426 lines
15 KiB
Fortran
426 lines
15 KiB
Fortran
SUBROUTINE TD04AD( ROWCOL, M, P, INDEX, DCOEFF, LDDCOE, UCOEFF,
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$ LDUCO1, LDUCO2, NR, A, LDA, B, LDB, C, LDC, D,
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$ LDD, TOL, IWORK, DWORK, LDWORK, INFO )
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C
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C SLICOT RELEASE 5.0.
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C
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C Copyright (c) 2002-2009 NICONET e.V.
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C
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C This program is free software: you can redistribute it and/or
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C modify it under the terms of the GNU General Public License as
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C published by the Free Software Foundation, either version 2 of
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C the License, or (at your option) any later version.
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C
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C This program is distributed in the hope that it will be useful,
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C but WITHOUT ANY WARRANTY; without even the implied warranty of
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C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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C GNU General Public License for more details.
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C
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C You should have received a copy of the GNU General Public License
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C along with this program. If not, see
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C <http://www.gnu.org/licenses/>.
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C
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C PURPOSE
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C
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C To find a minimal state-space representation (A,B,C,D) for a
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C proper transfer matrix T(s) given as either row or column
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C polynomial vectors over denominator polynomials, possibly with
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C uncancelled common terms.
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C
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C ARGUMENTS
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C
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C Mode Parameters
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C
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C ROWCOL CHARACTER*1
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C Indicates whether the transfer matrix T(s) is given as
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C rows or columns over common denominators as follows:
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C = 'R': T(s) is given as rows over common denominators;
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C = 'C': T(s) is given as columns over common denominators.
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C
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C Input/Output Parameters
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C
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C M (input) INTEGER
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C The number of system inputs. M >= 0.
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C
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C P (input) INTEGER
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C The number of system outputs. P >= 0.
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C
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C INDEX (input) INTEGER array, dimension (porm), where porm = P,
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C if ROWCOL = 'R', and porm = M, if ROWCOL = 'C'.
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C This array must contain the degrees of the denominator
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C polynomials in D(s).
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C
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C DCOEFF (input) DOUBLE PRECISION array, dimension (LDDCOE,kdcoef),
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C where kdcoef = MAX(INDEX(I)) + 1.
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C The leading porm-by-kdcoef part of this array must contain
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C the coefficients of each denominator polynomial.
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C DCOEFF(I,K) is the coefficient in s**(INDEX(I)-K+1) of the
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C I-th denominator polynomial in D(s), where
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C K = 1,2,...,kdcoef.
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C
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C LDDCOE INTEGER
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C The leading dimension of array DCOEFF.
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C LDDCOE >= MAX(1,P) if ROWCOL = 'R';
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C LDDCOE >= MAX(1,M) if ROWCOL = 'C'.
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C
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C UCOEFF (input) DOUBLE PRECISION array, dimension
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C (LDUCO1,LDUCO2,kdcoef)
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C The leading P-by-M-by-kdcoef part of this array must
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C contain the numerator matrix U(s); if ROWCOL = 'C', this
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C array is modified internally but restored on exit, and the
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C remainder of the leading MAX(M,P)-by-MAX(M,P)-by-kdcoef
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C part is used as internal workspace.
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C UCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
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C of polynomial (I,J) of U(s), where K = 1,2,...,kdcoef;
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C if ROWCOL = 'R' then iorj = I, otherwise iorj = J.
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C Thus for ROWCOL = 'R', U(s) =
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C diag(s**INDEX(I))*(UCOEFF(.,.,1)+UCOEFF(.,.,2)/s+...).
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C
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C LDUCO1 INTEGER
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C The leading dimension of array UCOEFF.
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C LDUCO1 >= MAX(1,P) if ROWCOL = 'R';
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C LDUCO1 >= MAX(1,M,P) if ROWCOL = 'C'.
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C
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C LDUCO2 INTEGER
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C The second dimension of array UCOEFF.
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C LDUCO2 >= MAX(1,M) if ROWCOL = 'R';
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C LDUCO2 >= MAX(1,M,P) if ROWCOL = 'C'.
