## SB02OY

### Constructing the extended Hamiltonian or symplectic matrix pairs for linear-quadratic optimization problems, and compressing them to 2N-by-2N matrices

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

Purpose

      To construct the extended matrix pairs for the computation of the
      solution of the algebraic matrix Riccati equations arising in the
      problems of optimal control, both discrete and continuous-time,
      and of spectral factorization, both discrete and continuous-time.
      These matrix pairs, of dimension 2N + M, are given by

            discrete-time                   continuous-time

      |A   0   B|     |E   0   0|    |A   0   B|     |E   0   0|
      |Q  -E'  L| - z |0  -A'  0|,   |Q   A'  L| - s |0  -E'  0|.   (1)
      |L'  0   R|     |0  -B'  0|    |L'  B'  R|     |0   0   0|

      After construction, these pencils are compressed to a form
      (see [1])

         lambda x A  - B ,
                   f    f

      where A  and B  are 2N-by-2N matrices.
             f      f
                               -1
      Optionally, matrix G = BR  B' may be given instead of B and R;
      then, for L = 0, 2N-by-2N matrix pairs are directly constructed as

          discrete-time            continuous-time

      |A   0 |     |E   G |    |A  -G |     |E   0 |
      |      | - z |      |,   |      | - s |      |.               (2)
      |Q  -E'|     |0  -A'|    |Q   A'|     |0  -E'|

      Similar pairs are obtained for non-zero L, if SLICOT Library
      routine SB02MT is called before SB02OY.
      Other options include the case with E identity matrix, L a zero
      matrix, or Q and/or R given in a factored form, Q = C'C, R = D'D.
      For spectral factorization problems, there are minor differences
      (e.g., B is replaced by C').
      The second matrix in (2) is not constructed in the continuous-time
      case if E is specified as being an identity matrix.

Specification

          SUBROUTINE SB02OY( TYPE, DICO, JOBB, FACT, UPLO, JOBL, JOBE, N, M,
         $                   P, A, LDA, B, LDB, Q, LDQ, R, LDR, L, LDL, E,
         $                   LDE, AF, LDAF, BF, LDBF, TOL, IWORK, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, FACT, JOBB, JOBE, JOBL, TYPE, UPLO
          INTEGER           INFO, LDA, LDAF, LDB, LDBF, LDE, LDL, LDQ, LDR,
         $                  LDWORK, M, N, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), AF(LDAF,*), B(LDB,*), BF(LDBF,*),
         $                  DWORK(*), E(LDE,*), L(LDL,*), Q(LDQ,*), R(LDR,*)

Arguments

Mode Parameters

      TYPE    CHARACTER*1
              Specifies the type of problem to be addressed as follows:
              = 'O':  Optimal control problem;
              = 'S':  Spectral factorization problem.

      DICO    CHARACTER*1
              Specifies the type of linear system considered as follows:
              = 'C':  Continuous-time system;
              = 'D':  Discrete-time system.

      JOBB    CHARACTER*1
              Specifies whether or not the matrix G is given, instead
              of the matrices B and R, as follows:
              = 'B':  B and R are given;
              = 'G':  G is given.
              For JOBB = 'G', a 2N-by-2N matrix pair is directly
              obtained assuming L = 0 (see the description of JOBL).

      FACT    CHARACTER*1
              Specifies whether or not the matrices Q and/or R (if
              JOBB = 'B') are factored, as follows:
              = 'N':  Not factored, Q and R are given;
              = 'C':  C is given, and Q = C'C;
              = 'D':  D is given, and R = D'D (if TYPE = 'O'), or
                      R = D + D' (if TYPE = 'S');
              = 'B':  Both factors C and D are given, Q = C'C, R = D'D
                      (or R = D + D').

      UPLO    CHARACTER*1
              If JOBB = 'G', or FACT = 'N', specifies which triangle of
              the matrices G and Q (if FACT = 'N'), or Q and R (if
              JOBB = 'B'), is stored, as follows:
              = 'U':  Upper triangle is stored;
              = 'L':  Lower triangle is stored.

