1: *> \brief <b> ZHEEV_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 ZHEEV_2STAGE + dependencies
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheev_2stage.f">
13: *> [TGZ]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheev_2stage.f">
15: *> [ZIP]</a>
16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheev_2stage.f">
17: *> [TXT]</a>
18: *> \endhtmlonly
19: *
20: * Definition:
21: * ===========
22: *
23: * SUBROUTINE ZHEEV_2STAGE( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
24: * RWORK, INFO )
25: *
26: * IMPLICIT NONE
27: *
28: * .. Scalar Arguments ..
29: * CHARACTER JOBZ, UPLO
30: * INTEGER INFO, LDA, LWORK, N
31: * ..
32: * .. Array Arguments ..
33: * DOUBLE PRECISION RWORK( * ), W( * )
34: * COMPLEX*16 A( LDA, * ), WORK( * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZHEEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
44: *> complex Hermitian matrix A using the 2stage technique for
45: *> the reduction to tridiagonal.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] JOBZ
52: *> \verbatim
53: *> JOBZ is CHARACTER*1
54: *> = 'N': Compute eigenvalues only;
55: *> = 'V': Compute eigenvalues and eigenvectors.
56: *> Not available in this release.
57: *> \endverbatim
58: *>
59: *> \param[in] UPLO
60: *> \verbatim
61: *> UPLO is CHARACTER*1
62: *> = 'U': Upper triangle of A is stored;
63: *> = 'L': Lower triangle of A is stored.
64: *> \endverbatim
65: *>
66: *> \param[in] N
67: *> \verbatim
68: *> N is INTEGER
69: *> The order of the matrix A. N >= 0.
70: *> \endverbatim
71: *>
72: *> \param[in,out] A
73: *> \verbatim
74: *> A is COMPLEX*16 array, dimension (LDA, N)
75: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
76: *> leading N-by-N upper triangular part of A contains the
77: *> upper triangular part of the matrix A. If UPLO = 'L',
78: *> the leading N-by-N lower triangular part of A contains
79: *> the lower triangular part of the matrix A.
80: *> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
81: *> orthonormal eigenvectors of the matrix A.
82: *> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
83: *> or the upper triangle (if UPLO='U') of A, including the
84: *> diagonal, is destroyed.
85: *> \endverbatim
86: *>
87: *> \param[in] LDA
88: *> \verbatim
89: *> LDA is INTEGER
90: *> The leading dimension of the array A. LDA >= max(1,N).
91: *> \endverbatim
92: *>
93: *> \param[out] W
94: *> \verbatim
95: *> W is DOUBLE PRECISION array, dimension (N)
96: *> If INFO = 0, the eigenvalues in ascending order.
97: *> \endverbatim
98: *>
99: *> \param[out] WORK
100: *> \verbatim
101: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
102: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
103: *> \endverbatim
104: *>
105: *> \param[in] LWORK
106: *> \verbatim
107: *> LWORK is INTEGER
108: *> The length of the array WORK. LWORK >= 1, when N <= 1;
109: *> otherwise
110: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
111: *> LWORK = MAX(1, dimension) where
112: *> dimension = max(stage1,stage2) + (KD+1)*N + N
113: *> = N*KD + N*max(KD+1,FACTOPTNB)
114: *> + max(2*KD*KD, KD*NTHREADS)
115: *> + (KD+1)*N + N
116: *> where KD is the blocking size of the reduction,
117: *> FACTOPTNB is the blocking used by the QR or LQ
118: *> algorithm, usually FACTOPTNB=128 is a good choice
119: *> NTHREADS is the number of threads used when
120: *> openMP compilation is enabled, otherwise =1.
121: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
122: *>
123: *> If LWORK = -1, then a workspace query is assumed; the routine
124: *> only calculates the optimal size of the WORK array, returns
125: *> this value as the first entry of the WORK array, and no error
126: *> message related to LWORK is issued by XERBLA.
