1: *> \brief <b> ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER 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 ZHBEVD_2STAGE + dependencies
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhbevd_2stage.f">
13: *> [TGZ]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhbevd_2stage.f">
15: *> [ZIP]</a>
16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhbevd_2stage.f">
17: *> [TXT]</a>
18: *> \endhtmlonly
19: *
20: * Definition:
21: * ===========
22: *
23: * SUBROUTINE ZHBEVD_2STAGE( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
24: * WORK, LWORK, RWORK, LRWORK, IWORK,
25: * LIWORK, INFO )
26: *
27: * IMPLICIT NONE
28: *
29: * .. Scalar Arguments ..
30: * CHARACTER JOBZ, UPLO
31: * INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
32: * ..
33: * .. Array Arguments ..
34: * INTEGER IWORK( * )
35: * DOUBLE PRECISION RWORK( * ), W( * )
36: * COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
37: * ..
38: *
39: *
40: *> \par Purpose:
41: * =============
42: *>
43: *> \verbatim
44: *>
45: *> ZHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
46: *> a complex Hermitian band matrix A using the 2stage technique for
47: *> the reduction to tridiagonal. If eigenvectors are desired, it
48: *> uses a divide and conquer algorithm.
49: *>
50: *> The divide and conquer algorithm makes very mild assumptions about
51: *> floating point arithmetic. It will work on machines with a guard
52: *> digit in add/subtract, or on those binary machines without guard
53: *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
54: *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
55: *> without guard digits, but we know of none.
56: *> \endverbatim
57: *
58: * Arguments:
59: * ==========
60: *
61: *> \param[in] JOBZ
62: *> \verbatim
63: *> JOBZ is CHARACTER*1
64: *> = 'N': Compute eigenvalues only;
65: *> = 'V': Compute eigenvalues and eigenvectors.
66: *> Not available in this release.
67: *> \endverbatim
68: *>
69: *> \param[in] UPLO
70: *> \verbatim
71: *> UPLO is CHARACTER*1
72: *> = 'U': Upper triangle of A is stored;
73: *> = 'L': Lower triangle of A is stored.
74: *> \endverbatim
75: *>
76: *> \param[in] N
77: *> \verbatim
78: *> N is INTEGER
79: *> The order of the matrix A. N >= 0.
80: *> \endverbatim
81: *>
82: *> \param[in] KD
83: *> \verbatim
84: *> KD is INTEGER
85: *> The number of superdiagonals of the matrix A if UPLO = 'U',
86: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
87: *> \endverbatim
88: *>
89: *> \param[in,out] AB
90: *> \verbatim
91: *> AB is COMPLEX*16 array, dimension (LDAB, N)
92: *> On entry, the upper or lower triangle of the Hermitian band
93: *> matrix A, stored in the first KD+1 rows of the array. The
94: *> j-th column of A is stored in the j-th column of the array AB
95: *> as follows:
96: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
97: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
98: *>
99: *> On exit, AB is overwritten by values generated during the
100: *> reduction to tridiagonal form. If UPLO = 'U', the first
101: *> superdiagonal and the diagonal of the tridiagonal matrix T
102: *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
103: *> the diagonal and first subdiagonal of T are returned in the
104: *> first two rows of AB.
105: *> \endverbatim
106: *>
107: *> \param[in] LDAB
108: *> \verbatim
109: *> LDAB is INTEGER
110: *> The leading dimension of the array AB. LDAB >= KD + 1.
111: *> \endverbatim
112: *>
113: *> \param[out] W
114: *> \verbatim
115: *> W is DOUBLE PRECISION array, dimension (N)
116: *> If INFO = 0, the eigenvalues in ascending order.
117: *> \endverbatim
118: *>
119: *> \param[out] Z
120: *> \verbatim
121: *> Z is COMPLEX*16 array, dimension (LDZ, N)
122: *> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
123: *> eigenvectors of the matrix A, with the i-th column of Z
124: *> holding the eigenvector associated with W(i).
