 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ dsyev_2stage()

 subroutine dsyev_2stage ( character JOBZ, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) W, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:
DSYEV_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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order. [out] WORK WORK is DOUBLE PRECISION 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).
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 dsyev_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  DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
196 * ..
197 *
198 * =====================================================================
199 *
200 * .. Parameters ..
201  DOUBLE PRECISION ZERO, ONE
202  parameter( zero = 0.0d0, one = 1.0d0 )
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  DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
209  \$ SMLNUM
210 * ..
211 * .. External Functions ..
212  LOGICAL LSAME
213  INTEGER ILAENV2STAGE
214  DOUBLE PRECISION DLAMCH, DLANSY
215  EXTERNAL lsame, dlamch, dlansy, ilaenv2stage
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL dlascl, dorgtr, dscal, dsteqr, dsterf,
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, 'DSYTRD_2STAGE', jobz, n, -1, -1, -1 )
245  ib = ilaenv2stage( 2, 'DSYTRD_2STAGE', jobz, n, kd, -1, -1 )
246  lhtrd = ilaenv2stage( 3, 'DSYTRD_2STAGE', jobz, n, kd, ib, -1 )
247  lwtrd = ilaenv2stage( 4, 'DSYTRD_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( 'DSYEV_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 = dlamch( 'Safe minimum' )
279  eps = dlamch( '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 = dlansy( '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 dlascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
298 *
299 * Call DSYTRD_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 dsytrd_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 DSTERF. For eigenvectors, first call
312 * DORGTR to generate the orthogonal matrix, then call DSTEQR.
313 *
314  IF( .NOT.wantz ) THEN
315  CALL dsterf( 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 dorgtr( uplo, n, a, lda, work( indtau ), work( indwrk ),
321  \$ llwork, iinfo )
322  CALL dsteqr( 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 dscal( 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 DSYEV_2STAGE
344 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEQR
Definition: dsteqr.f:131
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
DORGTR
Definition: dorgtr.f:123
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
subroutine dsytrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
DSYTRD_2STAGE
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