Annotation of rpl/lapack/lapack/zheevr_2stage.f, revision 1.2
1.1 bertrand 1: *> \brief <b> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
2: *
3: * @precisions fortran z -> s d c
4: *
5: * =========== DOCUMENTATION ===========
6: *
7: * Online html documentation available at
8: * http://www.netlib.org/lapack/explore-html/
9: *
10: *> \htmlonly
11: *> Download ZHEEVR_2STAGE + dependencies
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_2stage.f">
13: *> [TGZ]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevr_2stage.f">
15: *> [ZIP]</a>
16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr_2stage.f">
17: *> [TXT]</a>
18: *> \endhtmlonly
19: *
20: * Definition:
21: * ===========
22: *
23: * SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
24: * IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
25: * WORK, LWORK, RWORK, LRWORK, IWORK,
26: * LIWORK, INFO )
27: *
28: * IMPLICIT NONE
29: *
30: * .. Scalar Arguments ..
31: * CHARACTER JOBZ, RANGE, UPLO
32: * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
33: * $ M, N
34: * DOUBLE PRECISION ABSTOL, VL, VU
35: * ..
36: * .. Array Arguments ..
37: * INTEGER ISUPPZ( * ), IWORK( * )
38: * DOUBLE PRECISION RWORK( * ), W( * )
39: * COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
40: * ..
41: *
42: *
43: *> \par Purpose:
44: * =============
45: *>
46: *> \verbatim
47: *>
48: *> ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
49: *> of a complex Hermitian matrix A using the 2stage technique for
50: *> the reduction to tridiagonal. Eigenvalues and eigenvectors can
51: *> be selected by specifying either a range of values or a range of
52: *> indices for the desired eigenvalues.
53: *>
54: *> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
55: *> to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute
56: *> eigenspectrum using Relatively Robust Representations. ZSTEMR
57: *> computes eigenvalues by the dqds algorithm, while orthogonal
58: *> eigenvectors are computed from various "good" L D L^T representations
59: *> (also known as Relatively Robust Representations). Gram-Schmidt
60: *> orthogonalization is avoided as far as possible. More specifically,
61: *> the various steps of the algorithm are as follows.
62: *>
63: *> For each unreduced block (submatrix) of T,
64: *> (a) Compute T - sigma I = L D L^T, so that L and D
65: *> define all the wanted eigenvalues to high relative accuracy.
66: *> This means that small relative changes in the entries of D and L
67: *> cause only small relative changes in the eigenvalues and
68: *> eigenvectors. The standard (unfactored) representation of the
69: *> tridiagonal matrix T does not have this property in general.
70: *> (b) Compute the eigenvalues to suitable accuracy.
71: *> If the eigenvectors are desired, the algorithm attains full
72: *> accuracy of the computed eigenvalues only right before
73: *> the corresponding vectors have to be computed, see steps c) and d).
74: *> (c) For each cluster of close eigenvalues, select a new
75: *> shift close to the cluster, find a new factorization, and refine
76: *> the shifted eigenvalues to suitable accuracy.
77: *> (d) For each eigenvalue with a large enough relative separation compute
78: *> the corresponding eigenvector by forming a rank revealing twisted
79: *> factorization. Go back to (c) for any clusters that remain.
80: *>
81: *> The desired accuracy of the output can be specified by the input
82: *> parameter ABSTOL.
83: *>
84: *> For more details, see DSTEMR's documentation and:
85: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
86: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
87: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
88: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
89: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
90: *> 2004. Also LAPACK Working Note 154.
91: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
92: *> tridiagonal eigenvalue/eigenvector problem",
93: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
94: *> UC Berkeley, May 1997.
95: *>
96: *>
97: *> Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested
98: *> on machines which conform to the ieee-754 floating point standard.
99: *> ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN on non-ieee machines and
100: *> when partial spectrum requests are made.
101: *>
102: *> Normal execution of ZSTEMR may create NaNs and infinities and
103: *> hence may abort due to a floating point exception in environments
104: *> which do not handle NaNs and infinities in the ieee standard default
105: *> manner.
106: *> \endverbatim
107: *
108: * Arguments:
109: * ==========
110: *
111: *> \param[in] JOBZ
112: *> \verbatim
113: *> JOBZ is CHARACTER*1
114: *> = 'N': Compute eigenvalues only;
115: *> = 'V': Compute eigenvalues and eigenvectors.
116: *> Not available in this release.
