1: *> \brief <b> DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2: *
3: * @precisions fortran d -> s
4: *
5: * =========== DOCUMENTATION ===========
6: *
7: * Online html documentation available at
8: * http://www.netlib.org/lapack/explore-html/
9: *
10: *> \htmlonly
11: *> Download DSBEVD_2STAGE + dependencies
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevd_2stage.f">
13: *> [TGZ]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevd_2stage.f">
15: *> [ZIP]</a>
16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevd_2stage.f">
17: *> [TXT]</a>
18: *> \endhtmlonly
19: *
20: * Definition:
21: * ===========
22: *
23: * SUBROUTINE DSBEVD_2STAGE( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
24: * WORK, LWORK, IWORK, LIWORK, INFO )
25: *
26: * IMPLICIT NONE
27: *
28: * .. Scalar Arguments ..
29: * CHARACTER JOBZ, UPLO
30: * INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
31: * ..
32: * .. Array Arguments ..
33: * INTEGER IWORK( * )
34: * DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
44: *> a real symmetric band matrix A using the 2stage technique for
45: *> the reduction to tridiagonal. If eigenvectors are desired, it uses
46: *> a divide and conquer algorithm.
47: *>
48: *> The divide and conquer algorithm makes very mild assumptions about
49: *> floating point arithmetic. It will work on machines with a guard
50: *> digit in add/subtract, or on those binary machines without guard
51: *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
52: *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
53: *> without guard digits, but we know of none.
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] JOBZ
60: *> \verbatim
61: *> JOBZ is CHARACTER*1
62: *> = 'N': Compute eigenvalues only;
63: *> = 'V': Compute eigenvalues and eigenvectors.
64: *> Not available in this release.
65: *> \endverbatim
66: *>
67: *> \param[in] UPLO
68: *> \verbatim
69: *> UPLO is CHARACTER*1
70: *> = 'U': Upper triangle of A is stored;
71: *> = 'L': Lower triangle of A is stored.
72: *> \endverbatim
73: *>
74: *> \param[in] N
75: *> \verbatim
76: *> N is INTEGER
77: *> The order of the matrix A. N >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in] KD
81: *> \verbatim
82: *> KD is INTEGER
83: *> The number of superdiagonals of the matrix A if UPLO = 'U',
84: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
85: *> \endverbatim
86: *>
87: *> \param[in,out] AB
88: *> \verbatim
89: *> AB is DOUBLE PRECISION array, dimension (LDAB, N)
90: *> On entry, the upper or lower triangle of the symmetric band
91: *> matrix A, stored in the first KD+1 rows of the array. The
92: *> j-th column of A is stored in the j-th column of the array AB
93: *> as follows:
94: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
95: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
96: *>
97: *> On exit, AB is overwritten by values generated during the
98: *> reduction to tridiagonal form. If UPLO = 'U', the first
99: *> superdiagonal and the diagonal of the tridiagonal matrix T
100: *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
101: *> the diagonal and first subdiagonal of T are returned in the
102: *> first two rows of AB.
103: *> \endverbatim
104: *>
105: *> \param[in] LDAB
106: *> \verbatim
107: *> LDAB is INTEGER
108: *> The leading dimension of the array AB. LDAB >= KD + 1.
109: *> \endverbatim
110: *>
111: *> \param[out] W
112: *> \verbatim
113: *> W is DOUBLE PRECISION array, dimension (N)
114: *> If INFO = 0, the eigenvalues in ascending order.
115: *> \endverbatim
116: *>
117: *> \param[out] Z
118: *> \verbatim
119: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
120: *> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
121: *> eigenvectors of the matrix A, with the i-th column of Z
122: *> holding the eigenvector associated with W(i).
123: *> If JOBZ = 'N', then Z is not referenced.
124: *> \endverbatim
125: *>
126: *> \param[in] LDZ
127: *> \verbatim
128: *> LDZ is INTEGER
129: *> The leading dimension of the array Z. LDZ >= 1, and if
130: *> JOBZ = 'V', LDZ >= max(1,N).
131: *> \endverbatim
132: *>
133: *> \param[out] WORK
134: *> \verbatim
135: *> WORK is DOUBLE PRECISION array, dimension LWORK
136: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
137: *> \endverbatim
138: *>
139: *> \param[in] LWORK
140: *> \verbatim
141: *> LWORK is INTEGER
142: *> The length of the array WORK. LWORK >= 1, when N <= 1;
143: *> otherwise
144: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
145: *> LWORK = MAX(1, dimension) where
146: *> dimension = (2KD+1)*N + KD*NTHREADS + N
147: *> where KD is the size of the band.
