LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ssytrd_2stage()

 subroutine ssytrd_2stage ( character VECT, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, real, dimension( * ) HOUS2, integer LHOUS2, real, dimension( * ) WORK, integer LWORK, integer INFO )

SSYTRD_2STAGE

Purpose:
``` SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
tridiagonal form T by a orthogonal similarity transformation:
Q1**T Q2**T* 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 REAL array, dimension (LDA,N) On entry, the symmetric 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 orthogonal 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 orthogonal 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 REAL array, dimension (N-KD) The scalar factors of the elementary reflectors of the first stage (see Further Details).``` [out] HOUS2 ``` HOUS2 is REAL array, dimension LHOUS2, that store the Householder representation of the stage2 band to tridiagonal.``` [in] LHOUS2 ``` LHOUS2 is INTEGER The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension) 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. LHOUS2 = MAX(1, dimension) where dimension = 4*N if VECT='N' not available now if VECT='H'``` [out] WORK ` WORK is REAL 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```
Date
November 2017
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 227 of file ssytrd_2stage.f.

227 *
228  IMPLICIT NONE
229 *
230 * -- LAPACK computational routine (version 3.8.0) --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 * November 2017
234 *
235 * .. Scalar Arguments ..
236  CHARACTER vect, uplo
237  INTEGER n, lda, lwork, lhous2, info
238 * ..
239 * .. Array Arguments ..
240  REAL d( * ), e( * )
241  REAL a( lda, * ), tau( * ),
242  \$ hous2( * ), work( * )
243 * ..
244 *
245 * =====================================================================
246 * ..
247 * .. Local Scalars ..
248  LOGICAL lquery, upper, wantq
249  INTEGER kd, ib, lwmin, lhmin, lwrk, ldab, wpos, abpos
250 * ..
251 * .. External Subroutines ..
252  EXTERNAL xerbla, ssytrd_sy2sb, ssytrd_sb2st
253 * ..
254 * .. External Functions ..
255  LOGICAL lsame
256  INTEGER ilaenv2stage
257  EXTERNAL lsame, ilaenv2stage
258 * ..
259 * .. Executable Statements ..
260 *
261 * Test the input parameters
262 *
263  info = 0
264  wantq = lsame( vect, 'V' )
265  upper = lsame( uplo, 'U' )
266  lquery = ( lwork.EQ.-1 ) .OR. ( lhous2.EQ.-1 )
267 *
268 * Determine the block size, the workspace size and the hous size.
269 *
270  kd = ilaenv2stage( 1, 'SSYTRD_2STAGE', vect, n, -1, -1, -1 )
271  ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', vect, n, kd, -1, -1 )
272  lhmin = ilaenv2stage( 3, 'SSYTRD_2STAGE', vect, n, kd, ib, -1 )
273  lwmin = ilaenv2stage( 4, 'SSYTRD_2STAGE', vect, n, kd, ib, -1 )
274 * WRITE(*,*),'SSYTRD_2STAGE N KD UPLO LHMIN LWMIN ',N, KD, UPLO,
275 * \$ LHMIN, LWMIN
276 *
277  IF( .NOT.lsame( vect, 'N' ) ) THEN
278  info = -1
279  ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
280  info = -2
281  ELSE IF( n.LT.0 ) THEN
282  info = -3
283  ELSE IF( lda.LT.max( 1, n ) ) THEN
284  info = -5
285  ELSE IF( lhous2.LT.lhmin .AND. .NOT.lquery ) THEN
286  info = -10
287  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
288  info = -12
289  END IF
290 *
291  IF( info.EQ.0 ) THEN
292  hous2( 1 ) = lhmin
293  work( 1 ) = lwmin
294  END IF
295 *
296  IF( info.NE.0 ) THEN
297  CALL xerbla( 'SSYTRD_2STAGE', -info )
298  RETURN
299  ELSE IF( lquery ) THEN
300  RETURN
301  END IF
302 *
303 * Quick return if possible
304 *
305  IF( n.EQ.0 ) THEN
306  work( 1 ) = 1
307  RETURN
308  END IF
309 *
310 * Determine pointer position
311 *
312  ldab = kd+1
313  lwrk = lwork-ldab*n
314  abpos = 1
315  wpos = abpos + ldab*n
316  CALL ssytrd_sy2sb( uplo, n, kd, a, lda, work( abpos ), ldab,
317  \$ tau, work( wpos ), lwrk, info )
318  IF( info.NE.0 ) THEN
319  CALL xerbla( 'SSYTRD_SY2SB', -info )
320  RETURN
321  END IF
322  CALL ssytrd_sb2st( 'Y', vect, uplo, n, kd,
323  \$ work( abpos ), ldab, d, e,
324  \$ hous2, lhous2, work( wpos ), lwrk, info )
325  IF( info.NE.0 ) THEN
326  CALL xerbla( 'SSYTRD_SB2ST', -info )
327  RETURN
328  END IF
329 *
330 *
331  hous2( 1 ) = lhmin
332  work( 1 ) = lwmin
333  RETURN
334 *
335 * End of SSYTRD_2STAGE
336 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:151
subroutine ssytrd_sy2sb(UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
SSYTRD_SY2SB
Definition: ssytrd_sy2sb.f:245
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