1: *> \brief <b> ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHEEVR + 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/zheevr.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22: * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
23: * RWORK, LRWORK, IWORK, LIWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
28: * $ M, N
29: * DOUBLE PRECISION ABSTOL, VL, VU
30: * ..
31: * .. Array Arguments ..
32: * INTEGER ISUPPZ( * ), IWORK( * )
33: * DOUBLE PRECISION RWORK( * ), W( * )
34: * COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZHEEVR computes selected eigenvalues and, optionally, eigenvectors
44: *> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
45: *> be selected by specifying either a range of values or a range of
46: *> indices for the desired eigenvalues.
47: *>
48: *> ZHEEVR first reduces the matrix A to tridiagonal form T with a call
49: *> to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute
50: *> eigenspectrum using Relatively Robust Representations. ZSTEMR
51: *> computes eigenvalues by the dqds algorithm, while orthogonal
52: *> eigenvectors are computed from various "good" L D L^T representations
53: *> (also known as Relatively Robust Representations). Gram-Schmidt
54: *> orthogonalization is avoided as far as possible. More specifically,
55: *> the various steps of the algorithm are as follows.
56: *>
57: *> For each unreduced block (submatrix) of T,
58: *> (a) Compute T - sigma I = L D L^T, so that L and D
59: *> define all the wanted eigenvalues to high relative accuracy.
60: *> This means that small relative changes in the entries of D and L
61: *> cause only small relative changes in the eigenvalues and
62: *> eigenvectors. The standard (unfactored) representation of the
63: *> tridiagonal matrix T does not have this property in general.
64: *> (b) Compute the eigenvalues to suitable accuracy.
65: *> If the eigenvectors are desired, the algorithm attains full
66: *> accuracy of the computed eigenvalues only right before
67: *> the corresponding vectors have to be computed, see steps c) and d).
68: *> (c) For each cluster of close eigenvalues, select a new
69: *> shift close to the cluster, find a new factorization, and refine
70: *> the shifted eigenvalues to suitable accuracy.
71: *> (d) For each eigenvalue with a large enough relative separation compute
72: *> the corresponding eigenvector by forming a rank revealing twisted
73: *> factorization. Go back to (c) for any clusters that remain.
74: *>
75: *> The desired accuracy of the output can be specified by the input
76: *> parameter ABSTOL.
77: *>
78: *> For more details, see DSTEMR's documentation and:
79: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
80: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
81: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
82: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
83: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
84: *> 2004. Also LAPACK Working Note 154.
85: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
86: *> tridiagonal eigenvalue/eigenvector problem",
87: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
88: *> UC Berkeley, May 1997.
89: *>
90: *>
91: *> Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested
92: *> on machines which conform to the ieee-754 floating point standard.
93: *> ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
94: *> when partial spectrum requests are made.
95: *>
96: *> Normal execution of ZSTEMR may create NaNs and infinities and
97: *> hence may abort due to a floating point exception in environments
98: *> which do not handle NaNs and infinities in the ieee standard default
99: *> manner.
100: *> \endverbatim
101: *
102: * Arguments:
103: * ==========
104: *
105: *> \param[in] JOBZ
106: *> \verbatim
107: *> JOBZ is CHARACTER*1
108: *> = 'N': Compute eigenvalues only;
109: *> = 'V': Compute eigenvalues and eigenvectors.
110: *> \endverbatim
111: *>
112: *> \param[in] RANGE
113: *> \verbatim
114: *> RANGE is CHARACTER*1
115: *> = 'A': all eigenvalues will be found.
116: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
117: *> will be found.
118: *> = 'I': the IL-th through IU-th eigenvalues will be found.
119: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
120: *> ZSTEIN are called
121: *> \endverbatim
122: *>
123: *> \param[in] UPLO
124: *> \verbatim
125: *> UPLO is CHARACTER*1
126: *> = 'U': Upper triangle of A is stored;
127: *> = 'L': Lower triangle of A is stored.
