1: *> \brief \b ZSTEMR
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZSTEMR + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zstemr.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zstemr.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zstemr.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZSTEMR( 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: * COMPLEX*16 Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZSTEMR 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.ZSTEMR 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: *>
94: *> 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
95: *> real symmetric tridiagonal form.
96: *>
97: *> (Any complex Hermitean tridiagonal matrix has real values on its diagonal
98: *> and potentially complex numbers on its off-diagonals. By applying a
99: *> similarity transform with an appropriate diagonal matrix
100: *> diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
101: *> matrix can be transformed into a real symmetric matrix and complex
102: *> arithmetic can be entirely avoided.)
103: *>
104: *> While the eigenvectors of the real symmetric tridiagonal matrix are real,
105: *> the eigenvectors of original complex Hermitean matrix have complex entries
106: *> in general.
107: *> Since LAPACK drivers overwrite the matrix data with the eigenvectors,
108: *> ZSTEMR accepts complex workspace to facilitate interoperability
109: *> with ZUNMTR or ZUPMTR.
110: *> \endverbatim
111: *
112: * Arguments:
113: * ==========
114: *
115: *> \param[in] JOBZ
116: *> \verbatim
117: *> JOBZ is CHARACTER*1
118: *> = 'N': Compute eigenvalues only;
119: *> = 'V': Compute eigenvalues and eigenvectors.
120: *> \endverbatim
121: *>
122: *> \param[in] RANGE
123: *> \verbatim
124: *> RANGE is CHARACTER*1
125: *> = 'A': all eigenvalues will be found.
126: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
127: *> will be found.
128: *> = 'I': the IL-th through IU-th eigenvalues will be found.
129: *> \endverbatim
130: *>
131: *> \param[in] N
132: *> \verbatim
133: *> N is INTEGER
134: *> The order of the matrix. N >= 0.
135: *> \endverbatim
136: *>
137: *> \param[in,out] D
138: *> \verbatim
139: *> D is DOUBLE PRECISION array, dimension (N)
140: *> On entry, the N diagonal elements of the tridiagonal matrix
141: *> T. On exit, D is overwritten.
142: *> \endverbatim
143: *>
144: *> \param[in,out] E
145: *> \verbatim
146: *> E is DOUBLE PRECISION array, dimension (N)
147: *> On entry, the (N-1) subdiagonal elements of the tridiagonal
148: *> matrix T in elements 1 to N-1 of E. E(N) need not be set on
149: *> input, but is used internally as workspace.
150: *> On exit, E is overwritten.
151: *> \endverbatim
152: *>
153: *> \param[in] VL
154: *> \verbatim
155: *> VL is DOUBLE PRECISION
156: *> \endverbatim
157: *>
158: *> \param[in] VU
159: *> \verbatim
160: *> VU is DOUBLE PRECISION
161: *>
162: *> If RANGE='V', the lower and upper bounds of the interval to
163: *> be searched for eigenvalues. VL < VU.
164: *> Not referenced if RANGE = 'A' or 'I'.
165: *> \endverbatim
166: *>
167: *> \param[in] IL
168: *> \verbatim
169: *> IL is INTEGER
170: *> \endverbatim
171: *>
172: *> \param[in] IU
173: *> \verbatim
174: *> IU is INTEGER
175: *>
176: *> If RANGE='I', the indices (in ascending order) of the
177: *> smallest and largest eigenvalues to be returned.
178: *> 1 <= IL <= IU <= N, if N > 0.
179: *> Not referenced if RANGE = 'A' or 'V'.
180: *> \endverbatim
181: *>
182: *> \param[out] M
183: *> \verbatim
184: *> M is INTEGER
185: *> The total number of eigenvalues found. 0 <= M <= N.
186: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
187: *> \endverbatim
188: *>
189: *> \param[out] W
190: *> \verbatim
191: *> W is DOUBLE PRECISION array, dimension (N)
192: *> The first M elements contain the selected eigenvalues in
193: *> ascending order.
194: *> \endverbatim
195: *>
196: *> \param[out] Z
197: *> \verbatim
198: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
199: *> If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
200: *> contain the orthonormal eigenvectors of the matrix T
201: *> corresponding to the selected eigenvalues, with the i-th
202: *> column of Z holding the eigenvector associated with W(i).
