dsbevd_2stage.f

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*> \brief <b> DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  @precisions fortran d -> s
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBEVD_2STAGE( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
*                                 WORK, LWORK, IWORK, LIWORK, INFO )
*
*       IMPLICIT NONE
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
*> a real symmetric band 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.
*> \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] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, AB is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the first
*>          superdiagonal and the diagonal of the tridiagonal matrix T
*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*>          the diagonal and first subdiagonal of T are returned in the
*>          first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension 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 = (2KD+1)*N + KD*NTHREADS + N
*>                                   where KD is the size of the band.
*>                                   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 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.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 2, 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.
*> \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.
*
*> \date December 2016
*
*> \ingroup doubleOTHEReigen
*
*> \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 dsbevd_2stage( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
     $                          WORK, LWORK, IWORK, LIWORK, INFO )
*
      IMPLICIT NONE
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( ldab, * ), W( * ), WORK( * ), Z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, LQUERY, WANTZ
      INTEGER            IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
     $                   llwork, lwmin, lhtrd, lwtrd, ib, indhous,
     $                   llwrk2
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   smlnum
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANSB
      EXTERNAL           lsame, dlamch, dlansb, ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgemm, dlacpy, dlascl, dscal, dstedc,
     $                   dsterf, xerbla, dsytrd_sb2st
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      lower = lsame( uplo, 'L' )
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
*
      info = 0
      IF( n.LE.1 ) THEN
         liwmin = 1
         lwmin = 1
      ELSE
         ib    = ilaenv( 18, 'DSYTRD_SB2ST', jobz, n, kd, -1, -1 )
         lhtrd = ilaenv( 19, 'DSYTRD_SB2ST', jobz, n, kd, ib, -1 )
         lwtrd = ilaenv( 20, 'DSYTRD_SB2ST', jobz, n, kd, ib, -1 )
         IF( wantz ) THEN
            liwmin = 3 + 5*n
            lwmin = 1 + 5*n + 2*n**2
         ELSE
            liwmin = 1
            lwmin = max( 2*n, n+lhtrd+lwtrd )
         END IF
      END IF
      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( kd.LT.0 ) THEN
         info = -4
      ELSE IF( ldab.LT.kd+1 ) THEN
         info = -6
      ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
         info = -9
      END IF
*
      IF( info.EQ.0 ) THEN
         work( 1 )  = lwmin
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -11
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -13
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DSBEVD_2STAGE', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
      IF( n.EQ.1 ) THEN
         w( 1 ) = ab( 1, 1 )
         IF( wantz )
     $      z( 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 = dlansb( 'M', uplo, n, kd, ab, ldab, 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 ) THEN
         IF( lower ) THEN
            CALL dlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
         ELSE
            CALL dlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
         END IF
      END IF
*
*     Call DSYTRD_SB2ST to reduce band symmetric matrix to tridiagonal form.
*
      inde    = 1
      indhous = inde + n
      indwrk  = indhous + lhtrd
      llwork  = lwork - indwrk + 1
      indwk2  = indwrk + n*n
      llwrk2  = lwork - indwk2 + 1
*
      CALL dsytrd_sb2st( "N", jobz, uplo, n, kd, ab, ldab, w,
     $                    work( inde ), work( indhous ), lhtrd, 
     $                    work( indwrk ), llwork, iinfo )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEDC.
*
      IF( .NOT.wantz ) THEN
         CALL dsterf( n, w, work( inde ), info )
      ELSE
         CALL dstedc( 'I', n, w, work( inde ), work( indwrk ), n,
     $                work( indwk2 ), llwrk2, iwork, liwork, info )
         CALL dgemm( 'N', 'N', n, n, n, one, z, ldz, work( indwrk ), n,
     $               zero, work( indwk2 ), n )
         CALL dlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( iscale.EQ.1 )
     $   CALL dscal( n, one / sigma, w, 1 )
*
      work( 1 ) = lwmin
      iwork( 1 ) = liwmin
      RETURN
*
*     End of DSBEVD_2STAGE
*
      END