LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ zheev_2stage()

 subroutine zheev_2stage ( character jobz, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info )

ZHEEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:
``` ZHEEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian matrix A using the 2stage technique for
the reduction to tridiagonal.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A. If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.```
Further Details:
```  All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196```

Definition at line 187 of file zheev_2stage.f.

189*
190 IMPLICIT NONE
191*
192* -- LAPACK driver routine --
193* -- LAPACK is a software package provided by Univ. of Tennessee, --
194* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195*
196* .. Scalar Arguments ..
197 CHARACTER JOBZ, UPLO
198 INTEGER INFO, LDA, LWORK, N
199* ..
200* .. Array Arguments ..
201 DOUBLE PRECISION RWORK( * ), W( * )
202 COMPLEX*16 A( LDA, * ), WORK( * )
203* ..
204*
205* =====================================================================
206*
207* .. Parameters ..
208 DOUBLE PRECISION ZERO, ONE
209 parameter( zero = 0.0d0, one = 1.0d0 )
210 COMPLEX*16 CONE
211 parameter( cone = ( 1.0d0, 0.0d0 ) )
212* ..
213* .. Local Scalars ..
214 LOGICAL LOWER, LQUERY, WANTZ
215 INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
216 \$ LLWORK, LWMIN, LHTRD, LWTRD, KD, IB, INDHOUS
217 DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
218 \$ SMLNUM
219* ..
220* .. External Functions ..
221 LOGICAL LSAME
222 INTEGER ILAENV2STAGE
223 DOUBLE PRECISION DLAMCH, ZLANHE
224 EXTERNAL lsame, dlamch, zlanhe, ilaenv2stage
225* ..
226* .. External Subroutines ..
227 EXTERNAL dscal, dsterf, xerbla, zlascl, zsteqr,
229* ..
230* .. Intrinsic Functions ..
231 INTRINSIC dble, max, sqrt
232* ..
233* .. Executable Statements ..
234*
235* Test the input parameters.
236*
237 wantz = lsame( jobz, 'V' )
238 lower = lsame( uplo, 'L' )
239 lquery = ( lwork.EQ.-1 )
240*
241 info = 0
242 IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
243 info = -1
244 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
245 info = -2
246 ELSE IF( n.LT.0 ) THEN
247 info = -3
248 ELSE IF( lda.LT.max( 1, n ) ) THEN
249 info = -5
250 END IF
251*
252 IF( info.EQ.0 ) THEN
253 kd = ilaenv2stage( 1, 'ZHETRD_2STAGE', jobz, n, -1, -1, -1 )
254 ib = ilaenv2stage( 2, 'ZHETRD_2STAGE', jobz, n, kd, -1, -1 )
255 lhtrd = ilaenv2stage( 3, 'ZHETRD_2STAGE', jobz, n, kd, ib, -1 )
256 lwtrd = ilaenv2stage( 4, 'ZHETRD_2STAGE', jobz, n, kd, ib, -1 )
257 lwmin = n + lhtrd + lwtrd
258 work( 1 ) = lwmin
259*
260 IF( lwork.LT.lwmin .AND. .NOT.lquery )
261 \$ info = -8
262 END IF
263*
264 IF( info.NE.0 ) THEN
265 CALL xerbla( 'ZHEEV_2STAGE ', -info )
266 RETURN
267 ELSE IF( lquery ) THEN
268 RETURN
269 END IF
270*
271* Quick return if possible
272*
273 IF( n.EQ.0 ) THEN
274 RETURN
275 END IF
276*
277 IF( n.EQ.1 ) THEN
278 w( 1 ) = dble( a( 1, 1 ) )
279 work( 1 ) = 1
280 IF( wantz )
281 \$ a( 1, 1 ) = cone
282 RETURN
283 END IF
284*
285* Get machine constants.
286*
287 safmin = dlamch( 'Safe minimum' )
288 eps = dlamch( 'Precision' )
289 smlnum = safmin / eps
290 bignum = one / smlnum
291 rmin = sqrt( smlnum )
292 rmax = sqrt( bignum )
293*
294* Scale matrix to allowable range, if necessary.
295*
296 anrm = zlanhe( 'M', uplo, n, a, lda, rwork )
297 iscale = 0
298 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
299 iscale = 1
300 sigma = rmin / anrm
301 ELSE IF( anrm.GT.rmax ) THEN
302 iscale = 1
303 sigma = rmax / anrm
304 END IF
305 IF( iscale.EQ.1 )
306 \$ CALL zlascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
307*
308* Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
309*
310 inde = 1
311 indtau = 1
312 indhous = indtau + n
313 indwrk = indhous + lhtrd
314 llwork = lwork - indwrk + 1
315*
316 CALL zhetrd_2stage( jobz, uplo, n, a, lda, w, rwork( inde ),
317 \$ work( indtau ), work( indhous ), lhtrd,
318 \$ work( indwrk ), llwork, iinfo )
319*
320* For eigenvalues only, call DSTERF. For eigenvectors, first call
321* ZUNGTR to generate the unitary matrix, then call ZSTEQR.
322*
323 IF( .NOT.wantz ) THEN
324 CALL dsterf( n, w, rwork( inde ), info )
325 ELSE
326 CALL zungtr( uplo, n, a, lda, work( indtau ), work( indwrk ),
327 \$ llwork, iinfo )
328 indwrk = inde + n
329 CALL zsteqr( jobz, n, w, rwork( inde ), a, lda,
330 \$ rwork( indwrk ), info )
331 END IF
332*
333* If matrix was scaled, then rescale eigenvalues appropriately.
334*
335 IF( iscale.EQ.1 ) THEN
336 IF( info.EQ.0 ) THEN
337 imax = n
338 ELSE
339 imax = info - 1
340 END IF
341 CALL dscal( imax, one / sigma, w, 1 )
342 END IF
343*
344* Set WORK(1) to optimal complex workspace size.
345*
346 work( 1 ) = lwmin
347*
348 RETURN
349*
350* End of ZHEEV_2STAGE
351*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zhetrd_2stage(vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
ZHETRD_2STAGE
integer function ilaenv2stage(ispec, name, opts, n1, n2, n3, n4)
ILAENV2STAGE
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function zlanhe(norm, uplo, n, a, lda, work)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition zlanhe.f:124
subroutine zlascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition zlascl.f:143
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
subroutine zsteqr(compz, n, d, e, z, ldz, work, info)
ZSTEQR
Definition zsteqr.f:132
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:86
subroutine zungtr(uplo, n, a, lda, tau, work, lwork, info)
ZUNGTR
Definition zungtr.f:123
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