## TC05AD

### Frequency response of a left/right polynomial matrix representation

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

Purpose

      To evaluate the transfer matrix T(s) of a left polynomial matrix
      representation [T(s) = inv(P(s))*Q(s)] or a right polynomial
      matrix representation [T(s) = Q(s)*inv(P(s))] at any specified
      complex frequency s = SVAL.

      This routine will calculate the standard frequency response
      matrix at frequency omega if SVAL is supplied as (0.0,omega).

Specification

          SUBROUTINE TC05AD( LERI, M, P, SVAL, INDEX, PCOEFF, LDPCO1,
         $                   LDPCO2, QCOEFF, LDQCO1, LDQCO2, RCOND, CFREQR,
         $                   LDCFRE, IWORK, DWORK, ZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         LERI
          INTEGER           INFO, LDCFRE, LDPCO1, LDPCO2, LDQCO1, LDQCO2, M,
         $                  P
          DOUBLE PRECISION  RCOND
          COMPLEX*16        SVAL
    C     .. Array Arguments ..
          INTEGER           INDEX(*), IWORK(*)
          DOUBLE PRECISION  DWORK(*), PCOEFF(LDPCO1,LDPCO2,*),
         $                  QCOEFF(LDQCO1,LDQCO2,*)
          COMPLEX*16        CFREQR(LDCFRE,*), ZWORK(*)

Arguments

Mode Parameters

      LERI    CHARACTER*1
              Indicates whether a left polynomial matrix representation
              or a right polynomial matrix representation is to be used
              to evaluate the transfer matrix as follows:
              = 'L':  A left matrix fraction is input;
              = 'R':  A right matrix fraction is input.

Input/Output Parameters

      M       (input) INTEGER
              The number of system inputs.  M >= 0.

      P       (input) INTEGER
              The number of system outputs.  P >= 0.

      SVAL    (input) COMPLEX*16
              The frequency at which the transfer matrix or the
              frequency respose matrix is to be evaluated.
              For a standard frequency response set the real part
              of SVAL to zero.

      INDEX   (input) INTEGER array, dimension (MAX(M,P))
              If LERI = 'L', INDEX(I), I = 1,2,...,P, must contain the
              maximum degree of the polynomials in the I-th row of the
              denominator matrix P(s) of the given left polynomial
              matrix representation.
              If LERI = 'R', INDEX(I), I = 1,2,...,M, must contain the
              maximum degree of the polynomials in the I-th column of
              the denominator matrix P(s) of the given right polynomial
              matrix representation.

      PCOEFF  (input) DOUBLE PRECISION array, dimension
              (LDPCO1,LDPCO2,kpcoef), where kpcoef = MAX(INDEX(I)) + 1.
              If LERI = 'L' then porm = P, otherwise porm = M.
              The leading porm-by-porm-by-kpcoef part of this array must
              contain the coefficients of the denominator matrix P(s).
              PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
              of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
              LERI = 'L' then iorj = I, otherwise iorj = J.
              Thus for LERI = 'L', P(s) =
              diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
              If LERI = 'R', PCOEFF is modified by the routine but
              restored on exit.

      LDPCO1  INTEGER
              The leading dimension of array PCOEFF.
              LDPCO1 >= MAX(1,P) if LERI = 'L',
              LDPCO1 >= MAX(1,M) if LERI = 'R'.

      LDPCO2  INTEGER
              The second dimension of array PCOEFF.
              LDPCO2 >= MAX(1,P) if LERI = 'L',
              LDPCO2 >= MAX(1,M) if LERI = 'R'.

      QCOEFF  (input) DOUBLE PRECISION array, dimension
              (LDQCO1,LDQCO2,kpcoef)
              If LERI = 'L' then porp = M, otherwise porp = P.
              The leading porm-by-porp-by-kpcoef part of this array must
              contain the coefficients of the numerator matrix Q(s).
              QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
              If LERI = 'R', QCOEFF is modified by the routine but
              restored on exit.

      LDQCO1  INTEGER
              The leading dimension of array QCOEFF.
              LDQCO1 >= MAX(1,P)   if LERI = 'L',
              LDQCO1 >= MAX(1,M,P) if LERI = 'R'.

      LDQCO2  INTEGER
              The second dimension of array QCOEFF.
              LDQCO2 >= MAX(1,M)   if LERI = 'L',
              LDQCO2 >= MAX(1,M,P) if LERI = 'R'.

      RCOND   (output) DOUBLE PRECISION
              The estimated reciprocal of the condition number of the
              denominator matrix P(SVAL).
              If RCOND is nearly zero, SVAL is approximately a system
              pole.

      CFREQR  (output) COMPLEX*16 array, dimension (LDCFRE,MAX(M,P))
              The leading porm-by-porp part of this array contains the
              frequency response matrix T(SVAL).

      LDCFRE  INTEGER
              The leading dimension of array CFREQR.
              LDCFRE >= MAX(1,P)   if LERI = 'L',
              LDCFRE >= MAX(1,M,P) if LERI = 'R'.

Workspace

      IWORK   INTEGER array, dimension (liwork)
              where liwork = P, if LERI = 'L',
                    liwork = M, if LERI = 'R'.

      DWORK   DOUBLE PRECISION array, dimension (ldwork)
              where ldwork = 2*P, if LERI = 'L',
                    ldwork = 2*M, if LERI = 'R'.

