LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ chetrd_2stage()

 subroutine chetrd_2stage ( character VECT, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( * ) HOUS2, integer LHOUS2, complex, dimension( * ) WORK, integer LWORK, integer INFO )

CHETRD_2STAGE

Purpose:
``` CHETRD_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 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 REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T.``` [out] E ``` E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T.``` [out] TAU ``` TAU is COMPLEX array, dimension (N-KD) The scalar factors of the elementary reflectors of the first stage (see Further Details).``` [out] HOUS2 ``` HOUS2 is COMPLEX 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 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 chetrd_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  REAL D( * ), E( * )
237  COMPLEX 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 ..
248  EXTERNAL xerbla, chetrd_he2hb, chetrd_hb2st
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, 'CHETRD_2STAGE', vect, n, -1, -1, -1 )
267  ib = ilaenv2stage( 2, 'CHETRD_2STAGE', vect, n, kd, -1, -1 )
268  lhmin = ilaenv2stage( 3, 'CHETRD_2STAGE', vect, n, kd, ib, -1 )
269  lwmin = ilaenv2stage( 4, 'CHETRD_2STAGE', vect, n, kd, ib, -1 )
270 * WRITE(*,*),'CHETRD_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( 'CHETRD_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 chetrd_he2hb( uplo, n, kd, a, lda, work( abpos ), ldab,
313  \$ tau, work( wpos ), lwrk, info )
314  IF( info.NE.0 ) THEN
315  CALL xerbla( 'CHETRD_HE2HB', -info )
316  RETURN
317  END IF
318  CALL chetrd_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( 'CHETRD_HB2ST', -info )
323  RETURN
324  END IF
325 *
326 *
327  hous2( 1 ) = lhmin
328  work( 1 ) = lwmin
329  RETURN
330 *
331 * End of CHETRD_2STAGE
332 *
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine chetrd_he2hb(UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
CHETRD_HE2HB
Definition: chetrd_he2hb.f:243
subroutine chetrd_hb2st(STAGE1, VECT, UPLO, N, KD, AB, LDAB, D, E, HOUS, LHOUS, WORK, LWORK, INFO)
CHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
Definition: chetrd_hb2st.F:230
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