## BD02AD

### Benchmark examples for time-invariant discrete-time dynamical systems

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

Purpose

      To generate benchmark examples for time-invariant,
      discrete-time dynamical systems

        E x_k+1 = A x_k + B u_k

            y_k = C x_k + D u_k

      E, A are real N-by-N matrices, B is N-by-M, C is P-by-N, and
      D is P-by-M. In many examples, E is the identity matrix and D is
      the zero matrix.

      This routine is an implementation of the benchmark library
      DTDSX (Version 1.0) described in [1].

Specification

          SUBROUTINE BD02AD( DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A,
         1                   LDA, B, LDB, C, LDC, D, LDD, NOTE, DWORK,
         2                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DEF
          CHARACTER*70      NOTE
          INTEGER           INFO, LDA, LDB, LDC, LDD, LDE, LDWORK, M, N, P
    C     .. Array Arguments ..
          LOGICAL           VEC(8)
          INTEGER           IPAR(*), NR(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DPAR(*),
         1                  DWORK(*), E(LDE,*)

Arguments

Mode Parameters

      DEF     CHARACTER*1
              Specifies the kind of values used as parameters when
              generating parameter-dependent and scalable examples
              (i.e., examples with NR(1) = 2, 3, or 4):
              = 'D':  Default values defined in [1] are used;
              = 'N':  Values set in DPAR and IPAR are used.
              This parameter is not referenced if NR(1) = 1.
              Note that the scaling parameter of examples with
              NR(1) = 3 or 4 is considered as a regular parameter in
              this context.

Input/Output Parameters

      NR      (input) INTEGER array, dimension (2)
              Specifies the index of the desired example according
              to [1].
              NR(1) defines the group:
                    1 : parameter-free problems of fixed size
                    2 : parameter-dependent problems of fixed size
                    3 : parameter-free problems of scalable size
                    4 : parameter-dependent problems of scalable size
              NR(2) defines the number of the benchmark example
              within a certain group according to [1].

      DPAR    (input/output) DOUBLE PRECISION array, dimension (7)
              On entry, if DEF = 'N' and the desired example depends on
              real parameters, then the array DPAR must contain the
              values for these parameters.
              For an explanation of the parameters see [1].
              For Example 2.1, DPAR(1), ..., DPAR(3) define the
              parameters 'tau', 'delta', 'K', respectively.
              On exit, if DEF = 'D' and the desired example depends on
              real parameters, then the array DPAR is overwritten by the
              default values given in [1].

      IPAR    (input/output) INTEGER array, dimension (1)
              On entry, if DEF = 'N' and the desired example depends on
              integer parameters, then the array IPAR must contain the
              values for these parameters.
              For an explanation of the parameters see [1].
              For Example 3.1, IPAR(1) defines the parameter 'n'.
              On exit, if DEF = 'D' and the desired example depends on
              integer parameters, then the array IPAR is overwritten by
              the default values given in [1].

      VEC     (output) LOGICAL array, dimension (8)
              Flag vector which displays the availabilty of the output
              data:
              VEC(1), ..., VEC(3) refer to N, M, and P, respectively,
              and are always .TRUE..
              VEC(4) is .TRUE. iff E is NOT the identity matrix.
              VEC(5), ..., VEC(7) refer to A, B, and C, respectively,
              and are always .TRUE..
              VEC(8) is .TRUE. iff D is NOT the zero matrix.

      N       (output) INTEGER
              The actual state dimension, i.e., the order of the
              matrices E and A.

      M       (output) INTEGER
              The number of columns in the matrices B and D.

      P       (output) INTEGER
              The number of rows in the matrices C and D.

      E       (output) DOUBLE PRECISION array, dimension (LDE,N)
              The leading N-by-N part of this array contains the
              matrix E.
              NOTE that this array is overwritten (by the identity
              matrix), if VEC(4) = .FALSE..

      LDE     INTEGER
              The leading dimension of array E.  LDE >= N.

      A       (output) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array contains the
              matrix A.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= N.

      B       (output) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array contains the
              matrix B.

      LDB     INTEGER
              The leading dimension of array B.  LDB >= N.

      C       (output) DOUBLE PRECISION array, dimension (LDC,N)
              The leading P-by-N part of this array contains the
              matrix C.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= P.

      D       (output) DOUBLE PRECISION array, dimension (LDD,M)
              The leading P-by-M part of this array contains the
              matrix D.
              NOTE that this array is overwritten (by the zero
              matrix), if VEC(8) = .FALSE..

      LDD     INTEGER
              The leading dimension of array D.  LDD >= P.

      NOTE    (output) CHARACTER*70
              String containing short information about the chosen
              example.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              NOTE that DWORK is not used in the current version
              of BD02AD.

      LDWORK  INTEGER
              LDWORK >= 1.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value; in particular, INFO = -3 or -4 indicates
                    that at least one of the parameters in DPAR or
                    IPAR, respectively, has an illegal value;
              = 1:  data file can not be opened or has wrong format.

