SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
     $                   LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
     $                   IWORK, PQ, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
     $                   PQ
      DOUBLE PRECISION   RDSCAL, RDSUM, SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), E( LDE, * ), F( LDF, * )
*     ..
*
*  Purpose
*  =======
*
*  DTGSY2 solves the generalized Sylvester equation:
*
*              A * R - L * B = scale * C                (1)
*              D * R - L * E = scale * F,
*
*  using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
*  (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
*  N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
*  must be in generalized Schur canonical form, i.e. A, B are upper
*  quasi triangular and D, E are upper triangular. The solution (R, L)
*  overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
*  chosen to avoid overflow.
*
*  In matrix notation solving equation (1) corresponds to solve
*  Z*x = scale*b, where Z is defined as
*
*         Z = [ kron(In, A)  -kron(B', Im) ]             (2)
*             [ kron(In, D)  -kron(E', Im) ],
*
*  Ik is the identity matrix of size k and X' is the transpose of X.
*  kron(X, Y) is the Kronecker product between the matrices X and Y.
*  In the process of solving (1), we solve a number of such systems
*  where Dim(In), Dim(In) = 1 or 2.
*
*  If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
*  which is equivalent to solve for R and L in
*
*              A' * R  + D' * L   = scale *  C           (3)
*              R  * B' + L  * E'  = scale * -F
*
*  This case is used to compute an estimate of Dif[(A, D), (B, E)] =
*  sigma_min(Z) using reverse communicaton with DLACON.
*
*  DTGSY2 also (IJOB >= 1) contributes to the computation in STGSYL
*  of an upper bound on the separation between to matrix pairs. Then
*  the input (A, D), (B, E) are sub-pencils of the matrix pair in
*  DTGSYL. See STGSYL for details.
*
*  Arguments
*  =========
*
*  TRANS   (input) CHARACTER
*          = 'N', solve the generalized Sylvester equation (1).
*          = 'T': solve the 'transposed' system (3).
*
*  IJOB    (input) INTEGER
*          Specifies what kind of functionality to be performed.
*          = 0: solve (1) only.
*          = 1: A contribution from this subsystem to a Frobenius
*               norm-based estimate of the separation between two matrix
*               pairs is computed. (look ahead strategy is used).
*          = 2: A contribution from this subsystem to a Frobenius
*               norm-based estimate of the separation between two matrix
*               pairs is computed. (DGECON on sub-systems is used.)
*          Not referenced if TRANS = 'T'.
*
*  M       (input) INTEGER
*          On entry, M specifies the order of A and D, and the row
*          dimension of C, F, R and L.
*
*  N       (input) INTEGER
*          On entry, N specifies the order of B and E, and the column
*          dimension of C, F, R and L.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA, M)
*          On entry, A contains an upper quasi triangular matrix.
*
*  LDA     (input) INTEGER
*          The leading dimension of the matrix A. LDA >= max(1, M).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB, N)
*          On entry, B contains an upper quasi triangular matrix.
*
*  LDB     (input) INTEGER
*          The leading dimension of the matrix B. LDB >= max(1, N).
*
*  C       (input/ output) DOUBLE PRECISION array, dimension (LDC, N)
*          On entry, C contains the right-hand-side of the first matrix
*          equation in (1).
*          On exit, if IJOB = 0, C has been overwritten by the
*          solution R.
*
*  LDC     (input) INTEGER
*          The leading dimension of the matrix C. LDC >= max(1, M).
*
*  D       (input) DOUBLE PRECISION array, dimension (LDD, M)
*          On entry, D contains an upper triangular matrix.
*
*  LDD     (input) INTEGER
*          The leading dimension of the matrix D. LDD >= max(1, M).
*
*  E       (input) DOUBLE PRECISION array, dimension (LDE, N)
*          On entry, E contains an upper triangular matrix.
*
*  LDE     (input) INTEGER
*          The leading dimension of the matrix E. LDE >= max(1, N).
*
*  F       (input/ output) DOUBLE PRECISION array, dimension (LDF, N)
*          On entry, F contains the right-hand-side of the second matrix
*          equation in (1).
*          On exit, if IJOB = 0, F has been overwritten by the
*          solution L.
*
*  LDF     (input) INTEGER
*          The leading dimension of the matrix F. LDF >= max(1, M).
*
*  SCALE   (output) DOUBLE PRECISION
*          On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
*          R and L (C and F on entry) will hold the solutions to a
*          slightly perturbed system but the input matrices A, B, D and
*          E have not been changed. If SCALE = 0, R and L will hold the
*          solutions to the homogeneous system with C = F = 0. Normally,
*          SCALE = 1.
