Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - solves 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**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.

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.

        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 (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+2)

        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 routine (version 3.3.1)             April 2011                            CTGSYL(3lapack)