TG01FD

Orthogonal reduction of a descriptor system to a SVD-like coordinate form

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

Purpose

  To compute for the descriptor system (A-lambda E,B,C) 
  the orthogonal transformation matrices Q and Z such that the
  transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is
  in a SVD-like coordinate form with 
              
               ( A11  A12 )             ( Er  0 )
      Q'*A*Z = (          ) ,  Q'*E*Z = (       ) ,
               ( A21  A22 )             (  0  0 )
  
  where Er is an upper triangular invertible matrix.
  Optionally, the A22 matrix can be further reduced to the form

               ( Ar  X )
         A22 = (       ) ,
               (  0  0 )

  with Ar an upper triangular invertible matrix, and X either a full
  or a zero matrix.  
  The left and/or right orthogonal transformations performed 
  to reduce E and A22 can be optionally accumulated.     

Specification
      SUBROUTINE TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, E, LDE, 
     $                   B, LDB, C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22, 
     $                   TOL, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOBA
      INTEGER            INFO, L, LDA, LDB, LDC, LDE, LDQ, LDWORK, 
     $                   LDZ, M, N, P, RNKA22, RANKE 
      DOUBLE PRECISION   TOL 
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ), 
     $                   DWORK( * ),  E( LDE, * ), Q( LDQ, * ),  
     $                   Z( LDZ, * )

Arguments

Mode Parameters

  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.

  JOBA    CHARACTER*1
          = 'N':  do not reduce A22;
          = 'R':  reduce A22 to a SVD-like upper triangular form.
          = 'T':  reduce A22 to an upper trapezoidal form.

Input/Output Parameters
  L       (input) INTEGER
          The number of rows of matrices A, B, and E.  L >= 0.

  N       (input) INTEGER
          The number of columns of matrices A, E, and C.  N >= 0.

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

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

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading L-by-N part of this array must
          contain the state dynamics matrix A.
          On exit, the leading L-by-N part of this array contains 
          the transformed matrix Q'*A*Z. If JOBA = 'T', this matrix
          is in the form

                        ( A11  *   *  )
               Q'*A*Z = (  *   Ar  X  ) ,
                        (  *   0   0  )

          where A11 is a RANKE-by-RANKE matrix and Ar is a 
          RNKA22-by-RNKA22 invertible upper triangular matrix.
          If JOBA = 'R' then A has the above form with X = 0. 

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

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

                   ( Er  0 )
          Q'*E*Z = (       ) ,
                   (  0  0 )
  
          where Er is a RANKE-by-RANKE upper triangular invertible
          matrix.

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

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading L-by-M part of this array must
          contain the input/state matrix B.
          On exit, the leading L-by-M part of this array contains
          the transformed matrix Q'*B.

  LDB     INTEGER
          The leading dimension of array B.  
          LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.

  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 array C.  LDC >= MAX(1,P).

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

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

  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 array Z.  
          LDZ >= 1,        if COMPZ = 'N';
          LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.

  RANKE   (output) INTEGER
          The estimated rank of matrix E, and thus also the order 
          of the invertible upper triangular submatrix Er.

  RNKA22  (output) INTEGER
          If JOBA = 'R' or 'T', then RNKA22 is the estimated rank of  
          matrix A22, and thus also the order of the invertible 
          upper triangular submatrix Ar.
          If JOBA = 'N', then RNKA22 is not referenced.

Tolerances
  TOL     DOUBLE PRECISION
          The tolerance to be used in determining the rank of E
          and of A22. If the user sets TOL > 0, then the given  
          value of TOL is used as a lower bound for the
          reciprocal condition numbers of leading submatrices
          of R or R22 in the QR decompositions E * P = Q * R of E 
          or A22 * P22 = Q22 * R22 of A22. 
          A submatrix whose estimated condition number is less than
          1/TOL is considered to be of full rank.  If the user sets
          TOL <= 0, then an implicitly computed, default tolerance,
          defined by  TOLDEF = L*N*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 >= MAX( 1, P, MIN(L,N)+MAX(3*N,M,L) ).

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal 
                value.

Method
  The routine computes a truncated QR factorization with column
  pivoting of E, in the form

                    ( E11 E12 )
        E * P = Q * (         )
                    (  0  E22 )

  and finds the largest RANKE-by-RANKE leading submatrix E11 whose
  estimated condition number is less than 1/TOL. RANKE defines thus 
  the rank of matrix E. Further E22, being negligible, is set to 
  zero, and an orthogonal matrix Y is determined such that

        ( E11 E12 ) = ( Er  0 ) * Y .

