SB02RU

Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems

Purpose

  To construct the 2n-by-2n Hamiltonian or symplectic matrix S
  associated to the linear-quadratic optimization problem, used to
  solve the continuous- or discrete-time algebraic Riccati equation,
  respectively.

  For a continuous-time problem, S is defined by

          ( op(A)   -G    )
      S = (               ),                                     (1)
          (  -Q   -op(A)' )

  and for a discrete-time problem by

                  -1              -1
          (  op(A)           op(A)  *G       )
      S = (        -1                   -1   ),                  (2)
          ( Q*op(A)     op(A)' + Q*op(A)  *G )

  or
                           -T             -T
          ( op(A) + G*op(A)  *Q   -G*op(A)   )
      S = (           -T                 -T  ),                  (3)
          (     -op(A)  *Q          op(A)    )

  where op(A) = A or A' (A**T), A, G, and Q are n-by-n matrices,
  with G and Q symmetric. Matrix A must be nonsingular in the
  discrete-time case.

      SUBROUTINE SB02RU( DICO, HINV, TRANA, UPLO, N, A, LDA, G, LDG, Q,
     $                   LDQ, S, LDS, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO, HINV, TRANA, UPLO
      INTEGER           INFO, LDA, LDG, LDQ, LDS, LDWORK, N
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), DWORK(*), G(LDG,*), Q(LDQ,*),
     $                  S(LDS,*)

Mode Parameters

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

  HINV    CHARACTER*1
          If DICO = 'D', specifies which of the matrices (2) or (3)
          is constructed, as follows:
          = 'D':  The matrix S in (2) is constructed;
          = 'I':  The (inverse) matrix S in (3) is constructed.
          HINV is not referenced if DICO = 'C'.

  TRANA   CHARACTER*1
          Specifies the form of op(A) to be used, as follows:
          = 'N':  op(A) = A    (No transpose);
          = 'T':  op(A) = A**T (Transpose);
          = 'C':  op(A) = A**T (Conjugate transpose = Transpose).

  UPLO    CHARACTER*1
          Specifies which triangle of the matrices G and Q is
          stored, as follows:
          = 'U':  Upper triangle is stored;
          = 'L':  Lower triangle is stored.

  N       (input) INTEGER
          The order of the matrices A, G, and Q.  N >= 0.

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

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

  G       (input/output) DOUBLE PRECISION array, dimension (LDG,N)
          On entry, 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 matrix G.
          On exit, if DICO = 'D', the leading N-by-N part of this
          array contains the symmetric matrix G fully stored.
          If DICO = 'C', this array is not modified on exit, and the
          strictly lower triangular part (if UPLO = 'U') or strictly
          upper triangular part (if UPLO = 'L') is not referenced.

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

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, 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 matrix Q.
          On exit, if DICO = 'D', the leading N-by-N part of this
          array contains the symmetric matrix Q fully stored.
          If DICO = 'C', this array is not modified on exit, and the
          strictly lower triangular part (if UPLO = 'U') or strictly
          upper triangular part (if UPLO = 'L') is not referenced.

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

  S       (output) DOUBLE PRECISION array, dimension (LDS,2*N)
          If INFO = 0, the leading 2N-by-2N part of this array
          contains the Hamiltonian or symplectic matrix of the
          problem.

  LDS     INTEGER
          The leading dimension of the array S.  LDS >= MAX(1,2*N).

  IWORK   INTEGER array, dimension (LIWORK), where
          LIWORK >= 0,   if DICO = 'C';
          LIWORK >= 2*N, if DICO = 'D'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if DICO = 'D', DWORK(1) returns the reciprocal
          condition number  RCOND  of the given matrix  A,  and
          DWORK(2) returns the reciprocal pivot growth factor
          norm(A)/norm(U) (see SLICOT Library routine MB02PD).
          If DWORK(2) is much less than 1, then the computed  S
          and  RCOND  could be unreliable. If 0 < INFO <= N, then
          DWORK(2) contains the reciprocal pivot growth factor for
          the leading INFO columns of  A.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 0,          if DICO = 'C';
          LDWORK >= MAX(2,6*N), if DICO = 'D'.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = i:  if the leading i-by-i (1 <= i <= N) upper triangular
                submatrix of A is singular in discrete-time case;
          = N+1:  if matrix A is numerically singular in discrete-
                time case.

  For a continuous-time problem, the 2n-by-2n Hamiltonian matrix (1)
  is constructed.
  For a discrete-time problem, the 2n-by-2n symplectic matrix (2) or
  (3) - the inverse of the matrix in (2) - is constructed.

  The discrete-time case needs the inverse of the matrix A, hence
  the routine should not be used when A is ill-conditioned.
                            3
  The algorithm requires 0(n ) floating point operations in the
  discrete-time case.

  This routine is a functionally extended and with improved accuracy
  version of the SLICOT Library routine SB02MU. Transposed problems
  can be dealt with as well. The LU factorization of  op(A)  (with
  no equilibration) and iterative refinement are used for solving
  the various linear algebraic systems involved.

Program Text

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