*
*  RDSUM   (input/output) DOUBLE PRECISION
*          On entry, the sum of squares of computed contributions to
*          the Dif-estimate under computation by DTGSYL, where the
*          scaling factor RDSCAL (see below) has been factored out.
*          On exit, the corresponding sum of squares updated with the
*          contributions from the current sub-system.
*          If TRANS = 'T' RDSUM is not touched.
*          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
*
*  RDSCAL  (input/output) DOUBLE PRECISION
*          On entry, scaling factor used to prevent overflow in RDSUM.
*          On exit, RDSCAL is updated w.r.t. the current contributions
*          in RDSUM.
*          If TRANS = 'T', RDSCAL is not touched.
*          NOTE: RDSCAL only makes sense when DTGSY2 is called by
*                DTGSYL.
*
*  IWORK   (workspace) INTEGER array, dimension (M+N+2)
*
*  PQ      (output) INTEGER
*          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
*          8-by-8) solved by this routine.
*
*  INFO    (output) INTEGER
*          On exit, if INFO is set to
*            =0: Successful exit
*            <0: If INFO = -i, the i-th argument had an illegal value.
*            >0: The matrix pairs (A, D) and (B, E) have common or very
*                close eigenvalues.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*     Umea University, S-901 87 Umea, Sweden.
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            LDZ
      PARAMETER          ( LDZ = 8 )
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
     $                   K, MB, NB, P, Q, ZDIM
      DOUBLE PRECISION   ALPHA, SCALOC
*     ..
*     .. Local Arrays ..
      INTEGER            IPIV( LDZ ), JPIV( LDZ )
      DOUBLE PRECISION   RHS( LDZ ), Z( LDZ, LDZ )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMM, DGEMV, DGER, DGESC2,
     $                   DGETC2, DLATDF, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Decode and test input parameters
*
      INFO = 0
      IERR = 0
      NOTRAN = LSAME( TRANS, 'N' )
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -1
      ELSE IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
         INFO = -2
      ELSE IF( M.LE.0 ) THEN
         INFO = -3
      ELSE IF( N.LE.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
         INFO = -10
      ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
         INFO = -12
      ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
         INFO = -14
      ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSY2', -INFO )
         RETURN
      END IF
*
*     Determine block structure of A
*
      PQ = 0
      P = 0
      I = 1
   10 CONTINUE
      IF( I.GT.M )
     $   GO TO 20
      P = P + 1
      IWORK( P ) = I
      IF( I.EQ.M )
     $   GO TO 20
      IF( A( I+1, I ).NE.ZERO ) THEN
         I = I + 2
      ELSE
         I = I + 1
      END IF
      GO TO 10
   20 CONTINUE
      IWORK( P+1 ) = M + 1
*
*     Determine block structure of B
*
      Q = P + 1
      J = 1
   30 CONTINUE
      IF( J.GT.N )
     $   GO TO 40
      Q = Q + 1
      IWORK( Q ) = J
      IF( J.EQ.N )
     $   GO TO 40
      IF( B( J+1, J ).NE.ZERO ) THEN
         J = J + 2
      ELSE
         J = J + 1
      END IF
      GO TO 30
   40 CONTINUE
      IWORK( Q+1 ) = N + 1
      PQ = P*( Q-P-1 )
*
      IF( NOTRAN ) THEN
*
*        Solve (I, J) - subsystem
*           A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
*           D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
*        for I = P, P - 1, ..., 1; J = 1, 2, ..., Q
*
         SCALE = ONE
         SCALOC = ONE
         DO 120 J = P + 2, Q
            JS = IWORK( J )
            JSP1 = JS + 1
            JE = IWORK( J+1 ) - 1
            NB = JE - JS + 1
            DO 110 I = P, 1, -1
*
               IS = IWORK( I )
               ISP1 = IS + 1
               IE = IWORK( I+1 ) - 1
               MB = IE - IS + 1
               ZDIM = MB*NB*2
*
               IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
*                 Build a 2-by-2 system Z * x = RHS
*
                  Z( 1, 1 ) = A( IS, IS )
                  Z( 2, 1 ) = D( IS, IS )
                  Z( 1, 2 ) = -B( JS, JS )
                  Z( 2, 2 ) = -E( JS, JS )
*
*                 Set up right hand side(s)
*
                  RHS( 1 ) = C( IS, JS )
                  RHS( 2 ) = F( IS, JS )
*
*                 Solve Z * x = RHS
*