## BB01AD

### Benchmark examples for continuous-time algebraic Riccati equations

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

Purpose

      To generate the benchmark examples for the numerical solution of
      continuous-time algebraic Riccati equations (CAREs) of the form

        0 = Q + A'X + XA - XGX

      corresponding to the Hamiltonian matrix

             (  A   G  )
         H = (       T ).
             (  Q  -A  )

      A,G,Q,X are real N-by-N matrices, Q and G are symmetric and may
      be given in factored form

                    -1 T                         T
       (I)   G = B R  B  ,           (II)   Q = C W C .

      Here, C is P-by-N, W P-by-P, B N-by-M, and R M-by-M, where W
      and R are symmetric. In linear-quadratic optimal control problems,
      usually W is positive semidefinite and R positive definite.  The
      factorized form can be used if the CARE is solved using the
      deflating subspaces of the extended Hamiltonian pencil

                   (  A   0   B  )       (  I   0   0  )
                   (       T     )       (             )
         H - s K = (  Q   A   0  )  -  s (  0  -I   0  ) ,
                   (       T     )       (             )
                   (  0   B   R  )       (  0   0   0  )

      where I and 0 denote the identity and zero matrix, respectively,
      of appropriate dimensions.

      NOTE: the formulation of the CARE and the related matrix (pencils)
            used here does not include CAREs as they arise in robust
            control (H_infinity optimization).

Specification

          SUBROUTINE BB01AD(DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P,
         1                  A, LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX,
         2                  DWORK, LDWORK, INFO)
    C     .. Scalar Arguments ..
          INTEGER          INFO, LDA, LDB, LDC, LDG, LDQ, LDWORK, LDX, M, N,
         $                 P
          CHARACTER        DEF
    C     .. Array Arguments ..
          INTEGER          IPAR(4), NR(2)
          DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DPAR(*), DWORK(*),
         1                 G(*), Q(*), X(LDX,*)
          CHARACTER        CHPAR*(*)
          LOGICAL          BPAR(6), VEC(9)

Arguments

Mode Parameters

      DEF     CHARACTER
              This parameter specifies if the default parameters are
              to be used or not.
              = 'N' or 'n' : The parameters given in the input vectors
                             xPAR (x = 'D', 'I', 'B', 'CH') are used.
              = 'D' or 'd' : The default parameters for the example
                             are used.
              This parameter is not meaningful if NR(1) = 1.

Input/Output Parameters

      NR      (input) INTEGER array, dimension (2)
              This array determines the example for which CAREX returns
              data. NR(1) is the group of examples.
              NR(1) = 1 : parameter-free problems of fixed size.
              NR(1) = 2 : parameter-dependent problems of fixed size.
              NR(1) = 3 : parameter-free problems of scalable size.
              NR(1) = 4 : parameter-dependent problems of scalable size.
              NR(2) is the number of the example in group NR(1).
              Let NEXi be the number of examples in group i. Currently,
              NEX1 = 6, NEX2 = 9, NEX3 = 2, NEX4 = 4.
              1 <= NR(1) <= 4;
              1 <= NR(2) <= NEXi , where i = NR(1).

      DPAR    (input/output) DOUBLE PRECISION array, dimension (7)
              Double precision parameter vector. For explanation of the
              parameters see [1].
              DPAR(1)           : defines the parameters
                                  'delta' for NR(1) = 3,
                                  'q' for NR(1).NR(2) = 4.1,
                                  'a' for NR(1).NR(2) = 4.2, and
                                  'mu' for NR(1).NR(2) = 4.3.
              DPAR(2)           : defines parameters
                                  'r' for NR(1).NR(2) = 4.1,
                                  'b' for NR(1).NR(2) = 4.2, and
                                  'delta' for NR(1).NR(2) = 4.3.
              DPAR(3)           : defines parameters
                                  'c' for NR(1).NR(2) = 4.2 and
                                  'kappa' for NR(1).NR(2) = 4.3.
              DPAR(j), j=4,5,6,7: These arguments are only used to
                                  generate Example 4.2 and define in
                                  consecutive order the intervals
                                  ['beta_1', 'beta_2'],
                                  ['gamma_1', 'gamma_2'].
              NOTE that if DEF = 'D' or 'd', the values of DPAR entries
              on input are ignored and, on output, they are overwritten
              with the default parameters.

