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

NAME

       LAPACK-3 - solves the generalized Sylvester equation

SYNOPSIS

       SUBROUTINE DTGSY2( TRANS,  IJOB,  M,  N,  A,  LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF,
                          SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO )

           CHARACTER      TRANS

           INTEGER        IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, PQ

           DOUBLE         PRECISION RDSCAL, RDSUM, SCALE

           INTEGER        IWORK( * )

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

PURPOSE

       DTGSY2 solves the generalized Sylvester equation:
                    A * R - L * B = scale * C                (1)
                    D * R - L * E = scale * F,
        using Level 1 and 2 BLAS. 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 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 solving equation (1) corresponds to solve
        Z*x = scale*b, where Z is defined as
               Z = [ kron(In, A)  -kron(B**T, Im) ]             (2)
                   [ kron(In, D)  -kron(E**T, Im) ],
        Ik is the identity matrix of size k and X**T is the transpose of X.
        kron(X, Y) is the Kronecker product between the matrices X and Y.
        In the process of solving (1), we solve a number of such systems
        where Dim(In), Dim(In) = 1 or 2.
        If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
        which is equivalent to solve for R and L in
                    A**T * R  + D**T * L   = scale *  C           (3)
                    R  * B**T + L  * E**T  = scale * -F
        This case is used to compute an estimate of Dif[(A, D), (B, E)] =
        sigma_min(Z) using reverse communicaton with DLACON.
        DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
        of an upper bound on the separation between to matrix pairs. Then
        the input (A, D), (B, E) are sub-pencils of the matrix pair in
        DTGSYL. See DTGSYL for details.

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: A contribution from this subsystem to a Frobenius
                norm-based estimate of the separation between two matrix
                pairs is computed. (look ahead strategy is used).
                = 2: A contribution from this subsystem to a Frobenius
                norm-based estimate of the separation between two matrix
                pairs is computed. (DGECON on sub-systems is used.)
                Not referenced if TRANS = 'T'.

        M       (input) INTEGER
                On entry, M specifies the order of A and D, and the row
                dimension of C, F, R and L.

        N       (input) INTEGER
                On entry, N specifies the order of B and E, and the column
                dimension of C, F, R and L.

        A       (input) DOUBLE PRECISION array, dimension (LDA, M)
                On entry, A contains an upper quasi triangular matrix.

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

        B       (input) DOUBLE PRECISION array, dimension (LDB, N)
                On entry, B contains an upper quasi triangular matrix.

        LDB     (input) INTEGER
                The leading dimension of the matrix 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).
                On exit, if IJOB = 0, C has been overwritten by the
                solution R.

        LDC     (input) INTEGER
                The leading dimension of the matrix C. LDC >= max(1, M).

        D       (input) DOUBLE PRECISION array, dimension (LDD, M)
                On entry, D contains an upper triangular matrix.

        LDD     (input) INTEGER
                The leading dimension of the matrix D. LDD >= max(1, M).

        E       (input) DOUBLE PRECISION array, dimension (LDE, N)
                On entry, E contains an upper triangular matrix.

        LDE     (input) INTEGER
                The leading dimension of the matrix 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).
                On exit, if IJOB = 0, F has been overwritten by the
                solution L.

        LDF     (input) INTEGER
                The leading dimension of the matrix F. LDF >= max(1, M).

        SCALE   (output) DOUBLE PRECISION
                On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
                R and L (C and F on entry) will hold the solutions 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. Normally,
                SCALE = 1.

        RDSUM   (input/output) DOUBLE PRECISION
                On entry, the sum of squares of computed contributions to
                the Dif-estimate under computation by DTGSYL, where the
                scaling factor RDSCAL (see below) has been factored out.
                On exit, the corresponding sum of squares updated with the
                contributions from the current sub-system.
                If TRANS = 'T' RDSUM is not touched.
                NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.

        RDSCAL  (input/output) DOUBLE PRECISION
                On entry, scaling factor used to prevent overflow in RDSUM.
                On exit, RDSCAL is updated w.r.t. the current contributions
                in RDSUM.
                If TRANS = 'T', RDSCAL is not touched.
                NOTE: RDSCAL only makes sense when DTGSY2 is called by
                DTGSYL.

        IWORK   (workspace) INTEGER array, dimension (M+N+2)

        PQ      (output) INTEGER
                On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
                8-by-8) solved by this routine.

        INFO    (output) INTEGER
                On exit, if INFO is set to
                =0: Successful exit
                <0: If INFO = -i, the i-th argument had an illegal value.
                >0: The matrix pairs (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.

 LAPACK auxiliary routine (version 3.3.1)   April 2011                            DTGSY2(3lapack)