          SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, 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, LDA, LDB, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSYGS2 reduces a real symmetric-definite generalized eigenproblem
    *  to standard form.
    *
    *  If ITYPE = 1, the problem is A*x = lambda*B*x,
    *  and A is overwritten by inv(U')*A*inv(U) or inv(L)*A*inv(L')
    *
    *  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` or L'*A*L.
    *
    *  B must have been previously factorized as U'*U or L*L' by DPOTRF.
    *
    *  Arguments
    *  =========
    *
    *  ITYPE   (input) INTEGER
    *          = 1: compute inv(U')*A*inv(U) or inv(L)*A*inv(L');
    *          = 2 or 3: compute U*A*U' or L'*A*L.
    *
    *  UPLO    (input) CHARACTER*1
    *          Specifies whether the upper or lower triangular part of the
    *          symmetric matrix A is stored, and how B has been factorized.
    *          = 'U':  Upper triangular
    *          = 'L':  Lower triangular
    *
    *  N       (input) INTEGER
    *          The order of the matrices A and B.  N >= 0.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
    *          n by n upper triangular part of A contains the upper
    *          triangular part of the matrix A, and the strictly lower
    *          triangular part of A is not referenced.  If UPLO = 'L', the
    *          leading n by n lower triangular part of A contains the lower
    *          triangular part of the matrix A, and the strictly upper
    *          triangular part of A is not referenced.
    *
    *          On exit, if INFO = 0, the transformed matrix, stored in the
    *          same format as A.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
    *          The triangular factor from the Cholesky factorization of B,
    *          as returned by DPOTRF.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B.  LDB >= max(1,N).
    *
    *  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            K
          DOUBLE PRECISION   AKK, BKK, CT
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DSCAL, DSYR2, DTRMV, DTRSV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. 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
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -5
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -7
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DSYGS2', -INFO )
             RETURN
          END IF
    *
          IF( ITYPE.EQ.1 ) THEN
             IF( UPPER ) THEN
    *
    *           Compute inv(U')*A*inv(U)
    *
                DO 10 K = 1, N
    *
    *              Update the upper triangle of A(k:n,k:n)
    *
                   AKK = A( K, K )
                   BKK = B( K, K )
                   AKK = AKK / BKK**2
                   A( K, K ) = AKK
                   IF( K.LT.N ) THEN
                      CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
                      CT = -HALF*AKK
                      CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
         $                        LDA )
                      CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
         $                        B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
                      CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
         $                        LDA )
                      CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
         $                        B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
                   END IF
       10       CONTINUE
             ELSE
    *
    *           Compute inv(L)*A*inv(L')
    *
                DO 20 K = 1, N
    *
    *              Update the lower triangle of A(k:n,k:n)
    *
                   AKK = A( K, K )
                   BKK = B( K, K )
                   AKK = AKK / BKK**2
                   A( K, K ) = AKK
                   IF( K.LT.N ) THEN
                      CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
                      CT = -HALF*AKK
                      CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
                      CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
         $                        B( K+1, K ), 1, A( K+1, K+1 ), LDA )
                      CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
                      CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
         $                        B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
                   END IF
       20       CONTINUE
             END IF
          ELSE
             IF( UPPER ) THEN
    *
    *           Compute U*A*U'
    *
                DO 30 K = 1, N
    *
    *              Update the upper triangle of A(1:k,1:k)
    *
                   AKK = A( K, K )
                   BKK = B( K, K )
                   CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
         $                     LDB, A( 1, K ), 1 )
                   CT = HALF*AKK
                   CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
                   CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ), 1,
         $                     A, LDA )
                   CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
                   CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
                   A( K, K ) = AKK*BKK**2
       30       CONTINUE
             ELSE
    *
    *           Compute L'*A*L
    *
                DO 40 K = 1, N
    *
    *              Update the lower triangle of A(1:k,1:k)
    *
                   AKK = A( K, K )
                   BKK = B( K, K )
                   CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B, LDB,
         $                     A( K, 1 ), LDA )
                   CT = HALF*AKK
                   CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
                   CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
         $                     LDB, A, LDA )
                   CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
                   CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
                   A( K, K ) = AKK*BKK**2
       40       CONTINUE
             END IF
          END IF
          RETURN
    *
    *     End of DSYGS2
    *
          END