      IPAR    (input/output) INTEGER array, dimension (4)
              On input, IPAR(1) determines the actual state dimension,
              i.e., the order of the matrix A as follows, where
              NO = NR(1).NR(2).
              NR(1) = 1 or 2.1-2.8: IPAR(1) is ignored.
              NO = 2.9            : IPAR(1) = 1 generates the CARE for
                                    optimal state feedback (default);
                                    IPAR(1) = 2 generates the Kalman
                                    filter CARE.
              NO = 3.1            : IPAR(1) is the number of vehicles
                                    (parameter 'l' in the description
                                     in [1]).
              NO = 3.2, 4.1 or 4.2: IPAR(1) is the order of the matrix
                                    A.
              NO = 4.3 or 4.4     : IPAR(1) determines the dimension of
                                    the second-order system, i.e., the
                                    order of the stiffness matrix for
                                    Examples 4.3 and 4.4 (parameter 'l'
                                    in the description in [1]).

              The order of the output matrix A is N = 2*IPAR(1) for
              Example 4.3 and N = 2*IPAR(1)-1 for Examples 3.1 and 4.4.
              NOTE that IPAR(1) is overwritten for Examples 1.1-2.8. For
              the other examples, IPAR(1) is overwritten if the default
              parameters are to be used.
              On output, IPAR(1) contains the order of the matrix A.

              On input, IPAR(2) is the number of colums in the matrix B
              in (I) (in control problems, the number of inputs of the
              system). Currently, IPAR(2) is fixed or determined by
              IPAR(1) for all examples and thus is not referenced on
              input.
              On output, IPAR(2) is the number of columns of the
              matrix B from (I).
              NOTE that currently IPAR(2) is overwritten and that
              rank(G) <= IPAR(2).

              On input, IPAR(3) is the number of rows in the matrix C
              in (II) (in control problems, the number of outputs of the
              system). Currently, IPAR(3) is fixed or determined by
              IPAR(1) for all examples and thus is not referenced on
              input.
              On output, IPAR(3) contains the number of rows of the
              matrix C in (II).
              NOTE that currently IPAR(3) is overwritten and that
              rank(Q) <= IPAR(3).

              On input, if NR(1) = NR(2) = 4, and other data file than
              that used by default is desired, then IPAR(4) is the
              length of the character string in CHPAR specifying the
              file name.

      BPAR    (input) BOOLEAN array, dimension (6)
              This array defines the form of the output of the examples
              and the storage mode of the matrices G and Q.
              BPAR(1) = .TRUE.  : G is returned.
              BPAR(1) = .FALSE. : G is returned in factored form, i.e.,
                                  B and R from (I) are returned.
              BPAR(2) = .TRUE.  : The matrix returned in array G (i.e.,
                                  G if BPAR(1) = .TRUE. and R if
                                  BPAR(1) = .FALSE.) is stored as full
                                  matrix.
              BPAR(2) = .FALSE. : The matrix returned in array G is
                                  provided in packed storage mode.
              BPAR(3) = .TRUE.  : If BPAR(2) = .FALSE., the matrix
                                  returned in array G is stored in upper
                                  packed mode, i.e., the upper triangle
                                  of a symmetric n-by-n matrix is stored
                                  by columns, e.g., the matrix entry
                                  G(i,j) is stored in the array entry
                                  G(i+j*(j-1)/2) for i <= j.
                                  Otherwise, this entry is ignored.
              BPAR(3) = .FALSE. : If BPAR(2) = .FALSE., the matrix
                                  returned in array G is stored in lower
                                  packed mode, i.e., the lower triangle
                                  of a symmetric n-by-n matrix is stored
                                  by columns, e.g., the matrix entry
                                  G(i,j) is stored in the array entry
                                  G(i+(2*n-j)*(j-1)/2) for j <= i.
                                  Otherwise, this entry is ignored.
              BPAR(4) = .TRUE.  : Q is returned.
              BPAR(4) = .FALSE. : Q is returned in factored form, i.e.,
                                  C and W from (II) are returned.
              BPAR(5) = .TRUE.  : The matrix returned in array Q (i.e.,
                                  Q if BPAR(4) = .TRUE. and W if
                                  BPAR(4) = .FALSE.) is stored as full
                                  matrix.
              BPAR(5) = .FALSE. : The matrix returned in array Q is
                                  provided in packed storage mode.
              BPAR(6) = .TRUE.  : If BPAR(5) = .FALSE., the matrix
                                  returned in array Q is stored in upper
                                  packed mode (see above).
                                  Otherwise, this entry is ignored.
              BPAR(6) = .FALSE. : If BPAR(5) = .FALSE., the matrix
                                  returned in array Q is stored in lower
                                  packed mode (see above).
                                  Otherwise, this entry is ignored.
              NOTE that there are no default values for BPAR.  If all
              entries are declared to be .TRUE., then matrices G and Q
              are returned in conventional storage mode, i.e., as
              N-by-N arrays where the array element Z(I,J) contains the
              matrix entry Z_{i,j}.