128: *> \endverbatim
129: *>
130: *> \param[in] N
131: *> \verbatim
132: *> N is INTEGER
133: *> The order of the matrix A. N >= 0.
134: *> \endverbatim
135: *>
136: *> \param[in,out] A
137: *> \verbatim
138: *> A is COMPLEX*16 array, dimension (LDA, N)
139: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
140: *> leading N-by-N upper triangular part of A contains the
141: *> upper triangular part of the matrix A. If UPLO = 'L',
142: *> the leading N-by-N lower triangular part of A contains
143: *> the lower triangular part of the matrix A.
144: *> On exit, the lower triangle (if UPLO='L') or the upper
145: *> triangle (if UPLO='U') of A, including the diagonal, is
146: *> destroyed.
147: *> \endverbatim
148: *>
149: *> \param[in] LDA
150: *> \verbatim
151: *> LDA is INTEGER
152: *> The leading dimension of the array A. LDA >= max(1,N).
153: *> \endverbatim
154: *>
155: *> \param[in] VL
156: *> \verbatim
157: *> VL is DOUBLE PRECISION
158: *> If RANGE='V', the lower bound of the interval to
159: *> be searched for eigenvalues. VL < VU.
160: *> Not referenced if RANGE = 'A' or 'I'.
161: *> \endverbatim
162: *>
163: *> \param[in] VU
164: *> \verbatim
165: *> VU is DOUBLE PRECISION
166: *> If RANGE='V', the upper bound of the interval to
167: *> be searched for eigenvalues. VL < VU.
168: *> Not referenced if RANGE = 'A' or 'I'.
169: *> \endverbatim
170: *>
171: *> \param[in] IL
172: *> \verbatim
173: *> IL is INTEGER
174: *> If RANGE='I', the index of the
175: *> smallest eigenvalue to be returned.
176: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
177: *> Not referenced if RANGE = 'A' or 'V'.
178: *> \endverbatim
179: *>
180: *> \param[in] IU
181: *> \verbatim
182: *> IU is INTEGER
183: *> If RANGE='I', the index of the
184: *> largest eigenvalue to be returned.
185: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
186: *> Not referenced if RANGE = 'A' or 'V'.
187: *> \endverbatim
188: *>
189: *> \param[in] ABSTOL
190: *> \verbatim
191: *> ABSTOL is DOUBLE PRECISION
192: *> The absolute error tolerance for the eigenvalues.
193: *> An approximate eigenvalue is accepted as converged
194: *> when it is determined to lie in an interval [a,b]
195: *> of width less than or equal to
196: *>
197: *> ABSTOL + EPS * max( |a|,|b| ) ,
198: *>
199: *> where EPS is the machine precision. If ABSTOL is less than
200: *> or equal to zero, then EPS*|T| will be used in its place,
201: *> where |T| is the 1-norm of the tridiagonal matrix obtained
202: *> by reducing A to tridiagonal form.
203: *>
204: *> See "Computing Small Singular Values of Bidiagonal Matrices
205: *> with Guaranteed High Relative Accuracy," by Demmel and
206: *> Kahan, LAPACK Working Note #3.
207: *>
208: *> If high relative accuracy is important, set ABSTOL to
209: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
210: *> eigenvalues are computed to high relative accuracy when
211: *> possible in future releases. The current code does not
212: *> make any guarantees about high relative accuracy, but
213: *> furutre releases will. See J. Barlow and J. Demmel,
214: *> "Computing Accurate Eigensystems of Scaled Diagonally
215: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
216: *> of which matrices define their eigenvalues to high relative
217: *> accuracy.
218: *> \endverbatim
219: *>
220: *> \param[out] M
221: *> \verbatim
222: *> M is INTEGER
223: *> The total number of eigenvalues found. 0 <= M <= N.
