          SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, ITYPE, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AP( * ), BP( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSPGST reduces a real symmetric-definite generalized eigenproblem
    *  to standard form, using packed storage.
    *
    *  If ITYPE = 1, the problem is A*x = lambda*B*x,
    *  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
    *
    *  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
    *  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
    *
    *  B must have been previously factorized as U**T*U or L*L**T by DPPTRF.
    *
    *  Arguments
    *  =========
    *
    *  ITYPE   (input) INTEGER
    *          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
    *          = 2 or 3: compute U*A*U**T or L**T*A*L.
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored and B is factored as
    *                  U**T*U;
    *          = 'L':  Lower triangle of A is stored and B is factored as
    *                  L*L**T.
    *
    *  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)*(2n-j)/2) = A(i,j) for j<=i<=n.
    *
    *          On exit, if INFO = 0, the transformed matrix, stored in the
    *          same format as A.
    *
    *  BP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          The triangular factor from the Cholesky factorization of B,
    *          stored in the same format as A, as returned by DPPTRF.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, HALF
          PARAMETER          ( ONE = 1.0D0, HALF = 0.5D0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            J, J1, J1J1, JJ, K, K1, K1K1, KK
          DOUBLE PRECISION   AJJ, AKK, BJJ, BKK, CT
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DSCAL, DSPMV, DSPR2, DTPMV, DTPSV,
         $                   XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DDOT
          EXTERNAL           LSAME, DDOT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
             INFO = -1
          ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DSPGST', -INFO )
             RETURN
          END IF
    *
          IF( ITYPE.EQ.1 ) THEN
             IF( UPPER ) THEN
    *
    *           Compute inv(U')*A*inv(U)
    *
    *           J1 and JJ are the indices of A(1,j) and A(j,j)
    *
                JJ = 0
                DO 10 J = 1, N
                   J1 = JJ + 1
                   JJ = JJ + J
    *
    *              Compute the j-th column of the upper triangle of A
    *
                   BJJ = BP( JJ )
                   CALL DTPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
         $                     AP( J1 ), 1 )
                   CALL DSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
         $                     AP( J1 ), 1 )
                   CALL DSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
                   AP( JJ ) = ( AP( JJ )-DDOT( J-1, AP( J1 ), 1, BP( J1 ),
         $                    1 ) ) / BJJ
       10       CONTINUE
             ELSE
    *
    *           Compute inv(L)*A*inv(L')
    *
    *           KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
    *
                KK = 1
                DO 20 K = 1, N
                   K1K1 = KK + N - K + 1
    *
    *              Update the lower triangle of A(k:n,k:n)
    *
                   AKK = AP( KK )
                   BKK = BP( KK )
                   AKK = AKK / BKK**2
                   AP( KK ) = AKK
                   IF( K.LT.N ) THEN
                      CALL DSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
                      CT = -HALF*AKK
                      CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
                      CALL DSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
         $                        BP( KK+1 ), 1, AP( K1K1 ) )
                      CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
                      CALL DTPSV( UPLO, 'No transpose', 'Non-unit', N-K,
         $                        BP( K1K1 ), AP( KK+1 ), 1 )
                   END IF
                   KK = K1K1
       20       CONTINUE
             END IF
          ELSE
             IF( UPPER ) THEN
    *
    *           Compute U*A*U'
    *
    *           K1 and KK are the indices of A(1,k) and A(k,k)
    *
                KK = 0
                DO 30 K = 1, N
                   K1 = KK + 1
                   KK = KK + K
    *
    *              Update the upper triangle of A(1:k,1:k)
    *
                   AKK = AP( KK )
                   BKK = BP( KK )
                   CALL DTPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
         $                     AP( K1 ), 1 )
                   CT = HALF*AKK
                   CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
                   CALL DSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
         $                     AP )
                   CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
                   CALL DSCAL( K-1, BKK, AP( K1 ), 1 )
                   AP( KK ) = AKK*BKK**2
       30       CONTINUE
             ELSE
    *
    *           Compute L'*A*L
    *
    *           JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
    *
                JJ = 1
                DO 40 J = 1, N
                   J1J1 = JJ + N - J + 1
    *
    *              Compute the j-th column of the lower triangle of A
    *
                   AJJ = AP( JJ )
                   BJJ = BP( JJ )
                   AP( JJ ) = AJJ*BJJ + DDOT( N-J, AP( JJ+1 ), 1,
         $                    BP( JJ+1 ), 1 )
                   CALL DSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
                   CALL DSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
         $                     ONE, AP( JJ+1 ), 1 )
                   CALL DTPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
         $                     BP( JJ ), AP( JJ ), 1 )
                   JJ = J1J1
       40       CONTINUE
             END IF
          END IF
          RETURN
    *
    *     End of DSPGST
    *
          END