dsbgvd.f

*> \brief \b DSBGVD
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
*                          Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), W( * ),
*      $                   WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
*> of a real generalized symmetric-definite banded eigenproblem, of the
*> form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
*> banded, and B is also positive definite.  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.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangles of A and B are stored;
*>          = 'L':  Lower triangles of A and B are stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B.  N >= 0.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
*> \endverbatim
*>
*> \param[in] KB
*> \verbatim
*>          KB is INTEGER
*>          The number of superdiagonals of the matrix B if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KB >= 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 ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*>
*>          On exit, the contents of AB are destroyed.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KA+1.
*> \endverbatim
*>
*> \param[in,out] BB
*> \verbatim
*>          BB is DOUBLE PRECISION array, dimension (LDBB, N)
*>          On entry, the upper or lower triangle of the symmetric band
*>          matrix B, stored in the first kb+1 rows of the array.  The
*>          j-th column of B is stored in the j-th column of the array BB
*>          as follows:
*>          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*>
*>          On exit, the factor S from the split Cholesky factorization
*>          B = S**T*S, as returned by DPBSTF.
*> \endverbatim
*>
*> \param[in] LDBB
*> \verbatim
*>          LDBB is INTEGER
*>          The leading dimension of the array BB.  LDBB >= KB+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 matrix Z of
*>          eigenvectors, with the i-th column of Z holding the
*>          eigenvector associated with W(i).  The eigenvectors are
*>          normalized so Z**T*B*Z = 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 (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK >= 1.
*>          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*>          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
*>
*>          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 LIWORK > 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 >= 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK >= 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, and i is:
*>             <= N:  the algorithm 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 DPBSTF
*>                    returned INFO = i: B is not positive definite.
*>                    The factorization of B could not be completed and
*>                    no eigenvalues or eigenvectors were computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hbgvd
*
*> \par Contributors:
*  ==================
*>
*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
      SUBROUTINE dsbgvd( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
     $                   Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- 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, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( ldab, * ), BB( ldbb, * ), W( * ),
     $                   work( * ), z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      parameter( one = 1.0d+0, zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          VECT
      INTEGER            IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2,
     $                   lwmin
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgemm, dlacpy, dpbstf, dsbgst, dsbtrd, dstedc,
     $                   dsterf, xerbla
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      upper = lsame( uplo, 'U' )
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
*
      info = 0
      IF( n.LE.1 ) THEN
         liwmin = 1
         lwmin = 1
      ELSE IF( wantz ) THEN
         liwmin = 3 + 5*n
         lwmin = 1 + 5*n + 2*n**2
      ELSE
         liwmin = 1
         lwmin = 2*n
      END IF
*
      IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
         info = -1
      ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( ka.LT.0 ) THEN
         info = -4
      ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
         info = -5
      ELSE IF( ldab.LT.ka+1 ) THEN
         info = -7
      ELSE IF( ldbb.LT.kb+1 ) THEN
         info = -9
      ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
         info = -12
      END IF
*
      IF( info.EQ.0 ) THEN
         work( 1 ) = lwmin
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -14
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -16
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DSBGVD', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Form a split Cholesky factorization of B.
*
      CALL dpbstf( uplo, n, kb, bb, ldbb, info )
      IF( info.NE.0 ) THEN
         info = n + info
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem.
*
      inde = 1
      indwrk = inde + n
      indwk2 = indwrk + n*n
      llwrk2 = lwork - indwk2 + 1
      CALL dsbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, z, ldz,
     $             work, iinfo )
*
*     Reduce to tridiagonal form.
*
      IF( wantz ) THEN
         vect = 'U'
      ELSE
         vect = 'N'
      END IF
      CALL dsbtrd( vect, uplo, n, ka, ab, ldab, w, work( inde ), z, ldz,
     $             work( indwrk ), 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
*
      work( 1 ) = lwmin
      iwork( 1 ) = liwmin
*
      RETURN
*
*     End of DSBGVD
*
      END