      JOBL    CHARACTER*1
              Specifies whether or not the matrix L is zero, as follows:
              = 'Z':  L is zero;
              = 'N':  L is nonzero.
              JOBL is not used if JOBB = 'G' and JOBL = 'Z' is assumed.
              Using SLICOT Library routine SB02MT to compute the
              corresponding A and Q in this case, before calling SB02OY,
              enables to obtain 2N-by-2N matrix pairs directly.

      JOBE    CHARACTER*1
              Specifies whether or not the matrix E is identity, as
              follows:
              = 'I':  E is the identity matrix;
              = 'N':  E is a general matrix.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A, Q, and E, and the number
              of rows of the matrices B and L.  N >= 0.

      M       (input) INTEGER
              If JOBB = 'B', M is the order of the matrix R, and the
              number of columns of the matrix B.  M >= 0.
              M is not used if JOBB = 'G'.

      P       (input) INTEGER
              If FACT = 'C' or 'D' or 'B', or if TYPE = 'S', P is the
              number of rows of the matrix C and/or D, respectively.
              P >= 0, and if JOBB = 'B' and TYPE = 'S', then P = M.
              Otherwise, P is not used.

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              state matrix A of the system.

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

      B       (input) DOUBLE PRECISION array, dimension (LDB,*)
              If JOBB = 'B', the leading N-by-M part of this array must
              contain the input matrix B of the system.
              If JOBB = 'G', the leading N-by-N upper triangular part
              (if UPLO = 'U') or lower triangular part (if UPLO = 'L')
              of this array must contain the upper triangular part or
              lower triangular part, respectively, of the matrix
                    -1
              G = BR  B'. The strictly lower triangular part (if
              UPLO = 'U') or strictly upper triangular part (if
              UPLO = 'L') is not referenced.

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

      Q       (input) DOUBLE PRECISION array, dimension (LDQ,N)
              If FACT = 'N' or 'D', the leading N-by-N upper triangular
              part (if UPLO = 'U') or lower triangular part (if UPLO =
              'L') of this array must contain the upper triangular part
              or lower triangular part, respectively, of the symmetric
              output weighting matrix Q. The strictly lower triangular
              part (if UPLO = 'U') or strictly upper triangular part (if
              UPLO = 'L') is not referenced.
              If FACT = 'C' or 'B', the leading P-by-N part of this
              array must contain the output matrix C of the system.

      LDQ     INTEGER
              The leading dimension of array Q.
              LDQ >= MAX(1,N) if FACT = 'N' or 'D',
              LDQ >= MAX(1,P) if FACT = 'C' or 'B'.

      R       (input) DOUBLE PRECISION array, dimension (LDR,M)
              If FACT = 'N' or 'C', the leading M-by-M upper triangular
              part (if UPLO = 'U') or lower triangular part (if UPLO =
              'L') of this array must contain the upper triangular part
              or lower triangular part, respectively, of the symmetric
              input weighting matrix R. The strictly lower triangular
              part (if UPLO = 'U') or strictly upper triangular part (if
              UPLO = 'L') is not referenced.
              If FACT = 'D' or 'B', the leading P-by-M part of this
              array must contain the direct transmission matrix D of the
              system.
              If JOBB = 'G', this array is not referenced.

      LDR     INTEGER
              The leading dimension of array R.
              LDR >= MAX(1,M) if JOBB = 'B' and FACT = 'N' or 'C';
              LDR >= MAX(1,P) if JOBB = 'B' and FACT = 'D' or 'B';
              LDR >= 1        if JOBB = 'G'.

      L       (input) DOUBLE PRECISION array, dimension (LDL,M)
              If JOBL = 'N' (and JOBB = 'B'), the leading N-by-M part of
              this array must contain the cross weighting matrix L.
              If JOBL = 'Z' or JOBB = 'G', this array is not referenced.

