SUBROUTINE TC05AD( LERI, M, P, SVAL, INDEX, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, RCOND, CFREQR,
$ LDCFRE, IWORK, DWORK, ZWORK, INFO )
C
C PURPOSE
C
C To evaluate the transfer matrix T(s) of a left polynomial matrix
C representation [T(s) = inv(P(s))*Q(s)] or a right polynomial
C matrix representation [T(s) = Q(s)*inv(P(s))] at any specified
C complex frequency s = SVAL.
C
C This routine will calculate the standard frequency response
C matrix at frequency omega if SVAL is supplied as (0.0,omega).
C
C ARGUMENTS
C
C Mode Parameters
C
C LERI CHARACTER*1
C Indicates whether a left polynomial matrix representation
C or a right polynomial matrix representation is to be used
C to evaluate the transfer matrix as follows:
C = 'L': A left matrix fraction is input;
C = 'R': A right matrix fraction is input.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C P (input) INTEGER
C The number of system outputs. P >= 0.
C
C SVAL (input) COMPLEX*16
C The frequency at which the transfer matrix or the
C frequency respose matrix is to be evaluated.
C For a standard frequency response set the real part
C of SVAL to zero.
C
C INDEX (input) INTEGER array, dimension (MAX(M,P))
C If LERI = 'L', INDEX(I), I = 1,2,...,P, must contain the
C maximum degree of the polynomials in the I-th row of the
C denominator matrix P(s) of the given left polynomial
C matrix representation.
C If LERI = 'R', INDEX(I), I = 1,2,...,M, must contain the
C maximum degree of the polynomials in the I-th column of
C the denominator matrix P(s) of the given right polynomial
C matrix representation.
C
C PCOEFF (input) DOUBLE PRECISION array, dimension
C (LDPCO1,LDPCO2,kpcoef), where kpcoef = MAX(INDEX(I)) + 1.
C If LERI = 'L' then porm = P, otherwise porm = M.
C The leading porm-by-porm-by-kpcoef part of this array must
C contain the coefficients of the denominator matrix P(s).
C PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
C of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
C LERI = 'L' then iorj = I, otherwise iorj = J.
C Thus for LERI = 'L', P(s) =
C diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
C If LERI = 'R', PCOEFF is modified by the routine but
C restored on exit.
C
C LDPCO1 INTEGER
C The leading dimension of array PCOEFF.
C LDPCO1 >= MAX(1,P) if LERI = 'L',
C LDPCO1 >= MAX(1,M) if LERI = 'R'.
C
C LDPCO2 INTEGER
C The second dimension of array PCOEFF.
C LDPCO2 >= MAX(1,P) if LERI = 'L',
C LDPCO2 >= MAX(1,M) if LERI = 'R'.
C
C QCOEFF (input) DOUBLE PRECISION array, dimension
C (LDQCO1,LDQCO2,kpcoef)
C If LERI = 'L' then porp = M, otherwise porp = P.
C The leading porm-by-porp-by-kpcoef part of this array must
C contain the coefficients of the numerator matrix Q(s).
C QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
C If LERI = 'R', QCOEFF is modified by the routine but
C restored on exit.
C
C LDQCO1 INTEGER
C The leading dimension of array QCOEFF.
C LDQCO1 >= MAX(1,P) if LERI = 'L',
C LDQCO1 >= MAX(1,M,P) if LERI = 'R'.
C
C LDQCO2 INTEGER
C The second dimension of array QCOEFF.
C LDQCO2 >= MAX(1,M) if LERI = 'L',
C LDQCO2 >= MAX(1,M,P) if LERI = 'R'.
C
C RCOND (output) DOUBLE PRECISION
C The estimated reciprocal of the condition number of the
C denominator matrix P(SVAL).
C If RCOND is nearly zero, SVAL is approximately a system
C pole.
C
C CFREQR (output) COMPLEX*16 array, dimension (LDCFRE,MAX(M,P))
C The leading porm-by-porp part of this array contains the
C frequency response matrix T(SVAL).
C
C LDCFRE INTEGER
C The leading dimension of array CFREQR.
C LDCFRE >= MAX(1,P) if LERI = 'L',
C LDCFRE >= MAX(1,M,P) if LERI = 'R'.