      CHPAR   (input/output) CHARACTER*255
              On input, this is the name of a data file supplied by the
              user.
              In the current version, only Example 4.4 allows a
              user-defined data file. This file must contain
              consecutively DOUBLE PRECISION vectors mu, delta, gamma,
              and kappa. The length of these vectors is determined by
              the input value for IPAR(1).
              If on entry, IPAR(1) = L, then mu and delta must each
              contain L DOUBLE PRECISION values, and gamma and kappa
              must each contain L-1 DOUBLE PRECISION values.
              On output, this string contains short information about
              the chosen example.

      VEC     (output) LOGICAL array, dimension (9)
              Flag vector which displays the availability of the output
              data:
              VEC(j), j=1,2,3, refer to N, M, and P, respectively, and
              are always .TRUE.
              VEC(4) refers to A and is always .TRUE.
              VEC(5) is .TRUE. if BPAR(1) = .FALSE., i.e., the factors B
              and R from (I) are returned.
              VEC(6) is .TRUE. if BPAR(4) = .FALSE., i.e., the factors C
              and W from (II) are returned.
              VEC(7) refers to G and is always .TRUE.
              VEC(8) refers to Q and is always .TRUE.
              VEC(9) refers to X and is .TRUE. if the exact solution
              matrix is available.
              NOTE that VEC(i) = .FALSE. for i = 1 to 9 if on exit
              INFO .NE. 0.

      N       (output) INTEGER
              The order of the matrices A, X, G if BPAR(1) = .TRUE., and
              Q if BPAR(4) = .TRUE.

      M       (output) INTEGER
              The number of columns in the matrix B (or the dimension of
              the control input space of the underlying dynamical
              system).

      P       (output) INTEGER
              The number of rows in the matrix C (or the dimension of
              the output space of the underlying dynamical system).

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

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

      B       (output) DOUBLE PRECISION array, dimension (LDB,M)
              If (BPAR(1) = .FALSE.), then the leading N-by-M part of
              this array contains the matrix B of the factored form (I)
              of G. Otherwise, B is used as workspace.

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

      C       (output) DOUBLE PRECISION array, dimension (LDC,N)
              If (BPAR(4) = .FALSE.), then the leading P-by-N part of
              this array contains the matrix C of the factored form (II)
              of Q. Otherwise, C is used as workspace.

      LDC     INTEGER
              The leading dimension of array C.
              LDC >= P, where P is the number of rows of the matrix C,
              i.e., the output value of IPAR(3). (For all examples,
              P <= N, where N equals the output value of the argument
              IPAR(1), i.e., LDC >= LDA is always safe.)

      G       (output) DOUBLE PRECISION array, dimension (NG)
              If (BPAR(2) = .TRUE.)  then NG = LDG*N.
              If (BPAR(2) = .FALSE.) then NG = N*(N+1)/2.
              If (BPAR(1) = .TRUE.), then array G contains the
              coefficient matrix G of the CARE.
              If (BPAR(1) = .FALSE.), then array G contains the 'control
              weighting matrix' R of G's factored form as in (I). (For
              all examples, M <= N.) The symmetric matrix contained in
              array G is stored according to BPAR(2) and BPAR(3).

      LDG     INTEGER
              If conventional storage mode is used for G, i.e.,
              BPAR(2) = .TRUE., then G is stored like a 2-dimensional
              array with leading dimension LDG. If packed symmetric
              storage mode is used, then LDG is not referenced.
              LDG >= N if BPAR(2) = .TRUE..

      Q       (output) DOUBLE PRECISION array, dimension (NQ)
              If (BPAR(5) = .TRUE.)  then NQ = LDQ*N.
              If (BPAR(5) = .FALSE.) then NQ = N*(N+1)/2.
              If (BPAR(4) = .TRUE.), then array Q contains the
              coefficient matrix Q of the CARE.
              If (BPAR(4) = .FALSE.), then array Q contains the 'output
              weighting matrix' W of Q's factored form as in (II).
              The symmetric matrix contained in array Q is stored
              according to BPAR(5) and BPAR(6).

      LDQ     INTEGER
              If conventional storage mode is used for Q, i.e.,
              BPAR(5) = .TRUE., then Q is stored like a 2-dimensional
              array with leading dimension LDQ. If packed symmetric
              storage mode is used, then LDQ is not referenced.
              LDQ >= N if BPAR(5) = .TRUE..