127: *> \endverbatim
128: *>
129: *> \param[out] RWORK
130: *> \verbatim
131: *> RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))
132: *> \endverbatim
133: *>
134: *> \param[out] INFO
135: *> \verbatim
136: *> INFO is INTEGER
137: *> = 0: successful exit
138: *> < 0: if INFO = -i, the i-th argument had an illegal value
139: *> > 0: if INFO = i, the algorithm failed to converge; i
140: *> off-diagonal elements of an intermediate tridiagonal
141: *> form did not converge to zero.
142: *> \endverbatim
143: *
144: * Authors:
145: * ========
146: *
147: *> \author Univ. of Tennessee
148: *> \author Univ. of California Berkeley
149: *> \author Univ. of Colorado Denver
150: *> \author NAG Ltd.
151: *
152: *> \date November 2017
153: *
154: *> \ingroup complex16HEeigen
155: *
156: *> \par Further Details:
157: * =====================
158: *>
159: *> \verbatim
160: *>
161: *> All details about the 2stage techniques are available in:
162: *>
163: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
164: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
165: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
166: *> of 2011 International Conference for High Performance Computing,
167: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
168: *> Article 8 , 11 pages.
169: *> http://doi.acm.org/10.1145/2063384.2063394
170: *>
171: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
172: *> An improved parallel singular value algorithm and its implementation
173: *> for multicore hardware, In Proceedings of 2013 International Conference
174: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
175: *> Denver, Colorado, USA, 2013.
176: *> Article 90, 12 pages.
177: *> http://doi.acm.org/10.1145/2503210.2503292
178: *>
179: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
180: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
181: *> calculations based on fine-grained memory aware tasks.
182: *> International Journal of High Performance Computing Applications.
183: *> Volume 28 Issue 2, Pages 196-209, May 2014.
184: *> http://hpc.sagepub.com/content/28/2/196
185: *>
186: *> \endverbatim
187: *
188: * =====================================================================
189: SUBROUTINE ZHEEV_2STAGE( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
190: $ RWORK, INFO )
191: *
192: IMPLICIT NONE
193: *
194: * -- LAPACK driver routine (version 3.8.0) --
195: * -- LAPACK is a software package provided by Univ. of Tennessee, --
196: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197: * November 2017
198: *
199: * .. Scalar Arguments ..
200: CHARACTER JOBZ, UPLO
201: INTEGER INFO, LDA, LWORK, N
202: * ..
203: * .. Array Arguments ..
204: DOUBLE PRECISION RWORK( * ), W( * )
205: COMPLEX*16 A( LDA, * ), WORK( * )
206: * ..
207: *
208: * =====================================================================
209: *
210: * .. Parameters ..
211: DOUBLE PRECISION ZERO, ONE
212: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
213: COMPLEX*16 CONE
214: PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
215: * ..
216: * .. Local Scalars ..
217: LOGICAL LOWER, LQUERY, WANTZ
218: INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
219: $ LLWORK, LWMIN, LHTRD, LWTRD, KD, IB, INDHOUS
220: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
221: $ SMLNUM
222: * ..
223: * .. External Functions ..
224: LOGICAL LSAME
225: INTEGER ILAENV2STAGE
226: DOUBLE PRECISION DLAMCH, ZLANHE
227: EXTERNAL LSAME, DLAMCH, ZLANHE, ILAENV2STAGE
228: * ..
229: * .. External Subroutines ..
230: EXTERNAL DSCAL, DSTERF, XERBLA, ZLASCL, ZSTEQR,
231: $ ZUNGTR, ZHETRD_2STAGE
232: * ..
233: * .. Intrinsic Functions ..
234: INTRINSIC DBLE, MAX, SQRT
235: * ..
236: * .. Executable Statements ..
237: *
238: * Test the input parameters.