117: *> \endverbatim
118: *>
119: *> \param[in] RANGE
120: *> \verbatim
121: *> RANGE is CHARACTER*1
122: *> = 'A': all eigenvalues will be found.
123: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
124: *> will be found.
125: *> = 'I': the IL-th through IU-th eigenvalues will be found.
126: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
127: *> ZSTEIN are called
128: *> \endverbatim
129: *>
130: *> \param[in] UPLO
131: *> \verbatim
132: *> UPLO is CHARACTER*1
133: *> = 'U': Upper triangle of A is stored;
134: *> = 'L': Lower triangle of A is stored.
135: *> \endverbatim
136: *>
137: *> \param[in] N
138: *> \verbatim
139: *> N is INTEGER
140: *> The order of the matrix A. N >= 0.
141: *> \endverbatim
142: *>
143: *> \param[in,out] A
144: *> \verbatim
145: *> A is COMPLEX*16 array, dimension (LDA, N)
146: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
147: *> leading N-by-N upper triangular part of A contains the
148: *> upper triangular part of the matrix A. If UPLO = 'L',
149: *> the leading N-by-N lower triangular part of A contains
150: *> the lower triangular part of the matrix A.
151: *> On exit, the lower triangle (if UPLO='L') or the upper
152: *> triangle (if UPLO='U') of A, including the diagonal, is
153: *> destroyed.
154: *> \endverbatim
155: *>
156: *> \param[in] LDA
157: *> \verbatim
158: *> LDA is INTEGER
159: *> The leading dimension of the array A. LDA >= max(1,N).
160: *> \endverbatim
161: *>
162: *> \param[in] VL
163: *> \verbatim
164: *> VL is DOUBLE PRECISION
165: *> If RANGE='V', the lower bound of the interval to
166: *> be searched for eigenvalues. VL < VU.
167: *> Not referenced if RANGE = 'A' or 'I'.
168: *> \endverbatim
169: *>
170: *> \param[in] VU
171: *> \verbatim
172: *> VU is DOUBLE PRECISION
173: *> If RANGE='V', the upper bound of the interval to
174: *> be searched for eigenvalues. VL < VU.
175: *> Not referenced if RANGE = 'A' or 'I'.
176: *> \endverbatim
177: *>
178: *> \param[in] IL
179: *> \verbatim
180: *> IL is INTEGER
181: *> If RANGE='I', the index of the
182: *> smallest eigenvalue to be returned.
183: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
184: *> Not referenced if RANGE = 'A' or 'V'.
185: *> \endverbatim
186: *>
187: *> \param[in] IU
188: *> \verbatim
189: *> IU is INTEGER
190: *> If RANGE='I', the index of the
191: *> largest eigenvalue to be returned.
192: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
193: *> Not referenced if RANGE = 'A' or 'V'.
194: *> \endverbatim
195: *>
196: *> \param[in] ABSTOL
197: *> \verbatim
198: *> ABSTOL is DOUBLE PRECISION
199: *> The absolute error tolerance for the eigenvalues.
200: *> An approximate eigenvalue is accepted as converged
201: *> when it is determined to lie in an interval [a,b]
202: *> of width less than or equal to
203: *>
204: *> ABSTOL + EPS * max( |a|,|b| ) ,
205: *>
206: *> where EPS is the machine precision. If ABSTOL is less than
207: *> or equal to zero, then EPS*|T| will be used in its place,
208: *> where |T| is the 1-norm of the tridiagonal matrix obtained
209: *> by reducing A to tridiagonal form.
210: *>
211: *> See "Computing Small Singular Values of Bidiagonal Matrices
212: *> with Guaranteed High Relative Accuracy," by Demmel and
213: *> Kahan, LAPACK Working Note #3.
214: *>
215: *> If high relative accuracy is important, set ABSTOL to
216: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
217: *> eigenvalues are computed to high relative accuracy when
218: *> possible in future releases. The current code does not
219: *> make any guarantees about high relative accuracy, but
220: *> furutre releases will. See J. Barlow and J. Demmel,
221: *> "Computing Accurate Eigensystems of Scaled Diagonally
222: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
223: *> of which matrices define their eigenvalues to high relative
224: *> accuracy.
225: *> \endverbatim
226: *>
227: *> \param[out] M
228: *> \verbatim
229: *> M is INTEGER
230: *> The total number of eigenvalues found. 0 <= M <= N.
231: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
232: *> \endverbatim
233: *>
234: *> \param[out] W
235: *> \verbatim
236: *> W is DOUBLE PRECISION array, dimension (N)
237: *> The first M elements contain the selected eigenvalues in
238: *> ascending order.
