BB04AD - SLICOT Library Routine Documentation

## BB04AD

### Benchmark examples of (generalized) discrete-time Lyapunov equations

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

Purpose

      To generate benchmark examples of (generalized) discrete-time
      Lyapunov equations

         T           T
        A  X  A  -  E  X E  =  Y .

      In some examples, the right hand side has the form

                 T
        Y  =  - B  B

      and the solution can be represented as a product of Cholesky
      factors

               T
        X  =  U  U .

      E, A, Y, X, and U are real N-by-N matrices, and B is M-by-N. Note
      that E can be the identity matrix. For some examples, B, X, or U
      are not provided.

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

Specification

          SUBROUTINE BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA,
         1                  Y, LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK,
         2                  LDWORK, INFO)
    C     .. Scalar Arguments ..
          CHARACTER         DEF
          CHARACTER*70      NOTE
          INTEGER           INFO, LDA, LDB, LDE, LDU, LDWORK, LDX, LDY, M, N
    C     .. Array Arguments ..
          LOGICAL           VEC(8)
          INTEGER           IPAR(*), NR(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), DPAR(*), DWORK(LDWORK),
         1                  E(LDE,*), U(LDU,*), X(LDX,*), Y(LDY,*)

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):
              DEF = 'D' or 'd': Default values are used.
              DEF = 'N' or '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 2
              On entry, if DEF = 'N' or '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 4.1, DPAR(1) and DPAR(2) define 'r' and 's',
              respectively.
              For Example 4.2, DPAR(1) and DPAR(2) define 'lambda' and
              's', respectively.
              For Examples 4.3 and 4.4, DPAR(1) defines the parameter
              't'.
              On exit, if DEF = 'D' or '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 of DIMENSION at least 1
              On entry, if DEF = 'N' or '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 Examples 4.1, 4.2, and 4.3, IPAR(1) defines 'n'.
              For Example 4.4, IPAR(1) defines 'q'.
              On exit, if DEF = 'D' or '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 availability of the output
              data:
              VEC(1) and VEC(2) refer to N and M, respectively, and are
              always .TRUE.
              VEC(3) is .TRUE. iff E is NOT the identity matrix.
              VEC(4) and VEC(5) refer to A and Y, respectively, and are
              always .TRUE.
              VEC(6) is .TRUE. iff B is provided.
              VEC(7) is .TRUE. iff the solution matrix X is provided.
              VEC(8) is .TRUE. iff the Cholesky factor U is provided.

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

      M       (output) INTEGER
              The number of rows in the matrix B. If B is not provided
              for the desired example, M = 0 is returned.

      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(3) = .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.

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

      LDY     INTEGER
              The leading dimension of array Y.  LDY >= N.

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

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

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

      LDX     INTEGER
              The leading dimension of array X.  LDX >= N.

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

      LDU     INTEGER
              The leading dimension of array U.  LDU >= N.

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

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)

      LDWORK  INTEGER
              The length of the array DWORK.
              For Examples 4.1 and 4.2., LDWORK >= 2*IPAR(1) is
              required.
              For the other examples, no workspace is needed, i.e.,
              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.

References

      [1]  D. Kressner, V. Mehrmann, and T. Penzl.
           DTLEX - a Collection of Benchmark Examples for Discrete-
           Time Lyapunov Equations.
           SLICOT Working Note 1999-7, 1999.