203: *> If JOBZ = 'N', then Z is not referenced.
204: *> Note: the user must ensure that at least max(1,M) columns are
205: *> supplied in the array Z; if RANGE = 'V', the exact value of M
206: *> is not known in advance and can be computed with a workspace
207: *> query by setting NZC = -1, see below.
208: *> \endverbatim
209: *>
210: *> \param[in] LDZ
211: *> \verbatim
212: *> LDZ is INTEGER
213: *> The leading dimension of the array Z. LDZ >= 1, and if
214: *> JOBZ = 'V', then LDZ >= max(1,N).
215: *> \endverbatim
216: *>
217: *> \param[in] NZC
218: *> \verbatim
219: *> NZC is INTEGER
220: *> The number of eigenvectors to be held in the array Z.
221: *> If RANGE = 'A', then NZC >= max(1,N).
222: *> If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
223: *> If RANGE = 'I', then NZC >= IU-IL+1.
224: *> If NZC = -1, then a workspace query is assumed; the
225: *> routine calculates the number of columns of the array Z that
226: *> are needed to hold the eigenvectors.
227: *> This value is returned as the first entry of the Z array, and
228: *> no error message related to NZC is issued by XERBLA.
229: *> \endverbatim
230: *>
231: *> \param[out] ISUPPZ
232: *> \verbatim
233: *> ISUPPZ is INTEGER ARRAY, dimension ( 2*max(1,M) )
234: *> The support of the eigenvectors in Z, i.e., the indices
235: *> indicating the nonzero elements in Z. The i-th computed eigenvector
236: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
237: *> ISUPPZ( 2*i ). This is relevant in the case when the matrix
238: *> is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
239: *> \endverbatim
240: *>
241: *> \param[in,out] TRYRAC
242: *> \verbatim
243: *> TRYRAC is LOGICAL
244: *> If TRYRAC.EQ..TRUE., indicates that the code should check whether
245: *> the tridiagonal matrix defines its eigenvalues to high relative
246: *> accuracy. If so, the code uses relative-accuracy preserving
247: *> algorithms that might be (a bit) slower depending on the matrix.
248: *> If the matrix does not define its eigenvalues to high relative
249: *> accuracy, the code can uses possibly faster algorithms.
250: *> If TRYRAC.EQ..FALSE., the code is not required to guarantee
251: *> relatively accurate eigenvalues and can use the fastest possible
252: *> techniques.
253: *> On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
254: *> does not define its eigenvalues to high relative accuracy.
255: *> \endverbatim
256: *>
257: *> \param[out] WORK
258: *> \verbatim
259: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
260: *> On exit, if INFO = 0, WORK(1) returns the optimal
261: *> (and minimal) LWORK.
262: *> \endverbatim
263: *>
264: *> \param[in] LWORK
265: *> \verbatim
266: *> LWORK is INTEGER
267: *> The dimension of the array WORK. LWORK >= max(1,18*N)
268: *> if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
269: *> If LWORK = -1, then a workspace query is assumed; the routine
270: *> only calculates the optimal size of the WORK array, returns
271: *> this value as the first entry of the WORK array, and no error
272: *> message related to LWORK is issued by XERBLA.
273: *> \endverbatim
274: *>
275: *> \param[out] IWORK
276: *> \verbatim
277: *> IWORK is INTEGER array, dimension (LIWORK)
278: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
279: *> \endverbatim
280: *>
281: *> \param[in] LIWORK
282: *> \verbatim
283: *> LIWORK is INTEGER
284: *> The dimension of the array IWORK. LIWORK >= max(1,10*N)
285: *> if the eigenvectors are desired, and LIWORK >= max(1,8*N)
286: *> if only the eigenvalues are to be computed.
287: *> If LIWORK = -1, then a workspace query is assumed; the
288: *> routine only calculates the optimal size of the IWORK array,
289: *> returns this value as the first entry of the IWORK array, and
290: *> no error message related to LIWORK is issued by XERBLA.
