TG01LD

Finite-infinite decomposition of a descriptor system

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

Purpose

  To compute orthogonal transformation matrices Q and Z which
  reduce the regular pole pencil A-lambda*E of the descriptor system
  (A-lambda*E,B,C) to the form (if JOB = 'F')

             ( Af  *  )             ( Ef  *  )
    Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (1)
             ( 0   Ai )             ( 0   Ei )

  or to the form (if JOB = 'I')

             ( Ai  *  )             ( Ei  *  )
    Q'*A*Z = (        ) ,  Q'*E*Z = (        ) ,                 (2)
             ( 0   Af )             ( 0   Ef )

  where the subpencil Af-lambda*Ef, with Ef nonsingular and upper
  triangular, contains the finite eigenvalues, and the subpencil
  Ai-lambda*Ei, with Ai nonsingular and upper triangular, contains
  the infinite eigenvalues. The subpencil Ai-lambda*Ei is in a
  staircase form (see METHOD). If JOBA = 'H', the submatrix Af
  is further reduced to an upper Hessenberg form.

Specification
      SUBROUTINE TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
     $                   E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF, ND,
     $                   NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOB, JOBA
      INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
     $                   N, ND, NF, NIBLCK, P
      DOUBLE PRECISION   TOL
C     .. Array Arguments ..
      INTEGER            IBLCK( * ), IWORK(*)
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),
     $                   Z( LDZ, * )

Arguments

Mode Parameters

  JOB     CHARACTER*1
          = 'F':  perform the finite-infinite separation;
          = 'I':  perform the infinite-finite separation.

  JOBA    CHARACTER*1
          = 'H':  reduce Af further to an upper Hessenberg form;
          = 'N':  keep Af unreduced.

  COMPQ   CHARACTER*1
          = 'N':  do not compute Q;
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal matrix Q is returned;
          = 'U':  Q must contain an orthogonal matrix Q1 on entry,
                  and the product Q1*Q is returned.

  COMPZ   CHARACTER*1
          = 'N':  do not compute Z;
          = 'I':  Z is initialized to the unit matrix, and the
                  orthogonal matrix Z is returned;
          = 'U':  Z must contain an orthogonal matrix Z1 on entry,
                  and the product Z1*Z is returned.

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 matrices A
          and E.  N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix B.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix C.  P >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the N-by-N state matrix A.
          On exit, the leading N-by-N part of this array contains
          the transformed state matrix Q'*A*Z,

                             ( Af  *  )                 ( Ai  *  )
                    Q'*A*Z = (        ) ,  or  Q'*A*Z = (        ) ,
                             ( 0   Ai )                 ( 0   Af )

          depending on JOB, with Af an NF-by-NF matrix, and Ai an
          (N-NF)-by-(N-NF) nonsingular and upper triangular matrix.
          If JOBA = 'H', Af is in an upper Hessenberg form.
          Otherwise, Af is unreduced.
          Ai has a block structure as in (3) or (4), where A0,0 is
          ND-by-ND and Ai,i , for i = 1, ..., NIBLCK, is
          IBLCK(i)-by-IBLCK(i).

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

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N part of this array must
          contain the N-by-N descriptor matrix E.
          On exit, the leading N-by-N part of this array contains
          the transformed descriptor matrix Q'*E*Z,

                             ( Ef  *  )                 ( Ei  *  )
                    Q'*E*Z = (        ) ,  or  Q'*E*Z = (        ) ,
                             ( 0   Ei )                 ( 0   Ef )

          depending on JOB, with Ef an NF-by-NF nonsingular matrix,
          and Ei an (N-NF)-by-(N-NF) nilpotent matrix in an upper
          block triangular form, as in (3) or (4).

  LDE     INTEGER
          The leading dimension of the array E.  LDE >= MAX(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,K),
          where K = M if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.
          On entry, the leading N-by-M part of this array must
          contain the N-by-M input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed input matrix Q'*B.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= MAX(1,N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the state/output matrix C.
          On exit, the leading P-by-N part of this array contains
          the transformed matrix C*Z.

  LDC     INTEGER
          The leading dimension of the array C.  LDC >= MAX(1,K),
          where K = P if JOB = 'F', and K = MAX(M,P) if JOB = 'I'.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          If COMPQ = 'N':  Q is not referenced.
          If COMPQ = 'I':  on entry, Q need not be set;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix Q,
                           where Q' is the product of Householder
                           transformations applied to A, E, and B on
                           the left.
          If COMPQ = 'U':  on entry, the leading N-by-N part of this
                           array must contain an orthogonal matrix
                           Q1;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix
                           Q1*Q.

