## TB01WD

### Orthogonal similarity transformation of system state-matrix to real Schur form

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

Purpose

      To reduce the system state matrix A to an 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.

Specification

          SUBROUTINE TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
         $                   WR, WI, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER          INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
         $                 WI(*), WR(*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the original 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.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the original state dynamics matrix A.
              On exit, the leading N-by-N part of this array contains
              the matrix U' * A * U in real Schur form. The elements
              below the first subdiagonal are set to zero.
              Note:  A matrix is in real Schur form if it is upper
                     quasi-triangular with 1-by-1 and 2-by-2 blocks.
                     2-by-2 blocks are standardized in the form
                              [  a  b  ]
                              [  c  a  ]
                     where b*c < 0. The eigenvalues of such a block
                     are a +- sqrt(bc).

      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).

      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. The columns of U are the Schur vectors of
              matrix A.

      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.  LWORK >= 3*N.
              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;
              > 0:  if INFO = i, the QR algorithm failed to compute
                    all the eigenvalues; elements i+1:N of WR and WI
                    contain those eigenvalues which have converged;
                    U contains the matrix which reduces A to its
                    partially converged Schur form.

Method

      Matrix A is reduced to a real Schur form using an orthogonal
      similarity transformation A <- U'*A*U. 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 10N  floating point operations.

Further Comments

      None

Example

Program Text

    *     TB01WD 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 ..
          INTEGER          I, INFO, J, M, N, P
    *     .. 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         TB01WD
    *     .. 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
          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 TB01WD( N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
         $                      WR, WI, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      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 (' TB01WD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01WD = ',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)
          END

Program Data

     TB01WD EXAMPLE PROGRAM DATA (Continuous system)
      5  2   3
      -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

     TB01WD EXAMPLE PROGRAM RESULTS

     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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