291: *> \endverbatim
292: *>
293: *> \param[out] INFO
294: *> \verbatim
295: *> INFO is INTEGER
296: *> On exit, INFO
297: *> = 0: successful exit
298: *> < 0: if INFO = -i, the i-th argument had an illegal value
299: *> > 0: if INFO = 1X, internal error in DLARRE,
300: *> if INFO = 2X, internal error in ZLARRV.
301: *> Here, the digit X = ABS( IINFO ) < 10, where IINFO is
302: *> the nonzero error code returned by DLARRE or
303: *> ZLARRV, respectively.
304: *> \endverbatim
305: *
306: * Authors:
307: * ========
308: *
309: *> \author Univ. of Tennessee
310: *> \author Univ. of California Berkeley
311: *> \author Univ. of Colorado Denver
312: *> \author NAG Ltd.
313: *
314: *> \date November 2013
315: *
316: *> \ingroup complex16OTHERcomputational
317: *
318: *> \par Contributors:
319: * ==================
320: *>
321: *> Beresford Parlett, University of California, Berkeley, USA \n
322: *> Jim Demmel, University of California, Berkeley, USA \n
323: *> Inderjit Dhillon, University of Texas, Austin, USA \n
324: *> Osni Marques, LBNL/NERSC, USA \n
325: *> Christof Voemel, University of California, Berkeley, USA \n
326: *
327: * =====================================================================
328: SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
329: $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
330: $ IWORK, LIWORK, INFO )
331: *
332: * -- LAPACK computational routine (version 3.5.0) --
333: * -- LAPACK is a software package provided by Univ. of Tennessee, --
334: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
335: * November 2013
336: *
337: * .. Scalar Arguments ..
338: CHARACTER JOBZ, RANGE
339: LOGICAL TRYRAC
340: INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
341: DOUBLE PRECISION VL, VU
342: * ..
343: * .. Array Arguments ..
344: INTEGER ISUPPZ( * ), IWORK( * )
345: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
346: COMPLEX*16 Z( LDZ, * )
347: * ..
348: *
349: * =====================================================================
350: *
351: * .. Parameters ..
352: DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
353: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
354: $ FOUR = 4.0D0,
355: $ MINRGP = 1.0D-3 )
356: * ..
357: * .. Local Scalars ..
358: LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
359: INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
360: $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
361: $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
362: $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
363: $ NZCMIN, OFFSET, WBEGIN, WEND
364: DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
365: $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
366: $ THRESH, TMP, TNRM, WL, WU
367: * ..
368: * ..
369: * .. External Functions ..
370: LOGICAL LSAME
371: DOUBLE PRECISION DLAMCH, DLANST
372: EXTERNAL LSAME, DLAMCH, DLANST
373: * ..
374: * .. External Subroutines ..
375: EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
376: $ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
377: * ..
378: * .. Intrinsic Functions ..
379: INTRINSIC MAX, MIN, SQRT
380:
381:
382: * ..
383: * .. Executable Statements ..
384: *
385: * Test the input parameters.
386: *
387: WANTZ = LSAME( JOBZ, 'V' )
388: ALLEIG = LSAME( RANGE, 'A' )
389: VALEIG = LSAME( RANGE, 'V' )
390: INDEIG = LSAME( RANGE, 'I' )
391: *
392: LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
393: ZQUERY = ( NZC.EQ.-1 )
394:
395: * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
396: * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
397: * Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
398: IF( WANTZ ) THEN
399: LWMIN = 18*N
400: LIWMIN = 10*N
401: ELSE
402: * need less workspace if only the eigenvalues are wanted
403: LWMIN = 12*N
404: LIWMIN = 8*N
405: ENDIF
406:
407: WL = ZERO
408: WU = ZERO
409: IIL = 0
410: IIU = 0
411: NSPLIT = 0
412:
413: IF( VALEIG ) THEN
414: * We do not reference VL, VU in the cases RANGE = 'I','A'
415: * The interval (WL, WU] contains all the wanted eigenvalues.
416: * It is either given by the user or computed in DLARRE.
