## TG01ID

### Orthogonal reduction of a descriptor system to the observability staircase form

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

Purpose

      To compute orthogonal transformation matrices Q and Z which
      reduce the N-th order descriptor system (A-lambda*E,B,C)
      to the form

                 ( Ano  * )             ( Eno  * )           ( Bno )
        Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,  Q'*B = (     ) ,
                 ( 0   Ao )             ( 0   Eo )           ( Bo  )

           C*Z = ( 0   Co ) ,

      where the NOBSV-th order descriptor system (Ao-lambda*Eo,Bo,Co)
      is a finite and/or infinite observable. The pencil
      Ano - lambda*Eno is regular of order N-NOBSV and contains the
      unobservable finite and/or infinite eigenvalues of the pencil
      A-lambda*E.

      For JOBOBS = 'O' or 'I', the pencil ( Eo-lambda*Ao ) has full
                                          (      Co      )
      column rank NOBSV for all finite lambda and is in a staircase form
      with
                      _      _            _      _
                    ( Ek,k   Ek,k-1   ... Ek,2   Ek,1   )
                    ( _      _            _      _      )
        ( Eo ) =    ( Ek-1,k Ek-1,k-1 ... Ek-1,2 Ek-1,1 ) ,  (1)
        ( Co )      (     ...         ... _      _      )
                    (  0       0      ... E1,2   E1,1   )
                    (                            _      )
                    (  0       0      ... 0      E0,1   )
                      _          _      _
                    ( Ak,k  ...  Ak,2   Ak,1 )
                    (       ...  _      _    )
          Ao      = (   0   ...  A2,2   A2,1 ) ,             (2)
                    (                   _    )
                    (   0   ...    0    A1,1 )
            _
      where Ei-1,i is a CTAU(i-1)-by-CTAU(i) full column rank matrix
                             _
      (with CTAU(0) = P) and Ai,i is a CTAU(i)-by-CTAU(i)
      upper triangular matrix.

      For JOBOBS = 'F', the pencil ( Ao-lambda*Eo ) has full
                                   (      Co      )
      column rank NOBSV for all finite lambda and is in a staircase form
      with
                      _      _            _      _
                    ( Ak,k   Ak,k-1   ... Ak,2   Ak,1   )
                    ( _      _            _      _      )
        ( Ao ) =    ( Ak-1,k Ak-1,k-1 ... Ak-1,2 Ak-1,1 ) ,  (3)
        ( Co )      (     ...         ... _      _      )
                    (  0       0      ... A1,2   A1,1   )
                    (                            _      )
                    (  0       0      ... 0      A0,1   )
                      _          _      _
                    ( Ek,k  ...  Ek,2   Ek,1 )
                    (       ...  _      _    )
          Eo      = (   0   ...  E2,2   E2,1 ) ,             (4)
                    (                   _    )
                    (   0   ...    0    E1,1 )
            _
      where Ai-1,i is a CTAU(i-1)-by-CTAU(i) full column rank matrix
                             _
      (with CTAU(0) = P) and Ei,i is a CTAU(i)-by-CTAU(i)
      upper triangular matrix.

      For JOBOBS = 'O', the (N-NOBSV)-by-(N-NOBSV) regular pencil
      Ano - lambda*Eno has the form

                          ( Afno - lambda*Efno         *          )
       Ano - lambda*Eno = (                                       ) ,
                          (        0           Aino - lambda*Eino )

      where:
        1) the NIUOBS-by-NIUOBS regular pencil Aino - lambda*Eino,
           with Aino upper triangular and nonsingular, contains the
           unobservable infinite eigenvalues of A - lambda*E;
        2) the (N-NOBSV-NIUOBS)-by-(N-NOBSV-NIUOBS) regular pencil
           Afno - lambda*Efno, with Efno upper triangular and
           nonsingular, contains the unobservable finite
           eigenvalues of A - lambda*E.