239: *> \endverbatim
240: *>
241: *> \param[out] Z
242: *> \verbatim
243: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
244: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
245: *> contain the orthonormal eigenvectors of the matrix A
246: *> corresponding to the selected eigenvalues, with the i-th
247: *> column of Z holding the eigenvector associated with W(i).
248: *> If JOBZ = 'N', then Z is not referenced.
249: *> Note: the user must ensure that at least max(1,M) columns are
250: *> supplied in the array Z; if RANGE = 'V', the exact value of M
251: *> is not known in advance and an upper bound must be used.
252: *> \endverbatim
253: *>
254: *> \param[in] LDZ
255: *> \verbatim
256: *> LDZ is INTEGER
257: *> The leading dimension of the array Z. LDZ >= 1, and if
258: *> JOBZ = 'V', LDZ >= max(1,N).
259: *> \endverbatim
260: *>
261: *> \param[out] ISUPPZ
262: *> \verbatim
263: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
264: *> The support of the eigenvectors in Z, i.e., the indices
265: *> indicating the nonzero elements in Z. The i-th eigenvector
266: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
267: *> ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal
268: *> matrix). The support of the eigenvectors of A is typically
269: *> 1:N because of the unitary transformations applied by ZUNMTR.
270: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
271: *> \endverbatim
272: *>
273: *> \param[out] WORK
274: *> \verbatim
275: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
276: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
277: *> \endverbatim
278: *>
279: *> \param[in] LWORK
280: *> \verbatim
281: *> LWORK is INTEGER
282: *> The dimension of the array WORK.
283: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
284: *> LWORK = MAX(1, 26*N, dimension) where
285: *> dimension = max(stage1,stage2) + (KD+1)*N + N
286: *> = N*KD + N*max(KD+1,FACTOPTNB)
287: *> + max(2*KD*KD, KD*NTHREADS)
288: *> + (KD+1)*N + N
289: *> where KD is the blocking size of the reduction,
290: *> FACTOPTNB is the blocking used by the QR or LQ
291: *> algorithm, usually FACTOPTNB=128 is a good choice
292: *> NTHREADS is the number of threads used when
293: *> openMP compilation is enabled, otherwise =1.
294: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
295: *>
296: *> If LWORK = -1, then a workspace query is assumed; the routine
297: *> only calculates the optimal sizes of the WORK, RWORK and
298: *> IWORK arrays, returns these values as the first entries of
299: *> the WORK, RWORK and IWORK arrays, and no error message
300: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
301: *> \endverbatim
302: *>
303: *> \param[out] RWORK
304: *> \verbatim
305: *> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
306: *> On exit, if INFO = 0, RWORK(1) returns the optimal
307: *> (and minimal) LRWORK.
308: *> \endverbatim
309: *>
310: *> \param[in] LRWORK
311: *> \verbatim
312: *> LRWORK is INTEGER
313: *> The length of the array RWORK. LRWORK >= max(1,24*N).
314: *>
315: *> If LRWORK = -1, then a workspace query is assumed; the
316: *> routine only calculates the optimal sizes of the WORK, RWORK
317: *> and IWORK arrays, returns these values as the first entries
318: *> of the WORK, RWORK and IWORK arrays, and no error message
319: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
320: *> \endverbatim
321: *>
322: *> \param[out] IWORK
323: *> \verbatim
324: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
325: *> On exit, if INFO = 0, IWORK(1) returns the optimal
326: *> (and minimal) LIWORK.
327: *> \endverbatim
328: *>
329: *> \param[in] LIWORK
330: *> \verbatim
331: *> LIWORK is INTEGER
332: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
333: *>
334: *> If LIWORK = -1, then a workspace query is assumed; the
335: *> routine only calculates the optimal sizes of the WORK, RWORK
336: *> and IWORK arrays, returns these values as the first entries
337: *> of the WORK, RWORK and IWORK arrays, and no error message
338: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
339: *> \endverbatim
340: *>
341: *> \param[out] INFO
342: *> \verbatim
343: *> INFO is INTEGER
344: *> = 0: successful exit
345: *> < 0: if INFO = -i, the i-th argument had an illegal value
346: *> > 0: Internal error
347: *> \endverbatim
348: *
349: * Authors:
350: * ========
351: *
352: *> \author Univ. of Tennessee
353: *> \author Univ. of California Berkeley
354: *> \author Univ. of Colorado Denver
355: *> \author NAG Ltd.
