Purpose
To extract from the (N+P)-by-(M+N) system pencil
( B A-lambda*I )
( D C )
an (NR+PR)-by-(M+NR) "reduced" system pencil,
( Br Ar-lambda*I ),
( Dr Cr )
having the same transmission zeros, but with Dr of full row rank.
Specification
SUBROUTINE AB08NY( FIRST, N, M, P, SVLMAX, ABCD, LDABCD, NINFZ,
$ NR, PR, DINFZ, NKRONL, INFZ, KRONL, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
LOGICAL FIRST
INTEGER DINFZ, INFO, LDABCD, LDWORK, M, N, NINFZ,
$ NKRONL, NR, P, PR
DOUBLE PRECISION SVLMAX, TOL
C .. Array Arguments ..
INTEGER INFZ(*), IWORK(*), KRONL(*)
DOUBLE PRECISION ABCD(LDABCD,*), DWORK(*)
Arguments
Mode Parameters
FIRST LOGICAL
Specifies if AB08NY is called first time, or it is called
for an already reduced system, with D of full column rank,
with the last M rows in upper triangular form:
FIRST = .TRUE. : first time called;
FIRST = .FALSE. : not first time called.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C, and the order of the square matrix A.
N >= 0.
M (input) INTEGER
The number of columns of the matrices B and D. M >= 0.
M <= P, if FIRST = .FALSE.
P (input) INTEGER
The number of rows of the matrices C and D. P >= 0.
SVLMAX (input) DOUBLE PRECISION
An estimate of the largest singular value of the original
matrix ABCD (for instance, the Frobenius norm of ABCD).
SVLMAX >= 0.
ABCD (input/output) DOUBLE PRECISION array, dimension
(LDABCD,M+N)
On entry, the leading (N+P)-by-(M+N) part of this array
must contain the compound matrix
( B A ),
( D C )
where A is an N-by-N matrix, B is an N-by-M matrix,
C is a P-by-N matrix, and D is a P-by-M matrix.
If FIRST = .FALSE., then D must be a full column rank
matrix, with the last M rows in an upper triangular form.
On exit, the leading (NR+PR)-by-(M+NR) part of this array
contains the reduced compound matrix
( Br Ar ),
( Dr Cr )
where Ar is an NR-by-NR matrix, Br is an NR-by-M matrix,
Cr is a PR-by-NR matrix, and Dr is a PR-by-M full row rank
left upper-trapezoidal matrix, with the first PR columns
in an upper triangular form.
LDABCD INTEGER
The leading dimension of the array ABCD.
LDABCD >= MAX(1,N+P).
NINFZ (input/output) INTEGER
On entry, the currently computed number of infinite zeros.
It should be initialized to zero on the first call.
NINFZ >= 0.
If FIRST = .FALSE., then NINFZ is not modified.
On exit, the number of infinite zeros.
NR (output) INTEGER
The order of the reduced matrix Ar; also, the number of
rows of the reduced matrix Br and the number of columns of
the reduced matrix Cr.
If Dr is invertible, NR is also the number of finite Smith
zeros.
PR (output) INTEGER
The normal rank of the transfer-function matrix of the
original system; also, the number of rows of the reduced
matrices Cr and Dr.
DINFZ (output) INTEGER
The maximal multiplicity of infinite zeros.
DINFZ = 0 if FIRST = .FALSE. .
NKRONL (output) INTEGER
The maximal dimension of left elementary Kronecker blocks.
INFZ (output) INTEGER array, dimension (N)
INFZ(i) contains the number of infinite zeros of degree i,
where i = 1,2,...,DINFZ.
INFZ is not referenced if FIRST = .FALSE. .
KRONL (output) INTEGER array, dimension (N+1)
KRONL(i) contains the number of left elementary Kronecker
blocks of dimension i-by-(i-1), where i = 1,2,...,NKRONL.
Tolerances
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.
NOTE that when SVLMAX > 0, the estimated ranks could be
less than those defined above (see SVLMAX).
If the user sets TOL to be less than or equal to zero,
then the tolerance is taken as (N+P)*(N+M)*EPS, where EPS
is the machine precision (see LAPACK Library Routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
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 >= 1, if MIN(P, MAX(N,M)) = 0; otherwise,
LDWORK >= MAX( MIN(P,M) + M + MAX(2*M,N) - 1,
MIN(P,N) + MAX(N + MAX( P, M), 3*P - 1 ) ).
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.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
References
[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.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( (P+N)*(M+N)*N ) floating point operations.Further Comments
The number of infinite zeros is computed (if FIRST = .TRUE.) as
DINFZ
NINFZ = Sum (INFZ(i)*i .
i=1
Note that each infinite zero of multiplicity k corresponds to an
infinite eigenvalue of multiplicity k+1.
The multiplicities of the infinite eigenvalues can be determined
from PR, DINFZ and INFZ(i), i = 1, ..., DINFZ, as follows:
DINFZ
- there are PR - Sum (INFZ(i)) simple infinite eigenvalues;
i=1
- there are INFZ(i) infinite eigenvalues with multiplicity i+1,
for i = 1, ..., DINFZ.
The left Kronecker indices are:
[ 0 0 ... 0 | 1 1 ... 1 | .... | NKRONL ... NKRONL ]
|<- KRONL(1) ->|<- KRONL(2) ->| |<- KRONL(NKRONL) ->|
Example
Program Text
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