SRC/ctgsy2.f(3)

NAME
SRC/ctgsy2.f

SYNOPSIS

  Functions/Subroutines
subroutine ctgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Function/Subroutine Documentation

  subroutine ctgsy2 (character trans, integer ijob, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( ldd, * ) d, integer ldd, complex, dimension( lde, * ) e, integer lde, complex, dimension( ldf, * ) f, integer ldf, real scale, real rdsum, real rdscal, integer info)

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

    !>
    !> CTGSY2 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. 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 solving equation (1) corresponds 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) ],
    !>
    !> Ik is the identity matrix of size k and X**H is the 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
    !> = sigma_min(Z) using reverse communication with CLACON.
    !>
    !> CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
    !> of an upper bound on the separation between to matrix pairs. Then
    !> the input (A, D), (B, E) are sub-pencils of two matrix pairs in
    !> CTGSYL.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': solve the generalized Sylvester equation (1).
    !>          = 'T': solve the 'transposed' system (3).
    !>

    IJOB

    !>          IJOB is 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. (SGECON on sub-systems is used.)
    !>          Not referenced if TRANS = 'T'.
    !>

    M

    !>          M is INTEGER
    !>          On entry, M specifies the order of A and D, and the row
    !>          dimension of C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          On entry, N specifies the order of B and E, and the column
    !>          dimension of C, F, R and L.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA, M)
    !>          On entry, A contains an upper triangular matrix.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, B contains an upper triangular matrix.
    !>

    LDB

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

    LDC

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

    D

    !>          D is COMPLEX array, dimension (LDD, M)
    !>          On entry, D contains an upper triangular matrix.
    !>

    LDD

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

    E

    !>          E is COMPLEX array, dimension (LDE, N)
    !>          On entry, E contains an upper triangular matrix.
    !>

    LDE

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

    LDF

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

    SCALE

    !>          SCALE is REAL
    !>          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

    !>          RDSUM is REAL
    !>          On entry, the sum of squares of computed contributions to
    !>          the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by
    !>          CTGSYL.
    !>

    RDSCAL

    !>          RDSCAL is REAL
    !>          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 CTGSY2 is called by
    !>          CTGSYL.
    !>

    INFO

    !>          INFO is INTEGER
    !>          On exit, if INFO is set to
    !>            =0: Successful exit
    !>            <0: If INFO = -i, input argument number i is illegal.
    !>            >0: The matrix pairs (A, D) and (B, E) have common or very
    !>                close eigenvalues.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 256 of file ctgsy2.f.

Author

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