LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssyev_2stage()

 subroutine ssyev_2stage ( character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO )

SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

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Purpose:
``` SSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
real symmetric 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 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. 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 REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] WORK ``` WORK is REAL array, dimension 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 + 2*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + 2*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] 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).
Denver, Colorado, USA, 2013.
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 181 of file ssyev_2stage.f.

183 *
184  IMPLICIT NONE
185 *
186 * -- LAPACK driver routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  CHARACTER JOBZ, UPLO
192  INTEGER INFO, LDA, LWORK, N
193 * ..
194 * .. Array Arguments ..
195  REAL A( LDA, * ), W( * ), WORK( * )
196 * ..
197 *
198 * =====================================================================
199 *
200 * .. Parameters ..
201  REAL ZERO, ONE
202  parameter( zero = 0.0e0, one = 1.0e0 )
203 * ..
204 * .. Local Scalars ..
205  LOGICAL LOWER, LQUERY, WANTZ
206  INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
207  \$ LLWORK, LWMIN, LHTRD, LWTRD, KD, IB, INDHOUS
208  REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
209  \$ SMLNUM
210 * ..
211 * .. External Functions ..
212  LOGICAL LSAME
213  INTEGER ILAENV2STAGE
214  REAL SLAMCH, SLANSY
215  EXTERNAL lsame, slamch, slansy, ilaenv2stage
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL slascl, sorgtr, sscal, ssteqr, ssterf,
220 * ..
221 * .. Intrinsic Functions ..
222  INTRINSIC max, sqrt
223 * ..
224 * .. Executable Statements ..
225 *
226 * Test the input parameters.
227 *
228  wantz = lsame( jobz, 'V' )
229  lower = lsame( uplo, 'L' )
230  lquery = ( lwork.EQ.-1 )
231 *
232  info = 0
233  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
234  info = -1
235  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
236  info = -2
237  ELSE IF( n.LT.0 ) THEN
238  info = -3
239  ELSE IF( lda.LT.max( 1, n ) ) THEN
240  info = -5
241  END IF
242 *
243  IF( info.EQ.0 ) THEN
244  kd = ilaenv2stage( 1, 'SSYTRD_2STAGE', jobz, n, -1, -1, -1 )
245  ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', jobz, n, kd, -1, -1 )
246  lhtrd = ilaenv2stage( 3, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
247  lwtrd = ilaenv2stage( 4, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
248  lwmin = 2*n + lhtrd + lwtrd
249  work( 1 ) = lwmin
250 *
251  IF( lwork.LT.lwmin .AND. .NOT.lquery )
252  \$ info = -8
253  END IF
254 *
255  IF( info.NE.0 ) THEN
256  CALL xerbla( 'SSYEV_2STAGE ', -info )
257  RETURN
258  ELSE IF( lquery ) THEN
259  RETURN
260  END IF
261 *
262 * Quick return if possible
263 *
264  IF( n.EQ.0 ) THEN
265  RETURN
266  END IF
267 *
268  IF( n.EQ.1 ) THEN
269  w( 1 ) = a( 1, 1 )
270  work( 1 ) = 2
271  IF( wantz )
272  \$ a( 1, 1 ) = one
273  RETURN
274  END IF
275 *
276 * Get machine constants.
277 *
278  safmin = slamch( 'Safe minimum' )
279  eps = slamch( 'Precision' )
280  smlnum = safmin / eps
281  bignum = one / smlnum
282  rmin = sqrt( smlnum )
283  rmax = sqrt( bignum )
284 *
285 * Scale matrix to allowable range, if necessary.
286 *
287  anrm = slansy( 'M', uplo, n, a, lda, work )
288  iscale = 0
289  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
290  iscale = 1
291  sigma = rmin / anrm
292  ELSE IF( anrm.GT.rmax ) THEN
293  iscale = 1
294  sigma = rmax / anrm
295  END IF
296  IF( iscale.EQ.1 )
297  \$ CALL slascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
298 *
299 * Call SSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
300 *
301  inde = 1
302  indtau = inde + n
303  indhous = indtau + n
304  indwrk = indhous + lhtrd
305  llwork = lwork - indwrk + 1
306 *
307  CALL ssytrd_2stage( jobz, uplo, n, a, lda, w, work( inde ),
308  \$ work( indtau ), work( indhous ), lhtrd,
309  \$ work( indwrk ), llwork, iinfo )
310 *
311 * For eigenvalues only, call SSTERF. For eigenvectors, first call
312 * SORGTR to generate the orthogonal matrix, then call SSTEQR.
313 *
314  IF( .NOT.wantz ) THEN
315  CALL ssterf( n, w, work( inde ), info )
316  ELSE
317 * Not available in this release, and argument checking should not
318 * let it getting here
319  RETURN
320  CALL sorgtr( uplo, n, a, lda, work( indtau ), work( indwrk ),
321  \$ llwork, iinfo )
322  CALL ssteqr( jobz, n, w, work( inde ), a, lda, work( indtau ),
323  \$ info )
324  END IF
325 *
326 * If matrix was scaled, then rescale eigenvalues appropriately.
327 *
328  IF( iscale.EQ.1 ) THEN
329  IF( info.EQ.0 ) THEN
330  imax = n
331  ELSE
332  imax = info - 1
333  END IF
334  CALL sscal( imax, one / sigma, w, 1 )
335  END IF
336 *
337 * Set WORK(1) to optimal workspace size.
338 *
339  work( 1 ) = lwmin
340 *
341  RETURN
342 *
343 * End of SSYEV_2STAGE
344 *
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:143
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 ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:131
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
SORGTR
Definition: sorgtr.f:123
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
subroutine ssytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
SSYTRD_2STAGE
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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