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CTGSYL(l)			       )			     CTGSYL(l)

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( * )

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 trans‐
       pose 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 (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 ref‐
	       erenced 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.

       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  recipro‐
	       cal  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,

       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.

LAPACK version 3.0		 15 June 2000			     CTGSYL(l)
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