      Note: The significance of the two diagonal blocks can be
            interchanged by calling the routine with the
            arguments A and E interchanged. In this case,
            Aino - lambda*Eino contains the unobservable zero
            eigenvalues of A - lambda*E, while Afno - lambda*Efno
            contains the unobservable nonzero finite and infinite
            eigenvalues of A - lambda*E.

      For JOBOBS = 'F', the pencil Ano - lambda*Eno has the form

         Ano - lambda*Eno = Afno - lambda*Efno ,

      where the regular pencil Afno - lambda*Efno, with Efno
      upper triangular and nonsingular, contains the unobservable
      finite eigenvalues of A - lambda*E.

      For JOBOBS = 'I', the pencil Ano - lambda*Eno has the form

         Ano - lambda*Eno = Aino - lambda*Eino ,

      where the regular pencil Aino - lambda*Eino, with Aino
      upper triangular and nonsingular, contains the unobservable
      nonzero finite and infinite eigenvalues of A - lambda*E.

      The left and/or right orthogonal transformations Q and Z
      performed to reduce the system matrices can be optionally
      accumulated.

      The reduced order descriptor system (Ao-lambda*Eo,Bo,Co) has
      the same transfer-function matrix as the original system
      (A-lambda*E,B,C).

Specification

          SUBROUTINE TG01ID( JOBOBS, COMPQ, COMPZ, N, M, P, A, LDA, E, LDE,
         $                   B, LDB, C, LDC, Q, LDQ, Z, LDZ, NOBSV, NIUOBS,
         $                   NLBLCK, CTAU, TOL, IWORK, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPZ, JOBOBS
          INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDZ,
         $                   M, N, NIUOBS, NLBLCK, NOBSV, P
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          INTEGER            CTAU( * ), IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, *  ),
         $                   DWORK( * ), E( LDE, * ), Q( LDQ, * ),
         $                   Z( LDZ, * )

Arguments

Mode Parameters

      JOBOBS   CHARACTER*1
              = 'O':  separate both finite and infinite unobservable
                      eigenvalues;
              = 'F':  separate only finite unobservable eigenvalues;
              = 'I':  separate only nonzero finite and infinite
                      unobservable eigenvalues.

      COMPQ   CHARACTER*1
              = 'N':  do not compute Q;
              = 'I':  Q is initialized to the unit matrix, and the
                      orthogonal matrix Q is returned;
              = 'U':  Q must contain an orthogonal matrix Q1 on entry,
                      and the product Q1*Q is returned.

      COMPZ   CHARACTER*1
              = 'N':  do not compute Z;
              = 'I':  Z is initialized to the unit matrix, and the
                      orthogonal matrix Z is returned;
              = 'U':  Z must contain an orthogonal matrix Z1 on entry,
                      and the product Z1*Z is returned.

Input/Output Parameters

      N       (input) INTEGER
              The dimension of the descriptor state vector; also the
              order of square matrices A and E, the number of rows of
              matrix B, and the number of columns of matrix C.  N >= 0.

      M       (input) INTEGER
              The dimension of descriptor system input vector; also the
              number of columns of matrix B.  M >= 0.

      P       (input) INTEGER
              The dimension of descriptor system output vector; also the
              number of rows of matrix 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 N-by-N state matrix A.
              On exit, the leading N-by-N part of this array contains
              the transformed state matrix Q'*A*Z,

                                 ( Ano  *  )
                        Q'*A*Z = (         ) ,
                                 ( 0    Ao )

              where Ao is NOBSV-by-NOBSV and Ano is
              (N-NOBSV)-by-(N-NOBSV).
              If JOBOBS = 'F', the matrix ( Ao ) is in the observability
                                          ( Co )
              staircase form (3).
              If JOBOBS = 'O' or 'I', the submatrix Ao is upper
              triangular.
              If JOBOBS = 'O', the submatrix Ano has the form

                              ( Afno   *  )
                        Ano = (           ) ,
                              (  0   Aino )

              where the NIUOBS-by-NIUOBS matrix Aino is nonsingular and
              upper triangular.
              If JOBOBS = 'I', Ano is nonsingular and upper triangular.

