          SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, 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, ITYPE, LDZ, LIWORK, LWORK, N
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
         $                   Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSPGVD 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, stored in packed format, 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
    *  =========
    *
    *  ITYPE   (input) 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
    *
    *  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.
    *
    *  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          On entry, the upper or lower triangle of the symmetric matrix
    *          A, packed columnwise in a linear array.  The j-th column of A
    *          is stored in the array AP as follows:
    *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    *
    *          On exit, the contents of AP are destroyed.
    *
    *  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          On entry, the upper or lower triangle of the symmetric matrix
    *          B, packed columnwise in a linear array.  The j-th column of B
    *          is stored in the array BP as follows:
    *          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    *          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
    *
    *          On exit, the triangular factor U or L from the Cholesky
    *          factorization B = U**T*U or B = L*L**T, in the same storage
    *          format as B.
    *
    *  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.  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 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 required LWORK.
    *
    *  LWORK   (input) 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 + 6*N + 2*N**2.
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the required 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 INFO = 0, IWORK(1) returns the required 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 required 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:  DPPTRF or DSPEVD returned an error code:
    *             <= N:  if INFO = i, DSPEVD 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
    *  ===============
    *
    *  Based on contributions by
    *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   TWO
          PARAMETER          ( TWO = 2.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, UPPER, WANTZ
          CHARACTER          TRANS
          INTEGER            J, LIWMIN, LWMIN, NEIG
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          DBLE, MAX
    *     ..
    *     .. 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( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
             INFO = -1
          ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
             INFO = -2
          ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
             INFO = -3
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
             INFO = -9
          END IF
    *
          IF( INFO.EQ.0 ) THEN
             IF( N.LE.1 ) THEN
                LIWMIN = 1
                LWMIN = 1
             ELSE
                IF( WANTZ ) THEN
                   LIWMIN = 3 + 5*N
                   LWMIN = 1 + 6*N + 2*N**2
                ELSE
                   LIWMIN = 1
                   LWMIN = 2*N
                END IF
             END IF
             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( 'DSPGVD', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Form a Cholesky factorization of BP.
    *
          CALL DPPTRF( UPLO, N, BP, INFO )
          IF( INFO.NE.0 ) THEN
             INFO = N + INFO
             RETURN
          END IF
    *
    *     Transform problem to standard eigenvalue problem and solve.
    *
          CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
          CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK,
         $             LIWORK, INFO )
          LWMIN = MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) )
          LIWMIN = MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) )
    *
          IF( WANTZ ) THEN
    *
    *        Backtransform eigenvectors to the original problem.
    *
             NEIG = N
             IF( INFO.GT.0 )
         $      NEIG = INFO - 1
             IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
    *
    *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
    *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
    *
                IF( UPPER ) THEN
                   TRANS = 'N'
                ELSE
                   TRANS = 'T'
                END IF
    *
                DO 10 J = 1, NEIG
                   CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
         $                     1 )
       10       CONTINUE
    *
             ELSE IF( ITYPE.EQ.3 ) THEN
    *
    *           For B*A*x=(lambda)*x;
    *           backtransform eigenvectors: x = L*y or U'*y
    *
                IF( UPPER ) THEN
                   TRANS = 'T'
                ELSE
                   TRANS = 'N'
                END IF
    *
                DO 20 J = 1, NEIG
                   CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
         $                     1 )
       20       CONTINUE
             END IF
          END IF
    *
          WORK( 1 ) = LWMIN
          IWORK( 1 ) = LIWMIN
    *
          RETURN
    *
    *     End of DSPGVD
    *
          END