dsbgvd.f

          SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
         $                   Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  JOBZ    (input) CHARACTER*1
    *          = 'N':  Compute eigenvalues only;
    *          = 'V':  Compute eigenvalues and eigenvectors.
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangles of A and B are stored;
    *          = 'L':  Lower triangles of A and B are stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrices A and B.  N >= 0.
    *
    *  KA      (input) INTEGER
    *          The number of superdiagonals of the matrix A if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
    *
    *  KB      (input) INTEGER
    *          The number of superdiagonals of the matrix B if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
    *
    *  AB      (input/output) 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.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KA+1.
    *
    *  BB      (input/output) 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.
    *
    *  LDBB    (input) INTEGER
    *          The leading dimension of the array BB.  LDBB >= KB+1.
    *
    *  W       (output) DOUBLE PRECISION array, dimension (N)
    *          If INFO = 0, the eigenvalues in ascending order.
    *
    *  Z       (output) 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.
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDZ >= 1, and if
    *          JOBZ = 'V', LDZ >= max(1,N).
    *
    *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.
    *          If N <= 1,               LWORK >= 1.
    *          If JOBZ = 'N' and N > 1, LWORK >= 3*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.
    *
    *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
    *          On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
    *
    *  LIWORK  (input) 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.
    *
    *  INFO    (output) 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.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
    *
    *  =====================================================================
    *
    *     .. 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( INDWRK ), 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