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

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading N-by-N part of this array must
              contain the N-by-N descriptor matrix E.
              On exit, the leading N-by-N part of this array contains
              the transformed state matrix Q'*E*Z,

                                 ( Eno  *  )
                        Q'*E*Z = (         ) ,
                                 ( 0    Eo )

              where Eo is NOBSV-by-NOBSV and Eno is
              (N-NOBSV)-by-(N-NOBSV).
              If JOBOBS = 'O' or 'I', the matrix ( Eo ) is in the
                                                 ( Co )
              observability staircase form (1).
              If JOBOBS = 'F', the submatrix Eo is upper triangular.
              If JOBOBS = 'O', the Eno matrix has the form

                              ( Efno   *  )
                        Eno = (           ) ,
                              (  0   Eino )

              where the NIUOBS-by-NIUOBS matrix Eino is nilpotent
              and the (N-NOBSV-NIUOBS)-by-(N-NOBSV-NIUOBS) matrix Efno
              is nonsingular and upper triangular.
              If JOBOBS = 'F', Eno is nonsingular and upper triangular.

      LDE     INTEGER
              The leading dimension of array E.  LDE >= MAX(1,N).

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

      LDB     INTEGER
              The leading dimension of array B.
              LDB >= MAX(1,N) if M > 0 or LDB >= 1 if M = 0.

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

                      C*Z = (  0   Co ) ,

              where Co is P-by-NOBSV.
              If JOBOBS = 'O' or 'I', the matrix ( Eo ) is in the
                                                 ( Co )
              observability staircase form (1).
              If JOBOBS = 'F', the matrix ( Ao ) is in the observability
                                          ( Co )
              staircase form (3).

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

      Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
              If COMPQ = 'N': Q is not referenced.
              If COMPQ = 'I': on entry, Q need not be set;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix Q,
                              where Q' is the product of transformations
                              which are applied to A, E, and B on
                              the left.
              If COMPQ = 'U': on entry, the leading N-by-N part of this
                              array must contain an orthogonal matrix
                              Qc;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix
                              Qc*Q.

      LDQ     INTEGER
              The leading dimension of array Q.
              LDQ >= 1,        if COMPQ = 'N';
              LDQ >= MAX(1,N), if COMPQ = 'U' or 'I'.

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
              If COMPZ = 'N': Z is not referenced.
              If COMPZ = 'I': on entry, Z need not be set;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix Z,
                              which is the product of transformations
                              applied to A, E, and C on the right.
              If COMPZ = 'U': on entry, the leading N-by-N part of this
                              array must contain an orthogonal matrix
                              Zc;
                              on exit, the leading N-by-N part of this
                              array contains the orthogonal matrix
                              Zc*Z.

      LDZ     INTEGER
              The leading dimension of array Z.
              LDZ >= 1,        if COMPZ = 'N';
              LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.

      NOBSV   (output) INTEGER
              The order of the reduced matrices Ao and Eo, and the
              number of columns of reduced matrix Co; also the order of
              observable part of the pair (C, A-lambda*E).

      NIUOBS  (output) INTEGER
              For JOBOBS = 'O', the order of the reduced matrices
              Aino and Eino; also the number of unobservable
              infinite eigenvalues of the pencil A - lambda*E.
              For JOBOBS = 'F' or 'I', NIUOBS has no significance
              and is set to zero.

      NLBLCK  (output) INTEGER
              For JOBOBS = 'O' or 'I', the number k, of full column rank
                     _
              blocks Ei-1,i in the staircase form of the pencil
              (Eo-lambda*Ao) (see (1) and (2)).
              (    Co      )
              For JOBOBS = 'F', the number k, of full column rank blocks
              _
              Ai-1,i in the staircase form of the pencil (Ao-lambda*Eo)
                                                         (     Co     )
              (see (3) and (4)).