  LDQ     INTEGER
          The leading dimension of the array Q.
          LDQ >= 1,        if COMPQ = 'N';
          LDQ >= MAX(1,N), if COMPQ = 'I' or 'U'.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          If COMPZ = 'N':  Z is not referenced.
          If COMPZ = 'I':  on entry, Z need not be set;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix Z,
                           which is the product of Householder
                           transformations applied to A, E, and C on
                           the right.
          If COMPZ = 'U':  on entry, the leading N-by-N part of this
                           array must contain an orthogonal matrix
                           Z1;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix
                           Z1*Z.

  LDZ     INTEGER
          The leading dimension of the array Z.
          LDZ >= 1,        if COMPZ = 'N';
          LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'.

  NF      (output) INTEGER.
          The order of the reduced matrices Af and Ef; also, the
          number of finite generalized eigenvalues of the pencil
          A-lambda*E.

  ND      (output) INTEGER.
          The number of non-dynamic infinite eigenvalues of the
          pair (A,E). Note: N-ND is the rank of the matrix E.

  NIBLCK  (output) INTEGER
          If ND > 0, the number of infinite blocks minus one.
          If ND = 0, then NIBLCK = 0.

  IBLCK   (output) INTEGER array, dimension (N)
          IBLCK(i) contains the dimension of the i-th block in the
          staircase form (3) or (4), with i = 1,2, ..., NIBLCK.

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 factorization with column pivoting whose estimated
          condition number is less than 1/TOL. If the user sets
          TOL <= 0, then an implicitly computed, default tolerance,
          TOLDEF = N**2*EPS,  is used instead, where EPS is the
          machine precision (see LAPACK Library routine DLAMCH).
          TOL < 1.

Workspace
  IWORK   INTEGER array, dimension (N)

  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, and if N > 0,
          LDWORK >= N + MAX(3*N,M,P).

          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;
          = 1:  the pencil A-lambda*E is not regular.

Method
  The subroutine is based on the reduction algorithm of [1].
  If JOB = 'F', the matrices Ai and Ei have the form

        ( A0,0  A0,k ... A0,1 )         ( 0  E0,k ... E0,1 )
   Ai = (  0    Ak,k ... Ak,1 ) ,  Ei = ( 0   0   ... Ek,1 ) ;   (3)
        (  :     :    .    :  )         ( :   :    .    :  )
        (  0     0   ... A1,1 )         ( 0   0   ...   0  )

  if JOB = 'I', the matrices Ai and Ei have the form

        ( A1,1 ... A1,k  A1,0 )         ( 0 ... E1,k  E1,0 )
   Ai = (  :    .    :    :   ) ,  Ei = ( :  .    :    :   ) ,   (4)
        (  :   ... Ak,k  Ak,0 )         ( : ...   0   Ek,0 )
        (  0   ...   0   A0,0 )         ( 0 ...   0     0  )

  where Ai,i , for i = 0, 1, ..., k, are nonsingular upper
  triangular matrices. A0,0 corresponds to the non-dynamic infinite
  modes of the system.

References
  [1] Misra, P., Van Dooren, P., and Varga, A.
      Computation of structural invariants of generalized
      state-space systems.
      Automatica, 30, pp. 1921-1936, 1994.

Numerical Aspects
  The algorithm is numerically backward stable and requires
  0( N**3 )  floating point operations.

Further Comments
  The number of infinite poles is computed as

                NIBLCK
     NINFP =     Sum  IBLCK(i) = N - ND - NF.
                 i=1

  The multiplicities of infinite poles can be computed as follows:
  there are IBLCK(k)-IBLCK(k+1) infinite poles of multiplicity
  k, for k = 1, ..., NIBLCK, where IBLCK(NIBLCK+1) = 0.
  Note that each infinite pole of multiplicity k corresponds to
  an infinite eigenvalue of multiplicity k+1.