356: *
357: *> \date June 2016
358: *
359: *> \ingroup complex16HEeigen
360: *
361: *> \par Contributors:
362: * ==================
363: *>
364: *> Inderjit Dhillon, IBM Almaden, USA \n
365: *> Osni Marques, LBNL/NERSC, USA \n
366: *> Ken Stanley, Computer Science Division, University of
367: *> California at Berkeley, USA \n
368: *> Jason Riedy, Computer Science Division, University of
369: *> California at Berkeley, USA \n
370: *>
371: *> \par Further Details:
372: * =====================
373: *>
374: *> \verbatim
375: *>
376: *> All details about the 2stage techniques are available in:
377: *>
378: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
379: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
380: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
381: *> of 2011 International Conference for High Performance Computing,
382: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
383: *> Article 8 , 11 pages.
384: *> http://doi.acm.org/10.1145/2063384.2063394
385: *>
386: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
387: *> An improved parallel singular value algorithm and its implementation
388: *> for multicore hardware, In Proceedings of 2013 International Conference
389: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
390: *> Denver, Colorado, USA, 2013.
391: *> Article 90, 12 pages.
392: *> http://doi.acm.org/10.1145/2503210.2503292
393: *>
394: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
395: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
396: *> calculations based on fine-grained memory aware tasks.
397: *> International Journal of High Performance Computing Applications.
398: *> Volume 28 Issue 2, Pages 196-209, May 2014.
399: *> http://hpc.sagepub.com/content/28/2/196
400: *>
401: *> \endverbatim
402: *
403: * =====================================================================
404: SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
405: $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
406: $ WORK, LWORK, RWORK, LRWORK, IWORK,
407: $ LIWORK, INFO )
408: *
409: IMPLICIT NONE
410: *
411: * -- LAPACK driver routine (version 3.7.0) --
412: * -- LAPACK is a software package provided by Univ. of Tennessee, --
413: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
414: * June 2016
415: *
416: * .. Scalar Arguments ..
417: CHARACTER JOBZ, RANGE, UPLO
418: INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
419: $ M, N
420: DOUBLE PRECISION ABSTOL, VL, VU
421: * ..
422: * .. Array Arguments ..
423: INTEGER ISUPPZ( * ), IWORK( * )
424: DOUBLE PRECISION RWORK( * ), W( * )
425: COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
426: * ..
427: *
428: * =====================================================================
429: *
430: * .. Parameters ..
431: DOUBLE PRECISION ZERO, ONE, TWO
432: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
433: * ..
434: * .. Local Scalars ..
435: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
436: $ WANTZ, TRYRAC
437: CHARACTER ORDER
438: INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
439: $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
440: $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
441: $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
442: $ LWMIN, NSPLIT, LHTRD, LWTRD, KD, IB, INDHOUS
443: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
444: $ SIGMA, SMLNUM, TMP1, VLL, VUU
445: * ..
446: * .. External Functions ..
447: LOGICAL LSAME
448: INTEGER ILAENV
449: DOUBLE PRECISION DLAMCH, ZLANSY
450: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
451: * ..
452: * .. External Subroutines ..
453: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
454: $ ZHETRD_2STAGE, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
455: * ..
456: * .. Intrinsic Functions ..
457: INTRINSIC DBLE, MAX, MIN, SQRT
458: * ..
459: * .. Executable Statements ..
460: *
461: * Test the input parameters.
