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CTGSYL(3S)							    CTGSYL(3S)

NAME
     CTGSYL - solve the generalized Sylvester equation

SYNOPSIS
     SUBROUTINE CTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E,
			LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO )

	 CHARACTER	TRANS

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

	 REAL		DIF, SCALE

	 INTEGER	IWORK( * )

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

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
     CTGSYL solves the generalized Sylvester equation:
		 A * R - L * B = scale * C	      (1)
		 D * R - L * E = scale * F

     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
     complex entries. 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 (1) is equivalent to solve Zx = scale*b, where Z is
     defined as

	    Z = [ kron(In, A)  -kron(B', Im) ]	      (2)
		[ kron(In, D)  -kron(E', Im) ],

     Here Ix is the identity matrix of size x and X' is the conjugate
     transpose of X. Kron(X, Y) is the Kronecker product between the matrices

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CTGSYL(3S)							    CTGSYL(3S)

     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 (TRANS = 'C') is used to compute an one-norm-based estimate of
     Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and
     (B,E), using CLACON.

     If IJOB >= 1, CTGSYL computes a Frobenius norm-based estimate of
     Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
     reciprocal of the smallest singular value of Z.

     This is a level-3 BLAS algorithm.

ARGUMENTS
     TRANS   (input) CHARACTER*1
	     = 'N': solve the generalized sylvester equation (1).
	     = 'C': solve the "conjugate transposed" system (3).

     IJOB    (input) INTEGER
	     Specifies what kind of functionality to be performed.  =0: solve
	     (1) only.
	     =1: The functionality of 0 and 3.
	     =2: The functionality of 0 and 4.
	     =3: Only an estimate of Dif[(A,D), (B,E)] is computed.  (look
	     ahead strategy is used).  =4: Only an estimate of Dif[(A,D),
	     (B,E)] is computed.  (CGECON on sub-systems is used).  Not
	     referenced if TRANS = 'C'.

     M	     (input) INTEGER
	     The order of the matrices A and D, and the row dimension of the
	     matrices C, F, R and L.

     N	     (input) INTEGER
	     The order of the matrices B and E, and the column dimension of
	     the matrices C, F, R and L.

     A	     (input) COMPLEX array, dimension (LDA, M)
	     The upper triangular matrix A.

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

     B	     (input) COMPLEX array, dimension (LDB, N)
	     The upper triangular matrix B.

									Page 2

CTGSYL(3S)							    CTGSYL(3S)

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

     C	     (input/output) COMPLEX array, dimension (LDC, N)
	     On entry, C contains the right-hand-side of the first matrix
	     equation in (1) or (3).  On exit, if IJOB = 0, 1 or 2, C has been
	     overwritten by the solution R. If IJOB = 3 or 4 and TRANS = 'N',
	     C holds R, the solution achieved during the computation of the
	     Dif-estimate.

     LDC     (input) INTEGER
	     The leading dimension of the array C. LDC >= max(1, M).

     D	     (input) COMPLEX array, dimension (LDD, M)
	     The upper triangular matrix D.

     LDD     (input) INTEGER
	     The leading dimension of the array D. LDD >= max(1, M).

     E	     (input) COMPLEX array, dimension (LDE, N)
	     The upper triangular matrix E.

     LDE     (input) INTEGER
	     The leading dimension of the array E. LDE >= max(1, N).

     F	     (input/output) COMPLEX array, dimension (LDF, N)
	     On entry, F contains the right-hand-side of the second matrix
	     equation in (1) or (3).  On exit, if IJOB = 0, 1 or 2, F has been
	     overwritten by the solution L. If IJOB = 3 or 4 and TRANS = 'N',
	     F holds L, the solution achieved during the computation of the
	     Dif-estimate.

     LDF     (input) INTEGER
	     The leading dimension of the array F. LDF >= max(1, M).

     DIF     (output) REAL
	     On exit DIF is the reciprocal of a lower bound of the reciprocal
	     of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D),
	     (B,E)] = sigma-min(Z), where Z as in (2).	IF IJOB = 0 or TRANS =
	     'C', DIF is not referenced.

     SCALE   (output) REAL
	     On exit SCALE is the scaling factor in (1) or (3).	 If 0 < SCALE
	     < 1, C and F hold the solutions R and L, resp., 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 homogenious system with C = F = 0.

     WORK    (workspace/output) COMPLEX array, dimension (LWORK)
	     IF IJOB = 0, WORK is not referenced.  Otherwise,

									Page 3

CTGSYL(3S)							    CTGSYL(3S)

     LWORK   (input) INTEGER
	     The dimension of the array WORK. LWORK > = 1.  If IJOB = 1 or 2
	     and TRANS = 'N', LWORK >= 2*M*N.

	     If LWORK = -1, then a workspace query is assumed; the routine
	     only calculates the optimal size of the WORK array, returns this
	     value as the first entry of the WORK array, and no error message
	     related to LWORK is issued by XERBLA.

     IWORK   (workspace) INTEGER array, dimension (M+N+2)
	     If IJOB = 0, IWORK is not referenced.

     INFO    (output) INTEGER
	     =0: successful exit
	     <0: If INFO = -i, the i-th argument had an illegal value.
	     >0: (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.

     [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
	 for Solving the Generalized Sylvester Equation and Estimating the
	 Separation between Regular Matrix Pairs, Report UMINF - 93.23,
	 Department of Computing Science, Umea University, S-901 87 Umea,
	 Sweden, December 1993, Revised April 1994, Also as LAPACK Working
	 Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
	 No 1, 1996.

     [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
	 Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
	 Appl., 15(4):1045-1060, 1994.

     [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
	 Condition Estimators for Solving the Generalized Sylvester
	 Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
	 July 1989, pp 745-751.

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

     This man page is available only online.

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