TG01JY

Irreducible descriptor representation (blocked version)

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

Purpose

  To find a reduced (controllable, observable, or irreducible)
  descriptor representation (Ar-lambda*Er,Br,Cr) for an original
  descriptor representation (A-lambda*E,B,C).
  The pencil Ar-lambda*Er is in an upper block Hessenberg form, with
  either Ar or Er upper triangular.

Specification
      SUBROUTINE TG01JY( JOB, SYSTYP, EQUIL, CKSING, RESTOR, N, M, P, A,
     $                   LDA, E, LDE, B, LDB, C, LDC, NR, INFRED, TOL,
     $                   IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         CKSING, EQUIL, JOB, RESTOR, SYSTYP
      INTEGER           INFO, LDA, LDB, LDC, LDE, LDWORK, M, N, NR, P
C     .. Array Arguments ..
      INTEGER           INFRED(*), IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
     $                  E(LDE,*), TOL(*)

Arguments

Mode Parameters

  JOB     CHARACTER*1
          Indicates whether the user wishes to remove the
          uncontrollable and/or unobservable parts as follows:
          = 'I':  Remove both the uncontrollable and unobservable
                  parts to get an irreducible descriptor
                  representation;
          = 'C':  Remove the uncontrollable part only to get a
                  controllable descriptor representation;
          = 'O':  Remove the unobservable part only to get an
                  observable descriptor representation.

  SYSTYP  CHARACTER*1
          Indicates the type of descriptor system algorithm
          to be applied according to the assumed
          transfer-function matrix as follows:
          = 'R':  Rational transfer-function matrix;
          = 'S':  Proper (standard) transfer-function matrix;
          = 'P':  Polynomial transfer-function matrix.

  EQUIL   CHARACTER*1
          Specifies whether the user wishes to preliminarily scale
          the system (A-lambda*E,B,C) as follows:
          = 'S':  Perform scaling;
          = 'N':  Do not perform scaling.

  CKSING  CHARACTER*1
          Specifies whether the user wishes to check if the pencil
          (A-lambda*E) is singular as follows:
          = 'C':  Check singularity;
          = 'N':  Do not check singularity.
          If the pencil is singular, the reduced system computed for
          CKSING = 'N' can be wrong.

  RESTOR  CHARACTER*1
          Specifies whether the user wishes to save the system
          matrices before each phase and restore them if no order
          reduction took place as follows:
          = 'R':  Save and restore;
          = 'N':  Do not save the matrices.

Input/Output Parameters
  N       (input) INTEGER
          The dimension of the descriptor state vector; also the
          order of square matrices A and E, the number of rows of
          matrix B, and the number of columns of matrix C.  N >= 0.

  M       (input) INTEGER
          The dimension of descriptor system input vector; also the
          number of columns of matrix B.  M >= 0.

  P       (input) INTEGER
          The dimension of descriptor system output vector; also the
          number of rows of matrix C.  P >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the original state matrix A.
          On exit, the leading NR-by-NR part of this array contains
          the reduced order state matrix Ar of an irreducible,
          controllable, or observable realization for the original
          system, depending on the value of JOB, JOB = 'I',
          JOB = 'C', or JOB = 'O', respectively.
          The matrix Ar is upper triangular if SYSTYP = 'P'.
          If SYSTYP = 'S' and JOB = 'C', the matrix [Br Ar]
          is in a controllable staircase form (see SLICOT Library
          routine TG01HD).
          If SYSTYP = 'S' and JOB = 'I' or 'O', the matrix ( Ar )
                                                           ( Cr )
          is in an observable staircase form (see TG01HD).
          The resulting Ar has INFRED(5) nonzero sub-diagonals.
          The block structure of staircase forms is contained
          in the leading INFRED(7) elements of IWORK.

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

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N part of this array must
          contain the original descriptor matrix E.
          On exit, the leading NR-by-NR part of this array contains
          the reduced order descriptor matrix Er of an irreducible,
          controllable, or observable realization for the original
          system, depending on the value of JOB, JOB = 'I',
          JOB = 'C', or JOB = 'O', respectively.
          The resulting Er has INFRED(6) nonzero sub-diagonals.
          If at least for one k = 1,...,4, INFRED(k) >= 0, then the
          resulting Er is structured being either upper triangular
          or block Hessenberg, in accordance to the last
          performed order reduction phase (see METHOD).
          The block structure of staircase forms is contained
          in the leading INFRED(7) elements of IWORK.

