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## ztgsyl()

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

Download ZTGSYL + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  TRANS

      !>          TRANS is CHARACTER*1
      !>          = 'N': solve the generalized sylvester equation (1).
      !>          = 'C': solve the  system (3).
      !>

  [in]

  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'.
      !>

  [in]

  M

      !>          M is INTEGER
      !>          The order of the matrices A and D, and the row dimension of
      !>          the matrices C, F, R and L.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrices B and E, and the column dimension
      !>          of the matrices C, F, R and L.
      !>

  [in]

  A

      !>          A is COMPLEX*16 array, dimension (LDA, M)
      !>          The upper triangular matrix A.
      !>

  [in]

  LDA

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

  [in]

  B

      !>          B is COMPLEX*16 array, dimension (LDB, N)
      !>          The upper triangular matrix B.
      !>

  [in]

  LDB

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

  [in,out]

  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.
      !>

  [in]

  LDC

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

  [in]

  D

      !>          D is COMPLEX*16 array, dimension (LDD, M)
      !>          The upper triangular matrix D.
      !>

  [in]

  LDD

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

  [in]

  E

      !>          E is COMPLEX*16 array, dimension (LDE, N)
      !>          The upper triangular matrix E.
      !>

  [in]

  LDE

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

  [in,out]

  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.
      !>

  [in]

  LDF

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  WORK

      !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (M+N+2)
      !>

  [out]

  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.