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<HTML>
<HEAD><TITLE>SB02MX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB02MX">SB02MX</A></H2>
<H3>
Conversion of optimal problems with coupling weighting terms to standard 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 compute the following matrices
-1
G = B*R *B',
- -1
A = A +/- op(B*R *L'),
- -1
Q = Q +/- L*R *L',
where A, B, Q, R, L, and G are N-by-N, N-by-M, N-by-N, M-by-M,
N-by-M, and N-by-N matrices, respectively, with Q, R and G
symmetric matrices, and op(W) is one of
op(W) = W or op(W) = W'.
When R is well-conditioned with respect to inversion, standard
algorithms for solving linear-quadratic optimization problems will
then also solve optimization problems with coupling weighting
matrix L. Moreover, a gain in efficiency is possible using matrix
G in the deflating subspace algorithms (see SLICOT Library routine
SB02OD) or in the Newton's algorithms (see SLICOT Library routine
SG02CD).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB02MX( JOBG, JOBL, FACT, UPLO, TRANS, FLAG, DEF, N, M,
$ A, LDA, B, LDB, Q, LDQ, R, LDR, L, LDL, IPIV,
$ OUFACT, G, LDG, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DEF, FACT, FLAG, JOBG, JOBL, TRANS, UPLO
INTEGER INFO, LDA, LDB, LDG, LDL, LDQ, LDR, LDWORK, M,
$ N, OUFACT
C .. Array Arguments ..
INTEGER IPIV(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), G(LDG,*),
$ L(LDL,*), Q(LDQ,*), R(LDR,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOBG CHARACTER*1
Specifies whether or not the matrix G is to be computed,
as follows:
= 'G': Compute G;
= 'N': Do not compute G.
JOBL CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z': L is zero;
= 'N': L is nonzero.
FACT CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N': Array R contains the matrix R;
= 'C': Array R contains the Cholesky factor of R;
= 'U': Array R contains the factors of the symmetric
indefinite UdU' or LdL' factorization of R.
UPLO CHARACTER*1
Specifies which triangle of the matrices R, Q (if
JOBL = 'N'), and G (if JOBG = 'G') is stored, as follows:
= 'U': Upper triangle is stored;
= 'L': Lower triangle is stored.
TRANS CHARACTER*1
Specifies the form of op(W) to be used in the matrix
multiplication, as follows:
= 'N': op(W) = W;
= 'T': op(W) = W';
= 'C': op(W) = W'.
FLAG CHARACTER*1
Specifies which sign is used, as follows:
= 'P': The plus sign is used;
= 'M': The minus sign is used.
DEF CHARACTER*1
If FACT = 'N', specifies whether or not it is assumed that
matrix R is positive definite, as follows:
= 'D': Matrix R is assumed positive definite;
= 'I': Matrix R is assumed indefinite.
Both values can be used to perform the computations,
irrespective to the R definiteness, but using the adequate
value will save some computational effort (see FURTHER
COMMENTS).
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrices A, Q, and G, and the number of
rows of the matrices B and L. N >= 0.
M (input) INTEGER
The order of the matrix R, and the number of columns of
the matrices B and L. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if JOBL = 'N', the leading N-by-N part of this
array must contain the matrix A.
On exit, if JOBL = 'N', and INFO = 0, the leading N-by-N
-
part of this array contains the matrix A.
If JOBL = 'Z', this array is not referenced.
LDA INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if JOBL = 'N';
LDA >= 1 if JOBL = 'Z'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, if OUFACT = 1, and INFO = 0, the leading N-by-M
-1
part of this array contains the matrix B*chol(R) .
On exit, B is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBL = '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.
On exit, if JOBL = 'N' and INFO = 0, the leading N-by-N
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array contains the upper
triangular part or lower triangular part, respectively, of
- -1
the symmetric matrix Q = Q +/- L*R *L'.
If JOBL = 'Z', this array is not referenced.
LDQ INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if JOBL = 'N';
LDQ >= 1 if JOBL = 'Z'.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', 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.