125: *> If JOBZ = 'N', then Z is not referenced.
126: *> \endverbatim
127: *>
128: *> \param[in] LDZ
129: *> \verbatim
130: *> LDZ is INTEGER
131: *> The leading dimension of the array Z. LDZ >= 1, and if
132: *> JOBZ = 'V', LDZ >= max(1,N).
133: *> \endverbatim
134: *>
135: *> \param[out] WORK
136: *> \verbatim
137: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
138: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
139: *> \endverbatim
140: *>
141: *> \param[in] LWORK
142: *> \verbatim
143: *> LWORK is INTEGER
144: *> The length of the array WORK. LWORK >= 1, when N <= 1;
145: *> otherwise
146: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
147: *> LWORK = MAX(1, dimension) where
148: *> dimension = (2KD+1)*N + KD*NTHREADS
149: *> where KD is the size of the band.
150: *> NTHREADS is the number of threads used when
151: *> openMP compilation is enabled, otherwise =1.
152: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
153: *>
154: *> If LWORK = -1, then a workspace query is assumed; the routine
155: *> only calculates the optimal sizes of the WORK, RWORK and
156: *> IWORK arrays, returns these values as the first entries of
157: *> the WORK, RWORK and IWORK arrays, and no error message
158: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
159: *> \endverbatim
160: *>
161: *> \param[out] RWORK
162: *> \verbatim
163: *> RWORK is DOUBLE PRECISION array,
164: *> dimension (LRWORK)
165: *> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
166: *> \endverbatim
167: *>
168: *> \param[in] LRWORK
169: *> \verbatim
170: *> LRWORK is INTEGER
171: *> The dimension of array RWORK.
172: *> If N <= 1, LRWORK must be at least 1.
173: *> If JOBZ = 'N' and N > 1, LRWORK must be at least N.
174: *> If JOBZ = 'V' and N > 1, LRWORK must be at least
175: *> 1 + 5*N + 2*N**2.
176: *>
177: *> If LRWORK = -1, then a workspace query is assumed; the
178: *> routine only calculates the optimal sizes of the WORK, RWORK
179: *> and IWORK arrays, returns these values as the first entries
180: *> of the WORK, RWORK and IWORK arrays, and no error message
181: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
182: *> \endverbatim
183: *>
184: *> \param[out] IWORK
185: *> \verbatim
186: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
187: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
188: *> \endverbatim
189: *>
190: *> \param[in] LIWORK
191: *> \verbatim
192: *> LIWORK is INTEGER
193: *> The dimension of array IWORK.
194: *> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
195: *> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
196: *>
197: *> If LIWORK = -1, then a workspace query is assumed; the
198: *> routine only calculates the optimal sizes of the WORK, RWORK
199: *> and IWORK arrays, returns these values as the first entries
200: *> of the WORK, RWORK and IWORK arrays, and no error message
201: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
202: *> \endverbatim
203: *>
204: *> \param[out] INFO
205: *> \verbatim
206: *> INFO is INTEGER
207: *> = 0: successful exit.
208: *> < 0: if INFO = -i, the i-th argument had an illegal value.
209: *> > 0: if INFO = i, the algorithm failed to converge; i
210: *> off-diagonal elements of an intermediate tridiagonal
211: *> form did not converge to zero.
212: *> \endverbatim
213: *
214: * Authors:
215: * ========
216: *
217: *> \author Univ. of Tennessee
218: *> \author Univ. of California Berkeley
219: *> \author Univ. of Colorado Denver
220: *> \author NAG Ltd.
221: *
222: *> \ingroup complex16OTHEReigen
223: *
224: *> \par Further Details:
225: * =====================
226: *>
227: *> \verbatim
228: *>
229: *> All details about the 2stage techniques are available in:
230: *>
231: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
232: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
233: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
234: *> of 2011 International Conference for High Performance Computing,
235: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
236: *> Article 8 , 11 pages.