      ZWORK   COMPLEX*16 array, dimension (lzwork),
              where lzwork = P*(P+2), if LERI = 'L',
                    lzwork = M*(M+2), if LERI = 'R'.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if P(SVAL) is exactly or nearly singular;
                    no frequency response is calculated.

Method

      The method for a left matrix fraction will be described here;
      right matrix fractions are dealt with by obtaining the dual left
      fraction and calculating its frequency response (see SLICOT
      Library routine TC01OD). The first step is to calculate the
      complex value P(SVAL) of the denominator matrix P(s) at the
      desired frequency SVAL. If P(SVAL) is approximately singular,
      SVAL is approximately a pole of this system and so the frequency
      response matrix T(SVAL) is not calculated; in this case, the
      routine returns with the Error Indicator (INFO) set to 1.
      Otherwise, the complex value Q(SVAL) of the numerator matrix Q(s)
      at frequency SVAL is calculated in a similar way to P(SVAL), and
      the desired response matrix T(SVAL) = inv(P(SVAL))*Q(SVAL) is
      found by solving the corresponding system of complex linear
      equations.

References

      None

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Further Comments

      None

Example

Program Text

    *     TC05AD EXAMPLE PROGRAM TEXT.
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          MMAX, PMAX, KPCMAX
          PARAMETER        ( MMAX = 20, PMAX = 20, KPCMAX = 20 )
          INTEGER          MAXMP
          PARAMETER        ( MAXMP = MAX( MMAX, PMAX ) )
          INTEGER          LDCFRE, LDPCO1, LDPCO2, LDQCO1, LDQCO2
          PARAMETER        ( LDCFRE = MAXMP, LDPCO1 = MAXMP,
         $                   LDPCO2 = MAXMP, LDQCO1 = MAXMP,
         $                   LDQCO2 = MAXMP )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 2*MAXMP )
          INTEGER          LZWORK
          PARAMETER        ( LZWORK = ( MAXMP )*( MAXMP+2 ) )
    *     .. Local Scalars ..
          COMPLEX*16       SVAL
          DOUBLE PRECISION RCOND
          INTEGER          I, INFO, J, K, KPCOEF, M, P, PORM, PORP
          CHARACTER*1      LERI
          LOGICAL          LLERI
    *     .. Local Arrays ..
          COMPLEX*16       CFREQR(LDCFRE,MAXMP), ZWORK(LZWORK)
          DOUBLE PRECISION DWORK(LDWORK), PCOEFF(LDPCO1,LDPCO2,KPCMAX),
         $                 QCOEFF(LDQCO1,LDQCO2,KPCMAX)
          INTEGER          INDEX(MAXMP), IWORK(MAXMP)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         TC05AD
    *     .. 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 = * ) M, P, SVAL, LERI
          LLERI = LSAME( LERI, 'L' )
          IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) M
          ELSE IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) P
          ELSE
             PORM = P
             IF ( .NOT.LLERI ) PORM = M
             READ ( NIN, FMT = * ) ( INDEX(I), I = 1,PORM )
             PORP = M
             IF ( .NOT.LLERI ) PORP = P
             KPCOEF = 0
             DO 20 I = 1, PORM
                KPCOEF = MAX( KPCOEF, INDEX(I) )
       20    CONTINUE
             KPCOEF = KPCOEF + 1
             IF ( KPCOEF.LE.0 .OR. KPCOEF.GT.KPCMAX ) THEN
                WRITE ( NOUT, FMT = 99993 ) KPCOEF
             ELSE
                READ ( NIN, FMT = * )
         $         ( ( ( PCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORM ),
         $                              I = 1,PORM )
                READ ( NIN, FMT = * )
         $         ( ( ( QCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORP ),
         $                              I = 1,PORM )
    *           Find the standard frequency response matrix of left pmr
    *           at 0.5*j.
                CALL TC05AD( LERI, M, P, SVAL, INDEX, PCOEFF, LDPCO1,
         $                   LDPCO2, QCOEFF, LDQCO1, LDQCO2, RCOND, CFREQR,
         $                   LDCFRE, IWORK, DWORK, ZWORK, INFO )
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99997 ) RCOND
                   DO 40 I = 1, PORM
                      WRITE ( NOUT, FMT = 99996 )
         $                  ( CFREQR(I,J), J = 1,PORP )
       40          CONTINUE
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TC05AD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TC05AD = ',I2)
    99997 FORMAT (' RCOND = ',F4.2,//' The frequency response matrix T(SVA',
         $       'L) is ')
    99996 FORMAT (20(' (',F5.2,',',F5.2,') ',:))
    99995 FORMAT (/' M is out of range.',/' M = ',I5)
    99994 FORMAT (/' P is out of range.',/' P = ',I5)
    99993 FORMAT (/' KPCOEF is out of range.',/' KPCOEF = ',I5)
          END

Program Data

     TC05AD EXAMPLE PROGRAM DATA
       2     2     (0.0,0.5)     L
       2     2
       2.0   3.0   1.0
       4.0  -1.0  -1.0
       5.0   7.0  -6.0
       3.0   2.0   2.0
       6.0  -1.0   5.0
       1.0   7.0   5.0
       1.0   1.0   1.0
       4.0   1.0  -1.0

Program Results

     TC05AD EXAMPLE PROGRAM RESULTS

     RCOND = 0.19

     The frequency response matrix T(SVAL) is
     (-0.25,-0.33)  ( 0.26,-0.45)
     (-1.48, 0.35)  (-2.25,-1.11)

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