## TG01LY

### Finite-infinite decomposition of a structured 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), with the A and E matrices in the form

            ( A11 A12 A13 )             ( E11  0  0 )
        A = ( A21 A22 A23 ) ,       E = (  0   0  0 ) ,              (1)
            ( A31  0   0  )             (  0   0  0 )

      where E11 and A22 are nonsingular and upper triangular matrices,
      to the form

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

      where the subpencil Af-lambda*Ef contains the finite eigenvalues
      and the subpencil Ai-lambda*Ei contains the infinite eigenvalues.
      The subpencil Ai-lambda*Ei is in a staircase form with the
      matrices Ai and Ei of form

            ( A0,0  A0,k ... A0,1 )         ( 0  E0,k ... E0,1 )
       Ai = (  0    Ak,k ... Ak,1 ) ,  Ei = ( 0   0   ... Ek,1 ) ,   (2)
            (  :     :   ...   :  )         ( :   :   ...   :  )
            (  0     0   ... A1,1 )         ( 0   0   ...   0  )

      where Ai,i, for i = 0, 1, ..., k, are nonsingular upper triangular
      matrices.

Specification

          SUBROUTINE TG01LY( COMPQ, COMPZ, N, M, P, RANKE, RNKA22, A, LDA,
         $                   E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NF,
         $                   NIBLCK, IBLCK, TOL, IWORK, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          LOGICAL            COMPQ, COMPZ
          INTEGER            INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M,
         $                   N, NF, NIBLCK, P, RANKE, RNKA22
          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

      COMPQ   LOGICAL
              Specify the option to accumulate or not the performed
              left transformations:
              COMPQ = .FALSE. : do not accumulate the transformations;
              COMPQ = .TRUE.  : accumulate the transformations; in this
                      case, Q must contain an orthogonal matrix Q1
                      on entry, and the product Q1*Q is returned.

      COMPZ   LOGICAL
              Specify the option to accumulate or not the performed
              right transformations:
              COMPZ = .FALSE. : do not accumulate the transformations;
              COMPZ = .TRUE.  : accumulate the transformations; in this
                      case, 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.

      RANKE   (input) INTEGER
              The rank of the matrix E; also, the order of the upper
              triangular matrix E11.  0 <= RANKE <= N.

      RNKA22  (input) DOUBLE PRECISION
              The order of the nonsingular submatrix A22 of A.
              0 <= RNKA22 <= N - RANKE.

      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 in the form (1).
              On exit, the leading N-by-N part of this array contains
              the transformed state matrix Q'*A*Z,

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

              where Af is NF-by-NF and Ai is (N-NF)-by-(N-NF).
              The submatrix Ai is in the staircase form (2), where A0,0
              is (N-RANKE)-by-(N-RANKE), 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 in the form (1).
              On exit, the leading N-by-N part of this array contains
              the transformed descriptor matrix Q'*E*Z,

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

              where Ef is an NF-by-NF nonsingular matrix and Ei is an
              (N-NF)-by-(N-NF) nilpotent matrix in the staircase
              form (2).

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              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,P).

      Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
              If COMPQ = .FALSE., Q is not referenced.
              If COMPQ = .TRUE., 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 = .FALSE.;
              LDQ >= MAX(1,N), if COMPQ = .TRUE. .

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
              If COMPZ = .FALSE., Z is not referenced.
              If COMPZ = .TRUE., 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 = .FALSE.;
              LDZ >= MAX(1,N), if COMPZ = .TRUE. .

      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.

      NIBLCK  (output) INTEGER
              If RANKE < N, the number of infinite blocks minus one.
              If RANKE = N, NIBLCK = 0.

      IBLCK   (output) INTEGER array, dimension (N)
              IBLCK(i) contains the dimension of the i-th block in the
              staircase form (2), where 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-RANKE)

      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 RANKE = N; otherwise,
              LDWORK >= MAX(4*(N-RANKE)-1, N-RANKE-RNKA22+MAX(N,M)).
              For optimal 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;
              = 1:  the pencil A-lambda*E is not regular.

Method

      The subroutine is based on the reduction algorithm of [1].

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
              Sum  IBLCK(i) = RANKE - 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

      None

Program Data

      None

Program Results

      None

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