGIN ) THEN
702: IBEGIN = IEND + 1
703: GO TO 39
704: END IF
705:
706: OFFSET = IWORK(IINDW+WBEGIN-1)-1
707: IFIRST = IWORK(IINDW+WBEGIN-1)
708: ILAST = IWORK(IINDW+WEND-1)
709: RTOL2 = FOUR * EPS
710: CALL DLARRJ( IN,
711: $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
712: $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
713: $ WORK( INDERR+WBEGIN-1 ),
714: $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
715: $ TNRM, IINFO )
716: IBEGIN = IEND + 1
717: WBEGIN = WEND + 1
718: 39 CONTINUE
719: ENDIF
720: *
721: * If matrix was scaled, then rescale eigenvalues appropriately.
722: *
723: IF( SCALE.NE.ONE ) THEN
724: CALL DSCAL( M, ONE / SCALE, W, 1 )
725: END IF
726:
727: END IF
728:
729: *
730: * If eigenvalues are not in increasing order, then sort them,
731: * possibly along with eigenvectors.
732: *
733: IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
734: IF( .NOT. WANTZ ) THEN
735: CALL DLASRT( 'I', M, W, IINFO )
736: IF( IINFO.NE.0 ) THEN
737: INFO = 3
738: RETURN
739: END IF
740: ELSE
741: DO 60 J = 1, M - 1
742: I = 0
743: TMP = W( J )
744: DO 50 JJ = J + 1, M
745: IF( W( JJ ).LT.TMP ) THEN
746: I = JJ
747: TMP = W( JJ )
748: END IF
749: 50 CONTINUE
750: IF( I.NE.0 ) THEN
751: W( I ) = W( J )
752: W( J ) = TMP
753: IF( WANTZ ) THEN
754: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
755: ITMP = ISUPPZ( 2*I-1 )
756: ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
757: ISUPPZ( 2*J-1 ) = ITMP
758: ITMP = ISUPPZ( 2*I )
759: ISUPPZ( 2*I ) = ISUPPZ( 2*J )
760: ISUPPZ( 2*J ) = ITMP
761: END IF
762: END IF
763: 60 CONTINUE
764: END IF
765: ENDIF
766: *
767: *
768: WORK( 1 ) = LWMIN
769: IWORK( 1 ) = LIWMIN
770: RETURN
771: *
772: * End of DSTEMR
773: *
774: END
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