SRC/ztgsyl.f(3)

NAME
SRC/ztgsyl.f

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine ztgsyl (character trans, integer ijob, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldc, * ) c, integer ldc, complex*16, dimension( ldd, * ) d, integer ldd, complex*16, dimension( lde, * ) e, integer lde, complex*16, dimension( ldf, * ) f, integer ldf, double precision scale, double precision dif, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

ZTGSYL

Purpose:

     ZTGSYL 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 ZLACON.
     If IJOB >= 1, ZTGSYL 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.

Parameters
    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.
                  (ZGECON 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*16 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*16 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*16 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*16 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*16 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*16 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 = 'C', DIF is not referenced.

    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, R and L will
              hold the solutions to the homogeneous system with C = F = 0.

    WORK

              WORK is COMPLEX*16 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.

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 292 of file ztgsyl.f.

Author

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