## AB13DX

### Maximum singular value of a transfer-function matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

To compute the maximum singular value of a given continuous-time
or discrete-time transfer-function matrix, either standard or in
the descriptor form,

-1
G(lambda) = C*( lambda*E - A ) *B + D ,

for a given complex value lambda, where lambda = j*omega, in the
continuous-time case, and lambda = exp(j*omega), in the
discrete-time case. The matrices A, E, B, C, and D are real
matrices of appropriate dimensions. Matrix A must be in an upper
Hessenberg form, and if JOBE ='G', the matrix E must be upper
triangular. The matrices B and C must correspond to the system
in (generalized) Hessenberg form.

Specification
DOUBLE PRECISION FUNCTION AB13DX( DICO, JOBE, JOBD, N, M, P,
\$                                  OMEGA, A, LDA, E, LDE, B, LDB,
\$                                  C, LDC, D, LDD, IWORK, DWORK,
\$                                  LDWORK, CWORK, LCWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          DICO, JOBD, JOBE
INTEGER            INFO, LCWORK, LDA, LDB, LDC, LDD, LDE, LDWORK,
\$                   M, N, P
DOUBLE PRECISION   OMEGA
C     .. Array Arguments ..
COMPLEX*16         CWORK(  * )
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
\$                   D( LDD, * ), DWORK(  * ), E( LDE, * )
INTEGER            IWORK(  * )

Function Value
AB13DX   DOUBLE PRECISION
The maximum singular value of G(lambda).

Arguments

Mode Parameters

DICO    CHARACTER*1
Specifies the type of the system, as follows:
= 'C':  continuous-time system;
= 'D':  discrete-time system.

JOBE    CHARACTER*1
Specifies whether E is an upper triangular or an identity
matrix, as follows:
= 'G':  E is a general upper triangular matrix;
= 'I':  E is the identity matrix.

JOBD    CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D':  D is present;
= 'Z':  D is assumed a zero matrix.

Input/Output Parameters
N       (input) INTEGER
The order of the system.  N >= 0.

M       (input) INTEGER
The column size of the matrix B.  M >= 0.

P       (input) INTEGER
The row size of the matrix C.  P >= 0.

OMEGA   (input) DOUBLE PRECISION
The frequency value for which the calculations should be
done.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the state dynamics matrix A in upper
Hessenberg form. The elements below the subdiagonal are
not referenced.
On exit, if M > 0, P > 0, OMEGA = 0, DICO = 'C', B <> 0,
and C <> 0, the leading N-by-N upper Hessenberg part of
this array contains the factors L and U from the LU
factorization of A (A = P*L*U); the unit diagonal elements
of L are not stored, L is lower bidiagonal, and P is
stored in IWORK (see SLICOT Library routine MB02SD).
Otherwise, this array is unchanged on exit.

LDA     INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

E       (input) DOUBLE PRECISION array, dimension (LDE,N)
If JOBE = 'G', the leading N-by-N upper triangular part of
this array must contain the upper triangular descriptor
matrix E of the system. The elements of the strict lower
triangular part of this array are not referenced.
If JOBE = 'I', then E is assumed to be the identity
matrix and is not referenced.

LDE     INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= 1,        if JOBE = 'I'.

B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, if M > 0, P > 0, OMEGA = 0, DICO = 'C', B <> 0,
C <> 0, and INFO = 0 or N+1, the leading N-by-M part of
this array contains the solution of the system A*X = B.
Otherwise, this array is unchanged on exit.

LDB     INTEGER
The leading dimension of the 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
system output matrix C.

LDC     INTEGER
The leading dimension of the array C.  LDC >= max(1,P).

D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, if JOBD = 'D', the leading P-by-M part of this
array must contain the direct transmission matrix D.
On exit, if (N = 0, or B = 0, or C = 0) and JOBD = 'D',
or (OMEGA = 0, DICO = 'C', JOBD = 'D', and INFO = 0 or
N+1), the contents of this array is destroyed.
Otherwise, this array is unchanged on exit.
This array is not referenced if JOBD = 'Z'.

LDD     INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1,        if JOBD = 'Z'.

Workspace
IWORK   INTEGER array, dimension (LIWORK), where
LIWORK = N, if N > 0, M > 0, P > 0, B <> 0, and C <> 0;
LIWORK = 0, otherwise.
This array contains the pivot indices in the LU
factorization of the matrix lambda*E - A; for 1 <= i <= N,
row i of the matrix was interchanged with row IWORK(i).

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK, and DWORK(2), ..., DWORK(MIN(P,M)) contain the
singular values of G(lambda), except for the first one,
which is returned in the function value AB13DX.
If (N = 0, or B = 0, or C = 0) and JOBD = 'Z', the last
MIN(P,M)-1 zero singular values of G(lambda) are not
stored in DWORK(2), ..., DWORK(MIN(P,M)).

LDWORK  INTEGER
The dimension of the array DWORK.
LDWORK >= MAX(1, LDW1 + LDW2 ),
LDW1 = P*M, if N > 0, B <> 0, C <> 0, OMEGA = 0,
DICO = 'C', and JOBD = 'Z';
LDW1 = 0,   otherwise;
LDW2 = MIN(P,M) + MAX(3*MIN(P,M) + MAX(P,M), 5*MIN(P,M)),
if (N = 0, or B = 0, or C = 0) and JOBD = 'D',
or (N > 0, B <> 0, C <> 0, OMEGA = 0, and
DICO = 'C');
LDW2 = 0,   if (N = 0, or B = 0, or C = 0) and JOBD = 'Z',
or MIN(P,M) = 0;
LDW2 = 6*MIN(P,M), otherwise.
For good performance, LDWORK must generally be larger.

CWORK   COMPLEX*16 array, dimension (LCWORK)
On exit, if INFO = 0, CWORK(1) contains the optimal
LCWORK.

LCWORK  INTEGER
The dimension of the array CWORK.
LCWORK >= 1, if N = 0, or B = 0, or C = 0, or (OMEGA = 0
and DICO = 'C') or MIN(P,M) = 0;
LCWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(P,M) + MAX(P,M)),
otherwise.
For good performance, LCWORK must generally be larger.

Error Indicator
INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
> 0:  if INFO = i, U(i,i) is exactly zero; the LU
factorization of the matrix lambda*E - A has been
completed, but the factor U is exactly singular,
i.e., the matrix lambda*E - A is exactly singular;
= N+1:  the SVD algorithm for computing singular values
did not converge.

Method
The routine implements standard linear algebra calculations,
taking problem structure into account. LAPACK Library routines
DGESVD and ZGESVD are used for finding the singular values.