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

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M),
          if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
          On entry, the leading N-by-M part of this array must
          contain the original input matrix B; if JOB = 'I',
          or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
          part is used as internal workspace.
          On exit, the leading NR-by-M part of this array contains
          the reduced input matrix Br of an irreducible,
          controllable, or observable realization for the original
          system, depending on the value of JOB, JOB = 'I',
          JOB = 'C', or JOB = 'O', respectively.
          If JOB = 'C', only the first IWORK(1) rows of B are
          nonzero.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the original output matrix C; if JOB = 'I',
          or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
          part is used as internal workspace.
          On exit, the leading P-by-NR part of this array contains
          the transformed state/output matrix Cr of an irreducible,
          controllable, or observable realization for the original
          system, depending on the value of JOB, JOB = 'I',
          JOB = 'C', or JOB = 'O', respectively.
          If JOB = 'I', or JOB = 'O', only the last IWORK(1) columns
          (in the first NR columns) of C are nonzero.

  LDC     INTEGER
          The leading dimension of array C.
          LDC >= MAX(1,M,P) if N > 0.
          LDC >= 1          if N = 0.

  NR      (output) INTEGER
          The order of the reduced descriptor representation
          (Ar-lambda*Er,Br,Cr) of an irreducible, controllable,
          or observable realization for the original system,
          depending on JOB = 'I', JOB = 'C', or JOB = 'O',
          respectively.

  INFRED  (output) INTEGER array, dimension 7
          This array contains information on performed reduction
          and on structure of resulting system matrices as follows:
          INFRED(k) >= 0 (k = 1, 2, 3, or 4) if Phase k of reduction
                         (see METHOD) has been performed. In this
                         case, INFRED(k) is the achieved order
                         reduction in Phase k.
          INFRED(k) < 0  (k = 1, 2, 3, or 4) if Phase k was not
                         performed.
          INFRED(5)  -   the number of nonzero sub-diagonals of A.
          INFRED(6)  -   the number of nonzero sub-diagonals of E.
          INFRED(7)  -   the number of blocks in the resulting
                         staircase form at last performed reduction
                         phase. The block dimensions are contained
                         in the first INFRED(7) elements of IWORK.

Tolerances
  TOL     DOUBLE PRECISION array, dimension 3
          TOL(1) is the tolerance to be used in rank determinations
          when transforming (A-lambda*E,B,C). If the user sets
          TOL(1) > 0, then the given value of TOL(1) is used as a
          lower bound for reciprocal condition numbers in rank
          determinations; a (sub)matrix whose estimated condition
          number is less than 1/TOL(1) is considered to be of full
          rank.  If the user sets TOL(1) <= 0, then an implicitly
          computed, default tolerance, defined by TOLDEF1 = N*N*EPS,
          is used instead, where EPS is the machine precision (see
          LAPACK Library routine DLAMCH).  TOL(1) < 1.
          TOL(2) is the tolerance to be used for checking pencil
          singularity when CKSING = 'C', or singularity of the
          matrices A and E when CKSING = 'N'. If the user sets
          TOL(2) > 0, then the given value of TOL(2) is used.
          If the user sets TOL(2) <= 0, then an implicitly
          computed, default tolerance, defined by  TOLDEF2 = 10*EPS,
          is used instead.  TOL(2) < 1.
          TOL(3) is the threshold value for magnitude of the matrix
          elements, if EQUIL = 'S': elements with magnitude less
          than or equal to TOL(3) are ignored for scaling. If the
          user sets TOL(3) >= 0, then the given value of TOL(3) is
          used. If the user sets TOL(3) < 0, then an implicitly
          computed, default threshold, defined by  THRESH = c*EPS,
          where c = MAX(norm_1(A,E,B,C)) is used instead.
          TOL(3) = 0 is not always a good choice.  TOL(3) < 1.
          TOL(3) is not used if EQUIL = 'N'.

Workspace
  IWORK   INTEGER array, dimension (2*N+MAX(M,P))
          On exit, if INFO = 0, the leading INFRED(7) elements of
          IWORK contain the orders of the diagonal blocks of
          Ar-lambda*Er.