462: *
463: IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
464: *
465: LOWER = LSAME( UPLO, 'L' )
466: WANTZ = LSAME( JOBZ, 'V' )
467: ALLEIG = LSAME( RANGE, 'A' )
468: VALEIG = LSAME( RANGE, 'V' )
469: INDEIG = LSAME( RANGE, 'I' )
470: *
471: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
472: $ ( LIWORK.EQ.-1 ) )
473: *
474: KD = ILAENV( 17, 'DSYTRD_2STAGE', JOBZ, N, -1, -1, -1 )
475: IB = ILAENV( 18, 'DSYTRD_2STAGE', JOBZ, N, KD, -1, -1 )
476: LHTRD = ILAENV( 19, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
477: LWTRD = ILAENV( 20, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
478: LWMIN = N + LHTRD + LWTRD
479: LRWMIN = MAX( 1, 24*N )
480: LIWMIN = MAX( 1, 10*N )
481: *
482: INFO = 0
483: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
484: INFO = -1
485: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
486: INFO = -2
487: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
488: INFO = -3
489: ELSE IF( N.LT.0 ) THEN
490: INFO = -4
491: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
492: INFO = -6
493: ELSE
494: IF( VALEIG ) THEN
495: IF( N.GT.0 .AND. VU.LE.VL )
496: $ INFO = -8
497: ELSE IF( INDEIG ) THEN
498: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
499: INFO = -9
500: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
501: INFO = -10
502: END IF
503: END IF
504: END IF
505: IF( INFO.EQ.0 ) THEN
506: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
507: INFO = -15
508: END IF
509: END IF
510: *
511: IF( INFO.EQ.0 ) THEN
512: WORK( 1 ) = LWMIN
513: RWORK( 1 ) = LRWMIN
514: IWORK( 1 ) = LIWMIN
515: *
516: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
517: INFO = -18
518: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
519: INFO = -20
520: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
521: INFO = -22
522: END IF
523: END IF
524: *
525: IF( INFO.NE.0 ) THEN
526: CALL XERBLA( 'ZHEEVR_2STAGE', -INFO )
527: RETURN
528: ELSE IF( LQUERY ) THEN
529: RETURN
530: END IF
531: *
532: * Quick return if possible
533: *
534: M = 0
535: IF( N.EQ.0 ) THEN
536: WORK( 1 ) = 1
537: RETURN
538: END IF
539: *
540: IF( N.EQ.1 ) THEN
541: WORK( 1 ) = 2
542: IF( ALLEIG .OR. INDEIG ) THEN
543: M = 1
544: W( 1 ) = DBLE( A( 1, 1 ) )
545: ELSE
546: IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
547: $ THEN
548: M = 1
549: W( 1 ) = DBLE( A( 1, 1 ) )
550: END IF
551: END IF
552: IF( WANTZ ) THEN
553: Z( 1, 1 ) = ONE
554: ISUPPZ( 1 ) = 1
555: ISUPPZ( 2 ) = 1
556: END IF
557: RETURN
558: END IF
559: *
560: * Get machine constants.
561: *
562: SAFMIN = DLAMCH( 'Safe minimum' )
563: EPS = DLAMCH( 'Precision' )
564: SMLNUM = SAFMIN / EPS
565: BIGNUM = ONE / SMLNUM
566: RMIN = SQRT( SMLNUM )
567: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
568: *
569: * Scale matrix to allowable range, if necessary.
570: *
571: ISCALE = 0
572: ABSTLL = ABSTOL
573: IF (VALEIG) THEN
574: VLL = VL
575: VUU = VU
576: END IF
577: ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
578: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
579: ISCALE = 1
580: SIGMA = RMIN / ANRM
581: ELSE IF( ANRM.GT.RMAX ) THEN
582: ISCALE = 1
583: SIGMA = RMAX / ANRM
584: END IF
585: IF( ISCALE.EQ.1 ) THEN
586: IF( LOWER ) THEN
587: DO 10 J = 1, N
588: CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
589: 10 CONTINUE
590: ELSE
591: DO 20 J = 1, N
592: CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
593: 20 CONTINUE
594: END IF
595: IF( ABSTOL.GT.0 )
596: $ ABSTLL = ABSTOL*SIGMA
597: IF( VALEIG ) THEN
598: VLL = VL*SIGMA
599: VUU = VU*SIGMA
600: END IF
601: END IF
602:
603: * Initialize indices into workspaces. Note: The IWORK indices are
604: * used only if DSTERF or ZSTEMR fail.
605:
606: * WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
607: * elementary reflectors used in ZHETRD.
608: INDTAU = 1
609: * INDWK is the starting offset of the remaining complex workspace,
610: * and LLWORK is the remaining complex workspace size.
611: INDHOUS = INDTAU + N
612: INDWK = INDHOUS + LHTRD
613: LLWORK = LWORK - INDWK + 1
614:
615: * RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
616: * entries.
617: INDRD = 1
618: * RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
619: * tridiagonal matrix from ZHETRD.
620: INDRE = INDRD + N
621: * RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
622: * -written by ZSTEMR (the DSTERF path copies the diagonal to W).
623: INDRDD = INDRE + N
624: * RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
625: * -written while computing the eigenvalues in DSTERF and ZSTEMR.
626: INDREE = INDRDD + N
627: * INDRWK is the starting offset of the left-over real workspace, and
628: * LLRWORK is the remaining workspace size.
629: INDRWK = INDREE + N
630: LLRWORK = LRWORK - INDRWK + 1
631:
632: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
633: * stores the block indices of each of the M<=N eigenvalues.