224: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
225: *> \endverbatim
226: *>
227: *> \param[out] W
228: *> \verbatim
229: *> W is DOUBLE PRECISION array, dimension (N)
230: *> The first M elements contain the selected eigenvalues in
231: *> ascending order.
232: *> \endverbatim
233: *>
234: *> \param[out] Z
235: *> \verbatim
236: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
237: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
238: *> contain the orthonormal eigenvectors of the matrix A
239: *> corresponding to the selected eigenvalues, with the i-th
240: *> column of Z holding the eigenvector associated with W(i).
241: *> If JOBZ = 'N', then Z is not referenced.
242: *> Note: the user must ensure that at least max(1,M) columns are
243: *> supplied in the array Z; if RANGE = 'V', the exact value of M
244: *> is not known in advance and an upper bound must be used.
245: *> \endverbatim
246: *>
247: *> \param[in] LDZ
248: *> \verbatim
249: *> LDZ is INTEGER
250: *> The leading dimension of the array Z. LDZ >= 1, and if
251: *> JOBZ = 'V', LDZ >= max(1,N).
252: *> \endverbatim
253: *>
254: *> \param[out] ISUPPZ
255: *> \verbatim
256: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
257: *> The support of the eigenvectors in Z, i.e., the indices
258: *> indicating the nonzero elements in Z. The i-th eigenvector
259: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
260: *> ISUPPZ( 2*i ).
261: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
262: *> \endverbatim
263: *>
264: *> \param[out] WORK
265: *> \verbatim
266: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
267: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
268: *> \endverbatim
269: *>
270: *> \param[in] LWORK
271: *> \verbatim
272: *> LWORK is INTEGER
273: *> The length of the array WORK. LWORK >= max(1,2*N).
274: *> For optimal efficiency, LWORK >= (NB+1)*N,
275: *> where NB is the max of the blocksize for ZHETRD and for
276: *> ZUNMTR as returned by ILAENV.
277: *>
278: *> If LWORK = -1, then a workspace query is assumed; the routine
279: *> only calculates the optimal sizes of the WORK, RWORK and
280: *> IWORK arrays, returns these values as the first entries of
281: *> the WORK, RWORK and IWORK arrays, and no error message
282: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
283: *> \endverbatim
284: *>
285: *> \param[out] RWORK
286: *> \verbatim
287: *> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
288: *> On exit, if INFO = 0, RWORK(1) returns the optimal
289: *> (and minimal) LRWORK.
290: *> \endverbatim
291: *>
292: *> \param[in] LRWORK
293: *> \verbatim
294: *> LRWORK is INTEGER
295: *> The length of the array RWORK. LRWORK >= max(1,24*N).
296: *>
297: *> If LRWORK = -1, then a workspace query is assumed; the
298: *> routine only calculates the optimal sizes of the WORK, RWORK
299: *> and IWORK arrays, returns these values as the first entries
300: *> of the WORK, RWORK and IWORK arrays, and no error message
301: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
302: *> \endverbatim
303: *>
304: *> \param[out] IWORK
305: *> \verbatim
306: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
307: *> On exit, if INFO = 0, IWORK(1) returns the optimal
308: *> (and minimal) LIWORK.
309: *> \endverbatim
310: *>
311: *> \param[in] LIWORK
312: *> \verbatim
313: *> LIWORK is INTEGER
314: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
315: *>
316: *> If LIWORK = -1, then a workspace query is assumed; the
317: *> routine only calculates the optimal sizes of the WORK, RWORK
318: *> and IWORK arrays, returns these values as the first entries
319: *> of the WORK, RWORK and IWORK arrays, and no error message
320: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
321: *> \endverbatim
322: *>
323: *> \param[out] INFO
324: *> \verbatim
325: *> INFO is INTEGER
326: *> = 0: successful exit
327: *> < 0: if INFO = -i, the i-th argument had an illegal value
328: *> > 0: Internal error
329: *> \endverbatim
330: *
331: * Authors:
332: * ========
333: *
334: *> \author Univ. of Tennessee
335: *> \author Univ. of California Berkeley
336: *> \author Univ. of Colorado Denver
337: *> \author NAG Ltd.
