## AB08NW

### Construction of a regular pencil such that its generalized eigenvalues are finite Smith zeros of a system (variant)

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

Purpose

      To extract from the system pencil

                        ( A-lambda*I B )
            S(lambda) = (              )
                        (      C     D )

      a regular pencil Af-lambda*Ef which has the finite Smith zeros of
      S(lambda) as generalized eigenvalues. The routine also computes
      the orders of the infinite Smith zeros and determines the singular
      and infinite Kronecker structure of the system pencil, i.e., the
      right and left Kronecker indices, and the multiplicities of the
      infinite eigenvalues.

Specification

          SUBROUTINE AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE, NKROL,
         $                   INFZ, KRONR, INFE, KRONL, E, LDE, TOL, IWORK,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         EQUIL
          INTEGER           DINFZ, INFO, LDA, LDB, LDC, LDD, LDE, LDWORK,
         $                  M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           INFE(*),  INFZ(*),  IWORK(*), KRONL(*), KRONR(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DWORK(*), E(LDE,*)

Arguments

Mode Parameters

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to balance the system
              matrix as follows:
              = 'S':  Perform balancing (scaling);
              = 'N':  Do not perform balancing.

Input/Output Parameters

      N       (input) INTEGER.
              The order of the square matrix A, the number of rows of
              the matrix B, and number of columns of the matrix C.
              N >= 0.

      M       (input) INTEGER.
              The number of columns of the matrix B.  M >= 0.

      P       (input) INTEGER.
              The number of rows of the 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 state dynamics matrix A of the system.
              On exit, the leading NFZ-by-NFZ part of this array
              contains the matrix Af of the reduced pencil.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input/state matrix B of the system.
              On exit, this matrix does not contain useful information.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= 1, and
              LDB >= MAX(1,N), if M > 0.

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the state/output matrix C of the system.
              On exit, this matrix does not contain useful information.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= MAX(1,P).

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading P-by-M part of this array must contain the
              direct transmission matrix D of the system.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= MAX(1,P).

      NFZ     (output) INTEGER
              The number of finite zeros.

      NRANK   (output) INTEGER
              The normal rank of the system pencil.

      NIZ     (output) INTEGER
              The number of infinite zeros.

      DINFZ   (output) INTEGER
              The maximal multiplicity of infinite Smith zeros.

      NKROR   (output) INTEGER
              The number of right Kronecker indices.

      NINFE   (output) INTEGER
              The number of elementary infinite blocks.

      NKROL   (output) INTEGER
              The number of left Kronecker indices.

      INFZ    (output) INTEGER array, dimension (N+1)
              The leading DINFZ elements of INFZ contain information
              on the infinite elementary divisors as follows:
              the system has INFZ(i) infinite elementary divisors in
              the Smith form of degree i, where i = 1,2,...,DINFZ.

      KRONR   (output) INTEGER array, dimension (N+1)
              The leading NKROR elements of this array contain the
              right Kronecker (column) indices.

      INFE    (output) INTEGER array, dimension (N+1)
              The leading NINFE elements of INFE contain the
              multiplicities of infinite eigenvalues.

      KRONL   (output) INTEGER array, dimension (N+1)
              The leading NKROL elements of this array contain the
              left Kronecker (row) indices.

      E       (output) DOUBLE PRECISION array, dimension (LDE,N)
              The leading NFZ-by-NFZ part of this array contains the
              matrix Ef of the reduced pencil.

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

Tolerances

      TOL     DOUBLE PRECISION
              A tolerance used in rank decisions to determine the
              effective rank, which is defined as the order of the
              largest leading (or trailing) triangular submatrix in the
              QR (or RQ) factorization with column (or row) pivoting
              whose estimated condition number is less than 1/TOL.
              If the user sets TOL <= 0, then an implicitly computed,
              default tolerance  TOLDEF = MAX(N+P,N+M)**2*EPS,  is used
              instead, where EPS is the machine precision (see LAPACK
              Library routine DLAMCH).  TOL < 1.

Workspace

      IWORK   INTEGER array, dimension (MAX(M,P))

      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 >= 1, if MAX(N,M,P) = 0; otherwise,
              LDWORK >= MAX( MIN(P,M) + M + MAX(2*M,N) - 1,
                             MIN(P,N) + MAX(N + MAX(P,M), 3*P - 1 ) ) +
                             MAX(P+N,M+N)*MAX(P+N,M+N).