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C
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C NR (output) INTEGER
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C The order of the resulting minimal realization, i.e. the
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C order of the state dynamics matrix A.
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C
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C A (output) DOUBLE PRECISION array, dimension (LDA,N),
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C porm
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C where N = SUM INDEX(I).
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C I=1
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C The leading NR-by-NR part of this array contains the upper
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C block Hessenberg state dynamics matrix A of a minimal
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C realization.
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C
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C LDA INTEGER
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C The leading dimension of array A. LDA >= MAX(1,N).
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C
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C B (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
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C The leading NR-by-M part of this array contains the
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C input/state matrix B of a minimal realization; the
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C remainder of the leading N-by-MAX(M,P) part is used as
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C internal workspace.
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C
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C LDB INTEGER
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C The leading dimension of array B. LDB >= MAX(1,N).
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C
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C C (output) DOUBLE PRECISION array, dimension (LDC,N)
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C The leading P-by-NR part of this array contains the
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C state/output matrix C of a minimal realization; the
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C remainder of the leading MAX(M,P)-by-N part is used as
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C internal workspace.
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C
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C LDC INTEGER
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C The leading dimension of array C. LDC >= MAX(1,M,P).
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C
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C D (output) DOUBLE PRECISION array, dimension (LDD,M),
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C if ROWCOL = 'R', and (LDD,MAX(M,P)) if ROWCOL = 'C'.
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C The leading P-by-M part of this array contains the direct
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C transmission matrix D; if ROWCOL = 'C', the remainder of
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C the leading MAX(M,P)-by-MAX(M,P) part is used as internal
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C workspace.
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C
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C LDD INTEGER
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C The leading dimension of array D.
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C LDD >= MAX(1,P) if ROWCOL = 'R';
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C LDD >= MAX(1,M,P) if ROWCOL = 'C'.
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C
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C Tolerances
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C
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C TOL DOUBLE PRECISION
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C The tolerance to be used in rank determination when
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C transforming (A, B, C). If the user sets TOL > 0, then
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C the given value of TOL is used as a lower bound for the
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C reciprocal condition number (see the description of the
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C argument RCOND in the SLICOT routine MB03OD); a
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C (sub)matrix whose estimated condition number is less than
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C 1/TOL is considered to be of full rank. If the user sets
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C TOL <= 0, then an implicitly computed, default tolerance
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C (determined by the SLICOT routine TB01UD) is used instead.
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C
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C Workspace
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C
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C IWORK INTEGER array, dimension (N+MAX(M,P))
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C On exit, if INFO = 0, the first nonzero elements of
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C IWORK(1:N) return the orders of the diagonal blocks of A.
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C
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C DWORK DOUBLE PRECISION array, dimension (LDWORK)
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C On exit, if INFO = 0, DWORK(1) returns the optimal value
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C of LDWORK.
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C
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C LDWORK INTEGER
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C The length of the array DWORK.
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C LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P)).
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C For optimum performance LDWORK should be larger.
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C
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C Error Indicator
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C
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C INFO INTEGER
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C = 0: successful exit;
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C < 0: if INFO = -i, the i-th argument had an illegal
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C value;
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C > 0: if INFO = i, then i is the first integer for which
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C ABS( DCOEFF(I,1) ) is so small that the calculations
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C would overflow (see SLICOT Library routine TD03AY);
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C that is, the leading coefficient of a polynomial is
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C nearly zero; no state-space representation is
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C calculated.
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C
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C METHOD
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C
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C The method for transfer matrices factorized by rows will be
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C described here: T(s) factorized by columns is dealt with by
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C operating on the dual T'(s). This description for T(s) is
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C actually the left polynomial matrix representation
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C
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C T(s) = inv(D(s))*U(s),
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C
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C where D(s) is diagonal with its (I,I)-th polynomial element of
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C degree INDEX(I). The first step is to check whether the leading
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C coefficient of any polynomial element of D(s) is approximately
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C zero; if so the routine returns with INFO > 0. Otherwise,
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C Wolovich's Observable Structure Theorem is used to construct a
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C state-space representation in observable companion form which
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C is equivalent to the above polynomial matrix representation.