237: *> http://doi.acm.org/10.1145/2063384.2063394
238: *>
239: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
240: *> An improved parallel singular value algorithm and its implementation
241: *> for multicore hardware, In Proceedings of 2013 International Conference
242: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
243: *> Denver, Colorado, USA, 2013.
244: *> Article 90, 12 pages.
245: *> http://doi.acm.org/10.1145/2503210.2503292
246: *>
247: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
248: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
249: *> calculations based on fine-grained memory aware tasks.
250: *> International Journal of High Performance Computing Applications.
251: *> Volume 28 Issue 2, Pages 196-209, May 2014.
252: *> http://hpc.sagepub.com/content/28/2/196
253: *>
254: *> \endverbatim
255: *
256: * =====================================================================
257: SUBROUTINE ZHBEVD_2STAGE( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
258: $ WORK, LWORK, RWORK, LRWORK, IWORK,
259: $ LIWORK, INFO )
260: *
261: IMPLICIT NONE
262: *
263: * -- LAPACK driver routine --
264: * -- LAPACK is a software package provided by Univ. of Tennessee, --
265: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266: *
267: * .. Scalar Arguments ..
268: CHARACTER JOBZ, UPLO
269: INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
270: * ..
271: * .. Array Arguments ..
272: INTEGER IWORK( * )
273: DOUBLE PRECISION RWORK( * ), W( * )
274: COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
275: * ..
276: *
277: * =====================================================================
278: *
279: * .. Parameters ..
280: DOUBLE PRECISION ZERO, ONE
281: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
282: COMPLEX*16 CZERO, CONE
283: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
284: $ CONE = ( 1.0D0, 0.0D0 ) )
285: * ..
286: * .. Local Scalars ..
287: LOGICAL LOWER, LQUERY, WANTZ
288: INTEGER IINFO, IMAX, INDE, INDWK2, INDRWK, ISCALE,
289: $ LLWORK, INDWK, LHTRD, LWTRD, IB, INDHOUS,
290: $ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
291: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
292: $ SMLNUM
293: * ..
294: * .. External Functions ..
295: LOGICAL LSAME
296: INTEGER ILAENV2STAGE
297: DOUBLE PRECISION DLAMCH, ZLANHB
298: EXTERNAL LSAME, DLAMCH, ZLANHB, ILAENV2STAGE
299: * ..
300: * .. External Subroutines ..
301: EXTERNAL DSCAL, DSTERF, XERBLA, ZGEMM, ZLACPY,
302: $ ZLASCL, ZSTEDC, ZHETRD_HB2ST
303: * ..
304: * .. Intrinsic Functions ..
305: INTRINSIC DBLE, SQRT
306: * ..
307: * .. Executable Statements ..
308: *
309: * Test the input parameters.
310: *
311: WANTZ = LSAME( JOBZ, 'V' )
312: LOWER = LSAME( UPLO, 'L' )
313: LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 .OR. LRWORK.EQ.-1 )
314: *
315: INFO = 0
316: IF( N.LE.1 ) THEN
317: LWMIN = 1
318: LRWMIN = 1
319: LIWMIN = 1
320: ELSE
321: IB = ILAENV2STAGE( 2, 'ZHETRD_HB2ST', JOBZ, N, KD, -1, -1 )
322: LHTRD = ILAENV2STAGE( 3, 'ZHETRD_HB2ST', JOBZ, N, KD, IB, -1 )
323: LWTRD = ILAENV2STAGE( 4, 'ZHETRD_HB2ST', JOBZ, N, KD, IB, -1 )
324: IF( WANTZ ) THEN
325: LWMIN = 2*N**2
326: LRWMIN = 1 + 5*N + 2*N**2
327: LIWMIN = 3 + 5*N
328: ELSE
329: LWMIN = MAX( N, LHTRD + LWTRD )
