 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ssyevd_2stage()

 subroutine ssyevd_2stage ( character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:
``` SSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A using the 2stage technique for
the reduction to tridiagonal. If eigenvectors are desired, it uses a
divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.```
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 dimension of the array WORK. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1 = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + 2*N+1 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 at least 1 + 6*N + 2*N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.``` [out] IWORK ``` IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.``` [in] LIWORK ``` LIWORK is INTEGER The dimension of the array IWORK. If N <= 1, LIWORK must be at least 1. If JOBZ = 'N' and N > 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK 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 and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).```
Date
November 2017
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified description of INFO. Sven, 16 Feb 05.
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 229 of file ssyevd_2stage.f.

229 *
230  IMPLICIT NONE
231 *
232 * -- LAPACK driver routine (version 3.8.0) --
233 * -- LAPACK is a software package provided by Univ. of Tennessee, --
234 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
235 * November 2017
236 *
237 * .. Scalar Arguments ..
238  CHARACTER jobz, uplo
239  INTEGER info, lda, liwork, lwork, n
240 * ..
241 * .. Array Arguments ..
242  INTEGER iwork( * )
243  REAL a( lda, * ), w( * ), work( * )
244 * ..
245 *
246 * =====================================================================
247 *
248 * .. Parameters ..
249  REAL zero, one
250  parameter( zero = 0.0e+0, one = 1.0e+0 )
251 * ..
252 * .. Local Scalars ..
253 *
254  LOGICAL lower, lquery, wantz
255  INTEGER iinfo, inde, indtau, indwk2, indwrk, iscale,
256  \$ liwmin, llwork, llwrk2, lwmin,
257  \$ lhtrd, lwtrd, kd, ib, indhous
258  REAL anrm, bignum, eps, rmax, rmin, safmin, sigma,
259  \$ smlnum
260 * ..
261 * .. External Functions ..
262  LOGICAL lsame
263  INTEGER ilaenv2stage
264  REAL slamch, slansy
265  EXTERNAL lsame, slamch, slansy, ilaenv2stage
266 * ..
267 * .. External Subroutines ..
268  EXTERNAL slacpy, slascl, sormtr, sscal, sstedc, ssterf,
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC max, sqrt
273 * ..
274 * .. Executable Statements ..
275 *
276 * Test the input parameters.
277 *
278  wantz = lsame( jobz, 'V' )
279  lower = lsame( uplo, 'L' )
280  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
281 *
282  info = 0
283  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
284  info = -1
285  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
286  info = -2
287  ELSE IF( n.LT.0 ) THEN
288  info = -3
289  ELSE IF( lda.LT.max( 1, n ) ) THEN
290  info = -5
291  END IF
292 *
293  IF( info.EQ.0 ) THEN
294  IF( n.LE.1 ) THEN
295  liwmin = 1
296  lwmin = 1
297  ELSE
298  kd = ilaenv2stage( 1, 'SSYTRD_2STAGE', jobz,
299  \$ n, -1, -1, -1 )
300  ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', jobz,
301  \$ n, kd, -1, -1 )
302  lhtrd = ilaenv2stage( 3, 'SSYTRD_2STAGE', jobz,
303  \$ n, kd, ib, -1 )
304  lwtrd = ilaenv2stage( 4, 'SSYTRD_2STAGE', jobz,
305  \$ n, kd, ib, -1 )
306  IF( wantz ) THEN
307  liwmin = 3 + 5*n
308  lwmin = 1 + 6*n + 2*n**2
309  ELSE
310  liwmin = 1
311  lwmin = 2*n + 1 + lhtrd + lwtrd
312  END IF
313  END IF
314  work( 1 ) = lwmin
315  iwork( 1 ) = liwmin
316 *
317  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
318  info = -8
319  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
320  info = -10
321  END IF
322  END IF
323 *
324  IF( info.NE.0 ) THEN
325  CALL xerbla( 'SSYEVD_2STAGE', -info )
326  RETURN
327  ELSE IF( lquery ) THEN
328  RETURN
329  END IF
330 *
331 * Quick return if possible
332 *
333  IF( n.EQ.0 )
334  \$ RETURN
335 *
336  IF( n.EQ.1 ) THEN
337  w( 1 ) = a( 1, 1 )
338  IF( wantz )
339  \$ a( 1, 1 ) = one
340  RETURN
341  END IF
342 *
343 * Get machine constants.
344 *
345  safmin = slamch( 'Safe minimum' )
346  eps = slamch( 'Precision' )
347  smlnum = safmin / eps
348  bignum = one / smlnum
349  rmin = sqrt( smlnum )
350  rmax = sqrt( bignum )
351 *
352 * Scale matrix to allowable range, if necessary.
353 *
354  anrm = slansy( 'M', uplo, n, a, lda, work )
355  iscale = 0
356  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
357  iscale = 1
358  sigma = rmin / anrm
359  ELSE IF( anrm.GT.rmax ) THEN
360  iscale = 1
361  sigma = rmax / anrm
362  END IF
363  IF( iscale.EQ.1 )
364  \$ CALL slascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
365 *
366 * Call SSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
367 *
368  inde = 1
369  indtau = inde + n
370  indhous = indtau + n
371  indwrk = indhous + lhtrd
372  llwork = lwork - indwrk + 1
373  indwk2 = indwrk + n*n
374  llwrk2 = lwork - indwk2 + 1
375 *
376  CALL ssytrd_2stage( jobz, uplo, n, a, lda, w, work( inde ),
377  \$ work( indtau ), work( indhous ), lhtrd,
378  \$ work( indwrk ), llwork, iinfo )
379 *
380 * For eigenvalues only, call SSTERF. For eigenvectors, first call
381 * SSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
382 * tridiagonal matrix, then call SORMTR to multiply it by the
383 * Householder transformations stored in A.
384 *
385  IF( .NOT.wantz ) THEN
386  CALL ssterf( n, w, work( inde ), info )
387  ELSE
388 * Not available in this release, and agrument checking should not
389 * let it getting here
390  RETURN
391  CALL sstedc( 'I', n, w, work( inde ), work( indwrk ), n,
392  \$ work( indwk2 ), llwrk2, iwork, liwork, info )
393  CALL sormtr( 'L', uplo, 'N', n, n, a, lda, work( indtau ),
394  \$ work( indwrk ), n, work( indwk2 ), llwrk2, iinfo )
395  CALL slacpy( 'A', n, n, work( indwrk ), n, a, lda )
396  END IF
397 *
398 * If matrix was scaled, then rescale eigenvalues appropriately.
399 *
400  IF( iscale.EQ.1 )
401  \$ CALL sscal( n, one / sigma, w, 1 )
402 *
403  work( 1 ) = lwmin
404  iwork( 1 ) = liwmin
405 *
406  RETURN
407 *
408 * End of SSYEVD_2STAGE
409 *
subroutine ssytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
SSYTRD_2STAGE
subroutine sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:174
subroutine sstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEDC
Definition: sstedc.f:190
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:145
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:151
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:88
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, or the element of largest absolute value of a real symmetric matrix.
Definition: slansy.f:124
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