417: WL = VL
418: WU = VU
419: ELSEIF( INDEIG ) THEN
420: * We do not reference IL, IU in the cases RANGE = 'V','A'
421: IIL = IL
422: IIU = IU
423: ENDIF
424: *
425: INFO = 0
426: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
427: INFO = -1
428: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
429: INFO = -2
430: ELSE IF( N.LT.0 ) THEN
431: INFO = -3
432: ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
433: INFO = -7
434: ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
435: INFO = -8
436: ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
437: INFO = -9
438: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
439: INFO = -13
440: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
441: INFO = -17
442: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
443: INFO = -19
444: END IF
445: *
446: * Get machine constants.
447: *
448: SAFMIN = DLAMCH( 'Safe minimum' )
449: EPS = DLAMCH( 'Precision' )
450: SMLNUM = SAFMIN / EPS
451: BIGNUM = ONE / SMLNUM
452: RMIN = SQRT( SMLNUM )
453: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
454: *
455: IF( INFO.EQ.0 ) THEN
456: WORK( 1 ) = LWMIN
457: IWORK( 1 ) = LIWMIN
458: *
459: IF( WANTZ .AND. ALLEIG ) THEN
460: NZCMIN = N
461: ELSE IF( WANTZ .AND. VALEIG ) THEN
462: CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
463: $ NZCMIN, ITMP, ITMP2, INFO )
464: ELSE IF( WANTZ .AND. INDEIG ) THEN
465: NZCMIN = IIU-IIL+1
466: ELSE
467: * WANTZ .EQ. FALSE.
468: NZCMIN = 0
469: ENDIF
470: IF( ZQUERY .AND. INFO.EQ.0 ) THEN
471: Z( 1,1 ) = NZCMIN
472: ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
473: INFO = -14
474: END IF
475: END IF
476:
477: IF( INFO.NE.0 ) THEN
478: *
479: CALL XERBLA( 'ZSTEMR', -INFO )
480: *
481: RETURN
482: ELSE IF( LQUERY .OR. ZQUERY ) THEN
483: RETURN
484: END IF
485: *
486: * Handle N = 0, 1, and 2 cases immediately
487: *
488: M = 0
489: IF( N.EQ.0 )
490: $ RETURN
491: *
492: IF( N.EQ.1 ) THEN
493: IF( ALLEIG .OR. INDEIG ) THEN
494: M = 1
495: W( 1 ) = D( 1 )
496: ELSE
497: IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
498: M = 1
499: W( 1 ) = D( 1 )
500: END IF
501: END IF
502: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
503: Z( 1, 1 ) = ONE
504: ISUPPZ(1) = 1
505: ISUPPZ(2) = 1
506: END IF
507: RETURN
508: END IF
509: *
510: IF( N.EQ.2 ) THEN
511: IF( .NOT.WANTZ ) THEN
512: CALL DLAE2( D(1), E(1), D(2), R1, R2 )
513: ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
514: CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
515: END IF
516: IF( ALLEIG.OR.
517: $ (VALEIG.AND.(R2.GT.WL).AND.
518: $ (R2.LE.WU)).OR.
519: $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
520: M = M+1
521: W( M ) = R2
522: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
523: Z( 1, M ) = -SN
524: Z( 2, M ) = CS
525: * Note: At most one of SN and CS can be zero.
526: IF (SN.NE.ZERO) THEN
527: IF (CS.NE.ZERO) THEN
528: ISUPPZ(2*M-1) = 1
529: ISUPPZ(2*M-1) = 2
530: ELSE
531: ISUPPZ(2*M-1) = 1
532: ISUPPZ(2*M-1) = 1
533: END IF
534: ELSE
535: ISUPPZ(2*M-1) = 2
536: ISUPPZ(2*M) = 2
537: END IF
538: ENDIF
539: ENDIF
540: IF( ALLEIG.OR.
541: $ (VALEIG.AND.(R1.GT.WL).AND.
542: $ (R1.LE.WU)).OR.
543: $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
544: M = M+1
545: W( M ) = R1
546: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
547: Z( 1, M ) = CS
548: Z( 2, M ) = SN
549: * Note: At most one of SN and CS can be zero.
