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<HTML>
<HEAD><TITLE>AB08MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB08MD">AB08MD</A></H2>
<H3>
Normal rank of the transfer-function matrix of a state space model
</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 normal rank of the transfer-function matrix of a
state-space model (A,B,C,D).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB08MD( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ RANK, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER INFO, LDA, LDB, LDC, LDD, LDWORK, M, N, P, RANK
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the compound
matrix (see METHOD) as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The number of state variables, i.e., the order of the
matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
state dynamics matrix A of the system.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input/state matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
state/output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
RANK (output) INTEGER
The normal rank of the transfer-function matrix.
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH).
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (2*N+MAX(M,P)+1)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= (N+P)*(N+M) +
MAX( MIN(P,M) + MAX(3*M-1,N), 1,
MIN(P,N) + MAX(3*P-1,N+P,N+M) )
For optimum performance LDWORK should be larger.
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.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine reduces the (N+P)-by-(M+N) compound matrix (B A)
(D C)
to one with the same invariant zeros and with D of full row rank.
The normal rank of the transfer-function matrix is the rank of D.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of
the Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is backward stable (see [2] and [1]).
</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>
<HR>
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