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.
Specification
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,*)
Arguments
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.
Input/Output Parameters
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).
Workspace
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'.
Error Indicator
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.
Method
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.Numerical Aspects
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.
Further Comments
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.Example
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