## TG01CD

### Orthogonal reduction of a descriptor system pair (A-lambda E,B) to the QR-coordinate form

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

Purpose

      To reduce the descriptor system pair (A-lambda E,B) to the
      QR-coordinate form by computing an orthogonal transformation
      matrix Q such that the transformed descriptor system pair
      (Q'*A-lambda Q'*E, Q'*B) has the descriptor matrix Q'*E
      in an upper trapezoidal form.
      The left orthogonal transformations performed to reduce E
      can be optionally accumulated.

Specification

          SUBROUTINE TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB, Q, LDQ,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ
          INTEGER            INFO, L, LDA, LDB, LDE, LDQ, LDWORK, M, N
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
         $                   E( LDE, * ), Q( LDQ, * )

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.

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 and E.  N >= 0.

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

      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 in upper trapezoidal form,
              i.e.

                       ( E11 )
                Q'*E = (     ) ,     if L >= N ,
                       (  0  )
              or

                Q'*E = ( E11 E12 ),  if L < N ,

              where E11 is an MIN(L,N)-by-MIN(L,N) upper triangular
              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.

      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'.

Workspace

      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, MIN(L,N) + MAX(L,N,M)).
              For optimum performance
              LWORK >= MAX(1, MIN(L,N) + MAX(L,N,M)*NB),
              where NB is the optimal blocksize.

Error Indicator

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

Method

      The routine computes the QR factorization of E to reduce it
      to the upper trapezoidal form.

      The transformations are also applied to the rest of system
      matrices

          A <- Q' * A ,  B <- Q' * B.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

    *     TG01CD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          LMAX, NMAX, MMAX
          PARAMETER        ( LMAX = 20, NMAX = 20, MMAX = 20)
          INTEGER          LDA, LDB, LDE, LDQ
          PARAMETER        ( LDA = LMAX, LDB = LMAX,
         $                   LDE = LMAX, LDQ = LMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MIN(LMAX,NMAX)+MAX(LMAX,NMAX,MMAX) )
    *     .. Local Scalars ..
          CHARACTER*1      COMPQ
          INTEGER          I, INFO, J, L, M, N
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX),
         $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,LMAX)
    *     .. External Subroutines ..
          EXTERNAL         TG01CD
    *     .. 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
          COMPQ = 'I'
          IF ( L.LT.0 .OR. L.GT.LMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) L
          ELSE
             IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
                WRITE ( NOUT, FMT = 99991 ) 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 = 99990 ) M
                ELSE
                   READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,L )
    *              Find the transformed descriptor system pair
    *              (A-lambda E,B).
                   CALL TG01CD( COMPQ, L, N, M, A, LDA, E, LDE, B, LDB,
         $                      Q, LDQ, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      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 = 99994 )
                      DO 30 I = 1, L
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       30             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, L
                         WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,L )
       40             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TG01CD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TG01CD = ',I2)
    99997 FORMAT (/' The transformed state dynamics matrix Q''*A is ')
    99996 FORMAT (/' The transformed descriptor matrix Q''*E is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (/' The transformed input/state matrix Q''*B is ')
    99993 FORMAT (/' The left transformation matrix Q is ')
    99992 FORMAT (/' L is out of range.',/' L = ',I5)
    99991 FORMAT (/' N is out of range.',/' N = ',I5)
    99990 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

    TG01CD EXAMPLE PROGRAM DATA
      4    4     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

Program Results

     TG01CD EXAMPLE PROGRAM RESULTS

     The transformed state dynamics matrix Q'*A is
      -0.6325  -0.9487   0.0000  -4.7434
      -0.8706  -0.2176  -0.7255  -0.3627
      -0.5203  -0.1301   0.3902   1.4307
      -0.7559  -0.1890   0.5669   2.0788

     The transformed descriptor matrix Q'*E is
      -3.1623  -9.1706  -5.6921  -2.8460
       0.0000  -1.3784  -1.3059  -1.3784
       0.0000   0.0000  -2.4279   0.0000
       0.0000   0.0000   0.0000   0.0000

     The transformed input/state matrix Q'*B is
      -0.3162  -0.9487
       0.6529  -0.2176
      -0.4336  -0.9538
       1.1339   0.3780

     The left transformation matrix Q is
      -0.3162   0.6529   0.3902   0.5669
       0.0000  -0.7255   0.3902   0.5669
      -0.9487  -0.2176  -0.1301  -0.1890
       0.0000   0.0000  -0.8238   0.5669

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