## TB01LD

### Orthogonal similarity reduction so that the leading part of system state-matrix has eigenvalues in a specified domain

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To reduce the system state matrix A to an ordered upper real
      Schur form by using an orthogonal similarity transformation
      A <-- U'*A*U and to apply the transformation to the matrices
      B and C: B <-- U'*B and C <-- C*U.
      The leading block of the resulting A has eigenvalues in a
      suitably defined domain of interest.

Specification

          SUBROUTINE TB01LD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA, B,
         $                   LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        DICO, JOBA, STDOM
          INTEGER          INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, NDIM, P
          DOUBLE PRECISION ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
         $                 WI(*), WR(*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

      STDOM   CHARACTER*1
              Specifies whether the domain of interest is of stability
              type (left part of complex plane or inside of a circle)
              or of instability type (right part of complex plane or
              outside of a circle) as follows:
              = 'S':  stability type domain;
              = 'U':  instability type domain.

      JOBA    CHARACTER*1
              Specifies the shape of the state dynamics matrix on entry
              as follows:
              = 'S':  A is in an upper real Schur form;
              = 'G':  A is a general square dense matrix.

Input/Output Parameters

      N       (input) INTEGER
              The order of the state-space representation,
              i.e. the order of the matrix A.  N >= 0.

      M       (input) INTEGER
              The number of system inputs, or of columns of B.  M >= 0.

      P       (input) INTEGER
              The number of system outputs, or of rows of C.  P >= 0.

      ALPHA   (input) DOUBLE PRECISION.
              Specifies the boundary of the domain of interest for the
              eigenvalues of A. For a continuous-time system
              (DICO = 'C'), ALPHA is the boundary value for the real
              parts of eigenvalues, while for a discrete-time system
              (DICO = 'D'), ALPHA >= 0 represents the boundary value
              for the moduli of eigenvalues.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the unreduced state dynamics matrix A.
              If JOBA = 'S' then A must be a matrix in real Schur form.
              On exit, the leading N-by-N part of this array contains
              the ordered real Schur matrix U' * A * U with the elements
              below the first subdiagonal set to zero.
              The leading NDIM-by-NDIM part of A has eigenvalues in the
              domain of interest and the trailing (N-NDIM)-by-(N-NDIM)
              part has eigenvalues outside the domain of interest.
              The domain of interest for lambda(A), the eigenvalues
              of A, is defined by the parameters ALPHA, DICO and STDOM
              as follows:
              For a continuous-time system (DICO = 'C'):
                Real(lambda(A)) < ALPHA if STDOM = 'S';
                Real(lambda(A)) > ALPHA if STDOM = 'U';
              For a discrete-time system (DICO = 'D'):
                Abs(lambda(A)) < ALPHA if STDOM = 'S';
                Abs(lambda(A)) > ALPHA if STDOM = 'U'.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input matrix B.
              On exit, the leading N-by-M part of this array contains
              the transformed input matrix U' * B.

      LDB     INTEGER
              The leading dimension of array B.  LDB >= MAX(1,N).

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the output matrix C.
              On exit, the leading P-by-N part of this array contains
              the transformed output matrix C * U.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= MAX(1,P).

      NDIM    (output) INTEGER
              The number of eigenvalues of A lying inside the domain of
              interest for eigenvalues.

      U       (output) DOUBLE PRECISION array, dimension (LDU,N)
              The leading N-by-N part of this array contains the
              orthogonal transformation matrix used to reduce A to the
              real Schur form and/or to reorder the diagonal blocks of
              real Schur form of A. The first NDIM columns of U form
              an orthogonal basis for the invariant subspace of A
              corresponding to the first NDIM eigenvalues.

      LDU     INTEGER
              The leading dimension of array U.  LDU >= max(1,N).

      WR, WI  (output) DOUBLE PRECISION arrays, dimension (N)
              WR and WI contain the real and imaginary parts,
              respectively, of the computed eigenvalues of A. The
              eigenvalues will be in the same order that they appear on
              the diagonal of the output real Schur form of A. Complex
              conjugate pairs of eigenvalues will appear consecutively
              with the eigenvalue having the positive imaginary part
              first.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of working array DWORK.
              LDWORK >= MAX(1,N)   if JOBA = 'S';
              LDWORK >= MAX(1,3*N) if JOBA = 'G'.
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0: successful exit;
              < 0: if INFO = -i, the i-th argument had an illegal
                   value;
              = 1: the QR algorithm failed to compute all the
                   eigenvalues of A;
              = 2: a failure occured during the ordering of the real
                   Schur form of A.