C
C Workspace
C
C IWORK INTEGER array, dimension (liwork)
C where liwork = P, if LERI = 'L',
C liwork = M, if LERI = 'R'.
C
C DWORK DOUBLE PRECISION array, dimension (ldwork)
C where ldwork = 2*P, if LERI = 'L',
C ldwork = 2*M, if LERI = 'R'.
C
C ZWORK COMPLEX*16 array, dimension (lzwork),
C where lzwork = P*(P+2), if LERI = 'L',
C lzwork = M*(M+2), if LERI = 'R'.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 1: if P(SVAL) is exactly or nearly singular;
C no frequency response is calculated.
C
C METHOD
C
C The method for a left matrix fraction will be described here;
C right matrix fractions are dealt with by obtaining the dual left
C fraction and calculating its frequency response (see SLICOT
C Library routine TC01OD). The first step is to calculate the
C complex value P(SVAL) of the denominator matrix P(s) at the
C desired frequency SVAL. If P(SVAL) is approximately singular,
C SVAL is approximately a pole of this system and so the frequency
C response matrix T(SVAL) is not calculated; in this case, the
C routine returns with the Error Indicator (INFO) set to 1.
C Otherwise, the complex value Q(SVAL) of the numerator matrix Q(s)
C at frequency SVAL is calculated in a similar way to P(SVAL), and
C the desired response matrix T(SVAL) = inv(P(SVAL))*Q(SVAL) is
C found by solving the corresponding system of complex linear
C equations.
C
C REFERENCES
C
C None
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(N ) operations.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Dec. 1996.
C Supersedes Release 2.0 routine TC01AD by T.W.C.Williams, Kingston
C Polytechnic, United Kingdom, March 1982.
C
C REVISIONS
C
C February 22, 1998 (changed the name of TC01MD).
C
C KEYWORDS
C
C Coprime matrix fraction, elementary polynomial operations,
C polynomial matrix, state-space representation, transfer matrix.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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(*)
C .. Local Scalars ..
LOGICAL LLERI
INTEGER I, IZWORK, IJ, INFO1, J, K, KPCOEF, LDZWOR,
$ MAXIND, MINMP, MPLIM, MWORK, PWORK
DOUBLE PRECISION CNORM
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL DLAMCH, LSAME, ZLANGE
C .. External Subroutines ..
EXTERNAL TC01OD, XERBLA, ZCOPY, ZGECON, ZGETRF, ZGETRS,
$ ZSWAP
C .. Intrinsic Functions ..
INTRINSIC DCMPLX, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
LLERI = LSAME( LERI, 'L' )
MPLIM = MAX( M, P )
C
C Test the input scalar arguments.
C
IF( .NOT.LLERI .AND. .NOT.LSAME( LERI, 'R' ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( ( LLERI .AND. LDPCO1.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LLERI .AND. LDPCO1.LT.MAX( 1, M ) ) ) THEN
INFO = -7
ELSE IF( ( LLERI .AND. LDPCO2.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LLERI .AND. LDPCO2.LT.MAX( 1, M ) ) ) THEN
INFO = -8
ELSE IF( ( LLERI .AND. LDQCO1.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LLERI .AND. LDQCO1.LT.MAX( 1, M, P ) ) ) THEN
INFO = -10
ELSE IF( ( LLERI .AND. LDQCO2.LT.MAX( 1, M ) ) .OR.
$ ( .NOT.LLERI .AND. LDQCO2.LT.MAX( 1, MPLIM ) ) ) THEN
INFO = -11
ELSE IF( ( LLERI .AND. LDCFRE.LT.MAX( 1, P ) ) .OR.
$ ( .NOT.LLERI .AND. LDCFRE.LT.MAX( 1, MPLIM ) ) ) THEN
INFO = -14
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TC05AD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( M.EQ.0 .OR. P.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
C
IF ( LLERI ) THEN
C
C Initialization for left matrix fraction.
C
PWORK = P
MWORK = M
ELSE
C
C Initialization for right matrix fraction: obtain dual system.
