DTGSY2(l)								 )								 DTGSY2(l)

NAME

       DTGSY2 - solve the generalized Sylvester equation

SYNOPSIS

       SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO )

	   CHARACTER	  TRANS

	   INTEGER	  IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, PQ

	   DOUBLE	  PRECISION RDSCAL, RDSUM, SCALE

	   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  per-
	       turbed  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 cur-
	       rent 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.

LAPACK version 3.0						   15 June 2000 							 DTGSY2(l)