338: *
339: *> \date June 2016
340: *
341: *> \ingroup complex16HEeigen
342: *
343: *> \par Contributors:
344: * ==================
345: *>
346: *> Inderjit Dhillon, IBM Almaden, USA \n
347: *> Osni Marques, LBNL/NERSC, USA \n
348: *> Ken Stanley, Computer Science Division, University of
349: *> California at Berkeley, USA \n
350: *> Jason Riedy, Computer Science Division, University of
351: *> California at Berkeley, USA \n
352: *>
353: * =====================================================================
354: SUBROUTINE ZHEEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
355: $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
356: $ RWORK, LRWORK, IWORK, LIWORK, INFO )
357: *
358: * -- LAPACK driver routine (version 3.6.1) --
359: * -- LAPACK is a software package provided by Univ. of Tennessee, --
360: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
361: * June 2016
362: *
363: * .. Scalar Arguments ..
364: CHARACTER JOBZ, RANGE, UPLO
365: INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
366: $ M, N
367: DOUBLE PRECISION ABSTOL, VL, VU
368: * ..
369: * .. Array Arguments ..
370: INTEGER ISUPPZ( * ), IWORK( * )
371: DOUBLE PRECISION RWORK( * ), W( * )
372: COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
373: * ..
374: *
375: * =====================================================================
376: *
377: * .. Parameters ..
378: DOUBLE PRECISION ZERO, ONE, TWO
379: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
380: * ..
381: * .. Local Scalars ..
382: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
383: $ WANTZ, TRYRAC
384: CHARACTER ORDER
385: INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
386: $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
387: $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
388: $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
389: $ LWKOPT, LWMIN, NB, NSPLIT
390: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
391: $ SIGMA, SMLNUM, TMP1, VLL, VUU
392: * ..
393: * .. External Functions ..
394: LOGICAL LSAME
395: INTEGER ILAENV
396: DOUBLE PRECISION DLAMCH, ZLANSY
397: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
398: * ..
399: * .. External Subroutines ..
400: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
401: $ ZHETRD, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
402: * ..
403: * .. Intrinsic Functions ..
404: INTRINSIC DBLE, MAX, MIN, SQRT
405: * ..
406: * .. Executable Statements ..
407: *
408: * Test the input parameters.
409: *
410: IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
411: *
412: LOWER = LSAME( UPLO, 'L' )
413: WANTZ = LSAME( JOBZ, 'V' )
414: ALLEIG = LSAME( RANGE, 'A' )
415: VALEIG = LSAME( RANGE, 'V' )
416: INDEIG = LSAME( RANGE, 'I' )
417: *
418: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
419: $ ( LIWORK.EQ.-1 ) )
420: *
421: LRWMIN = MAX( 1, 24*N )
422: LIWMIN = MAX( 1, 10*N )
423: LWMIN = MAX( 1, 2*N )
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( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
431: INFO = -3
432: ELSE IF( N.LT.0 ) THEN
433: INFO = -4
434: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
435: INFO = -6
436: ELSE
437: IF( VALEIG ) THEN
438: IF( N.GT.0 .AND. VU.LE.VL )
439: $ INFO = -8
440: ELSE IF( INDEIG ) THEN
441: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
442: INFO = -9
443: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
444: INFO = -10
445: END IF
446: END IF
447: END IF
448: IF( INFO.EQ.0 ) THEN
449: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
450: INFO = -15
451: END IF
452: END IF
453: *
454: IF( INFO.EQ.0 ) THEN
455: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
456: NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