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C The method is particularly easy here due to the diagonal form
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C of D(s). This state-space representation is not necessarily
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C controllable (as D(s) and U(s) are not necessarily relatively
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C left prime), but it is in theory completely observable; however,
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C its observability matrix may be poorly conditioned, so it is
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C treated as a general state-space representation and SLICOT
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C Library routine TB01PD is then called to separate out a minimal
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C realization from this general state-space representation by means
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C of orthogonal similarity transformations.
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C
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C REFERENCES
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C
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C [1] Patel, R.V.
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C Computation of Minimal-Order State-Space Realizations and
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C Observability Indices using Orthogonal Transformations.
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C Int. J. Control, 33, pp. 227-246, 1981.
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C
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C [2] Wolovich, W.A.
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C Linear Multivariable Systems, (Theorem 4.3.3).
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C Springer-Verlag, 1974.
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C
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C NUMERICAL ASPECTS
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C 3
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C The algorithm requires 0(N ) operations.
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C
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C CONTRIBUTOR
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C
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C V. Sima, Katholieke Univ. Leuven, Belgium, March 1998.
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C Supersedes Release 3.0 routine TD01OD.
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C
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C REVISIONS
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C
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C -
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C
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C KEYWORDS
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C
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C Controllability, elementary polynomial operations, minimal
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C realization, polynomial matrix, state-space representation,
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C transfer matrix.
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C
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C ******************************************************************
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C
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C .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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C .. Scalar Arguments ..
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CHARACTER ROWCOL
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INTEGER INFO, LDA, LDB, LDC, LDD, LDDCOE, LDUCO1,
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$ LDUCO2, LDWORK, M, NR, P
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DOUBLE PRECISION TOL
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C .. Array Arguments ..
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INTEGER INDEX(*), IWORK(*)
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DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
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$ DCOEFF(LDDCOE,*), DWORK(*),
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$ UCOEFF(LDUCO1,LDUCO2,*)
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C .. Local Scalars ..
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LOGICAL LROCOC, LROCOR
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INTEGER I, J, JSTOP, K, KDCOEF, MPLIM, MWORK, N, PWORK
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C .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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C .. External Subroutines ..
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EXTERNAL DLASET, DSWAP, TB01PD, TB01XD, TD03AY, XERBLA
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C .. Intrinsic Functions ..
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INTRINSIC MAX
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C .. Executable Statements ..
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C
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INFO = 0
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LROCOR = LSAME( ROWCOL, 'R' )
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LROCOC = LSAME( ROWCOL, 'C' )
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MPLIM = MAX( 1, M, P )
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C
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C Test the input scalar arguments.
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C
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IF( .NOT.LROCOR .AND. .NOT.LROCOC ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( P.LT.0 ) THEN
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INFO = -3
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ELSE IF( ( LROCOR .AND. LDDCOE.LT.MAX( 1, P ) ) .OR.
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$ ( LROCOC .AND. LDDCOE.LT.MAX( 1, M ) ) ) THEN
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INFO = -6
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ELSE IF( ( LROCOR .AND. LDUCO1.LT.MAX( 1, P ) ) .OR.
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$ ( LROCOC .AND. LDUCO1.LT.MPLIM ) ) THEN
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INFO = -8
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ELSE IF( ( LROCOR .AND. LDUCO2.LT.MAX( 1, M ) ) .OR.
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$ ( LROCOC .AND. LDUCO2.LT.MPLIM ) ) THEN
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INFO = -9
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END IF
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C
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N = 0
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IF ( INFO.EQ.0 ) THEN
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IF ( LROCOR ) THEN
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C
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C Initialization for T(s) given as rows over common
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C denominators.
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C
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PWORK = P
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MWORK = M
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ELSE
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C
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C Initialization for T(s) given as columns over common
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C denominators.
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C
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PWORK = M
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MWORK = P
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END IF
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C
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C Calculate N, the order of the resulting state-space
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C representation.