148: *> NTHREADS is the number of threads used when
149: *> openMP compilation is enabled, otherwise =1.
150: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
151: *>
152: *> If LWORK = -1, then a workspace query is assumed; the routine
153: *> only calculates the optimal sizes of the WORK and IWORK
154: *> arrays, returns these values as the first entries of the WORK
155: *> and IWORK arrays, and no error message related to LWORK or
156: *> LIWORK is issued by XERBLA.
157: *> \endverbatim
158: *>
159: *> \param[out] IWORK
160: *> \verbatim
161: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
162: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
163: *> \endverbatim
164: *>
165: *> \param[in] LIWORK
166: *> \verbatim
167: *> LIWORK is INTEGER
168: *> The dimension of the array IWORK.
169: *> If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
170: *> If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
171: *>
172: *> If LIWORK = -1, then a workspace query is assumed; the
173: *> routine only calculates the optimal sizes of the WORK and
174: *> IWORK arrays, returns these values as the first entries of
175: *> the WORK and IWORK arrays, and no error message related to
176: *> LWORK or LIWORK is issued by XERBLA.
177: *> \endverbatim
178: *>
179: *> \param[out] INFO
180: *> \verbatim
181: *> INFO is INTEGER
182: *> = 0: successful exit
183: *> < 0: if INFO = -i, the i-th argument had an illegal value
184: *> > 0: if INFO = i, the algorithm failed to converge; i
185: *> off-diagonal elements of an intermediate tridiagonal
186: *> form did not converge to zero.
187: *> \endverbatim
188: *
189: * Authors:
190: * ========
191: *
192: *> \author Univ. of Tennessee
193: *> \author Univ. of California Berkeley
194: *> \author Univ. of Colorado Denver
195: *> \author NAG Ltd.
196: *
197: *> \date December 2016
198: *
199: *> \ingroup doubleOTHEReigen
200: *
201: *> \par Further Details:
202: * =====================
203: *>
204: *> \verbatim
205: *>
206: *> All details about the 2stage techniques are available in:
207: *>
208: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
209: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
210: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
211: *> of 2011 International Conference for High Performance Computing,
212: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
213: *> Article 8 , 11 pages.
214: *> http://doi.acm.org/10.1145/2063384.2063394
215: *>
216: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
217: *> An improved parallel singular value algorithm and its implementation
218: *> for multicore hardware, In Proceedings of 2013 International Conference
219: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
220: *> Denver, Colorado, USA, 2013.
221: *> Article 90, 12 pages.
222: *> http://doi.acm.org/10.1145/2503210.2503292
223: *>
224: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
225: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
226: *> calculations based on fine-grained memory aware tasks.
227: *> International Journal of High Performance Computing Applications.
228: *> Volume 28 Issue 2, Pages 196-209, May 2014.
229: *> http://hpc.sagepub.com/content/28/2/196
230: *>
231: *> \endverbatim
232: *
233: * =====================================================================
234: SUBROUTINE DSBEVD_2STAGE( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
235: $ WORK, LWORK, IWORK, LIWORK, INFO )
236: *
237: IMPLICIT NONE
238: *
239: * -- LAPACK driver routine (version 3.7.0) --
240: * -- LAPACK is a software package provided by Univ. of Tennessee, --
241: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
242: * December 2016
243: *
244: * .. Scalar Arguments ..
245: CHARACTER JOBZ, UPLO
246: INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
247: * ..
248: * .. Array Arguments ..
249: INTEGER IWORK( * )
250: DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
251: * ..
252: *
253: * =====================================================================
254: *
255: * .. Parameters ..
256: DOUBLE PRECISION ZERO, ONE
257: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
258: * ..
259: * .. Local Scalars ..
260: LOGICAL LOWER, LQUERY, WANTZ
261: INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
262: $ LLWORK, LWMIN, LHTRD, LWTRD, IB, INDHOUS,
263: $ LLWRK2
264: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
265: $ SMLNUM
266: * ..
267: * .. External Functions ..
268: LOGICAL LSAME
269: INTEGER ILAENV
270: DOUBLE PRECISION DLAMCH, DLANSB
271: EXTERNAL LSAME, DLAMCH, DLANSB, ILAENV
272: * ..
273: * .. External Subroutines ..
274: EXTERNAL DGEMM, DLACPY, DLASCL, DSCAL, DSTEDC,
275: $ DSTERF, XERBLA, DSYTRD_SB2ST
276: * ..
277: * .. Intrinsic Functions ..
278: INTRINSIC SQRT
279: * ..