239: *
240: WANTZ = LSAME( JOBZ, 'V' )
241: LOWER = LSAME( UPLO, 'L' )
242: LQUERY = ( LWORK.EQ.-1 )
243: *
244: INFO = 0
245: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
246: INFO = -1
247: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
248: INFO = -2
249: ELSE IF( N.LT.0 ) THEN
250: INFO = -3
251: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
252: INFO = -5
253: END IF
254: *
255: IF( INFO.EQ.0 ) THEN
256: KD = ILAENV2STAGE( 1, 'ZHETRD_2STAGE', JOBZ, N, -1, -1, -1 )
257: IB = ILAENV2STAGE( 2, 'ZHETRD_2STAGE', JOBZ, N, KD, -1, -1 )
258: LHTRD = ILAENV2STAGE( 3, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 )
259: LWTRD = ILAENV2STAGE( 4, 'ZHETRD_2STAGE', JOBZ, N, KD, IB, -1 )
260: LWMIN = N + LHTRD + LWTRD
261: WORK( 1 ) = LWMIN
262: *
263: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY )
264: $ INFO = -8
265: END IF
266: *
267: IF( INFO.NE.0 ) THEN
268: CALL XERBLA( 'ZHEEV_2STAGE ', -INFO )
269: RETURN
270: ELSE IF( LQUERY ) THEN
271: RETURN
272: END IF
273: *
274: * Quick return if possible
275: *
276: IF( N.EQ.0 ) THEN
277: RETURN
278: END IF
279: *
280: IF( N.EQ.1 ) THEN
281: W( 1 ) = DBLE( A( 1, 1 ) )
282: WORK( 1 ) = 1
283: IF( WANTZ )
284: $ A( 1, 1 ) = CONE
285: RETURN
286: END IF
287: *
288: * Get machine constants.
289: *
290: SAFMIN = DLAMCH( 'Safe minimum' )
291: EPS = DLAMCH( 'Precision' )
292: SMLNUM = SAFMIN / EPS
293: BIGNUM = ONE / SMLNUM
294: RMIN = SQRT( SMLNUM )
295: RMAX = SQRT( BIGNUM )
296: *
297: * Scale matrix to allowable range, if necessary.
298: *
299: ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
300: ISCALE = 0
301: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
302: ISCALE = 1
303: SIGMA = RMIN / ANRM
304: ELSE IF( ANRM.GT.RMAX ) THEN
305: ISCALE = 1
306: SIGMA = RMAX / ANRM
307: END IF
308: IF( ISCALE.EQ.1 )
309: $ CALL ZLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
310: *
311: * Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
312: *
313: INDE = 1
314: INDTAU = 1
315: INDHOUS = INDTAU + N
316: INDWRK = INDHOUS + LHTRD
317: LLWORK = LWORK - INDWRK + 1
318: *
319: CALL ZHETRD_2STAGE( JOBZ, UPLO, N, A, LDA, W, RWORK( INDE ),
320: $ WORK( INDTAU ), WORK( INDHOUS ), LHTRD,
321: $ WORK( INDWRK ), LLWORK, IINFO )
322: *
323: * For eigenvalues only, call DSTERF. For eigenvectors, first call
324: * ZUNGTR to generate the unitary matrix, then call ZSTEQR.
325: *
326: IF( .NOT.WANTZ ) THEN
327: CALL DSTERF( N, W, RWORK( INDE ), INFO )
328: ELSE
329: CALL ZUNGTR( UPLO, N, A, LDA, WORK( INDTAU ), WORK( INDWRK ),
330: $ LLWORK, IINFO )
331: INDWRK = INDE + N
332: CALL ZSTEQR( JOBZ, N, W, RWORK( INDE ), A, LDA,
333: $ RWORK( INDWRK ), INFO )
334: END IF
335: *
336: * If matrix was scaled, then rescale eigenvalues appropriately.
337: *
338: IF( ISCALE.EQ.1 ) THEN
339: IF( INFO.EQ.0 ) THEN
340: IMAX = N
341: ELSE
342: IMAX = INFO - 1
343: END IF
344: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
345: END IF
346: *
347: * Set WORK(1) to optimal complex workspace size.
348: *
349: WORK( 1 ) = LWMIN
350: *
351: RETURN
352: *
353: * End of ZHEEV_2STAGE
354: *
355: END
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