LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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).
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,
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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