              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.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The routine extracts from the system matrix of a state space
      system, (A-lambda*I,B,C,D), a regular pencil Af-lambda*Ef, which
      has the finite zeros of the system as generalized eigenvalues.
      The procedure has the following main computational steps:

         (a) construct the (N+P)-by-(M+N) system pencil

              S(lambda) = (B  A)-lambda*( 0  I );
                          (D  C)        ( 0  0 )

         (b) reduce S(lambda) to S1(lambda) with the same finite zeros
             and right Kronecker structure, but with D of full row rank;

         (c) reduce the pencil S1(lambda) to S2(lambda) with the same
             finite zeros and with D square invertible;

         (d) perform a unitary transformation on the columns of
             S2(lambda) = (A-lambda*I   B), in order to reduce it to
                          (     C       D)

             (Af-lambda*Ef   X), with Y and Ef square invertible;
             (     0         Y)

         (e) compute the right and left Kronecker indices of the system
             matrix, which, together with the multiplicities of the
             finite and infinite eigenvalues, constitute the complete
             set of structural invariants under strict equivalence
             transformations of a linear system.

References

      [1] Svaricek, F.
          Computation of the Structural Invariants of Linear
          Multivariable Systems with an Extended Version of the
          Program ZEROS.
          System & Control Letters, 6, pp. 261-266, 1985.

      [2] Emami-Naeini, A. and Van Dooren, P.
          Computation of Zeros of Linear Multivariable Systems.
          Automatica, 18, pp. 415-430, 1982.

Numerical Aspects

      The algorithm is backward stable (see [2] and [1]).

Further Comments

      In order to compute the finite Smith zeros of the system
      explicitly, a call to this routine may be followed by a call to
      the LAPACK Library routine DGGEV.

Example

Program Text

    *     AB08NW EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          MPMAX
          PARAMETER        ( MPMAX = MAX( MMAX, PMAX ) )
          INTEGER          LDA, LDB, LDC, LDD, LDE, LDQ, LDZ
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX, LDE = NMAX, LDQ = 1, LDZ = 1 )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( MAX( MIN( PMAX, MMAX ) + MMAX +
         $                                      MAX( 2*MMAX, NMAX ) - 1,
         $                                      MIN( PMAX, NMAX ) +
         $                                      MAX( NMAX + MPMAX,
         $                                           3*PMAX - 1 ) ) +
         $                                 ( NMAX + MPMAX )**2, 8*NMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          DINFZ, I, INFO, J, M, N, NFZ, NINFE, NIZ, NKROL,
         $                 NKROR, NRANK, P
          CHARACTER*1      EQUIL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), ALFI(NMAX), ALFR(NMAX),
         $                 AS(LDA,NMAX), B(LDB,MMAX), BS(LDB,MMAX),
         $                 BETA(NMAX), C(LDC,NMAX), CS(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX),
         $                 Q(LDQ,1), Z(LDZ,1)
          INTEGER          INFE(NMAX+1), INFZ(NMAX+1), IWORK(MPMAX),
         $                 KRONL(NMAX+1), KRONR(NMAX+1)
    *     .. External Subroutines ..
          EXTERNAL         AB08NW, DLACPY, DGEGV
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX, MIN
    *     .. 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, EQUIL
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99972 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99971 ) 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 = 99970 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
                   READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *              Save the matrices A, B, and C.
                   CALL DLACPY( 'Full', N, N, A, LDA, AS, LDA )
                   CALL DLACPY( 'Full', N, M, B, LDB, BS, LDB )
                   CALL DLACPY( 'Full', P, N, C, LDC, CS, LDC )
    *              Check the observability and compute the ordered set of
    *              the observability indices (call the routine with M = 0).
                   CALL AB08NW( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
         $                      NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
         $                      TOL, IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
                      IF ( NFZ.EQ.0 ) THEN
                         WRITE ( NOUT, FMT = 99993 )
                      ELSE
                         WRITE ( NOUT, FMT = 99992 ) N - NFZ
                         WRITE ( NOUT, FMT = 99991 )
                         WRITE ( NOUT, FMT = 99990 )
                         DO 20 I = 1, NFZ
                            WRITE ( NOUT, FMT = 99989 )
         $                        ( A(I,J), J = 1,NFZ )
       20                CONTINUE
                      END IF
                   END IF
    *              Restore the matrices A and C.
                   CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
                   CALL DLACPY( 'Full', P, N, CS, LDC, C, LDC )
    *              Check the controllability and compute the ordered set of
    *              the controllability indices (call the routine with P = 0)
                   CALL AB08NW( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
         $                      NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
         $                      TOL, IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
                      IF ( NFZ.EQ.0 ) THEN
                         WRITE ( NOUT, FMT = 99987 )
                      ELSE
                         WRITE ( NOUT, FMT = 99986 ) N - NFZ
                         WRITE ( NOUT, FMT = 99985 )
                         WRITE ( NOUT, FMT = 99990 )
                         DO 40 I = 1, NFZ
                            WRITE ( NOUT, FMT = 99989 )
         $                        ( A(I,J), J = 1,NFZ )
       40                CONTINUE
                      END IF
                   END IF
    *              Restore the matrices A and B.
                   CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
                   CALL DLACPY( 'Full', N, M, BS, LDB, B, LDB )
    *              Compute the structural invariants of the given system.
                   CALL AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
         $                      NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
         $                      TOL, IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99984 ) NFZ
                      IF ( NFZ.GT.0 ) THEN
    *                    Compute the invariant zeros of the given system.
    *                    Workspace: need 8*NFZ.
                         WRITE ( NOUT, FMT = 99983 )
                         CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
         $                           LDA, E, LDE, ALFR, ALFI, BETA, Q,
         $                           LDQ, Z, LDZ, DWORK, LDWORK, INFO )
    *
                         IF ( INFO.NE.0 ) THEN
                            WRITE ( NOUT, FMT = 99997 ) INFO
                         ELSE
                            WRITE ( NOUT, FMT = 99981 )
                            DO 60 I = 1, NFZ
                               IF ( ALFI(I).EQ.ZERO ) THEN
                                  WRITE ( NOUT, FMT = 99980 )
         $                                ALFR(I)/BETA(I)
                               ELSE
                                  WRITE ( NOUT, FMT = 99979 )
         $                                ALFR(I)/BETA(I),
         $                                ALFI(I)/BETA(I)
                               END IF
       60                   CONTINUE
                            WRITE ( NOUT, FMT = 99982 )
                         END IF
                      END IF
                      WRITE ( NOUT, FMT = 99978 ) NIZ
                      IF ( NIZ.GT.0 ) THEN
                         DO 100 I = 1, DINFZ
                            WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
      100                CONTINUE
                      END IF
                      WRITE ( NOUT, FMT = 99976 ) NKROR
                      IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
         $                                      ( KRONR(I), I = 1,NKROR )
                      WRITE ( NOUT, FMT = 99974 ) NKROL
                      IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
         $                                      ( KRONL(I), I = 1,NKROL )
                      IF ( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99969 )
         $                                      ( INFE(I), I = 1,NINFE )
                   END IF
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' AB08NW EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB08NW = ',I2)
    99997 FORMAT (' INFO on exit from DGEGV = ',I2)
    99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
    99993 FORMAT (/' The system (A,C) is completely observable ')
    99992 FORMAT (/' The dimension of the observable subspace = ',I3)
    99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
         $       'e matrix AF. ')
    99990 FORMAT (/' The matrix AF is ')
    99989 FORMAT (20(1X,F8.4))
    99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
         $       ))
    99987 FORMAT (/' The system (A,B) is completely controllable ')
    99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
    99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
         $       ' matrix AF. ')
    99984 FORMAT (//' The number of finite invariant zeros = ',I3)
    99983 FORMAT (/' The finite invariant zeros are ')
    99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
         $       'ambda*EF - AF).')
    99981 FORMAT (/' real  part     imag  part ')
    99980 FORMAT (1X,F9.4)
    99979 FORMAT (1X,F9.4,6X,F9.4)
    99978 FORMAT (//' The number of infinite zeros = ',I3)
    99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
    99976 FORMAT (/' The number of right Kronecker indices = ',I3)
    99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
         $       /(20(I3,2X)))
    99974 FORMAT (/' The number of left Kronecker indices = ',I3)
    99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
         $       /(20(I3,2X)))
    99972 FORMAT (/' N is out of range.',/' N = ',I5)
    99971 FORMAT (/' M is out of range.',/' M = ',I5)
    99970 FORMAT (/' P is out of range.',/' P = ',I5)
    99969 FORMAT (/' Multiplicities of infinite eigenvalues',/(20(I3,2X)))
          END

