## TG01DD

### Orthogonal reduction of a descriptor system pair (C,A-lambda E) to the RQ-coordinate form

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

Purpose

      To reduce the descriptor system pair (C,A-lambda E) to the
      RQ-coordinate form by computing an orthogonal transformation
      matrix Z such that the transformed descriptor system pair
      (C*Z,A*Z-lambda E*Z) has the descriptor matrix E*Z in an upper
      trapezoidal form.
      The right orthogonal transformations performed to reduce E can
      be optionally accumulated.

Specification

          SUBROUTINE TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC, Z, LDZ,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPZ
          INTEGER            INFO, L, LDA, LDC, LDE, LDWORK, LDZ, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), DWORK( * ),
         $                   E( LDE, * ), Z( LDZ, * )

Arguments

Mode Parameters

      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

      L       (input) INTEGER
              The number of rows of matrices A and E.  L >= 0.

      N       (input) INTEGER
              The number of columns of matrices A, E, and C.  N >= 0.

      P       (input) INTEGER
              The number of rows of matrix C.  P >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading L-by-N part of this array must
              contain the state dynamics matrix A.
              On exit, the leading L-by-N part of this array contains
              the transformed matrix A*Z.

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

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading L-by-N part of this array must
              contain the descriptor matrix E.
              On exit, the leading L-by-N part of this array contains
              the transformed matrix E*Z in upper trapezoidal form,
              i.e.

                       ( E11 )
                 E*Z = (     ) ,  if L >= N ,
                       (  R  )
              or

                 E*Z = ( 0  R ),  if L < N ,

              where R is an MIN(L,N)-by-MIN(L,N) upper triangular
              matrix.

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

      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.

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

      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 Householder
                               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
                               Z1;
                               on exit, the leading N-by-N part of this
                               array contains the orthogonal matrix
                               Z1*Z.

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

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 >= MAX(1, MIN(L,N) + MAX(L,N,P)).
              For optimum performance
              LWORK >= MAX(1, MIN(L,N) + MAX(L,N,P)*NB),
              where NB is the optimal blocksize.

Error Indicator

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

Method

      The routine computes the RQ factorization of E to reduce it
      the upper trapezoidal form.

      The transformations are also applied to the rest of system
      matrices

          A <- A * Z,  C <- C * Z.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

    *     TG01DD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          LMAX, NMAX, PMAX
          PARAMETER        ( LMAX = 20, NMAX = 20, PMAX = 20)
          INTEGER          LDA, LDC, LDE, LDZ
          PARAMETER        ( LDA = LMAX, LDC = PMAX,
         $                   LDE = LMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,PMAX) )
    *     .. Local Scalars ..
          CHARACTER*1      COMPZ
          INTEGER          I, INFO, J, L, N, P
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01DD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) L, N, P
          COMPZ = 'I'
          IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) L
          ELSE
             IF( N.LT.0 .OR. N.GT.NMAX ) THEN
                WRITE ( NOUT, FMT = 99991 ) N
             ELSE
                READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,L )
                READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99990 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find the transformed descriptor system pair
    *              (A-lambda E,B).
                   CALL TG01DD( COMPZ, L, N, P, A, LDA, E, LDE, C, LDC,
         $                      Z, LDZ, DWORK, LDWORK, INFO )
    *
                   IF( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 )
                      DO 10 I = 1, L
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       10             CONTINUE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, L
                         WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99994 )
                      DO 30 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
       30             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
       40             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01DD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01DD = ',I2)
    99997 FORMAT (/' The transformed state dynamics matrix A*Z is ')
    99996 FORMAT (/' The transformed descriptor matrix E*Z is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (/' The transformed input/state matrix C*Z is ')
    99993 FORMAT (/' The right transformation matrix Z is ')
    99992 FORMAT (/' L is out of range.',/' L = ',I5)
    99991 FORMAT (/' N is out of range.',/' N = ',I5)
    99990 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

    TG01DD EXAMPLE PROGRAM DATA
      4    4     2    0.0
        -1     0     0     3
         0     0     1     2
         1     1     0     4
         0     0     0     0
         1     2     0     0
         0     1     0     1
         3     9     6     3
         0     0     2     0
        -1     0     1     0
         0     1    -1     1

Program Results

     TG01DD EXAMPLE PROGRAM RESULTS

     The transformed state dynamics matrix A*Z is
       0.4082   3.0773   0.6030   0.0000
       0.8165   1.7233   0.6030  -1.0000
       2.0412   2.8311   2.4121   0.0000
       0.0000   0.0000   0.0000   0.0000

     The transformed descriptor matrix E*Z is
       0.0000  -0.7385   2.1106   0.0000
       0.0000   0.7385   1.2060   0.0000
       0.0000   0.0000   9.9499  -6.0000
       0.0000   0.0000   0.0000  -2.0000

     The transformed input/state matrix C*Z is
      -0.8165   0.4924  -0.3015  -1.0000
       0.0000   0.7385   1.2060   1.0000

     The right transformation matrix Z is
       0.8165  -0.4924   0.3015   0.0000
      -0.4082  -0.1231   0.9045   0.0000
       0.0000   0.0000   0.0000  -1.0000
       0.4082   0.8616   0.3015   0.0000

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