!>
!> 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**H, Im) ]        (2)
!>            [ kron(In, D)  -kron(E**H, Im) ],
!>
!> Here Ix is the identity matrix of size x and X**H is the conjugate
!> transpose 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**H *y = scale*b
!> is solved for, which is equivalent to solve for R and L in
!>
!>             A**H * R + D**H * L = scale * C           (3)
!>             R * B**H + L * E**H = 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.
!> 

TRANS

!>          TRANS is CHARACTER*1
!>          = 'N': solve the generalized sylvester equation (1).
!>          = 'C': solve the  system (3).
!> 

IJOB

!>          IJOB is 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

!>          M is INTEGER
!>          The order of the matrices A and D, and the row dimension of
!>          the matrices C, F, R and L.
!> 

N

!>          N is INTEGER
!>          The order of the matrices B and E, and the column dimension
!>          of the matrices C, F, R and L.
!> 

A

!>          A is COMPLEX array, dimension (LDA, M)
!>          The upper triangular matrix A.
!> 

LDA

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

B

!>          B is COMPLEX array, dimension (LDB, N)
!>          The upper triangular matrix B.
!> 

LDB

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

C

!>          C is 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

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

D

!>          D is COMPLEX array, dimension (LDD, M)
!>          The upper triangular matrix D.
!> 

LDD

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

E

!>          E is COMPLEX array, dimension (LDE, N)
!>          The upper triangular matrix E.
!> 

LDE

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

F

!>          F is 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

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

DIF

!>          DIF is 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

!>          SCALE is 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 homogeneous system with C = F = 0.
!> 

WORK

!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is 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

!>          IWORK is INTEGER array, dimension (M+N+2)
!> 

INFO

!>          INFO is 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.