Example

Program Text

*     TG01LD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
     $                   LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = NMAX+MAX( 3*NMAX, MMAX, PMAX ) )
*     .. Local Scalars ..
      CHARACTER*1      COMPQ, COMPZ, JOB, JOBA
      INTEGER          I, INFO, J, M, N, ND, NF, NIBLCK, P
      DOUBLE PRECISION TOL
*     .. Local Arrays ..
      INTEGER          IBLCK(NMAX), IWORK(NMAX)
      DOUBLE PRECISION A(LDA,NMAX),   B(LDB,MMAX), C(LDC,NMAX),
     $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
     $                 Z(LDZ,NMAX)
*     .. External Subroutines ..
      EXTERNAL         TG01LD
*     .. Intrinsic Functions ..
      INTRINSIC        MAX
*     .. Executable Statements ..
*
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) N, M, P, JOB, JOBA, TOL
      COMPQ = 'I'
      COMPZ = 'I'
      IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99988 ) N
      ELSE
         READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
         READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
         IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
            WRITE ( NOUT, FMT = 99987 ) M
         ELSE
            READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
            IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
               WRITE ( NOUT, FMT = 99986 ) P
            ELSE
               READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Find the reduced descriptor system
*              (A-lambda E,B,C).
               CALL TG01LD( JOB, JOBA, COMPQ, COMPZ, N, M, P, A, LDA,
     $                      E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF,
     $                      ND, NIBLCK, IBLCK, TOL, IWORK, DWORK,
     $                      LDWORK, INFO )
*
               IF ( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99994 ) NF, ND
                  WRITE ( NOUT, FMT = 99989 ) NIBLCK + 1
                  IF ( NIBLCK.GT.0 ) THEN
                     WRITE ( NOUT, FMT = 99985 )
     $                     ( IBLCK(I), I = 1, NIBLCK ) 
                  END IF
                  WRITE ( NOUT, FMT = 99997 )
                  DO 10 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
   10             CONTINUE
                  WRITE ( NOUT, FMT = 99996 )
                  DO 20 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
   20             CONTINUE
                  WRITE ( NOUT, FMT = 99993 )
                  DO 30 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   30             CONTINUE
                  WRITE ( NOUT, FMT = 99992 )
                  DO 40 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
   40             CONTINUE
                  WRITE ( NOUT, FMT = 99991 )
                  DO 50 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
   50             CONTINUE
                  WRITE ( NOUT, FMT = 99990 )
                  DO 60 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
   60             CONTINUE
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' TG01LD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01LD = ',I2)
99997 FORMAT (/' The reduced state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The reduced descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Order of reduced system =', I5/
     $        ' Number of non-dynamic infinite eigenvalues =', I5)
99993 FORMAT (/' The reduced input/state matrix Q''*B is ')
99992 FORMAT (/' The reduced state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT ( ' Number of infinite blocks = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT ( ' Dimension of the blocks'/20I5)
      END
Program Data
TG01LD EXAMPLE PROGRAM DATA
  4     2     2     F     N     0.0    
    -1     0     0     3
     0     0     1     2
     1     1     0     4
     0     0     0     0
     1     2     0     0
     0     1     0     1
     3     9     6     3
     0     0     2     0
     1     0
     0     0
     0     1
     1     1
    -1     0     1     0
     0     1    -1     1
Program Results
 TG01LD EXAMPLE PROGRAM RESULTS

 Order of reduced system =    3
 Number of non-dynamic infinite eigenvalues =    1
 Number of infinite blocks =     1

 The reduced state dynamics matrix Q'*A*Z is 
   2.4497  -1.3995   0.2397  -4.0023
  -0.0680  -0.0030   0.1739  -1.6225
   0.3707   0.0161  -0.9482   0.1049
   0.0000   0.0000   0.0000   2.2913

 The reduced descriptor matrix Q'*E*Z is 
   9.9139   4.7725  -3.4725  -2.3836
   0.0000  -1.2024   2.0137   0.7926
   0.0000   0.0000   0.2929  -0.9914
   0.0000   0.0000   0.0000   0.0000

 The reduced input/state matrix Q'*B is 
  -0.2157  -0.9705
   0.3015   0.9516
   0.7595   0.0991
   1.1339   0.3780

 The reduced state/output matrix C*Z is 
   0.5345  -1.1134   0.3758   0.5774
  -1.0690   0.2784  -1.2026   0.5774

 The left transformation matrix Q is 
  -0.2157  -0.5088   0.6109   0.5669
  -0.1078  -0.2544  -0.7760   0.5669
  -0.9705   0.1413  -0.0495  -0.1890
   0.0000   0.8102   0.1486   0.5669

 The right transformation matrix Z is 
  -0.5345   0.6263   0.4617  -0.3299
  -0.8018  -0.5219  -0.2792  -0.0825
   0.0000  -0.4871   0.8375   0.2474
  -0.2673   0.3132  -0.0859   0.9073

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