### 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.

```
```  None
```
Example

Program Text

```*     TC05AD EXAMPLE PROGRAM TEXT.
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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 ..
*     .. 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)
```