      CTAU    (output) INTEGER array, dimension (N)
              CTAU(i), for i = 1, ..., NLBLCK, is the column dimension
                                            _         _
              of the full column rank block Ei-1,i or Ai-1,i in the
              staircase form (1) or (3) for JOBOBS = 'O' or 'I', or
              for JOBOBS = 'F', respectively.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used in rank determinations when
              transforming (A'-lambda*E',C')'. If the user sets TOL > 0,
              then the given value of TOL is used as a lower bound for
              reciprocal condition numbers in rank determinations; a
              (sub)matrix whose estimated condition number is less than
              1/TOL is considered to be of full rank.  If the user sets
              TOL <= 0, then an implicitly computed, default tolerance,
              defined by  TOLDEF = N*N*EPS,  is used instead, where EPS
              is the machine precision (see LAPACK Library routine
              DLAMCH).  TOL < 1.

Workspace

      IWORK   INTEGER array, dimension (P)

      DWORK   DOUBLE PRECISION array, dimension (MAX(N,2*P))

Error Indicator

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

Method

      The subroutine is based on the dual of the reduction
      algorithms of [1].

References

      [1] A. Varga
          Computation of Irreducible Generalized State-Space
          Realizations.
          Kybernetika, vol. 26, pp. 89-106, 1990.

Numerical Aspects

      The algorithm is numerically backward stable and requires
      0( N**3 )  floating point operations.

Further Comments

      If the system matrices A, E and C are badly scaled, it is
      generally recommendable to scale them with the SLICOT routine
      TG01AD, before calling TG01ID.

Example

Program Text

    *     TG01ID EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          LMAX, NMAX, MMAX, PMAX
          PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          MPMX
          PARAMETER        ( MPMX = MAX( MMAX, PMAX ) )
          INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
          PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = MAX(MMAX,PMAX),
         $                   LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( 1, NMAX, 2*PMAX ) )
    *     .. Local Scalars ..
          CHARACTER*1      COMPQ, COMPZ, JOBOBS
          INTEGER          I, INFO, J, M, N, NOBSV, NIUOBS, NLBLCK, P
          DOUBLE PRECISION TOL
    *     .. Local Arrays ..
          INTEGER          IWORK(MMAX), CTAU(NMAX)
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MPMX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
         $                 Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01ID
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. 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, TOL, JOBOBS
          COMPQ = 'I'
          COMPZ = 'I'
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99988 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99987 ) 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 = 99986 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find the transformed descriptor system (A-lambda E,B,C).
                   CALL TG01ID( JOBOBS, COMPQ, COMPZ, N, M, P, A, LDA,
         $                      E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ,
         $                      NOBSV, NIUOBS, NLBLCK, CTAU, TOL, IWORK,
         $                      DWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) NOBSV, NIUOBS
                      WRITE ( NOUT, FMT = 99985 )
                      WRITE ( NOUT, FMT = 99984 ) ( CTAU(I), I = 1,NLBLCK )
                      WRITE ( NOUT, FMT = 99997 )
                      DO 10 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       10             CONTINUE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 30 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       30             CONTINUE
                      WRITE ( NOUT, FMT = 99992 )
                      DO 40 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
       40             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 50 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
       50             CONTINUE
                      WRITE ( NOUT, FMT = 99990 )
                      DO 60 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
       60             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01ID EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01ID = ',I2)
    99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
    99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (' Dimension of observable part   =', I5/
         $        ' Number of unobservable infinite eigenvalues =', I5)
    99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
    99992 FORMAT (/' The transformed state/output matrix C*Z is ')
    99991 FORMAT (/' The left transformation matrix Q is ')
    99990 FORMAT (/' The right transformation matrix Z is ')
    99989 FORMAT (/' L is out of range.',/' L = ',I5)
    99988 FORMAT (/' N is out of range.',/' N = ',I5)
    99987 FORMAT (/' M is out of range.',/' M = ',I5)
    99986 FORMAT (/' P is out of range.',/' P = ',I5)
    99985 FORMAT (/' The staircase form column dimensions are ' )
    99984 FORMAT (10I5)
          END

