AB13DX

Maximum singular value of a transfer-function matrix

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.

      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(  * )

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

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.

  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'.

  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.

  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.

  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.

  None

Program Text

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