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<HTML>
<HEAD><TITLE>SB02MU - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB02MU">SB02MU</A></H2>
<H3>
Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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 strictly lower triangular part (if UPLO = 'U') or
strictly 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 strictly lower triangular part (if UPLO = 'U') or
strictly 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).
</PRE>
<B>Workspace</B>
<PRE>
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.
</PRE>
<B>Error Indicator</B>
<PRE>
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.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
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.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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