*> \brief <b> DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
*
* @precisions fortran d -> s
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYEV_2STAGE + dependencies
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*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyev_2stage.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEV_2STAGE( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
* INFO )
*
* IMPLICIT NONE
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
*> real symmetric matrix A using the 2stage technique for
*> the reduction to tridiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> Not available in this release.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup heev_2stage
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DSYEV_2STAGE( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
$ INFO )
*
IMPLICIT NONE
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE,
$ LLWORK, LWMIN, LHTRD, LWTRD, KD, IB, INDHOUS
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV2STAGE
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, DLAMCH, DLANSY, ILAENV2STAGE
* ..
* .. External Subroutines ..
EXTERNAL DLASCL, DORGTR, DSCAL, DSTEQR,
$ DSTERF,
$ XERBLA, DSYTRD_2STAGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
KD = ILAENV2STAGE( 1, 'DSYTRD_2STAGE', JOBZ, N, -1, -1,
$ -1 )
IB = ILAENV2STAGE( 2, 'DSYTRD_2STAGE', JOBZ, N, KD, -1,
$ -1 )
LHTRD = ILAENV2STAGE( 3, 'DSYTRD_2STAGE', JOBZ, N, KD, IB,
$ -1 )
LWTRD = ILAENV2STAGE( 4, 'DSYTRD_2STAGE', JOBZ, N, KD, IB,
$ -1 )
LWMIN = 2*N + LHTRD + LWTRD
WORK( 1 ) = LWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY )
$ INFO = -8
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYEV_2STAGE ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
RETURN
END IF
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
WORK( 1 ) = 2
IF( WANTZ )
$ A( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call DSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
*
INDE = 1
INDTAU = INDE + N
INDHOUS = INDTAU + N
INDWRK = INDHOUS + LHTRD
LLWORK = LWORK - INDWRK + 1
*
CALL DSYTRD_2STAGE( JOBZ, UPLO, N, A, LDA, W, WORK( INDE ),
$ WORK( INDTAU ), WORK( INDHOUS ), LHTRD,
$ WORK( INDWRK ), LLWORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* DORGTR to generate the orthogonal matrix, then call DSTEQR.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
* Not available in this release, and argument checking should not
* let it getting here
RETURN
CALL DORGTR( UPLO, N, A, LDA, WORK( INDTAU ),
$ WORK( INDWRK ),
$ LLWORK, IINFO )
CALL DSTEQR( JOBZ, N, W, WORK( INDE ), A, LDA,
$ WORK( INDTAU ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWMIN
*
RETURN
*
* End of DSYEV_2STAGE
*
END