  The overal transformation matrix Z results as Z = P * Y and the
  resulting transformed matrices Q'*A*Z and Q'*E*Z have the form

                       ( Er  0 )                      ( A11  A12 )
      E <- Q'* E * Z = (       ) ,  A <- Q' * A * Z = (          ) ,
                       (  0  0 )                      ( A21  A22 )

  where Er is an upper triangular invertible matrix.
  If JOBA = 'R' the same reduction is performed on A22 to obtain it
  in the form

               ( Ar  0 )
         A22 = (       ) ,
               (  0  0 )

  with Ar an upper triangular invertible matrix.    
  If JOBA = 'T' then A22 is row compressed using the QR factorization
  with column pivoting to the form

               ( Ar  X )
         A22 = (       ) 
               (  0  0 )

  with Ar an upper triangular invertible matrix.    

  The transformations are also applied to the rest of system 
  matrices 

       B <- Q' * B, C <- C * Z. 

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

Further Comments
  None
Example

Program Text

*     TG01FD EXAMPLE PROGRAM TEXT
*     RELEASE 4.5, Copyright (c) 2002-2005 NICONET Int. Society.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          LMAX, NMAX, MMAX, PMAX
      PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
      PARAMETER        ( LDA = LMAX, LDB = LMAX, LDC = PMAX, 
     $                   LDE = LMAX, LDQ = LMAX, LDZ = NMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = MAX( 1, PMAX,
     $                   MIN(LMAX,NMAX)+MAX( 3*NMAX, MMAX, LMAX ) ) )
*     .. Local Scalars ..
      CHARACTER*1      COMPQ, COMPZ, JOBA
      INTEGER          I, INFO, J, L, M, N, P, RANKE, RNKA22
      DOUBLE PRECISION TOL
*     .. Local Arrays ..
      INTEGER          IWORK(NMAX)
      DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
     $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX),
     $                 Z(LDZ,NMAX)
*     .. External Subroutines ..
      EXTERNAL         TG01FD
*     .. Intrinsic Functions ..
      INTRINSIC        MAX, MIN
*     .. Executable Statements ..
*
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) L, N, M, P, TOL
      COMPQ = 'I'
      COMPZ = 'I'
      JOBA = 'R'
      IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
         WRITE ( NOUT, FMT = 99989 ) L
      ELSE
         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,L )
            READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,L )
            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,L )
               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 transformed descriptor system
*                 (A-lambda E,B,C).
                  CALL TG01FD( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, 
     $                         E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, 
     $                         RANKE, RNKA22, TOL, IWORK, DWORK, LDWORK,
     $                         INFO )
*
                  IF ( INFO.NE.0 ) THEN
                     WRITE ( NOUT, FMT = 99998 ) INFO
                  ELSE
                     WRITE ( NOUT, FMT = 99994 ) RANKE, RNKA22
                     WRITE ( NOUT, FMT = 99997 )
                     DO 10 I = 1, L
                        WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
   10                CONTINUE
                     WRITE ( NOUT, FMT = 99996 )
                     DO 20 I = 1, L
                        WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
   20                CONTINUE
                     WRITE ( NOUT, FMT = 99993 )
                     DO 30 I = 1, L
                        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, L
                        WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
   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
      END IF
      STOP
*
99999 FORMAT (' TG01FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01FD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Rank of matrix E   =', I5/
     $        ' Rank of matrix A22 =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99989 FORMAT (/' L is out of range.',/' L = ',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)
      END
Program Data
TG01FD EXAMPLE PROGRAM DATA
  4    4     2     2     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
 TG01FD EXAMPLE PROGRAM RESULTS

 Rank of matrix E   =    3
 Rank of matrix A22 =    1

 The transformed state dynamics matrix Q'*A*Z is 
   2.0278   0.1078   3.9062  -2.1571
  -0.0980   0.2544   1.6053  -0.1269
   0.2713   0.7760  -0.3692  -0.4853
   0.0690  -0.5669  -2.1974   0.3086

 The transformed descriptor matrix Q'*E*Z is 
  10.1587   5.8230   1.3021   0.0000
   0.0000  -2.4684  -0.1896   0.0000
   0.0000   0.0000   1.0338   0.0000
   0.0000   0.0000   0.0000   0.0000

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

 The transformed state/output matrix C*Z is 
   0.3651  -1.0000  -0.4472  -0.8165
  -1.0954   1.0000  -0.8944   0.0000

 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 
  -0.3651   0.0000   0.4472   0.8165
  -0.9129   0.0000   0.0000  -0.4082
   0.0000  -1.0000   0.0000   0.0000
  -0.1826   0.0000  -0.8944   0.4082

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