457: LWKOPT = MAX( ( NB+1 )*N, LWMIN )
458: WORK( 1 ) = LWKOPT
459: RWORK( 1 ) = LRWMIN
460: IWORK( 1 ) = LIWMIN
461: *
462: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
463: INFO = -18
464: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
465: INFO = -20
466: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
467: INFO = -22
468: END IF
469: END IF
470: *
471: IF( INFO.NE.0 ) THEN
472: CALL XERBLA( 'ZHEEVR', -INFO )
473: RETURN
474: ELSE IF( LQUERY ) THEN
475: RETURN
476: END IF
477: *
478: * Quick return if possible
479: *
480: M = 0
481: IF( N.EQ.0 ) THEN
482: WORK( 1 ) = 1
483: RETURN
484: END IF
485: *
486: IF( N.EQ.1 ) THEN
487: WORK( 1 ) = 2
488: IF( ALLEIG .OR. INDEIG ) THEN
489: M = 1
490: W( 1 ) = DBLE( A( 1, 1 ) )
491: ELSE
492: IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
493: $ THEN
494: M = 1
495: W( 1 ) = DBLE( A( 1, 1 ) )
496: END IF
497: END IF
498: IF( WANTZ ) THEN
499: Z( 1, 1 ) = ONE
500: ISUPPZ( 1 ) = 1
501: ISUPPZ( 2 ) = 1
502: END IF
503: RETURN
504: END IF
505: *
506: * Get machine constants.
507: *
508: SAFMIN = DLAMCH( 'Safe minimum' )
509: EPS = DLAMCH( 'Precision' )
510: SMLNUM = SAFMIN / EPS
511: BIGNUM = ONE / SMLNUM
512: RMIN = SQRT( SMLNUM )
513: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
514: *
515: * Scale matrix to allowable range, if necessary.
516: *
517: ISCALE = 0
518: ABSTLL = ABSTOL
519: IF (VALEIG) THEN
520: VLL = VL
521: VUU = VU
522: END IF
523: ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
524: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
525: ISCALE = 1
526: SIGMA = RMIN / ANRM
527: ELSE IF( ANRM.GT.RMAX ) THEN
528: ISCALE = 1
529: SIGMA = RMAX / ANRM
530: END IF
531: IF( ISCALE.EQ.1 ) THEN
532: IF( LOWER ) THEN
533: DO 10 J = 1, N
534: CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
535: 10 CONTINUE
536: ELSE
537: DO 20 J = 1, N
538: CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
539: 20 CONTINUE
540: END IF
541: IF( ABSTOL.GT.0 )
542: $ ABSTLL = ABSTOL*SIGMA
543: IF( VALEIG ) THEN
544: VLL = VL*SIGMA
545: VUU = VU*SIGMA
546: END IF
547: END IF
548:
549: * Initialize indices into workspaces. Note: The IWORK indices are
550: * used only if DSTERF or ZSTEMR fail.
551:
552: * WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
553: * elementary reflectors used in ZHETRD.
554: INDTAU = 1
555: * INDWK is the starting offset of the remaining complex workspace,
556: * and LLWORK is the remaining complex workspace size.
557: INDWK = INDTAU + N
558: LLWORK = LWORK - INDWK + 1
559:
560: * RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
561: * entries.
562: INDRD = 1
563: * RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
564: * tridiagonal matrix from ZHETRD.
565: INDRE = INDRD + N
566: * RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
567: * -written by ZSTEMR (the DSTERF path copies the diagonal to W).
568: INDRDD = INDRE + N
569: * RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
570: * -written while computing the eigenvalues in DSTERF and ZSTEMR.
571: INDREE = INDRDD + N
572: * INDRWK is the starting offset of the left-over real workspace, and
573: * LLRWORK is the remaining workspace size.
574: INDRWK = INDREE + N
575: LLRWORK = LRWORK - INDRWK + 1
576:
577: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
578: * stores the block indices of each of the M<=N eigenvalues.
579: INDIBL = 1
580: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
581: * stores the starting and finishing indices of each block.
582: INDISP = INDIBL + N
583: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
584: * that corresponding to eigenvectors that fail to converge in
585: * DSTEIN. This information is discarded; if any fail, the driver
586: * returns INFO > 0.