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C
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KDCOEF = 0
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C
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DO 10 I = 1, PWORK
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KDCOEF = MAX( KDCOEF, INDEX(I) )
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N = N + INDEX(I)
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10 CONTINUE
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C
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KDCOEF = KDCOEF + 1
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C
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IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -12
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -14
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ELSE IF( LDC.LT.MPLIM ) THEN
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INFO = -16
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ELSE IF( ( LROCOR .AND. LDD.LT.MAX( 1, P ) ) .OR.
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$ ( LROCOC .AND. LDD.LT.MPLIM ) ) THEN
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INFO = -18
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ELSE IF( LDWORK.LT.MAX( 1, N + MAX( N, 3*M, 3*P ) ) ) THEN
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INFO = -22
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END IF
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END IF
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C
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IF ( INFO.NE.0 ) THEN
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C
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C Error return.
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C
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CALL XERBLA( 'TD04AD', -INFO )
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RETURN
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END IF
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C
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C Quick return if possible.
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C
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IF ( MAX( N, M, P ).EQ.0 ) THEN
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NR = 0
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DWORK(1) = ONE
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RETURN
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END IF
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C
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IF ( LROCOC ) THEN
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C
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C Initialize the remainder of the leading
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C MPLIM-by-MPLIM-by-KDCOEF part of U(s) to zero.
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C
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IF ( P.LT.M ) THEN
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C
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DO 20 K = 1, KDCOEF
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CALL DLASET( 'Full', M-P, MPLIM, ZERO, ZERO,
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$ UCOEFF(P+1,1,K), LDUCO1 )
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20 CONTINUE
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C
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ELSE IF ( P.GT.M ) THEN
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C
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DO 30 K = 1, KDCOEF
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CALL DLASET( 'Full', MPLIM, P-M, ZERO, ZERO,
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$ UCOEFF(1,M+1,K), LDUCO1 )
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30 CONTINUE
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C
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END IF
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C
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IF ( MPLIM.NE.1 ) THEN
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C
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C Non-scalar T(s) factorized by columns: transpose it (i.e.
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C U(s)).
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C
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JSTOP = MPLIM - 1
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C
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DO 50 K = 1, KDCOEF
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C
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DO 40 J = 1, JSTOP
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CALL DSWAP( MPLIM-J, UCOEFF(J+1,J,K), 1,
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$ UCOEFF(J,J+1,K), LDUCO1 )
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40 CONTINUE
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C
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50 CONTINUE
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C
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END IF
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END IF
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C
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C Construct non-minimal state-space representation (by Wolovich's
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C Structure Theorem) which has transfer matrix T(s) or T'(s) as
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C appropriate ...
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C
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CALL TD03AY( MWORK, PWORK, INDEX, DCOEFF, LDDCOE, UCOEFF, LDUCO1,
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$ LDUCO2, N, A, LDA, B, LDB, C, LDC, D, LDD, INFO )
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IF ( INFO.GT.0 )
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$ RETURN
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C
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C and then separate out a minimal realization from this.
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C
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C Workspace: need N + MAX(N, 3*MWORK, 3*PWORK).
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C
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CALL TB01PD( 'Minimal', 'Scale', N, MWORK, PWORK, A, LDA, B, LDB,
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$ C, LDC, NR, TOL, IWORK, DWORK, LDWORK, INFO )
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C
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IF ( LROCOC ) THEN
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C
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C If T(s) originally factorized by columns, find dual of minimal
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C state-space representation, and reorder the rows and columns
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C to get an upper block Hessenberg state dynamics matrix.
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C
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K = IWORK(1)+IWORK(2)-1
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CALL TB01XD( 'D', NR, MWORK, PWORK, K, NR-1, A, LDA, B, LDB,
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$ C, LDC, D, LDD, INFO )
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IF ( MPLIM.NE.1 ) THEN
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C
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C Also, retranspose U(s) if this is non-scalar.
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C
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DO 70 K = 1, KDCOEF
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C
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DO 60 J = 1, JSTOP
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CALL DSWAP( MPLIM-J, UCOEFF(J+1,J,K), 1,
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$ UCOEFF(J,J+1,K), LDUCO1 )
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60 CONTINUE
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C
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70 CONTINUE
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C
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END IF
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END IF
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C
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RETURN
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C *** Last line of TD04AD ***
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END
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