On entry, if FACT = '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 Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if FACT = 'U', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the factors of
the UdU' or LdL' factorization, respectively, of the
symmetric indefinite input weighting matrix R (as produced
by LAPACK routine DSYTRF).
If FACT = 'N' and DEF = 'D', the strictly lower triangular
part (if UPLO = 'U') or strictly upper triangular part
(if UPLO = 'L') of this array is used as workspace (filled
in by symmetry). If FACT = 'N' and DEF = 'I', the strictly
lower triangular part (if UPLO = 'U') or strictly upper
triangular part (if UPLO = 'L') is unchanged.
On exit, if OUFACT = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix.
On exit, if OUFACT = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix.
On exit R is unchanged if FACT = 'C' or 'U'.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,M).
L (input/output) DOUBLE PRECISION array, dimension (LDL,M)
On entry, if JOBL = 'N', the leading N-by-M part of this
array must contain the matrix L.
On exit, if JOBL = 'N', OUFACT = 1, and INFO = 0, the
leading N-by-M part of this array contains the matrix
-1
L*chol(R) .
On exit, L is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
L is not referenced if JOBL = 'Z'.
LDL INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1 if JOBL = 'Z'.
IPIV (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R,
as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'C'.
OUFACT (output) INTEGER
Information about the factorization finally used.
OUFACT = 0: no factorization of R has been used (M = 0);
OUFACT = 1: Cholesky factorization of R has been used;
OUFACT = 2: UdU' (if UPLO = 'U') or LdL' (if UPLO = 'L')
factorization of R has been used.
G (output) DOUBLE PRECISION array, dimension (LDG,N)
If JOBG = 'G', and INFO = 0, the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array contains the upper
triangular part (if UPLO = 'U') or lower triangular part
-1
(if UPLO = 'L'), respectively, of the matrix G = B*R B'.
If JOBG = 'N', this array is not referenced.
LDG INTEGER
The leading dimension of array G.
LDG >= MAX(1,N) if JOBG = 'G';
LDG >= 1 if JOBG = 'N'.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (M)
If FACT = 'C' or FACT = 'U', this array is not referenced.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK; if FACT = 'N' and LDWORK is set
as specified below, DWORK(2) contains the reciprocal
condition number of the given matrix R. DWORK(2) is set to
zero if M = 0.
On exit, if LDWORK = -2 on input or INFO = -26, then
DWORK(1) returns the minimal value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= 1 if FACT = 'C' or (FACT = 'U' and
JOBG = 'N' and JOBL = 'Z');
LDWORK >= MAX(2,3*M) if FACT = 'N' and JOBG = 'N' and
JOBL = 'Z';
LDWORK >= MAX(2,3*M,N*M) if FACT = 'N' and (JOBG = 'G' or
JOBL = 'N');
LDWORK >= MAX(1,N*M) if FACT = 'U' and (JOBG = 'G' or
JOBL = 'N').
For optimum performance LDWORK should be larger than 3*M,
if FACT = 'N'.
If LDWORK = -1, an optimal 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 is issued by XERBLA.
If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message 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 i-th element (1 <= i <= M) of the d factor is
exactly zero; the UdU' (or LdL') factorization has
been completed, but the block diagonal matrix d is
exactly singular;
= M+1: if the matrix R is numerically singular.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE> - -
The matrices G, and/or A and Q are evaluated using the given or
computed symmetric factorization of R.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The routine should not be used when R is ill-conditioned.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
Using argument TRANS allows to avoid the transposition of matrix A
needed to solve optimal filtering/estimation problems by the same
routines solving optimal control problems.
If DEF is set to 'D', but R is indefinite, the computational
effort for factorization will be approximately double, since
Cholesky factorization, tried first, will fail, and symmetric
indefinite factorization will then be used.
If DEF is set to 'I', but R is positive definite, the
computational effort will be slightly higher than that when using
Cholesky factorization. It is recommended to use DEF = 'D' also if
the definiteness is not known, but M is (much) smaller than N.
</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>
<HR>
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