      X       (output) DOUBLE PRECISION array, dimension (LDX,IPAR(1))
              If an exact solution is available (NR = 1.1, 1.2, 2.1,
              2.3-2.6, 3.2), then the leading N-by-N part of this array
              contains the solution matrix X in conventional storage
              mode. Otherwise, X is not referenced.

      LDX     INTEGER
              The leading dimension of array X.  LDX >= 1, and
              LDX >= N if NR = 1.1, 1.2, 2.1, 2.3-2.6, 3.2.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= N*MAX(4,N).

Error Indicator

      INFO    INTEGER
              = 0 : successful exit;
              < 0 : if INFO = -i, the i-th argument had an illegal
                    value;
              = 1 : data file could not be opened or had wrong format;
              = 2 : division by zero;
              = 3 : G can not be computed as in (I) due to a singular R
                    matrix.

References

      [1] Abels, J. and Benner, P.
          CAREX - A Collection of Benchmark Examples for Continuous-Time
          Algebraic Riccati Equations (Version 2.0).
          SLICOT Working Note 1999-14, November 1999. Available from
          http://www.win.tue.nl/niconet/NIC2/reports.html.

      This is an updated and extended version of

      [2] Benner, P., Laub, A.J., and Mehrmann, V.
          A Collection of Benchmark Examples for the Numerical Solution
          of Algebraic Riccati Equations I: Continuous-Time Case.
          Technical Report SPC 95_22, Fak. f. Mathematik,
          TU Chemnitz-Zwickau (Germany), October 1995.

Numerical Aspects

      If the original data as taken from the literature is given via
      matrices G and Q, but factored forms are requested as output, then
      these factors are obtained from Cholesky or LDL' decompositions of
      G and Q, i.e., the output data will be corrupted by roundoff
      errors.

Further Comments

      Some benchmark examples read data from the data files provided
      with the collection.