References

      [1]  Kressner, D., Mehrmann, V. and Penzl, T.
           DTDSX - a Collection of Benchmark Examples for State-Space
           Realizations of Discrete-Time Dynamical Systems.
           SLICOT Working Note 1998-10. 1998.

Numerical Aspects

      None

Further Comments

      None

Example

Program Text

    C     BD02AD EXAMPLE PROGRAM TEXT
    C
    C     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        (NIN = 5, NOUT = 6)
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        (NMAX = 421, MMAX = 211, PMAX = 211)
          INTEGER          LDA, LDB, LDC, LDD, LDE, LDWORK
          PARAMETER        (LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
         1                  LDE = NMAX, LDWORK = 1)
    C     .. Local Scalars ..
          CHARACTER        DEF
          INTEGER          I, INFO, J, LDPAR, LIPAR, M, N, P
          CHARACTER*70     NOTE
    C     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         1                 D(LDD,MMAX), DPAR(7), DWORK(LDWORK), E(LDE,NMAX)
          INTEGER          NR(2), IPAR(7)
          LOGICAL          VEC(8)
    C     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    C     .. External Subroutines ..
          EXTERNAL         BD02AD
    C     .. Executable Statements ..
          WRITE (NOUT, FMT = 99999)
    C     Skip the heading in the data file and read the data.
          READ (NIN, FMT = '()')
          READ (NIN, FMT = *) DEF
          READ (NIN, FMT = *) (NR(I), I = 1, 2)
          IF (LSAME(DEF,'N')) THEN
            READ (NIN, FMT = *) LDPAR
            IF (LDPAR .GT. 0)  READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
            READ (NIN, FMT = *) LIPAR
            IF (LIPAR .GT. 0)  READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
          END IF
    C     Generate benchmark example
          CALL BD02AD(DEF, NR, DPAR, IPAR, VEC, N, M, P, E, LDE, A, LDA,
         1            B, LDB, C, LDC, D, LDD, NOTE, DWORK, LDWORK, INFO)
    C
          IF (INFO .NE. 0) THEN
            WRITE (NOUT, FMT = 99998) INFO
          ELSE
            WRITE (NOUT, FMT = *) NOTE
            WRITE (NOUT, FMT = 99997) N
            WRITE (NOUT, FMT = 99996) M
            WRITE (NOUT, FMT = 99995) P
            IF (VEC(4)) THEN
              WRITE (NOUT, FMT = 99994)
              DO 10  I = 1, N
                WRITE (NOUT, FMT = 99987) (E(I,J), J = 1, N)
    10        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99993)
            END IF
            WRITE (NOUT,FMT = 99992)
            DO 20  I = 1, N
              WRITE (NOUT, FMT = 99987) (A(I,J), J = 1, N)
    20      CONTINUE
            WRITE (NOUT,FMT = 99991)
            DO 30  I = 1, N
              WRITE (NOUT, FMT = 99987) (B(I,J), J = 1, M)
    30      CONTINUE
            WRITE (NOUT,FMT = 99990)
            DO 40  I = 1, P
              WRITE (NOUT, FMT = 99987) (C(I,J), J = 1, N)
    40      CONTINUE
            IF (VEC(8)) THEN
              WRITE (NOUT,FMT = 99989)
              DO 50  I = 1, P
                WRITE (NOUT, FMT = 99987) (D(I,J), J = 1, M)
    50        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99988)
            END IF
          END IF
    C
    99999 FORMAT (' BD02AD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' INFO on exit from BD02AD = ', I3)
    99997 FORMAT (/' Order of matrix A:               N  = ', I3)
    99996 FORMAT (' Number of columns in matrix B:   M  = ', I3)
    99995 FORMAT (' Number of rows in matrix C:      P  = ', I3)
    99994 FORMAT (/' E  = ')
    99993 FORMAT (/' E is the identity matrix.')
    99992 FORMAT (' A  = ')
    99991 FORMAT (' B  = ')
    99990 FORMAT (' C  = ')
    99989 FORMAT (' D  = ')
    99988 FORMAT (' D is of zeros.')
    99987 FORMAT (20(1X,F8.4))
    C
          END

Program Data

    BD02AD EXAMPLE PROGRAM DATA
    D
    1 1

Program Results

     BD02AD EXAMPLE PROGRAM RESULTS

     Laub 1979, Ex. 2: uncontrollable-unobservable data

     Order of matrix A:               N  =   2
     Number of columns in matrix B:   M  =   1
     Number of rows in matrix C:      P  =   1

     E is the identity matrix.
     A  =
       4.0000   3.0000
      -4.5000  -3.5000
     B  =
       1.0000
      -1.0000
     C  =
       3.0000   2.0000
     D is of zeros.

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