 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssygv_2stage()

 subroutine ssygv_2stage ( integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO )

SSYGV_2STAGE

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Purpose:
``` SSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
Here A and B are assumed to be symmetric and B is also
positive definite.
This routine use the 2stage technique for the reduction to tridiagonal
which showed higher performance on recent architecture and for large
sizes N>2000.```
Parameters
 [in] ITYPE ``` ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*x``` [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 triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.``` [in] N ``` N is INTEGER The order of the matrices A and B. 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 matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is REAL array, dimension (LDB, N) On entry, the symmetric positive definite matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B. On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= 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 (MAX(1,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: SPOTRF or SSYEV returned an error code: <= N: if INFO = i, SSYEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.```
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 224 of file ssygv_2stage.f.

226 *
227  IMPLICIT NONE
228 *
229 * -- LAPACK driver routine --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 *
233 * .. Scalar Arguments ..
234  CHARACTER JOBZ, UPLO
235  INTEGER INFO, ITYPE, LDA, LDB, LWORK, N
236 * ..
237 * .. Array Arguments ..
238  REAL A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
239 * ..
240 *
241 * =====================================================================
242 *
243 * .. Parameters ..
244  REAL ONE
245  parameter( one = 1.0e+0 )
246 * ..
247 * .. Local Scalars ..
248  LOGICAL LQUERY, UPPER, WANTZ
249  CHARACTER TRANS
250  INTEGER NEIG, LWMIN, LHTRD, LWTRD, KD, IB
251 * ..
252 * .. External Functions ..
253  LOGICAL LSAME
254  INTEGER ILAENV2STAGE
255  EXTERNAL lsame, ilaenv2stage
256 * ..
257 * .. External Subroutines ..
258  EXTERNAL spotrf, ssygst, strmm, strsm, xerbla,
259  \$ ssyev_2stage
260 * ..
261 * .. Intrinsic Functions ..
262  INTRINSIC max
263 * ..
264 * .. Executable Statements ..
265 *
266 * Test the input parameters.
267 *
268  wantz = lsame( jobz, 'V' )
269  upper = lsame( uplo, 'U' )
270  lquery = ( lwork.EQ.-1 )
271 *
272  info = 0
273  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
274  info = -1
275  ELSE IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
276  info = -2
277  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
278  info = -3
279  ELSE IF( n.LT.0 ) THEN
280  info = -4
281  ELSE IF( lda.LT.max( 1, n ) ) THEN
282  info = -6
283  ELSE IF( ldb.LT.max( 1, n ) ) THEN
284  info = -8
285  END IF
286 *
287  IF( info.EQ.0 ) THEN
288  kd = ilaenv2stage( 1, 'SSYTRD_2STAGE', jobz, n, -1, -1, -1 )
289  ib = ilaenv2stage( 2, 'SSYTRD_2STAGE', jobz, n, kd, -1, -1 )
290  lhtrd = ilaenv2stage( 3, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
291  lwtrd = ilaenv2stage( 4, 'SSYTRD_2STAGE', jobz, n, kd, ib, -1 )
292  lwmin = 2*n + lhtrd + lwtrd
293  work( 1 ) = lwmin
294 *
295  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
296  info = -11
297  END IF
298  END IF
299 *
300  IF( info.NE.0 ) THEN
301  CALL xerbla( 'SSYGV_2STAGE ', -info )
302  RETURN
303  ELSE IF( lquery ) THEN
304  RETURN
305  END IF
306 *
307 * Quick return if possible
308 *
309  IF( n.EQ.0 )
310  \$ RETURN
311 *
312 * Form a Cholesky factorization of B.
313 *
314  CALL spotrf( uplo, n, b, ldb, info )
315  IF( info.NE.0 ) THEN
316  info = n + info
317  RETURN
318  END IF
319 *
320 * Transform problem to standard eigenvalue problem and solve.
321 *
322  CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
323  CALL ssyev_2stage( jobz, uplo, n, a, lda, w, work, lwork, info )
324 *
325  IF( wantz ) THEN
326 *
327 * Backtransform eigenvectors to the original problem.
328 *
329  neig = n
330  IF( info.GT.0 )
331  \$ neig = info - 1
332  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
333 *
334 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
335 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
336 *
337  IF( upper ) THEN
338  trans = 'N'
339  ELSE
340  trans = 'T'
341  END IF
342 *
343  CALL strsm( 'Left', uplo, trans, 'Non-unit', n, neig, one,
344  \$ b, ldb, a, lda )
345 *
346  ELSE IF( itype.EQ.3 ) THEN
347 *
348 * For B*A*x=(lambda)*x;
349 * backtransform eigenvectors: x = L*y or U**T*y
350 *
351  IF( upper ) THEN
352  trans = 'T'
353  ELSE
354  trans = 'N'
355  END IF
356 *
357  CALL strmm( 'Left', uplo, trans, 'Non-unit', n, neig, one,
358  \$ b, ldb, a, lda )
359  END IF
360  END IF
361 *
362  work( 1 ) = lwmin
363  RETURN
364 *
365 * End of SSYGV_2STAGE
366 *
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 spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:107
subroutine ssygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
SSYGST
Definition: ssygst.f:127
subroutine ssyev_2stage(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO)
SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matr...
Definition: ssyev_2stage.f:183
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:177
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181
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