587: INDIFL = INDISP + N
588: * INDIWO is the offset of the remaining integer workspace.
589: INDIWO = INDIFL + N
590:
591: *
592: * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
593: *
594: CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDRD ), RWORK( INDRE ),
595: $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
596: *
597: * If all eigenvalues are desired
598: * then call DSTERF or ZSTEMR and ZUNMTR.
599: *
600: TEST = .FALSE.
601: IF( INDEIG ) THEN
602: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
603: TEST = .TRUE.
604: END IF
605: END IF
606: IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
607: IF( .NOT.WANTZ ) THEN
608: CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
609: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
610: CALL DSTERF( N, W, RWORK( INDREE ), INFO )
611: ELSE
612: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
613: CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
614: *
615: IF (ABSTOL .LE. TWO*N*EPS) THEN
616: TRYRAC = .TRUE.
617: ELSE
618: TRYRAC = .FALSE.
619: END IF
620: CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
621: $ RWORK( INDREE ), VL, VU, IL, IU, M, W,
622: $ Z, LDZ, N, ISUPPZ, TRYRAC,
623: $ RWORK( INDRWK ), LLRWORK,
624: $ IWORK, LIWORK, INFO )
625: *
626: * Apply unitary matrix used in reduction to tridiagonal
627: * form to eigenvectors returned by ZSTEIN.
628: *
629: IF( WANTZ .AND. INFO.EQ.0 ) THEN
630: INDWKN = INDWK
631: LLWRKN = LWORK - INDWKN + 1
632: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
633: $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
634: $ LLWRKN, IINFO )
635: END IF
636: END IF
637: *
638: *
639: IF( INFO.EQ.0 ) THEN
640: M = N
641: GO TO 30
642: END IF
643: INFO = 0
644: END IF
645: *
646: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
647: * Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
648: *
649: IF( WANTZ ) THEN
650: ORDER = 'B'
651: ELSE
652: ORDER = 'E'
653: END IF
654:
655: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
656: $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
657: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
658: $ IWORK( INDIWO ), INFO )
659: *
660: IF( WANTZ ) THEN
661: CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
662: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
663: $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
664: $ INFO )
665: *
666: * Apply unitary matrix used in reduction to tridiagonal
667: * form to eigenvectors returned by ZSTEIN.
668: *
669: INDWKN = INDWK
670: LLWRKN = LWORK - INDWKN + 1
671: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
672: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
673: END IF
674: *
675: * If matrix was scaled, then rescale eigenvalues appropriately.
676: *
677: 30 CONTINUE
678: IF( ISCALE.EQ.1 ) THEN
679: IF( INFO.EQ.0 ) THEN
680: IMAX = M
681: ELSE
682: IMAX = INFO - 1
683: END IF
684: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
685: END IF
686: *
687: * If eigenvalues are not in order, then sort them, along with
688: * eigenvectors.
689: *
690: IF( WANTZ ) THEN
691: DO 50 J = 1, M - 1
692: I = 0
693: TMP1 = W( J )
694: DO 40 JJ = J + 1, M
695: IF( W( JJ ).LT.TMP1 ) THEN
696: I = JJ
697: TMP1 = W( JJ )
698: END IF
699: 40 CONTINUE
700: *
701: IF( I.NE.0 ) THEN
702: ITMP1 = IWORK( INDIBL+I-1 )
703: W( I ) = W( J )
704: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
705: W( J ) = TMP1
706: IWORK( INDIBL+J-1 ) = ITMP1
707: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
708: END IF
709: 50 CONTINUE
710: END IF
711: *
712: * Set WORK(1) to optimal workspace size.
713: *
714: WORK( 1 ) = LWKOPT
715: RWORK( 1 ) = LRWMIN
716: IWORK( 1 ) = LIWMIN
717: *
718: RETURN
719: *
720: * End of ZHEEVR
721: *
722: END
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