  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,x,y,8*N), if EQUIL = 'S',
          LDWORK >= MAX(1,x,y),     if EQUIL = 'N',
          where x = MAX(2*(z+MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'R'
                x = MAX(  2*(MAX(M,P)+N-1),N*N+4*N), if RESTOR = 'N'
                y = 2*N*N+10*N+MAX(N,23), if CKSING = 'C',
                y = 0,                    if CKSING = 'N',
                z = 2*N*N+N*M+N*P, if JOB  = 'I',
                z = 0,             if JOB <> 'I'.
          For good performance, LDWORK should be generally larger.
          If RESTOR = 'R', or
          LDWORK >= MAX(1,2*N*N+N*M+N*P+2*(MAX(M,P)+N-1),
          more accurate results are to be expected by considering
          only those reductions phases (see METHOD), where effective
          order reduction occurs. This is achieved by saving the
          system matrices before each phase and restoring them if no
          order reduction took place. Actually, if JOB = 'I' and
          RESTOR = 'N', then the saved matrices are those obtained
          after orthogonally triangularizing the matrix A (if
          SYSTYP = 'R' or 'P'), or the matrix E (if SYSTYP = 'R'
          or 'S').

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA. The optimal workspace includes the
          extra space for improving the accuracy.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the given pencil A - lambda*E is numerically
                singular and the reduced system is not computed.
                This error can be returned only if CKSING = 'C'.

Method
  The subroutine is based on the reduction algorithms of [1], but
  with a different ordering of the phases.
  The order reduction is performed in 4 phases:
  Phase 1: Eliminate all infinite and finite nonzero uncontrollable
           eigenvalues. The resulting matrix ( Br Er ) is in a
           controllable staircase form (see TG01HD), and Ar is
           upper triangular.
           This phase is performed if JOB = 'I' or 'C' and
           SYSTYP = 'R' or 'P'.
  Phase 2: Eliminate all infinite and finite nonzero unobservable
           eigenvalues. The resulting matrix ( Er ) is in an
                                             ( Cr )
           observable staircase form (see SLICOT Library routine
           TG01ID), and Ar is upper triangular.
           This phase is performed if JOB = 'I' or 'O' and
           SYSTYP = 'R' or 'P'.
  Phase 3: Eliminate all finite uncontrollable eigenvalues.
           The resulting matrix ( Br Ar ) is in a controllable
           staircase form (see TG01HD), and Er is upper triangular.
           This phase is performed if JOB = 'I' or 'C' and
           SYSTYP = 'R' or 'S'.
  Phase 4: Eliminate all finite unobservable eigenvalues.
           The resulting matrix ( Ar ) is in an observable
                                ( Cr )
           staircase form (see TG01ID), and Er is upper triangular.
           This phase is performed if JOB = 'I' or 'O' and
           SYSTYP = 'R' or 'S'.
  The routine checks the singularity of the matrices A and/or E
  (depending on JOB and SYSTYP) and skips the unnecessary phases.
  See FURTHER COMMENTS.

References
  [1] A. Varga
      Computation of Irreducible Generalized State-Space
      Realizations.
      Kybernetika, vol. 26, pp. 89-106, 1990.

Numerical Aspects
  The algorithm is numerically backward stable and requires
  0( N**3 )  floating point operations.

Further Comments
  If the pencil A-lambda*E has no zero eigenvalues, then an
  irreducible realization is computed skipping Phases 3 and 4
  (equivalent to setting: JOB = 'I' and SYSTYP = 'P').

Example

Program Text

*     TG01JY EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDE
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
     $                   LDE = NMAX )
      INTEGER          LDWORK, LIWORK
      PARAMETER        ( LDWORK = 2*NMAX*NMAX + 
     $                            MAX( 2*( NMAX*( NMAX + MMAX + PMAX ) +
     $                                 MAX( MMAX, PMAX ) + NMAX - 1 ),
     $                                 10*NMAX + MAX( NMAX, 23 ) ),
     $                   LIWORK = 2*NMAX + MAX( MMAX, PMAX ) )
*     .. Local Scalars ..
      CHARACTER        CKSING, EQUIL, JOB, RESTOR, SYSTYP
      INTEGER          I, INFO, J, M, N, NR, P
*     .. Local Arrays ..
      INTEGER          INFRED(7), IWORK(LIWORK)
      DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
     $                 DWORK(LDWORK), E(LDE,NMAX), TOL(3)