Example

Program Text

    *     BB01AD EXAMPLE PROGRAM TEXT
    *
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          MMAX, NMAX, PMAX
          PARAMETER        ( MMAX = 100, NMAX = 100, PMAX = 100 )
          INTEGER          LDA, LDB, LDC, LDG, LDQ, LDX
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDG = NMAX, LDQ = NMAX, LDX = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX*MAX( 4, NMAX ) )
    *     .. Local Scalars ..
          CHARACTER        DEF
          INTEGER          I, INFO, ISYMM, J, LBPAR, LDPAR, LIPAR, M, N, P
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA, NMAX), B(LDB,MMAX), C(LDC, NMAX),
         $                 DPAR(7), DWORK(LDWORK), G(LDG, NMAX),
         $                 Q(LDQ, NMAX), X(LDX, NMAX)
          INTEGER          IPAR(4), NR(2)
          LOGICAL          BPAR(6), VEC(9)
          CHARACTER        CHPAR*255
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         BB01AD, MA02DD
    *     .. 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 = * ) DEF
          READ( NIN, FMT = * ) ( NR(I), I = 1, 2 )
          IF( LSAME( DEF, 'N' ) ) THEN
            READ( NIN, FMT = * ) LBPAR
            IF( LBPAR.GT.0 )  READ( NIN, FMT = * ) ( BPAR(I), I = 1, LBPAR )
            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
    *     Generate benchmark example
          CALL BB01AD( DEF, NR, DPAR, IPAR, BPAR, CHPAR, VEC, N, M, P, A,
         $             LDA, B, LDB, C, LDC, G, LDG, Q, LDQ, X, LDX, DWORK,
         $             LDWORK, INFO )
    *
          IF( INFO.NE.0 ) THEN
            WRITE( NOUT, FMT = 99998 ) INFO
          ELSE
            WRITE( NOUT, FMT = * ) CHPAR(1:70)
            WRITE( NOUT, FMT = 99997 ) N
            WRITE( NOUT, FMT = 99996 ) M
            WRITE( NOUT, FMT = 99995 ) P
            WRITE( NOUT, FMT = 99994 )
            DO 10  I = 1, N
              WRITE( NOUT, FMT = 99979 ) ( A(I,J), J = 1, N )
       10   CONTINUE
            IF( VEC(5) ) THEN
              WRITE( NOUT, FMT = 99993 )
              DO 20  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( B(I,J), J = 1, M )
       20     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99992 )
            END IF
            IF( VEC(6) ) THEN
              WRITE( NOUT,FMT = 99991 )
              DO 30  I = 1, P
                WRITE( NOUT, FMT = 99979 ) ( C(I,J), J = 1, N )
       30     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99990 )
            END IF
            IF( .NOT.VEC(5) ) THEN
              WRITE( NOUT, FMT = 99989 )
              IF( .NOT.BPAR(2) ) THEN
                ISYMM = ( N * ( N + 1 ) ) / 2
                CALL DCOPY( ISYMM, G, 1, DWORK, 1 )
                IF( BPAR(3) ) THEN
                  CALL MA02DD( 'Unpack', 'Upper', N, G, LDG, DWORK )
                ELSE
                  CALL MA02DD( 'Unpack', 'Lower', N, G, LDG, DWORK )
                END IF
              END IF
              DO 40  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N )
       40     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99988 )
            END IF
            IF( .NOT.VEC(6) ) THEN
              IF( .NOT. BPAR(5) ) THEN
                ISYMM = ( N * ( N + 1 ) ) / 2
                CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
                IF( BPAR(6) ) THEN
                  CALL MA02DD( 'Unpack', 'Upper', N, Q, LDQ, DWORK )
                ELSE
                  CALL MA02DD( 'Unpack', 'Lower', N, Q, LDQ, DWORK )
                END IF
              END IF
              WRITE( NOUT, FMT = 99987 )
              DO 50  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N )
       50     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99986 )
            END IF
            IF( VEC(6) ) THEN
              IF( .NOT.BPAR(5) ) THEN
                ISYMM = ( P * ( P + 1 ) ) / 2
                CALL DCOPY( ISYMM, Q, 1, DWORK, 1 )
                IF( BPAR(6) ) THEN
                  CALL MA02DD( 'Unpack', 'Upper', P, Q, LDQ, DWORK )
                ELSE
                  CALL MA02DD( 'Unpack', 'Lower', P, Q, LDQ, DWORK )
                END IF
              END IF
              WRITE( NOUT, FMT = 99985 )
              DO 60  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( Q(I,J), J = 1, N )
       60     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99984 )
            END IF
            IF( VEC(5) ) THEN
              IF( .NOT.BPAR(2) ) THEN
                ISYMM = ( M * ( M + 1 ) ) / 2
                CALL DCOPY( ISYMM, G, 1, DWORK, 1 )
                IF( BPAR(3) ) THEN
                  CALL MA02DD( 'Unpack', 'Upper', M, G, LDG, DWORK )
                ELSE
                  CALL MA02DD( 'Unpack', 'Lower', M, G, LDG, DWORK )
                END IF
              END IF
              WRITE( NOUT, FMT = 99983 )
              DO 70  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( G(I,J), J = 1, N )
       70     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99982 )
            END IF
            IF( VEC(9) ) THEN
              WRITE( NOUT, FMT = 99981 )
              DO 80  I = 1, N
                WRITE( NOUT, FMT = 99979 ) ( X(I,J), J = 1, N )
       80     CONTINUE
            ELSE
              WRITE( NOUT, FMT = 99980 )
            END IF
          END IF
          STOP
    *
    99999 FORMAT (' BB01AD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' INFO on exit from BB03AD = ', 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 (' A  = ')
    99993 FORMAT (' B  = ')
    99992 FORMAT (' B is not provided.')
    99991 FORMAT (' C  = ')
    99990 FORMAT (' C is not provided.')
    99989 FORMAT (' G  = ')
    99988 FORMAT (' G is not provided.')
    99987 FORMAT (' Q  = ')
    99986 FORMAT (' Q is not provided.')
    99985 FORMAT (' W  = ')
    99984 FORMAT (' W is not provided.')
    99983 FORMAT (' R  = ')
    99982 FORMAT (' R is not provided.')
    99981 FORMAT (' X  = ')
    99980 FORMAT (' X is not provided.')
    99979 FORMAT (20(1X,F8.4))
    *
          END

Program Data

    BB01AD EXAMPLE PROGRAM DATA
    N
    2 3
    6
    .T. .T. .T. .F. .F. .T.
    1
    .1234
    0

Program Results

     BB01AD EXAMPLE PROGRAM RESULTS

     Kenney/Laub/Wette 1989, Ex.2: ARE ill conditioned for EPS -> oo

     Order of matrix A:              N  =   2
     Number of columns in matrix B:  M  =   1
     Number of rows in matrix C:     P  =   2
     A  =
       0.0000   0.1234
       0.0000   0.0000
     B is not provided.
     C  =
       1.0000   0.0000
       0.0000   1.0000
     G  =
       0.0000   0.0000
       0.0000   1.0000
     Q is not provided.
     W  =
       1.0000   0.0000
       0.0000   1.0000
     R is not provided.
     X  =
       9.0486   1.0000
       1.0000   1.1166

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