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 "conjugate transposed" 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 homogenious 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.

November 2011

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.