ZTGSY2(l)								 )								 ZTGSY2(l)

NAME

       ZTGSY2 - solve 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,

SYNOPSIS

       SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO )

	   CHARACTER	  TRANS

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

	   DOUBLE	  PRECISION RDSCAL, RDSUM, SCALE

	   COMPLEX*16	  A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * )

PURPOSE

       ZTGSY2 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. A, B,
       D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form).

       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 Zx = 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.

       If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b is solved for, 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 ZLACON.

       ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL of an upper bound on the separation between  to  matrix  pairs.  Then  the
       input (A, D), (B, E) are sub-pencils of two matrix pairs in ZTGSYL.

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) COMPLEX*16 array, dimension (LDA, M)
	       On entry, A contains an upper triangular matrix.

       LDA     (input) INTEGER
	       The leading dimension of the matrix A. LDA >= max(1, M).

       B       (input) COMPLEX*16 array, dimension (LDB, N)
	       On entry, B contains an upper triangular matrix.

       LDB     (input) INTEGER
	       The leading dimension of the matrix B. LDB >= max(1, N).

       C       (input/ output) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 ZTGSYL, 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 ZTGSY2 is called by ZTGSYL.

       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 ZTGSY2 is called by ZTGSYL.

       INFO    (output) INTEGER
	       On exit, if INFO is set to =0: Successful exit
	       <0: If INFO = -i, input argument number i is illegal.
	       >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 							 ZTGSY2(l)