## TB01WX

### Orthogonal similarity transformation of a standard system to one with state matrix in a Hessenberg form

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

Purpose

      To reduce the system state matrix A to an upper Hessenberg 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 TB01WX( COMPU, N, M, P, A, LDA, B, LDB, C, LDC, U, LDU,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        COMPU
          INTEGER          INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*)

Arguments

Mode Parameters

      COMPU   CHARACTER*1
              = 'N':  do not compute U;
              = 'I':  U is initialized to the unit matrix, and the
                      orthogonal matrix U is returned;
              = 'U':  U must contain an orthogonal matrix U1 on entry,
                      and the product U1*U is returned.

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 Hessenberg form. The elements
              below the first subdiagonal are set to zero.

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

      U       (input/output) DOUBLE PRECISION array, dimension (LDU,*)
              On entry, if COMPU = 'U', the leading N-by-N part of this
              array must contain the given matrix U1. Otherwise, this
              array need not be set on input.
              On exit, if COMPU <> 'N', the leading N-by-N part of this
              array contains the orthogonal transformation matrix used
              to reduce A to the Hessenberg form (U1*U if COMPU = 'U').
              If COMPU = 'N', this array is not referenced.

      LDU     INTEGER
              The leading dimension of the array U.
              LDU >= 1,        if COMPU =  'N';
              LDU >= max(1,N), if COMPU <> 'N'.

Workspace

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

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= 1, and if N > 0,
              LDWORK >= N - 1 + MAX(N,M,P).
              For optimum performance LDWORK should be larger.

              If LDWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the DWORK
              array, returns this value as the first entry of the DWORK
              array, and no error message related to LDWORK is issued by
              XERBLA.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      Matrix A is reduced to the Hessenberg 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      2
      The algorithm requires about 5N /3 + N (M+P) floating point
                                               3
      operations, if COMPU = 'N'. Otherwise, 2N /3 additional operations
      are needed.

Further Comments

      None

Example

Program Text

    *     TB01WX 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 = NMAX - 1 + MAX( NMAX, MMAX, PMAX ) )
    *     .. Local Scalars ..
          CHARACTER        COMPU
          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)
    *     .. External Subroutines ..
          EXTERNAL         TB01WX
    *     .. 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, COMPU
          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 TB01WX( COMPU, N, M, P, A, LDA, B, LDB, C, LDC, U,
         $                      LDU, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      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 (' TB01WX EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01WX = ',I2)
    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

     TB01WX EXAMPLE PROGRAM DATA (Continuous system)
      5  2   3     I
      -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

     TB01WX EXAMPLE PROGRAM RESULTS

     The transformed state dynamics matrix U'*A*U is
      -0.0416  -6.3778   1.4826  -1.9856   1.2630
       1.4911  -2.8851  -0.4353   0.8984  -0.5714
       0.0000  -2.1254   1.6804  -4.9686  -1.7731
       0.0000   0.0000   2.1880  -3.3545  -2.6069
       0.0000   0.0000   0.0000   0.7554  -2.1424

     The transformed input/state matrix U'*B is
       0.0000   0.0000
      -3.0996   0.0000
      -0.6488   0.0000
       0.8689   1.7872
      -0.5527   2.8098

     The transformed state/output matrix C*U is
       1.0000   0.0000   0.0000   0.0000   0.0000
       0.0000   0.3655  -0.4962   0.6645  -0.4227
       0.0000   0.0000  -0.8461  -0.4498   0.2861

     The similarity transformation matrix U is
       1.0000   0.0000   0.0000   0.0000   0.0000
       0.0000  -0.9308  -0.1948   0.2609  -0.1660
       0.0000   0.3655  -0.4962   0.6645  -0.4227
       0.0000   0.0000  -0.8461  -0.4498   0.2861
       0.0000   0.0000   0.0000   0.5367   0.8438

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