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DTGSYL(1)		 LAPACK routine (version 3.2)		     DTGSYL(1)

NAME
       DTGSYL - solves the generalized Sylvester equation

SYNOPSIS
       SUBROUTINE DTGSYL( 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

	   DOUBLE	  PRECISION DIF, SCALE

	   INTEGER	  IWORK( * )

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

PURPOSE
       DTGSYL 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  real  entries.  (A,  D)  and (B, E) must be in generalized (real)
       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 (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  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  = 'T', DTGSYL solves the transposed system Z'*y = scale*b, 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 = 'T') 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 DLACON.
       If IJOB >= 1,  DTGSYL  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.	 See  [1-2]  for  more
       information.
       This is a level 3 BLAS algorithm.

ARGUMENTS
       TRANS   (input) CHARACTER*1
	       =  '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: 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 IJOB  = 1 is used).  =4:	Only  an  estimate  of
	       Dif[(A,D), (B,E)] is computed.  ( DGECON on sub-systems is used
	       ).  Not referenced if TRANS = 'T'.

       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) DOUBLE PRECISION array, dimension (LDA, M)
	       The upper quasi triangular matrix A.

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

       B       (input) DOUBLE PRECISION array, dimension (LDB, N)
	       The upper quasi triangular matrix B.

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

       C       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
	       On exit DIF is the reciprocal of a lower bound of the  recipro‚Äź
	       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 = 'T', DIF is not touched.

       SCALE   (output) DOUBLE PRECISION
	       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, C and F hold the solutions
	       R  and  L, respectively, to the homogeneous system with C = F =
	       0. Normally, SCALE = 1.

       WORK	 (workspace/output)   DOUBLE   PRECISION   array,    dimension
       (MAX(1,LWORK))
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK. LWORK > = 1.  If IJOB = 1 or 2
	       and TRANS = 'N', LWORK >= max(1,2*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+6)

       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 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 routine (version 3.2)	 November 2008			     DTGSYL(1)
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