stgsyl.f(3)

NAME
stgsyl.f -

SYNOPSIS

  Functions/Subroutines

subroutine stgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
STGSYL

Function/Subroutine Documentation

  subroutine stgsyl (characterTRANS, integerIJOB, integerM, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( ldc, * )C, integerLDC, real, dimension( ldd, * )D, integerLDD, real, dimension( lde, * )E, integerLDE, real, dimension( ldf, * )F, integerLDF, realSCALE, realDIF, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

STGSYL

Purpose:

     STGSYL 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', STGSYL 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 SLACON.

     If IJOB >= 1, STGSYL 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.
                   ( SGECON 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
              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 REAL
              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 REAL 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.

Date:
    November 2011

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 298 of file stgsyl.f.

Author

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