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DTGSY2(3S)							    DTGSY2(3S)

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, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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),

									Page 1

DTGSY2(3S)							    DTGSY2(3S)

     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).

									Page 2

DTGSY2(3S)							    DTGSY2(3S)

     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)

									Page 3

DTGSY2(3S)							    DTGSY2(3S)

     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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.

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