550: IF (SN.NE.ZERO) THEN
551: IF (CS.NE.ZERO) THEN
552: ISUPPZ(2*M-1) = 1
553: ISUPPZ(2*M-1) = 2
554: ELSE
555: ISUPPZ(2*M-1) = 1
556: ISUPPZ(2*M-1) = 1
557: END IF
558: ELSE
559: ISUPPZ(2*M-1) = 2
560: ISUPPZ(2*M) = 2
561: END IF
562: ENDIF
563: ENDIF
564: ELSE
565:
566: * Continue with general N
567:
568: INDGRS = 1
569: INDERR = 2*N + 1
570: INDGP = 3*N + 1
571: INDD = 4*N + 1
572: INDE2 = 5*N + 1
573: INDWRK = 6*N + 1
574: *
575: IINSPL = 1
576: IINDBL = N + 1
577: IINDW = 2*N + 1
578: IINDWK = 3*N + 1
579: *
580: * Scale matrix to allowable range, if necessary.
581: * The allowable range is related to the PIVMIN parameter; see the
582: * comments in DLARRD. The preference for scaling small values
583: * up is heuristic; we expect users' matrices not to be close to the
584: * RMAX threshold.
585: *
586: SCALE = ONE
587: TNRM = DLANST( 'M', N, D, E )
588: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
589: SCALE = RMIN / TNRM
590: ELSE IF( TNRM.GT.RMAX ) THEN
591: SCALE = RMAX / TNRM
592: END IF
593: IF( SCALE.NE.ONE ) THEN
594: CALL DSCAL( N, SCALE, D, 1 )
595: CALL DSCAL( N-1, SCALE, E, 1 )
596: TNRM = TNRM*SCALE
597: IF( VALEIG ) THEN
598: * If eigenvalues in interval have to be found,
599: * scale (WL, WU] accordingly
600: WL = WL*SCALE
601: WU = WU*SCALE
602: ENDIF
603: END IF
604: *
605: * Compute the desired eigenvalues of the tridiagonal after splitting
606: * into smaller subblocks if the corresponding off-diagonal elements
607: * are small
608: * THRESH is the splitting parameter for DLARRE
609: * A negative THRESH forces the old splitting criterion based on the
610: * size of the off-diagonal. A positive THRESH switches to splitting
611: * which preserves relative accuracy.
612: *
613: IF( TRYRAC ) THEN
614: * Test whether the matrix warrants the more expensive relative approach.
615: CALL DLARRR( N, D, E, IINFO )
616: ELSE
617: * The user does not care about relative accurately eigenvalues
618: IINFO = -1
619: ENDIF
620: * Set the splitting criterion
621: IF (IINFO.EQ.0) THEN
622: THRESH = EPS
623: ELSE
624: THRESH = -EPS
625: * relative accuracy is desired but T does not guarantee it
626: TRYRAC = .FALSE.
627: ENDIF
628: *
629: IF( TRYRAC ) THEN
630: * Copy original diagonal, needed to guarantee relative accuracy
631: CALL DCOPY(N,D,1,WORK(INDD),1)
632: ENDIF
633: * Store the squares of the offdiagonal values of T
634: DO 5 J = 1, N-1
635: WORK( INDE2+J-1 ) = E(J)**2
636: 5 CONTINUE
637:
638: * Set the tolerance parameters for bisection
639: IF( .NOT.WANTZ ) THEN
640: * DLARRE computes the eigenvalues to full precision.
641: RTOL1 = FOUR * EPS
642: RTOL2 = FOUR * EPS
643: ELSE
644: * DLARRE computes the eigenvalues to less than full precision.
645: * ZLARRV will refine the eigenvalue approximations, and we only
646: * need less accurate initial bisection in DLARRE.