Numerical Aspects

      None

Further Comments

      None

Example

Program Text

    C     BB04AD EXAMPLE PROGRAM TEXT
    C
    C     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        (NIN = 5, NOUT = 6)
          INTEGER          NMAX, MMAX
          PARAMETER        (NMAX = 100, MMAX = 100)
          INTEGER          LDE, LDA, LDY, LDB, LDX, LDU, LDWORK
          PARAMETER        (LDE = NMAX, LDA = NMAX, LDY = NMAX, LDB = MMAX,
         1                  LDX = NMAX, LDU = NMAX, LDWORK = 2*NMAX)
    C     .. Local Scalars ..
          CHARACTER        DEF
          INTEGER          INFO, N, M, I, J, LDPAR, LIPAR
          CHARACTER*70     NOTE
    C     .. Local Arrays ..
          DOUBLE PRECISION E(LDE,NMAX), A(LDA, NMAX), Y(LDY, NMAX),
         1                 B(LDB,NMAX), X(LDX, NMAX), U(LDU, NMAX),
         2                 DPAR(2), DWORK(LDWORK)
          INTEGER          NR(2), IPAR(1)
          LOGICAL          VEC(8)
    C     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    C     .. External Subroutines ..
          EXTERNAL         BB04AD
    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 BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA, Y,
         1            LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK, LDWORK,
         2            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
            IF (VEC(3)) THEN
              WRITE (NOUT, FMT = 99995)
              DO 10  I = 1, N
                WRITE (NOUT, FMT = 99985) (E(I,J), J = 1, N)
    10        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99994)
            END IF
            WRITE (NOUT,FMT = 99993)
            DO 20  I = 1, N
              WRITE (NOUT, FMT = 99985) (A(I,J), J = 1, N)
    20      CONTINUE
            IF (VEC(6)) THEN
              WRITE (NOUT,FMT = 99992)
              DO 30  I = 1, M
                WRITE (NOUT, FMT = 99985) (B(I,J), J = 1, N)
    30        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99991)
            END IF
            WRITE (NOUT,FMT = 99990)
            DO 40  I = 1, N
              WRITE (NOUT, FMT = 99985) (Y(I,J), J = 1, N)
    40      CONTINUE
            IF (VEC(7)) THEN
              WRITE (NOUT, FMT = 99989)
              DO 50  I = 1, N
                WRITE (NOUT, FMT = 99985) (X(I,J), J = 1, N)
    50        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99988)
            END IF
            IF (VEC(8)) THEN
              WRITE (NOUT, FMT = 99987)
              DO 60  I = 1, N
                WRITE (NOUT, FMT = 99985) (U(I,J), J = 1, N)
    60        CONTINUE
            ELSE
              WRITE (NOUT, FMT = 99986)
            END IF
          END IF
    C
    99999 FORMAT (' BB04AD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' INFO on exit from BB04AD = ', I3)
    99997 FORMAT (/' Order of matrix A:            N  = ', I3)
    99996 FORMAT (' Number of rows in matrix B:   M  = ', I3)
    99995 FORMAT (/' E  = ')
    99994 FORMAT (/' E is the identity matrix.')
    99993 FORMAT (' A  = ')
    99992 FORMAT (' B  = ')
    99991 FORMAT (' B is not provided.')
    99990 FORMAT (' Y  = ')
    99989 FORMAT (' X  = ')
    99988 FORMAT (' X is not provided.')
    99987 FORMAT (' U  = ')
    99986 FORMAT (' U is not provided.')
    99985 FORMAT (20(1X,F8.4))
    C
          END

Program Data

    BB04AD EXAMPLE PROGRAM DATA
    N
    4 1
    2
    .15D1
    .15D1
    1
    5

Program Results

     BB04AD EXAMPLE PROGRAM RESULTS

     DTLEX: Example 4.1

     Order of matrix A:            N  =   5
     Number of rows in matrix B:   M  =   1

     E is the identity matrix.
     A  =
       0.4562   0.0308   0.1990   0.0861   0.0217
       0.0637   0.5142  -0.1828   0.0096  -0.1148
       0.3139   0.1287   0.3484   0.1653  -0.1975
       0.1500   0.0053  -0.1838   0.2501  -0.0687
       0.0568  -0.1006  -0.3735  -0.0202   0.2285
     B  =
       0.3086   0.0247  -0.4691   0.1728  -0.3704
     Y  =
      -0.0953  -0.0076   0.1448  -0.0533   0.1143
      -0.0076  -0.0006   0.0116  -0.0043   0.0091
       0.1448   0.0116  -0.2201   0.0811  -0.1738
      -0.0533  -0.0043   0.0811  -0.0299   0.0640
       0.1143   0.0091  -0.1738   0.0640  -0.1372
     X  =
       0.0953   0.0076  -0.1448   0.0533  -0.1143
       0.0076   0.0006  -0.0116   0.0043  -0.0091
      -0.1448  -0.0116   0.2201  -0.0811   0.1738
       0.0533   0.0043  -0.0811   0.0299  -0.0640
      -0.1143  -0.0091   0.1738  -0.0640   0.1372
     U is not provided.

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