Program Data

     AB08NW EXAMPLE PROGRAM DATA
       6     2     3     0.0     N
       1.0   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   3.0   0.0   0.0   0.0
       0.0   0.0   0.0  -4.0   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   3.0
       0.0  -1.0
      -1.0   0.0
       1.0  -1.0
       0.0   0.0
       0.0   1.0
      -1.0  -1.0
       1.0   0.0   0.0   1.0   0.0   0.0
       0.0   1.0   0.0   1.0   0.0   1.0
       0.0   0.0   1.0   0.0   0.0   1.0
       0.0   0.0
       0.0   0.0
       0.0   0.0

Program Results

     AB08NW EXAMPLE PROGRAM RESULTS

     The left Kronecker indices of (A,C) are
      1    2    2

     The dimension of the observable subspace =   5

     The output decoupling zeros are the eigenvalues of the matrix AF.

     The matrix AF is
      -1.0000

     The right Kronecker indices of (A,B) are
      2    3

     The dimension of the controllable subspace =   5

     The input decoupling zeros are the eigenvalues of the matrix AF.

     The matrix AF is
      -4.0000

     The number of finite invariant zeros =   2

     The finite invariant zeros are

     real  part     imag  part
        2.0000
       -1.0000

     which correspond to the generalized eigenvalues of (lambda*EF - AF).

     The number of infinite zeros =   2
       2 infinite zero(s) of order   1

     The number of right Kronecker indices =   0

     The number of left Kronecker indices =   1

     The left Kronecker (row) indices of (A,B,C,D) are
      2

     Multiplicities of infinite eigenvalues
      2    2

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