647: * Note: these settings do only affect the subset case and DLARRE
648: RTOL1 = SQRT(EPS)
649: RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
650: ENDIF
651: CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
652: $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
653: $ IWORK( IINSPL ), M, W, WORK( INDERR ),
654: $ WORK( INDGP ), IWORK( IINDBL ),
655: $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
656: $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
657: IF( IINFO.NE.0 ) THEN
658: INFO = 10 + ABS( IINFO )
659: RETURN
660: END IF
661: * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
662: * part of the spectrum. All desired eigenvalues are contained in
663: * (WL,WU]
664:
665:
666: IF( WANTZ ) THEN
667: *
668: * Compute the desired eigenvectors corresponding to the computed
669: * eigenvalues
670: *
671: CALL ZLARRV( N, WL, WU, D, E,
672: $ PIVMIN, IWORK( IINSPL ), M,
673: $ 1, M, MINRGP, RTOL1, RTOL2,
674: $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
675: $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
676: $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
677: IF( IINFO.NE.0 ) THEN
678: INFO = 20 + ABS( IINFO )
679: RETURN
680: END IF
681: ELSE
682: * DLARRE computes eigenvalues of the (shifted) root representation
683: * ZLARRV returns the eigenvalues of the unshifted matrix.
684: * However, if the eigenvectors are not desired by the user, we need
685: * to apply the corresponding shifts from DLARRE to obtain the
686: * eigenvalues of the original matrix.
687: DO 20 J = 1, M
688: ITMP = IWORK( IINDBL+J-1 )
689: W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
690: 20 CONTINUE
691: END IF
692: *
693:
694: IF ( TRYRAC ) THEN
695: * Refine computed eigenvalues so that they are relatively accurate
696: * with respect to the original matrix T.
697: IBEGIN = 1
698: WBEGIN = 1
699: DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
700: IEND = IWORK( IINSPL+JBLK-1 )
701: IN = IEND - IBEGIN + 1
702: WEND = WBEGIN - 1
703: * check if any eigenvalues have to be refined in this block
704: 36 CONTINUE
705: IF( WEND.LT.M ) THEN
706: IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
707: WEND = WEND + 1
708: GO TO 36
709: END IF
710: END IF
711: IF( WEND.LT.WBEGIN ) THEN
712: IBEGIN = IEND + 1
713: GO TO 39
714: END IF
715:
716: OFFSET = IWORK(IINDW+WBEGIN-1)-1
717: IFIRST = IWORK(IINDW+WBEGIN-1)
718: ILAST = IWORK(IINDW+WEND-1)
719: RTOL2 = FOUR * EPS
720: CALL DLARRJ( IN,
721: $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
722: $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
723: $ WORK( INDERR+WBEGIN-1 ),
724: $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
725: $ TNRM, IINFO )
726: IBEGIN = IEND + 1
727: WBEGIN = WEND + 1
728: 39 CONTINUE
729: ENDIF
730: *
731: * If matrix was scaled, then rescale eigenvalues appropriately.
732: *
733: IF( SCALE.NE.ONE ) THEN
734: CALL DSCAL( M, ONE / SCALE, W, 1 )
735: END IF
736: END IF
737: *
738: * If eigenvalues are not in increasing order, then sort them,
739: * possibly along with eigenvectors.
740: *
741: IF( NSPLIT.GT.1 .OR. N.EQ.2 ) THEN
742: IF( .NOT. WANTZ ) THEN
743: CALL DLASRT( 'I', M, W, IINFO )
744: IF( IINFO.NE.0 ) THEN
745: INFO = 3
746: RETURN
747: END IF
748: ELSE
749: DO 60 J = 1, M - 1
750: I = 0
751: TMP = W( J )
752: DO 50 JJ = J + 1, M
753: IF( W( JJ ).LT.TMP ) THEN
754: I = JJ
755: TMP = W( JJ )
756: END IF
757: 50 CONTINUE
758: IF( I.NE.0 ) THEN
759: W( I ) = W( J )
760: W( J ) = TMP
761: IF( WANTZ ) THEN
762: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
763: ITMP = ISUPPZ( 2*I-1 )
764: ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
765: ISUPPZ( 2*J-1 ) = ITMP
766: ITMP = ISUPPZ( 2*I )
767: ISUPPZ( 2*I ) = ISUPPZ( 2*J )
768: ISUPPZ( 2*J ) = ITMP
769: END IF
770: END IF
771: 60 CONTINUE
772: END IF
773: ENDIF
774: *
775: *
776: WORK( 1 ) = LWMIN
777: IWORK( 1 ) = LIWMIN
778: RETURN
779: *
780: * End of ZSTEMR
781: *
782: END
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