C
PWORK = M
MWORK = P
IF ( MPLIM.GT.1 )
$ CALL TC01OD( 'R', M, P, KPCOEF, PCOEFF, LDPCO1, LDPCO2,
$ QCOEFF, LDQCO1, LDQCO2, INFO )
END IF
C
LDZWOR = PWORK
IZWORK = LDZWOR*LDZWOR + 1
MAXIND = 0
C
DO 10 I = 1, PWORK
IF ( INDEX(I).GT.MAXIND ) MAXIND = INDEX(I)
10 CONTINUE
C
KPCOEF = MAXIND + 1
C
C Calculate the complex denominator matrix P(SVAL), row by row.
C
DO 50 I = 1, PWORK
IJ = I
C
DO 20 J = 1, PWORK
ZWORK(IJ) = DCMPLX( PCOEFF(I,J,1), ZERO )
IJ = IJ + PWORK
20 CONTINUE
C
C Possibly non-constant row: finish evaluating it.
C
DO 40 K = 2, INDEX(I) + 1
C
IJ = I
C
DO 30 J = 1, PWORK
ZWORK(IJ) = ( SVAL*ZWORK(IJ) ) +
$ DCMPLX( PCOEFF(I,J,K), ZERO )
IJ = IJ + PWORK
30 CONTINUE
C
40 CONTINUE
C
50 CONTINUE
C
C Check if this P(SVAL) is singular: if so, don't compute T(SVAL).
C Note that DWORK is not actually referenced in ZLANGE routine.
C
CNORM = ZLANGE( '1-norm', PWORK, PWORK, ZWORK, LDZWOR, DWORK )
C
CALL ZGETRF( PWORK, PWORK, ZWORK, LDZWOR, IWORK, INFO )
C
IF ( INFO.GT.0 ) THEN
C
C Singular matrix. Set INFO and RCOND for error return.
C
INFO = 1
RCOND = ZERO
ELSE
C
C Estimate the reciprocal condition of P(SVAL).
C Workspace: ZWORK: PWORK*PWORK + 2*PWORK, DWORK: 2*PWORK.
C
CALL ZGECON( '1-norm', PWORK, ZWORK, LDZWOR, CNORM, RCOND,
$ ZWORK(IZWORK), DWORK, INFO )
C
IF ( RCOND.LE.DLAMCH( 'Epsilon' ) ) THEN
C
C Nearly singular matrix. Set INFO for error return.
C
INFO = 1
ELSE
C
C Calculate the complex numerator matrix Q(SVAL), row by row.
C
DO 90 I = 1, PWORK
C
DO 60 J = 1, MWORK
CFREQR(I,J) = DCMPLX( QCOEFF(I,J,1), ZERO )
60 CONTINUE
C
C Possibly non-constant row: finish evaluating it.
C
DO 80 K = 2, INDEX(I) + 1
C
DO 70 J = 1, MWORK
CFREQR(I,J) = ( SVAL*CFREQR(I,J) ) +
$ DCMPLX( QCOEFF(I,J,K), ZERO )
70 CONTINUE
C
80 CONTINUE
C
90 CONTINUE
C
C Now calculate frequency response T(SVAL).
C
CALL ZGETRS( 'No transpose', PWORK, MWORK, ZWORK, LDZWOR,
$ IWORK, CFREQR, LDCFRE, INFO )
END IF
END IF
C
C For right matrix fraction, return to original (dual of the dual)
C system.
C
IF ( ( .NOT.LLERI ) .AND. ( MPLIM.NE.1 ) ) THEN
CALL TC01OD( 'L', MWORK, PWORK, KPCOEF, PCOEFF, LDPCO1,
$ LDPCO2, QCOEFF, LDQCO1, LDQCO2, INFO1 )
C
IF ( INFO.EQ.0 ) THEN
C
C Also, transpose T(SVAL) here if this was successfully
C calculated.
C
MINMP = MIN( M, P )
C
DO 100 J = 1, MPLIM
IF ( J.LT.MINMP ) THEN
CALL ZSWAP( MINMP-J, CFREQR(J+1,J), 1, CFREQR(J,J+1),
$ LDCFRE )
ELSE IF ( J.GT.P ) THEN
CALL ZCOPY( P, CFREQR(1,J), 1, CFREQR(J,1), LDCFRE )
ELSE IF ( J.GT.M ) THEN
CALL ZCOPY( M, CFREQR(J,1), LDCFRE, CFREQR(1,J), 1 )
END IF
100 CONTINUE
C
END IF
END IF
C
RETURN
C *** Last line of TC05AD ***
END