634: INDIBL = 1
635: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
636: * stores the starting and finishing indices of each block.
637: INDISP = INDIBL + N
638: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
639: * that corresponding to eigenvectors that fail to converge in
640: * ZSTEIN. This information is discarded; if any fail, the driver
641: * returns INFO > 0.
642: INDIFL = INDISP + N
643: * INDIWO is the offset of the remaining integer workspace.
644: INDIWO = INDIFL + N
645:
646: *
647: * Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
648: *
649: CALL ZHETRD_2STAGE( JOBZ, UPLO, N, A, LDA, RWORK( INDRD ),
650: $ RWORK( INDRE ), WORK( INDTAU ),
651: $ WORK( INDHOUS ), LHTRD,
652: $ WORK( INDWK ), LLWORK, IINFO )
653: *
654: * If all eigenvalues are desired
655: * then call DSTERF or ZSTEMR and ZUNMTR.
656: *
657: TEST = .FALSE.
658: IF( INDEIG ) THEN
659: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
660: TEST = .TRUE.
661: END IF
662: END IF
663: IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
664: IF( .NOT.WANTZ ) THEN
665: CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
666: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
667: CALL DSTERF( N, W, RWORK( INDREE ), INFO )
668: ELSE
669: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
670: CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
671: *
672: IF (ABSTOL .LE. TWO*N*EPS) THEN
673: TRYRAC = .TRUE.
674: ELSE
675: TRYRAC = .FALSE.
676: END IF
677: CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
678: $ RWORK( INDREE ), VL, VU, IL, IU, M, W,
679: $ Z, LDZ, N, ISUPPZ, TRYRAC,
680: $ RWORK( INDRWK ), LLRWORK,
681: $ IWORK, LIWORK, INFO )
682: *
683: * Apply unitary matrix used in reduction to tridiagonal
684: * form to eigenvectors returned by ZSTEMR.
685: *
686: IF( WANTZ .AND. INFO.EQ.0 ) THEN
687: INDWKN = INDWK
688: LLWRKN = LWORK - INDWKN + 1
689: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
690: $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
691: $ LLWRKN, IINFO )
692: END IF
693: END IF
694: *
695: *
696: IF( INFO.EQ.0 ) THEN
697: M = N
698: GO TO 30
699: END IF
700: INFO = 0
701: END IF
702: *
703: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
704: * Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
705: *
706: IF( WANTZ ) THEN
707: ORDER = 'B'
708: ELSE
709: ORDER = 'E'
710: END IF
711:
712: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
713: $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
714: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
715: $ IWORK( INDIWO ), INFO )
716: *
717: IF( WANTZ ) THEN
718: CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
719: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
720: $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
721: $ INFO )
722: *
723: * Apply unitary matrix used in reduction to tridiagonal
724: * form to eigenvectors returned by ZSTEIN.
725: *
726: INDWKN = INDWK
727: LLWRKN = LWORK - INDWKN + 1
728: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
729: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
730: END IF
731: *
732: * If matrix was scaled, then rescale eigenvalues appropriately.
733: *
734: 30 CONTINUE
735: IF( ISCALE.EQ.1 ) THEN
736: IF( INFO.EQ.0 ) THEN
737: IMAX = M
738: ELSE
739: IMAX = INFO - 1
740: END IF
741: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
742: END IF
743: *
744: * If eigenvalues are not in order, then sort them, along with
745: * eigenvectors.
746: *
747: IF( WANTZ ) THEN
748: DO 50 J = 1, M - 1
749: I = 0
750: TMP1 = W( J )
751: DO 40 JJ = J + 1, M
752: IF( W( JJ ).LT.TMP1 ) THEN
753: I = JJ
754: TMP1 = W( JJ )
755: END IF
756: 40 CONTINUE
757: *
758: IF( I.NE.0 ) THEN
759: ITMP1 = IWORK( INDIBL+I-1 )
760: W( I ) = W( J )
761: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
762: W( J ) = TMP1
763: IWORK( INDIBL+J-1 ) = ITMP1
764: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
765: END IF
766: 50 CONTINUE
767: END IF
768: *
769: * Set WORK(1) to optimal workspace size.
770: *
771: WORK( 1 ) = LWMIN
772: RWORK( 1 ) = LRWMIN
773: IWORK( 1 ) = LIWMIN
774: *
775: RETURN
776: *
777: * End of ZHEEVR_2STAGE
778: *
779: END
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