280: * .. Executable Statements ..
281: *
282: * Test the input parameters.
283: *
284: WANTZ = LSAME( JOBZ, 'V' )
285: LOWER = LSAME( UPLO, 'L' )
286: LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
287: *
288: INFO = 0
289: IF( N.LE.1 ) THEN
290: LIWMIN = 1
291: LWMIN = 1
292: ELSE
293: IB = ILAENV( 18, 'DSYTRD_SB2ST', JOBZ, N, KD, -1, -1 )
294: LHTRD = ILAENV( 19, 'DSYTRD_SB2ST', JOBZ, N, KD, IB, -1 )
295: LWTRD = ILAENV( 20, 'DSYTRD_SB2ST', JOBZ, N, KD, IB, -1 )
296: IF( WANTZ ) THEN
297: LIWMIN = 3 + 5*N
298: LWMIN = 1 + 5*N + 2*N**2
299: ELSE
300: LIWMIN = 1
301: LWMIN = MAX( 2*N, N+LHTRD+LWTRD )
302: END IF
303: END IF
304: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
305: INFO = -1
306: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
307: INFO = -2
308: ELSE IF( N.LT.0 ) THEN
309: INFO = -3
310: ELSE IF( KD.LT.0 ) THEN
311: INFO = -4
312: ELSE IF( LDAB.LT.KD+1 ) THEN
313: INFO = -6
314: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
315: INFO = -9
316: END IF
317: *
318: IF( INFO.EQ.0 ) THEN
319: WORK( 1 ) = LWMIN
320: IWORK( 1 ) = LIWMIN
321: *
322: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
323: INFO = -11
324: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
325: INFO = -13
326: END IF
327: END IF
328: *
329: IF( INFO.NE.0 ) THEN
330: CALL XERBLA( 'DSBEVD_2STAGE', -INFO )
331: RETURN
332: ELSE IF( LQUERY ) THEN
333: RETURN
334: END IF
335: *
336: * Quick return if possible
337: *
338: IF( N.EQ.0 )
339: $ RETURN
340: *
341: IF( N.EQ.1 ) THEN
342: W( 1 ) = AB( 1, 1 )
343: IF( WANTZ )
344: $ Z( 1, 1 ) = ONE
345: RETURN
346: END IF
347: *
348: * Get machine constants.
349: *
350: SAFMIN = DLAMCH( 'Safe minimum' )
351: EPS = DLAMCH( 'Precision' )
352: SMLNUM = SAFMIN / EPS
353: BIGNUM = ONE / SMLNUM
354: RMIN = SQRT( SMLNUM )
355: RMAX = SQRT( BIGNUM )
356: *
357: * Scale matrix to allowable range, if necessary.
358: *
359: ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
360: ISCALE = 0
361: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
362: ISCALE = 1
363: SIGMA = RMIN / ANRM
364: ELSE IF( ANRM.GT.RMAX ) THEN
365: ISCALE = 1
366: SIGMA = RMAX / ANRM
367: END IF
368: IF( ISCALE.EQ.1 ) THEN
369: IF( LOWER ) THEN
370: CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
371: ELSE
372: CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
373: END IF
374: END IF
375: *
376: * Call DSYTRD_SB2ST to reduce band symmetric matrix to tridiagonal form.
377: *
378: INDE = 1
379: INDHOUS = INDE + N
380: INDWRK = INDHOUS + LHTRD
381: LLWORK = LWORK - INDWRK + 1
382: INDWK2 = INDWRK + N*N
383: LLWRK2 = LWORK - INDWK2 + 1
384: *
385: CALL DSYTRD_SB2ST( "N", JOBZ, UPLO, N, KD, AB, LDAB, W,
386: $ WORK( INDE ), WORK( INDHOUS ), LHTRD,
387: $ WORK( INDWRK ), LLWORK, IINFO )
388: *
389: * For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
390: *
391: IF( .NOT.WANTZ ) THEN
392: CALL DSTERF( N, W, WORK( INDE ), INFO )
393: ELSE
394: CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
395: $ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
396: CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N,
397: $ ZERO, WORK( INDWK2 ), N )
398: CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
399: END IF
400: *
401: * If matrix was scaled, then rescale eigenvalues appropriately.
402: *
403: IF( ISCALE.EQ.1 )
404: $ CALL DSCAL( N, ONE / SIGMA, W, 1 )
405: *
406: WORK( 1 ) = LWMIN
407: IWORK( 1 ) = LIWMIN
408: RETURN
409: *
410: * End of DSBEVD_2STAGE
411: *
412: END
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