Method

      Matrix A is reduced to an ordered upper real Schur form using an
      orthogonal similarity transformation A <-- U'*A*U. This
      transformation is determined so that the leading block of the
      resulting A has eigenvalues in a suitably defined domain of
      interest. Then, the transformation is applied to the matrices B
      and C: B <-- U'*B and C <-- C*U.

Numerical Aspects

                                      3
      The algorithm requires about 14N  floating point operations.

Further Comments

      None

Example

Program Text

    *     TB01LD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDU
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDU = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      DICO, JOBA, STDOM
          INTEGER          I, INFO, J, M, N, NDIM, P
          DOUBLE PRECISION ALPHA
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), U(LDU,NMAX), WI(NMAX), WR(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TB01LD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, P, ALPHA, DICO, STDOM, JOBA
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99989 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99988 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find the transformed ssr for (A,B,C).
                   CALL TB01LD( DICO, STDOM, JOBA, N, M, P, ALPHA,
         $                      A, LDA, B, LDB, C, LDC, NDIM, U, LDU,
         $                      WR, WI, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99987 ) NDIM
                      WRITE ( NOUT, FMT = 99997 )
                      DO 10 I = 1, N
                         WRITE ( NOUT, FMT = 99994 ) WR(I), WI(I)
       10             CONTINUE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40             CONTINUE
                      WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TB01LD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01LD = ',I2)
    99997 FORMAT (/' The eigenvalues of state dynamics matrix A are ')
    99996 FORMAT (/' The transformed state dynamics matrix U''*A*U is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
    99993 FORMAT (/' The transformed input/state matrix U''*B is ')
    99992 FORMAT (/' The transformed state/output matrix C*U is ')
    99991 FORMAT (/' The similarity transformation matrix U is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' P is out of range.',/' P = ',I5)
    99987 FORMAT (/' The number of eigenvalues in the domain of interest =',
         $        I5 )
          END

Program Data

     TB01LD EXAMPLE PROGRAM DATA (Continuous system)
      5     2     3    -1.0      C     U     G
      -0.04165    4.9200   -4.9200         0         0
     -1.387944   -3.3300         0         0         0
        0.5450         0         0   -0.5450         0
             0         0    4.9200  -0.04165    4.9200
             0         0         0 -1.387944   -3.3300
             0         0
        3.3300         0
             0         0
             0         0
             0    3.3300
         1     0     0     0     0
         0     0     1     0     0
         0     0     0     1     0

Program Results

     TB01LD EXAMPLE PROGRAM RESULTS

     The number of eigenvalues in the domain of interest =    2

     The eigenvalues of state dynamics matrix A are
     ( -0.7483,   2.9940 )
     ( -0.7483,  -2.9940 )
     ( -1.6858,   2.0311 )
     ( -1.6858,  -2.0311 )
     ( -1.8751,   0.0000 )

     The transformed state dynamics matrix U'*A*U is
      -0.7483  -8.6406   0.0000   0.0000   1.1745
       1.0374  -0.7483   0.0000   0.0000  -2.1164
       0.0000   0.0000  -1.6858   5.5669   0.0000
       0.0000   0.0000  -0.7411  -1.6858   0.0000
       0.0000   0.0000   0.0000   0.0000  -1.8751

     The transformed input/state matrix U'*B is
      -0.5543   0.5543
      -1.6786   1.6786
      -0.8621  -0.8621
       2.1912   2.1912
      -1.5555   1.5555

     The transformed state/output matrix C*U is
       0.6864  -0.0987   0.6580   0.2589  -0.1381
      -0.0471   0.6873   0.0000   0.0000  -0.7249
      -0.6864   0.0987   0.6580   0.2589   0.1381

     The similarity transformation matrix U is
       0.6864  -0.0987   0.6580   0.2589  -0.1381
      -0.1665  -0.5041  -0.2589   0.6580  -0.4671
      -0.0471   0.6873   0.0000   0.0000  -0.7249
      -0.6864   0.0987   0.6580   0.2589   0.1381
       0.1665   0.5041  -0.2589   0.6580   0.4671

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