LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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◆ zhetrd_2stage()

 subroutine zhetrd_2stage ( character vect, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) hous2, integer lhous2, complex*16, dimension( * ) work, integer lwork, integer info )

ZHETRD_2STAGE

Purpose:
``` ZHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
tridiagonal form T by a unitary similarity transformation:
Q1**H Q2**H* A * Q2 * Q1 = T.```
Parameters
 [in] VECT ``` VECT is CHARACTER*1 = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N); = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] D ``` D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T.``` [out] E ``` E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T.``` [out] TAU ``` TAU is COMPLEX*16 array, dimension (N-KD) The scalar factors of the elementary reflectors of the first stage (see Further Details).``` [out] HOUS2 ``` HOUS2 is COMPLEX*16 array, dimension (LHOUS2) Stores the Householder representation of the stage2 band to tridiagonal.``` [in] LHOUS2 ``` LHOUS2 is INTEGER The dimension of the array HOUS2. If LWORK = -1, or LHOUS2 = -1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. If VECT='N', LHOUS2 = max(1, 4*n); if VECT='V', option not yet available.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK = MAX(1, dimension) If LWORK = -1, or LHOUS2=-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. LWORK = MAX(1, dimension) where dimension = max(stage1,stage2) + (KD+1)*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

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 222 of file zhetrd_2stage.f.

224*
225 IMPLICIT NONE
226*
227* -- LAPACK computational routine --
228* -- LAPACK is a software package provided by Univ. of Tennessee, --
229* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
230*
231* .. Scalar Arguments ..
232 CHARACTER VECT, UPLO
233 INTEGER N, LDA, LWORK, LHOUS2, INFO
234* ..
235* .. Array Arguments ..
236 DOUBLE PRECISION D( * ), E( * )
237 COMPLEX*16 A( LDA, * ), TAU( * ),
238 \$ HOUS2( * ), WORK( * )
239* ..
240*
241* =====================================================================
242* ..
243* .. Local Scalars ..
244 LOGICAL LQUERY, UPPER, WANTQ
245 INTEGER KD, IB, LWMIN, LHMIN, LWRK, LDAB, WPOS, ABPOS
246* ..
247* .. External Subroutines ..
249* ..
250* .. External Functions ..
251 LOGICAL LSAME
252 INTEGER ILAENV2STAGE
253 EXTERNAL lsame, ilaenv2stage
254* ..
255* .. Executable Statements ..
256*
257* Test the input parameters
258*
259 info = 0
260 wantq = lsame( vect, 'V' )
261 upper = lsame( uplo, 'U' )
262 lquery = ( lwork.EQ.-1 ) .OR. ( lhous2.EQ.-1 )
263*
264* Determine the block size, the workspace size and the hous size.
265*
266 kd = ilaenv2stage( 1, 'ZHETRD_2STAGE', vect, n, -1, -1, -1 )
267 ib = ilaenv2stage( 2, 'ZHETRD_2STAGE', vect, n, kd, -1, -1 )
268 lhmin = ilaenv2stage( 3, 'ZHETRD_2STAGE', vect, n, kd, ib, -1 )
269 lwmin = ilaenv2stage( 4, 'ZHETRD_2STAGE', vect, n, kd, ib, -1 )
270* WRITE(*,*),'ZHETRD_2STAGE N KD UPLO LHMIN LWMIN ',N, KD, UPLO,
271* \$ LHMIN, LWMIN
272*
273 IF( .NOT.lsame( vect, 'N' ) ) THEN
274 info = -1
275 ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
276 info = -2
277 ELSE IF( n.LT.0 ) THEN
278 info = -3
279 ELSE IF( lda.LT.max( 1, n ) ) THEN
280 info = -5
281 ELSE IF( lhous2.LT.lhmin .AND. .NOT.lquery ) THEN
282 info = -10
283 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
284 info = -12
285 END IF
286*
287 IF( info.EQ.0 ) THEN
288 hous2( 1 ) = lhmin
289 work( 1 ) = lwmin
290 END IF
291*
292 IF( info.NE.0 ) THEN
293 CALL xerbla( 'ZHETRD_2STAGE', -info )
294 RETURN
295 ELSE IF( lquery ) THEN
296 RETURN
297 END IF
298*
299* Quick return if possible
300*
301 IF( n.EQ.0 ) THEN
302 work( 1 ) = 1
303 RETURN
304 END IF
305*
306* Determine pointer position
307*
308 ldab = kd+1
309 lwrk = lwork-ldab*n
310 abpos = 1
311 wpos = abpos + ldab*n
312 CALL zhetrd_he2hb( uplo, n, kd, a, lda, work( abpos ), ldab,
313 \$ tau, work( wpos ), lwrk, info )
314 IF( info.NE.0 ) THEN
315 CALL xerbla( 'ZHETRD_HE2HB', -info )
316 RETURN
317 END IF
318 CALL zhetrd_hb2st( 'Y', vect, uplo, n, kd,
319 \$ work( abpos ), ldab, d, e,
320 \$ hous2, lhous2, work( wpos ), lwrk, info )
321 IF( info.NE.0 ) THEN
322 CALL xerbla( 'ZHETRD_HB2ST', -info )
323 RETURN
324 END IF
325*
326*
327 hous2( 1 ) = lhmin
328 work( 1 ) = lwmin
329 RETURN
330*
331* End of ZHETRD_2STAGE
332*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zhetrd_hb2st(stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
subroutine zhetrd_he2hb(uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
ZHETRD_HE2HB
integer function ilaenv2stage(ispec, name, opts, n1, n2, n3, n4)
ILAENV2STAGE
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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