330: LRWMIN = N
331: LIWMIN = 1
332: END IF
333: END IF
334: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
335: INFO = -1
336: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
337: INFO = -2
338: ELSE IF( N.LT.0 ) THEN
339: INFO = -3
340: ELSE IF( KD.LT.0 ) THEN
341: INFO = -4
342: ELSE IF( LDAB.LT.KD+1 ) THEN
343: INFO = -6
344: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
345: INFO = -9
346: END IF
347: *
348: IF( INFO.EQ.0 ) THEN
349: WORK( 1 ) = LWMIN
350: RWORK( 1 ) = LRWMIN
351: IWORK( 1 ) = LIWMIN
352: *
353: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
354: INFO = -11
355: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
356: INFO = -13
357: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
358: INFO = -15
359: END IF
360: END IF
361: *
362: IF( INFO.NE.0 ) THEN
363: CALL XERBLA( 'ZHBEVD_2STAGE', -INFO )
364: RETURN
365: ELSE IF( LQUERY ) THEN
366: RETURN
367: END IF
368: *
369: * Quick return if possible
370: *
371: IF( N.EQ.0 )
372: $ RETURN
373: *
374: IF( N.EQ.1 ) THEN
375: W( 1 ) = DBLE( AB( 1, 1 ) )
376: IF( WANTZ )
377: $ Z( 1, 1 ) = CONE
378: RETURN
379: END IF
380: *
381: * Get machine constants.
382: *
383: SAFMIN = DLAMCH( 'Safe minimum' )
384: EPS = DLAMCH( 'Precision' )
385: SMLNUM = SAFMIN / EPS
386: BIGNUM = ONE / SMLNUM
387: RMIN = SQRT( SMLNUM )
388: RMAX = SQRT( BIGNUM )
389: *
390: * Scale matrix to allowable range, if necessary.
391: *
392: ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
393: ISCALE = 0
394: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
395: ISCALE = 1
396: SIGMA = RMIN / ANRM
397: ELSE IF( ANRM.GT.RMAX ) THEN
398: ISCALE = 1
399: SIGMA = RMAX / ANRM
400: END IF
401: IF( ISCALE.EQ.1 ) THEN
402: IF( LOWER ) THEN
403: CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
404: ELSE
405: CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
406: END IF
407: END IF
408: *
409: * Call ZHBTRD_HB2ST to reduce Hermitian band matrix to tridiagonal form.
410: *
411: INDE = 1
412: INDRWK = INDE + N
413: LLRWK = LRWORK - INDRWK + 1
414: INDHOUS = 1
415: INDWK = INDHOUS + LHTRD
416: LLWORK = LWORK - INDWK + 1
417: INDWK2 = INDWK + N*N
418: LLWK2 = LWORK - INDWK2 + 1
419: *
420: CALL ZHETRD_HB2ST( "N", JOBZ, UPLO, N, KD, AB, LDAB, W,
421: $ RWORK( INDE ), WORK( INDHOUS ), LHTRD,
422: $ WORK( INDWK ), LLWORK, IINFO )
423: *
424: * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
425: *
426: IF( .NOT.WANTZ ) THEN
427: CALL DSTERF( N, W, RWORK( INDE ), INFO )
428: ELSE
429: CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ),
430: $ LLWK2, RWORK( INDRWK ), LLRWK, IWORK, LIWORK,
431: $ INFO )
432: CALL ZGEMM( 'N', 'N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO,
433: $ WORK( INDWK2 ), N )
434: CALL ZLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
435: END IF
436: *
437: * If matrix was scaled, then rescale eigenvalues appropriately.
438: *
439: IF( ISCALE.EQ.1 ) THEN
440: IF( INFO.EQ.0 ) THEN
441: IMAX = N
442: ELSE
443: IMAX = INFO - 1
444: END IF
445: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
446: END IF
447: *
448: WORK( 1 ) = LWMIN
449: RWORK( 1 ) = LRWMIN
450: IWORK( 1 ) = LIWMIN
451: RETURN
452: *
453: * End of ZHBEVD_2STAGE
454: *
455: END
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