SB02MU

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

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

  and for a discrete-time problem by

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

  or

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

  where 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 SB02MU( DICO, HINV, UPLO, N, A, LDA, G, LDG, Q, LDQ, S,
     $                   LDS, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO, HINV, 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'.

  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/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the matrix A.
          On exit, if DICO = 'D', and INFO = 0, the leading N-by-N
                                                  -1
          part of this array contains the matrix A  .
          Otherwise, the array A is unchanged on exit.

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

  G       (input) DOUBLE PRECISION array, dimension (LDG,N)
          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. The stricly
          lower triangular part (if UPLO = 'U') or stricly upper
          triangular part (if UPLO = 'L') is not referenced.

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

  Q       (input) DOUBLE PRECISION array, dimension (LDQ,N)
          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. The stricly
          lower triangular part (if UPLO = 'U') or stricly upper
          triangular part (if UPLO = 'L') is not referenced.

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

  IWORK   INTEGER array, dimension (2*N)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK; if DICO = 'D', DWORK(2) returns the reciprocal
          condition number of the given matrix  A.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 1          if DICO = 'C';
          LDWORK >= MAX(2,4*N) if DICO = 'D'.
          For optimum performance LDWORK should be larger, if
          DICO = 'D'.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

  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.

  None

Program Text

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