*     .. External Subroutines ..
      EXTERNAL         TG01JY
*     .. 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 = * ) N, M, P, TOL(1), TOL(2), TOL(3), JOB,
     $                      SYSTYP, EQUIL, CKSING, RESTOR
      IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99988 ) N
      ELSE
         READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
         READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
         IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
            WRITE ( NOUT, FMT = 99987 ) M
         ELSE
            READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
            IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
               WRITE ( NOUT, FMT = 99986 ) P
            ELSE
               READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Find the irreducible descriptor system (Ar-lambda Er,Br,Cr).
               CALL TG01JY( JOB, SYSTYP, EQUIL, CKSING, RESTOR, N, M, P,
     $                      A, LDA, E, LDE, B, LDB, C, LDC, NR, INFRED,
     $                      TOL, IWORK, DWORK, LDWORK, INFO )
*
               IF ( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99994 ) NR
                  WRITE ( NOUT, FMT = 99991 )
                  DO 10 I = 1, 4
                     IF( INFRED(I).GE.0 )
     $                  WRITE ( NOUT, FMT = 99990 ) I, INFRED(I)
   10             CONTINUE
                  WRITE ( NOUT, FMT = 99997 )
                  DO 20 I = 1, NR
                     WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
   20             CONTINUE
                  WRITE ( NOUT, FMT = 99996 )
                  DO 30 I = 1, NR
                     WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,NR )
   30             CONTINUE
                  WRITE ( NOUT, FMT = 99993 )
                  DO 40 I = 1, NR
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   40             CONTINUE
                  WRITE ( NOUT, FMT = 99992 )
                  DO 50 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
   50             CONTINUE
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' TG01JY EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01JY = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Ar is ')
99996 FORMAT (/' The reduced descriptor matrix Er is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5 )
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' Achieved order reductions in different phases')
99990 FORMAT (' Phase',I2,':', I3, ' elliminated eigenvalue(s)' )
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
      END
Program Data
TG01JY EXAMPLE PROGRAM DATA
  9    2    2     0.0    0.0    0.0    I  R  N  N  N
    -2    -3     0     0     0     0     0     0     0
     1     0     0     0     0     0     0     0     0
     0     0    -2    -3     0     0     0     0     0
     0     0     1     0     0     0     0     0     0
     0     0     0     0     1     0     0     0     0
     0     0     0     0     0     1     0     0     0
     0     0     0     0     0     0     1     0     0
     0     0     0     0     0     0     0     1     0
     0     0     0     0     0     0     0     0     1
     1     0     0     0     0     0     0     0     0
     0     1     0     0     0     0     0     0     0
     0     0     1     0     0     0     0     0     0
     0     0     0     1     0     0     0     0     0
     0     0     0     0     0     0     0     0     0
     0     0     0     0     1     0     0     0     0
     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     1     0     0
     0     0     0     0     0     0     0     1     0
     1     0
     0     0
     0     1
     0     0
    -1     0
     0     0
     0    -1
     0     0
     0     0
     1     0     1    -3     0     1     0     2     0
     0     1     1     3     0     1     0     0     1

Program Results
 TG01JY EXAMPLE PROGRAM RESULTS

 Order of reduced system =    7

 Achieved order reductions in different phases
 Phase 1:  0 elliminated eigenvalue(s)
 Phase 2:  2 elliminated eigenvalue(s)

 The reduced state dynamics matrix Ar is 
   1.0000  -0.0393  -0.0980   0.1066  -0.0781   0.2330  -0.0777
   0.0000   1.0312   0.2717  -0.2609   0.1533  -0.6758   0.3553
   0.0000   0.0000   1.3887  -0.6699   0.4281  -1.6389   0.7615
   0.0000   0.0000   0.0000   1.2147  -0.2423   0.9792  -0.4788
   0.0000   0.0000   0.0000   0.0000   1.0545  -0.5035   0.2788
   0.0000   0.0000   0.0000   0.0000   0.0000   1.6355  -0.4323
   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000

 The reduced descriptor matrix Er is 
   0.4100   0.2590   0.5080   0.3109  -0.0705  -0.1429   0.1477
  -0.7629  -0.3464   0.0992   0.3007  -0.0619  -0.2483   0.0152
   0.1120  -0.2124  -0.4184   0.1288  -0.0569   0.4213   0.6182
   0.0000   0.1122  -0.0039  -0.2771   0.0758  -0.0975  -0.3923
   0.0000   0.0000   0.3708   0.4290  -0.1006  -0.1402   0.2699
   0.0000   0.0000   0.0000   0.0000   0.9458  -0.2211   0.2378
   0.0000   0.0000   0.0000   0.5711   0.2648   0.5948  -0.5000

 The reduced input/state matrix Br is 
   0.5597  -0.2363
   0.4843   0.0498
   0.4727   0.1491
  -0.1802  -1.1574
  -0.5995  -0.1556
  -0.1729  -0.3999
   0.0000   0.2500

 The reduced state/output matrix Cr is 
   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   4.0000
   0.0000   0.0000   0.0000   0.0000   0.0000   3.1524  -1.7500

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