      LDL     INTEGER
              The leading dimension of array L.
              LDL >= MAX(1,N) if JOBL = 'N';
              LDL >= 1        if JOBL = 'Z' or JOBB = 'G'.

      E       (input) DOUBLE PRECISION array, dimension (LDE,N)
              If JOBE = 'N', the leading N-by-N part of this array must
              contain the matrix E of the descriptor system.
              If JOBE = 'I', E is taken as identity and this array is
              not referenced.

      LDE     INTEGER
              The leading dimension of array E.
              LDE >= MAX(1,N) if JOBE = 'N';
              LDE >= 1        if JOBE = 'I'.

      AF      (output) DOUBLE PRECISION array, dimension (LDAF,*)
              The leading 2N-by-2N part of this array contains the
              matrix A  in the matrix pencil.
                      f
              Array AF must have 2*N+M columns if JOBB = 'B', and 2*N
              columns, otherwise.

      LDAF    INTEGER
              The leading dimension of array AF.
              LDAF >= MAX(1,2*N+M) if JOBB = 'B',
              LDAF >= MAX(1,2*N)   if JOBB = 'G'.

      BF      (output) DOUBLE PRECISION array, dimension (LDBF,2*N)
              If DICO = 'D' or JOBB = 'B' or JOBE = 'N', the leading
              2N-by-2N part of this array contains the matrix B  in the
                                                               f
              matrix pencil.
              The last M zero columns are never constructed.
              If DICO = 'C' and JOBB = 'G' and JOBE = 'I', this array
              is not referenced.

      LDBF    INTEGER
              The leading dimension of array BF.
              LDBF >= MAX(1,2*N+M) if JOBB = 'B',
              LDBF >= MAX(1,2*N)   if JOBB = 'G' and ( DICO = 'D' or
                                                       JOBE = 'N' ),
              LDBF >= 1            if JOBB = 'G' and ( DICO = 'C' and
                                                       JOBE = 'I' ).

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance to be used to test for near singularity of
              the original matrix pencil, specifically of the triangular
              factor obtained during the reduction process. If the user
              sets TOL > 0, then the given value of TOL is used as a
              lower bound for the reciprocal condition number of that
              matrix; a matrix whose estimated condition number is less
              than 1/TOL is considered to be nonsingular. If the user
              sets TOL <= 0, then a default tolerance, defined by
              TOLDEF = EPS, is used instead, where EPS is the machine
              precision (see LAPACK Library routine DLAMCH).
              This parameter is not referenced if JOBB = 'G'.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK >= M if JOBB = 'B',
              LIWORK >= 1 if JOBB = 'G'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK. If JOBB = 'B', DWORK(2) returns the reciprocal
              of the condition number of the M-by-M lower triangular
              matrix obtained after compression.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 1                  if JOBB = 'G',
              LDWORK >= MAX(1,2*N + M,3*M) if JOBB = 'B'.
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if the computed extended matrix pencil is singular,
                    possibly due to rounding errors.

Method

      The extended matrix pairs are constructed, taking various options
      into account. If JOBB = 'B', the problem order is reduced from
      2N+M to 2N (see [1]).

References

      [1] Van Dooren, P.
          A Generalized Eigenvalue Approach for Solving Riccati
          Equations.
          SIAM J. Sci. Stat. Comp., 2, pp. 121-135, 1981.

      [2] Mehrmann, V.
          The Autonomous Linear Quadratic Control Problem. Theory and
          Numerical Solution.
          Lect. Notes in Control and Information Sciences, vol. 163,
          Springer-Verlag, Berlin, 1991.

      [3] Sima, V.
          Algorithms for Linear-Quadratic Optimization.
          Pure and Applied Mathematics: A Series of Monographs and
          Textbooks, vol. 200, Marcel Dekker, Inc., New York, 1996.

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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