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STGSY2(l)					)					STGSY2(l)

NAME
       STGSY2 - solve the generalized Sylvester equation

SYNOPSIS
       SUBROUTINE STGSY2( 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

	   REAL 	  RDSCAL, RDSUM, SCALE

	   INTEGER	  IWORK( * )

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

PURPOSE
       STGSY2 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  Kro-
       necker  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 SLACON.

       STGSY2 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 STGSYL. See STGSYL for details.

ARGUMENTS
       TRANS   (input) CHARACTER
	       = 'N', solve the generalized Sylvester equation (1).  =	'T':  solve  the  'trans-
	       posed' 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. (SGECON 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL
	       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 matri-
	       ces A, B, D and E have not been changed. If SCALE = 0, R and L will hold the solu-
	       tions to the homogeneous system with C = F = 0. Normally, SCALE = 1.

       RDSUM   (input/output) REAL
	       On entry, the sum of squares of computed contributions to the  Dif-estimate  under
	       computation  by	STGSYL, where the scaling factor RDSCAL (see below) has been fac-
	       tored out.  On exit, the corresponding sum of squares updated with  the	contribu-
	       tions  from  the  current sub-system.  If TRANS = 'T' RDSUM is not touched.  NOTE:
	       RDSUM only makes sense when STGSY2 is called by STGSYL.

       RDSCAL  (input/output) REAL
	       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 STGSY2 is called by STGSYL.

       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 				STGSY2(l)
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