Program Data

    TG01ID EXAMPLE PROGRAM DATA
      7    2     3     0.0    O
         2     0     0     0     0     0     0
         0     1     0     0     0     1     0
         2     0     0     2     0     0     0
         0     0     1     0     1     0     1
        -1     1     0    -1     0     1     0
         3     0     0     3     0     0     0
         1     0     1     1     1     0     1
         0     0     0     0     0     0     1
         0     0     0     0     0     0     3
         1     0     0     0     0     1     0
         0     0     0     0     1     0     2
         0     0     0     0     0    -1     0
         0     1     0     0     0     0     0
         0     0     1     1     0     0     0
         1     0
         0    -1
         0     1
         1     0
         0    -1
         0     1
         1     0
         2     0     0     0     0     0     1
         1     0     0     0     0     0     2
         0     0     0     0     0     0     3

Program Results

     TG01ID EXAMPLE PROGRAM RESULTS

     Dimension of observable part   =    3
     Number of unobservable infinite eigenvalues =    1

     The staircase form column dimensions are
        2    1

     The transformed state dynamics matrix Q'*A*Z is
       0.2177   0.2414   0.5742   0.4342   0.0000  -0.4342   0.4666
       0.2022   0.2242   0.5334  -0.2924  -0.7723   0.2924   0.4334
      -0.5892  -0.6533  -1.5540   0.8520  -0.2651  -0.8520  -1.2627
       0.0000   0.0000   0.0000   3.7417   0.3780  -3.7417   0.0000
       0.0000   0.0000   0.0000   0.0000   1.7862   0.0000   0.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   2.0000   0.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000

     The transformed descriptor matrix Q'*E*Z is
       1.0000   0.0000   0.0000   0.4342   0.0000   0.0000   1.8016
       0.0000   1.1937  -0.1496  -0.2924   0.3861   0.5461   0.2819
       0.0000   0.0000  -1.0260   0.8520   0.1325   0.1874  -0.8214
       0.0000   0.0000   0.0000   0.0000  -1.1339  -0.5345   0.0000
       0.0000   0.0000   0.0000   0.0000  -0.1333   0.3770   2.3752
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000
       0.0000   0.0000   0.0000   0.0000  -0.1728   0.4887  -1.8325

     The transformed input/state matrix Q'*B is
       0.4666   0.0000
       0.4334   0.5461
      -1.2627   0.1874
       0.0000  -1.6036
       0.0000  -0.9803
       1.0000   0.0000
       0.0000   0.3665

     The transformed state/output matrix C*Z is
       0.0000   0.0000   0.0000   0.0000   0.0000   2.0000   1.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   1.0000   2.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   3.0000

     The left transformation matrix Q is
       0.0000   0.0000   0.0000   0.0000   0.0000   1.0000   0.0000
       0.0000   0.0000   0.0000   0.0000   0.7917   0.0000  -0.6108
       0.0000   0.5461   0.1874  -0.5345   0.3770   0.0000   0.4887
       0.9008   0.1410  -0.4107   0.0000   0.0000   0.0000   0.0000
       0.0000  -0.5461  -0.1874   0.2673   0.4713   0.0000   0.6108
       0.0000  -0.5461  -0.1874  -0.8018  -0.0943   0.0000  -0.1222
      -0.4342   0.2924  -0.8520   0.0000   0.0000   0.0000   0.0000

     The right transformation matrix Z is
       0.0000   0.0000   0.0000   0.0000   0.0000   1.0000   0.0000
       0.0000  -0.6519   0.2740   0.0000   0.7071   0.0000   0.0000
      -0.4342   0.3491   0.8304   0.0000   0.0000   0.0000   0.0000
       0.0000   0.0000   0.0000  -1.0000   0.0000   0.0000   0.0000
       0.9008   0.1683   0.4003   0.0000   0.0000   0.0000   0.0000
       0.0000   0.6519  -0.2740   0.0000   0.7071   0.0000   0.0000
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000

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