SRC/dtgsyl.f(3)

NAME
SRC/dtgsyl.f

SYNOPSIS

  Functions/Subroutines
subroutine dtgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
DTGSYL

Function/Subroutine Documentation

  subroutine dtgsyl (character trans, integer ijob, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( ldd, * ) d, integer ldd, double precision, dimension( lde, * ) e, integer lde, double precision, dimension( ldf, * ) f, integer ldf, double precision scale, double precision dif, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DTGSYL

Purpose:

    !>
    !> DTGSYL 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 real entries. (A, D) and (B, E) must be in
    !> generalized (real) 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 (1) is equivalent to solve  Zx = scale b, where
    !> Z is defined as
    !>
    !>            Z = [ kron(In, A)  -kron(B**T, Im) ]         (2)
    !>                [ kron(In, D)  -kron(E**T, Im) ].
    !>
    !> Here 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.
    !>
    !> If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
    !> 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 (TRANS = 'T') 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 DLACON.
    !>
    !> If IJOB >= 1, DTGSYL 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. See [1-2] for more
    !> information.
    !>
    !> This is a level 3 BLAS algorithm.
    !>

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: 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 IJOB  = 1 is used).
    !>          = 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
    !>               ( DGECON on sub-systems is used ).
    !>          Not referenced if TRANS = 'T'.
    !>

    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 DOUBLE PRECISION array, dimension (LDA, M)
    !>          The upper quasi triangular matrix A.
    !>

    LDA

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

    B

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

    LDB

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

    C

    !>          C is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
    !>          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 = 'T', DIF is not touched.
    !>

    SCALE

    !>          SCALE is DOUBLE PRECISION
    !>          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, C and F hold the
    !>          solutions R and L, respectively, to the homogeneous system
    !>          with C = F = 0. Normally, SCALE = 1.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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+6)
    !>

    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 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.

References:

    !>
    !>  [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.
    !>

Definition at line 296 of file dtgsyl.f.

Author

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